Add files using upload-large-folder tool

lib/python3.12/site-packages/cloudpickle/__init__.py

@@ -0,0 +1,18 @@
+from . import cloudpickle
+from .cloudpickle import *  # noqa
+
+__doc__ = cloudpickle.__doc__
+
+__version__ = "3.1.2"
+
+__all__ = [  # noqa
+    "__version__",
+    "Pickler",
+    "CloudPickler",
+    "dumps",
+    "loads",
+    "dump",
+    "load",
+    "register_pickle_by_value",
+    "unregister_pickle_by_value",
+]

lib/python3.12/site-packages/cloudpickle/cloudpickle.py

@@ -0,0 +1,1552 @@
+"""Pickler class to extend the standard pickle.Pickler functionality
+
+The main objective is to make it natural to perform distributed computing on
+clusters (such as PySpark, Dask, Ray...) with interactively defined code
+(functions, classes, ...) written in notebooks or console.
+
+In particular this pickler adds the following features:
+- serialize interactively-defined or locally-defined functions, classes,
+  enums, typevars, lambdas and nested functions to compiled byte code;
+- deal with some other non-serializable objects in an ad-hoc manner where
+  applicable.
+
+This pickler is therefore meant to be used for the communication between short
+lived Python processes running the same version of Python and libraries. In
+particular, it is not meant to be used for long term storage of Python objects.
+
+It does not include an unpickler, as standard Python unpickling suffices.
+
+This module was extracted from the `cloud` package, developed by `PiCloud, Inc.
+<https://web.archive.org/web/20140626004012/http://www.picloud.com/>`_.
+
+Copyright (c) 2012-now, CloudPickle developers and contributors.
+Copyright (c) 2012, Regents of the University of California.
+Copyright (c) 2009 `PiCloud, Inc. <https://web.archive.org/web/20140626004012/http://www.picloud.com/>`_.
+All rights reserved.
+
+Redistribution and use in source and binary forms, with or without
+modification, are permitted provided that the following conditions
+are met:
+    * Redistributions of source code must retain the above copyright
+      notice, this list of conditions and the following disclaimer.
+    * Redistributions in binary form must reproduce the above copyright
+      notice, this list of conditions and the following disclaimer in the
+      documentation and/or other materials provided with the distribution.
+    * Neither the name of the University of California, Berkeley nor the
+      names of its contributors may be used to endorse or promote
+      products derived from this software without specific prior written
+      permission.
+
+THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED
+TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
+PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
+LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+"""
+
+import _collections_abc
+from collections import ChainMap, OrderedDict
+import abc
+import builtins
+import copyreg
+import dataclasses
+import dis
+from enum import Enum
+import io
+import itertools
+import logging
+import opcode
+import pickle
+from pickle import _getattribute as _pickle_getattribute
+import platform
+import struct
+import sys
+import threading
+import types
+import typing
+import uuid
+import warnings
+import weakref
+
+# The following import is required to be imported in the cloudpickle
+# namespace to be able to load pickle files generated with older versions of
+# cloudpickle. See: tests/test_backward_compat.py
+from types import CellType  # noqa: F401
+
+
+# cloudpickle is meant for inter process communication: we expect all
+# communicating processes to run the same Python version hence we favor
+# communication speed over compatibility:
+DEFAULT_PROTOCOL = pickle.HIGHEST_PROTOCOL
+
+# Names of modules whose resources should be treated as dynamic.
+_PICKLE_BY_VALUE_MODULES = set()
+
+# Track the provenance of reconstructed dynamic classes to make it possible to
+# reconstruct instances from the matching singleton class definition when
+# appropriate and preserve the usual "isinstance" semantics of Python objects.
+_DYNAMIC_CLASS_TRACKER_BY_CLASS = weakref.WeakKeyDictionary()
+_DYNAMIC_CLASS_TRACKER_BY_ID = weakref.WeakValueDictionary()
+_DYNAMIC_CLASS_TRACKER_LOCK = threading.Lock()
+
+PYPY = platform.python_implementation() == "PyPy"
+
+builtin_code_type = None
+if PYPY:
+    # builtin-code objects only exist in pypy
+    builtin_code_type = type(float.__new__.__code__)
+
+_extract_code_globals_cache = weakref.WeakKeyDictionary()
+
+
+def _get_or_create_tracker_id(class_def):
+    with _DYNAMIC_CLASS_TRACKER_LOCK:
+        class_tracker_id = _DYNAMIC_CLASS_TRACKER_BY_CLASS.get(class_def)
+        if class_tracker_id is None:
+            class_tracker_id = uuid.uuid4().hex
+            _DYNAMIC_CLASS_TRACKER_BY_CLASS[class_def] = class_tracker_id
+            _DYNAMIC_CLASS_TRACKER_BY_ID[class_tracker_id] = class_def
+    return class_tracker_id
+
+
+def _lookup_class_or_track(class_tracker_id, class_def):
+    if class_tracker_id is not None:
+        with _DYNAMIC_CLASS_TRACKER_LOCK:
+            class_def = _DYNAMIC_CLASS_TRACKER_BY_ID.setdefault(
+                class_tracker_id, class_def
+            )
+            _DYNAMIC_CLASS_TRACKER_BY_CLASS[class_def] = class_tracker_id
+    return class_def
+
+
+def register_pickle_by_value(module):
+    """Register a module to make its functions and classes picklable by value.
+
+    By default, functions and classes that are attributes of an importable
+    module are to be pickled by reference, that is relying on re-importing
+    the attribute from the module at load time.
+
+    If `register_pickle_by_value(module)` is called, all its functions and
+    classes are subsequently to be pickled by value, meaning that they can
+    be loaded in Python processes where the module is not importable.
+
+    This is especially useful when developing a module in a distributed
+    execution environment: restarting the client Python process with the new
+    source code is enough: there is no need to re-install the new version
+    of the module on all the worker nodes nor to restart the workers.
+
+    Note: this feature is considered experimental. See the cloudpickle
+    README.md file for more details and limitations.
+    """
+    if not isinstance(module, types.ModuleType):
+        raise ValueError(f"Input should be a module object, got {str(module)} instead")
+    # In the future, cloudpickle may need a way to access any module registered
+    # for pickling by value in order to introspect relative imports inside
+    # functions pickled by value. (see
+    # https://github.com/cloudpipe/cloudpickle/pull/417#issuecomment-873684633).
+    # This access can be ensured by checking that module is present in
+    # sys.modules at registering time and assuming that it will still be in
+    # there when accessed during pickling. Another alternative would be to
+    # store a weakref to the module. Even though cloudpickle does not implement
+    # this introspection yet, in order to avoid a possible breaking change
+    # later, we still enforce the presence of module inside sys.modules.
+    if module.__name__ not in sys.modules:
+        raise ValueError(
+            f"{module} was not imported correctly, have you used an "
+            "`import` statement to access it?"
+        )
+    _PICKLE_BY_VALUE_MODULES.add(module.__name__)
+
+
+def unregister_pickle_by_value(module):
+    """Unregister that the input module should be pickled by value."""
+    if not isinstance(module, types.ModuleType):
+        raise ValueError(f"Input should be a module object, got {str(module)} instead")
+    if module.__name__ not in _PICKLE_BY_VALUE_MODULES:
+        raise ValueError(f"{module} is not registered for pickle by value")
+    else:
+        _PICKLE_BY_VALUE_MODULES.remove(module.__name__)
+
+
+def list_registry_pickle_by_value():
+    return _PICKLE_BY_VALUE_MODULES.copy()
+
+
+def _is_registered_pickle_by_value(module):
+    module_name = module.__name__
+    if module_name in _PICKLE_BY_VALUE_MODULES:
+        return True
+    while True:
+        parent_name = module_name.rsplit(".", 1)[0]
+        if parent_name == module_name:
+            break
+        if parent_name in _PICKLE_BY_VALUE_MODULES:
+            return True
+        module_name = parent_name
+    return False
+
+
+if sys.version_info >= (3, 14):
+    def _getattribute(obj, name):
+        return _pickle_getattribute(obj, name.split('.'))
+else:
+    def _getattribute(obj, name):
+        return _pickle_getattribute(obj, name)[0]
+
+
+def _whichmodule(obj, name):
+    """Find the module an object belongs to.
+
+    This function differs from ``pickle.whichmodule`` in two ways:
+    - it does not mangle the cases where obj's module is __main__ and obj was
+      not found in any module.
+    - Errors arising during module introspection are ignored, as those errors
+      are considered unwanted side effects.
+    """
+    module_name = getattr(obj, "__module__", None)
+
+    if module_name is not None:
+        return module_name
+    # Protect the iteration by using a copy of sys.modules against dynamic
+    # modules that trigger imports of other modules upon calls to getattr or
+    # other threads importing at the same time.
+    for module_name, module in sys.modules.copy().items():
+        # Some modules such as coverage can inject non-module objects inside
+        # sys.modules
+        if (
+            module_name == "__main__"
+            or module_name == "__mp_main__"
+            or module is None
+            or not isinstance(module, types.ModuleType)
+        ):
+            continue
+        try:
+            if _getattribute(module, name) is obj:
+                return module_name
+        except Exception:
+            pass
+    return None
+
+
+def _should_pickle_by_reference(obj, name=None):
+    """Test whether an function or a class should be pickled by reference
+
+    Pickling by reference means by that the object (typically a function or a
+    class) is an attribute of a module that is assumed to be importable in the
+    target Python environment. Loading will therefore rely on importing the
+    module and then calling `getattr` on it to access the function or class.
+
+    Pickling by reference is the only option to pickle functions and classes
+    in the standard library. In cloudpickle the alternative option is to
+    pickle by value (for instance for interactively or locally defined
+    functions and classes or for attributes of modules that have been
+    explicitly registered to be pickled by value.
+    """
+    if isinstance(obj, types.FunctionType) or issubclass(type(obj), type):
+        module_and_name = _lookup_module_and_qualname(obj, name=name)
+        if module_and_name is None:
+            return False
+        module, name = module_and_name
+        return not _is_registered_pickle_by_value(module)
+
+    elif isinstance(obj, types.ModuleType):
+        # We assume that sys.modules is primarily used as a cache mechanism for
+        # the Python import machinery. Checking if a module has been added in
+        # is sys.modules therefore a cheap and simple heuristic to tell us
+        # whether we can assume that a given module could be imported by name
+        # in another Python process.
+        if _is_registered_pickle_by_value(obj):
+            return False
+        return obj.__name__ in sys.modules
+    else:
+        raise TypeError(
+            "cannot check importability of {} instances".format(type(obj).__name__)
+        )
+
+
+def _lookup_module_and_qualname(obj, name=None):
+    if name is None:
+        name = getattr(obj, "__qualname__", None)
+    if name is None:  # pragma: no cover
+        # This used to be needed for Python 2.7 support but is probably not
+        # needed anymore. However we keep the __name__ introspection in case
+        # users of cloudpickle rely on this old behavior for unknown reasons.
+        name = getattr(obj, "__name__", None)
+
+    module_name = _whichmodule(obj, name)
+
+    if module_name is None:
+        # In this case, obj.__module__ is None AND obj was not found in any
+        # imported module. obj is thus treated as dynamic.
+        return None
+
+    if module_name == "__main__":
+        return None
+
+    # Note: if module_name is in sys.modules, the corresponding module is
+    # assumed importable at unpickling time. See #357
+    module = sys.modules.get(module_name, None)
+    if module is None:
+        # The main reason why obj's module would not be imported is that this
+        # module has been dynamically created, using for example
+        # types.ModuleType. The other possibility is that module was removed
+        # from sys.modules after obj was created/imported. But this case is not
+        # supported, as the standard pickle does not support it either.
+        return None
+
+    try:
+        obj2 = _getattribute(module, name)
+    except AttributeError:
+        # obj was not found inside the module it points to
+        return None
+    if obj2 is not obj:
+        return None
+    return module, name
+
+
+def _extract_code_globals(co):
+    """Find all globals names read or written to by codeblock co."""
+    out_names = _extract_code_globals_cache.get(co)
+    if out_names is None:
+        # We use a dict with None values instead of a set to get a
+        # deterministic order and avoid introducing non-deterministic pickle
+        # bytes as a results.
+        out_names = {name: None for name in _walk_global_ops(co)}
+
+        # Declaring a function inside another one using the "def ..." syntax
+        # generates a constant code object corresponding to the one of the
+        # nested function's As the nested function may itself need global
+        # variables, we need to introspect its code, extract its globals, (look
+        # for code object in it's co_consts attribute..) and add the result to
+        # code_globals
+        if co.co_consts:
+            for const in co.co_consts:
+                if isinstance(const, types.CodeType):
+                    out_names.update(_extract_code_globals(const))
+
+        _extract_code_globals_cache[co] = out_names
+
+    return out_names
+
+
+def _find_imported_submodules(code, top_level_dependencies):
+    """Find currently imported submodules used by a function.
+
+    Submodules used by a function need to be detected and referenced for the
+    function to work correctly at depickling time. Because submodules can be
+    referenced as attribute of their parent package (``package.submodule``), we
+    need a special introspection technique that does not rely on GLOBAL-related
+    opcodes to find references of them in a code object.
+
+    Example:
+    ```
+    import concurrent.futures
+    import cloudpickle
+    def func():
+        x = concurrent.futures.ThreadPoolExecutor
+    if __name__ == '__main__':
+        cloudpickle.dumps(func)
+    ```
+    The globals extracted by cloudpickle in the function's state include the
+    concurrent package, but not its submodule (here, concurrent.futures), which
+    is the module used by func. Find_imported_submodules will detect the usage
+    of concurrent.futures. Saving this module alongside with func will ensure
+    that calling func once depickled does not fail due to concurrent.futures
+    not being imported
+    """
+
+    subimports = []
+    # check if any known dependency is an imported package
+    for x in top_level_dependencies:
+        if (
+            isinstance(x, types.ModuleType)
+            and hasattr(x, "__package__")
+            and x.__package__
+        ):
+            # check if the package has any currently loaded sub-imports
+            prefix = x.__name__ + "."
+            # A concurrent thread could mutate sys.modules,
+            # make sure we iterate over a copy to avoid exceptions
+            for name in list(sys.modules):
+                # Older versions of pytest will add a "None" module to
+                # sys.modules.
+                if name is not None and name.startswith(prefix):
+                    # check whether the function can address the sub-module
+                    tokens = set(name[len(prefix) :].split("."))
+                    if not tokens - set(code.co_names):
+                        subimports.append(sys.modules[name])
+    return subimports
+
+
+# relevant opcodes
+STORE_GLOBAL = opcode.opmap["STORE_GLOBAL"]
+DELETE_GLOBAL = opcode.opmap["DELETE_GLOBAL"]
+LOAD_GLOBAL = opcode.opmap["LOAD_GLOBAL"]
+GLOBAL_OPS = (STORE_GLOBAL, DELETE_GLOBAL, LOAD_GLOBAL)
+HAVE_ARGUMENT = dis.HAVE_ARGUMENT
+EXTENDED_ARG = dis.EXTENDED_ARG
+
+
+_BUILTIN_TYPE_NAMES = {}
+for k, v in types.__dict__.items():
+    if type(v) is type:
+        _BUILTIN_TYPE_NAMES[v] = k
+
+
+def _builtin_type(name):
+    if name == "ClassType":  # pragma: no cover
+        # Backward compat to load pickle files generated with cloudpickle
+        # < 1.3 even if loading pickle files from older versions is not
+        # officially supported.
+        return type
+    return getattr(types, name)
+
+
+def _walk_global_ops(code):
+    """Yield referenced name for global-referencing instructions in code."""
+    for instr in dis.get_instructions(code):
+        op = instr.opcode
+        if op in GLOBAL_OPS:
+            yield instr.argval
+
+
+def _extract_class_dict(cls):
+    """Retrieve a copy of the dict of a class without the inherited method."""
+    # Hack to circumvent non-predictable memoization caused by string interning.
+    # See the inline comment in _class_setstate for details.
+    clsdict = {"".join(k): cls.__dict__[k] for k in sorted(cls.__dict__)}
+
+    if len(cls.__bases__) == 1:
+        inherited_dict = cls.__bases__[0].__dict__
+    else:
+        inherited_dict = {}
+        for base in reversed(cls.__bases__):
+            inherited_dict.update(base.__dict__)
+    to_remove = []
+    for name, value in clsdict.items():
+        try:
+            base_value = inherited_dict[name]
+            if value is base_value:
+                to_remove.append(name)
+        except KeyError:
+            pass
+    for name in to_remove:
+        clsdict.pop(name)
+    return clsdict
+
+
+def is_tornado_coroutine(func):
+    """Return whether `func` is a Tornado coroutine function.
+
+    Running coroutines are not supported.
+    """
+    warnings.warn(
+        "is_tornado_coroutine is deprecated in cloudpickle 3.0 and will be "
+        "removed in cloudpickle 4.0. Use tornado.gen.is_coroutine_function "
+        "directly instead.",
+        category=DeprecationWarning,
+    )
+    if "tornado.gen" not in sys.modules:
+        return False
+    gen = sys.modules["tornado.gen"]
+    if not hasattr(gen, "is_coroutine_function"):
+        # Tornado version is too old
+        return False
+    return gen.is_coroutine_function(func)
+
+
+def subimport(name):
+    # We cannot do simply: `return __import__(name)`: Indeed, if ``name`` is
+    # the name of a submodule, __import__ will return the top-level root module
+    # of this submodule. For instance, __import__('os.path') returns the `os`
+    # module.
+    __import__(name)
+    return sys.modules[name]
+
+
+def dynamic_subimport(name, vars):
+    mod = types.ModuleType(name)
+    mod.__dict__.update(vars)
+    mod.__dict__["__builtins__"] = builtins.__dict__
+    return mod
+
+
+def _get_cell_contents(cell):
+    try:
+        return cell.cell_contents
+    except ValueError:
+        # Handle empty cells explicitly with a sentinel value.
+        return _empty_cell_value
+
+
+def instance(cls):
+    """Create a new instance of a class.
+
+    Parameters
+    ----------
+    cls : type
+        The class to create an instance of.
+
+    Returns
+    -------
+    instance : cls
+        A new instance of ``cls``.
+    """
+    return cls()
+
+
+@instance
+class _empty_cell_value:
+    """Sentinel for empty closures."""
+
+    @classmethod
+    def __reduce__(cls):
+        return cls.__name__
+
+
+def _make_function(code, globals, name, argdefs, closure):
+    # Setting __builtins__ in globals is needed for nogil CPython.
+    globals["__builtins__"] = __builtins__
+    return types.FunctionType(code, globals, name, argdefs, closure)
+
+
+def _make_empty_cell():
+    if False:
+        # trick the compiler into creating an empty cell in our lambda
+        cell = None
+        raise AssertionError("this route should not be executed")
+
+    return (lambda: cell).__closure__[0]
+
+
+def _make_cell(value=_empty_cell_value):
+    cell = _make_empty_cell()
+    if value is not _empty_cell_value:
+        cell.cell_contents = value
+    return cell
+
+
+def _make_skeleton_class(
+    type_constructor, name, bases, type_kwargs, class_tracker_id, extra
+):
+    """Build dynamic class with an empty __dict__ to be filled once memoized
+
+    If class_tracker_id is not None, try to lookup an existing class definition
+    matching that id. If none is found, track a newly reconstructed class
+    definition under that id so that other instances stemming from the same
+    class id will also reuse this class definition.
+
+    The "extra" variable is meant to be a dict (or None) that can be used for
+    forward compatibility shall the need arise.
+    """
+    # We need to intern the keys of the type_kwargs dict to avoid having
+    # different pickles for the same dynamic class depending on whether it was
+    # dynamically created or reconstructed from a pickled stream.
+    type_kwargs = {sys.intern(k): v for k, v in type_kwargs.items()}
+
+    skeleton_class = types.new_class(
+        name, bases, {"metaclass": type_constructor}, lambda ns: ns.update(type_kwargs)
+    )
+
+    return _lookup_class_or_track(class_tracker_id, skeleton_class)
+
+
+def _make_skeleton_enum(
+    bases, name, qualname, members, module, class_tracker_id, extra
+):
+    """Build dynamic enum with an empty __dict__ to be filled once memoized
+
+    The creation of the enum class is inspired by the code of
+    EnumMeta._create_.
+
+    If class_tracker_id is not None, try to lookup an existing enum definition
+    matching that id. If none is found, track a newly reconstructed enum
+    definition under that id so that other instances stemming from the same
+    class id will also reuse this enum definition.
+
+    The "extra" variable is meant to be a dict (or None) that can be used for
+    forward compatibility shall the need arise.
+    """
+    # enums always inherit from their base Enum class at the last position in
+    # the list of base classes:
+    enum_base = bases[-1]
+    metacls = enum_base.__class__
+    classdict = metacls.__prepare__(name, bases)
+
+    for member_name, member_value in members.items():
+        classdict[member_name] = member_value
+    enum_class = metacls.__new__(metacls, name, bases, classdict)
+    enum_class.__module__ = module
+    enum_class.__qualname__ = qualname
+
+    return _lookup_class_or_track(class_tracker_id, enum_class)
+
+
+def _make_typevar(name, bound, constraints, covariant, contravariant, class_tracker_id):
+    tv = typing.TypeVar(
+        name,
+        *constraints,
+        bound=bound,
+        covariant=covariant,
+        contravariant=contravariant,
+    )
+    return _lookup_class_or_track(class_tracker_id, tv)
+
+
+def _decompose_typevar(obj):
+    return (
+        obj.__name__,
+        obj.__bound__,
+        obj.__constraints__,
+        obj.__covariant__,
+        obj.__contravariant__,
+        _get_or_create_tracker_id(obj),
+    )
+
+
+def _typevar_reduce(obj):
+    # TypeVar instances require the module information hence why we
+    # are not using the _should_pickle_by_reference directly
+    module_and_name = _lookup_module_and_qualname(obj, name=obj.__name__)
+
+    if module_and_name is None:
+        return (_make_typevar, _decompose_typevar(obj))
+    elif _is_registered_pickle_by_value(module_and_name[0]):
+        return (_make_typevar, _decompose_typevar(obj))
+
+    return (getattr, module_and_name)
+
+
+def _get_bases(typ):
+    if "__orig_bases__" in getattr(typ, "__dict__", {}):
+        # For generic types (see PEP 560)
+        # Note that simply checking `hasattr(typ, '__orig_bases__')` is not
+        # correct.  Subclasses of a fully-parameterized generic class does not
+        # have `__orig_bases__` defined, but `hasattr(typ, '__orig_bases__')`
+        # will return True because it's defined in the base class.
+        bases_attr = "__orig_bases__"
+    else:
+        # For regular class objects
+        bases_attr = "__bases__"
+    return getattr(typ, bases_attr)
+
+
+def _make_dict_keys(obj, is_ordered=False):
+    if is_ordered:
+        return OrderedDict.fromkeys(obj).keys()
+    else:
+        return dict.fromkeys(obj).keys()
+
+
+def _make_dict_values(obj, is_ordered=False):
+    if is_ordered:
+        return OrderedDict((i, _) for i, _ in enumerate(obj)).values()
+    else:
+        return {i: _ for i, _ in enumerate(obj)}.values()
+
+
+def _make_dict_items(obj, is_ordered=False):
+    if is_ordered:
+        return OrderedDict(obj).items()
+    else:
+        return obj.items()
+
+
+# COLLECTION OF OBJECTS __getnewargs__-LIKE METHODS
+# -------------------------------------------------
+
+
+def _class_getnewargs(obj):
+    type_kwargs = {}
+    if "__module__" in obj.__dict__:
+        type_kwargs["__module__"] = obj.__module__
+
+    __dict__ = obj.__dict__.get("__dict__", None)
+    if isinstance(__dict__, property):
+        type_kwargs["__dict__"] = __dict__
+
+    return (
+        type(obj),
+        obj.__name__,
+        _get_bases(obj),
+        type_kwargs,
+        _get_or_create_tracker_id(obj),
+        None,
+    )
+
+
+def _enum_getnewargs(obj):
+    members = {e.name: e.value for e in obj}
+    return (
+        obj.__bases__,
+        obj.__name__,
+        obj.__qualname__,
+        members,
+        obj.__module__,
+        _get_or_create_tracker_id(obj),
+        None,
+    )
+
+
+# COLLECTION OF OBJECTS RECONSTRUCTORS
+# ------------------------------------
+def _file_reconstructor(retval):
+    return retval
+
+
+# COLLECTION OF OBJECTS STATE GETTERS
+# -----------------------------------
+
+
+def _function_getstate(func):
+    # - Put func's dynamic attributes (stored in func.__dict__) in state. These
+    #   attributes will be restored at unpickling time using
+    #   f.__dict__.update(state)
+    # - Put func's members into slotstate. Such attributes will be restored at
+    #   unpickling time by iterating over slotstate and calling setattr(func,
+    #   slotname, slotvalue)
+    slotstate = {
+        # Hack to circumvent non-predictable memoization caused by string interning.
+        # See the inline comment in _class_setstate for details.
+        "__name__": "".join(func.__name__),
+        "__qualname__": "".join(func.__qualname__),
+        "__annotations__": func.__annotations__,
+        "__kwdefaults__": func.__kwdefaults__,
+        "__defaults__": func.__defaults__,
+        "__module__": func.__module__,
+        "__doc__": func.__doc__,
+        "__closure__": func.__closure__,
+    }
+
+    f_globals_ref = _extract_code_globals(func.__code__)
+    f_globals = {k: func.__globals__[k] for k in f_globals_ref if k in func.__globals__}
+
+    if func.__closure__ is not None:
+        closure_values = list(map(_get_cell_contents, func.__closure__))
+    else:
+        closure_values = ()
+
+    # Extract currently-imported submodules used by func. Storing these modules
+    # in a smoke _cloudpickle_subimports attribute of the object's state will
+    # trigger the side effect of importing these modules at unpickling time
+    # (which is necessary for func to work correctly once depickled)
+    slotstate["_cloudpickle_submodules"] = _find_imported_submodules(
+        func.__code__, itertools.chain(f_globals.values(), closure_values)
+    )
+    slotstate["__globals__"] = f_globals
+
+    # Hack to circumvent non-predictable memoization caused by string interning.
+    # See the inline comment in _class_setstate for details.
+    state = {"".join(k): v for k, v in func.__dict__.items()}
+    return state, slotstate
+
+
+def _class_getstate(obj):
+    clsdict = _extract_class_dict(obj)
+    clsdict.pop("__weakref__", None)
+
+    if issubclass(type(obj), abc.ABCMeta):
+        # If obj is an instance of an ABCMeta subclass, don't pickle the
+        # cache/negative caches populated during isinstance/issubclass
+        # checks, but pickle the list of registered subclasses of obj.
+        clsdict.pop("_abc_cache", None)
+        clsdict.pop("_abc_negative_cache", None)
+        clsdict.pop("_abc_negative_cache_version", None)
+        registry = clsdict.pop("_abc_registry", None)
+        if registry is None:
+            # The abc caches and registered subclasses of a
+            # class are bundled into the single _abc_impl attribute
+            clsdict.pop("_abc_impl", None)
+            (registry, _, _, _) = abc._get_dump(obj)
+
+            clsdict["_abc_impl"] = [subclass_weakref() for subclass_weakref in registry]
+        else:
+            # In the above if clause, registry is a set of weakrefs -- in
+            # this case, registry is a WeakSet
+            clsdict["_abc_impl"] = [type_ for type_ in registry]
+
+    if "__slots__" in clsdict:
+        # pickle string length optimization: member descriptors of obj are
+        # created automatically from obj's __slots__ attribute, no need to
+        # save them in obj's state
+        if isinstance(obj.__slots__, str):
+            clsdict.pop(obj.__slots__)
+        else:
+            for k in obj.__slots__:
+                clsdict.pop(k, None)
+
+    clsdict.pop("__dict__", None)  # unpicklable property object
+
+    if sys.version_info >= (3, 14):
+        # PEP-649/749: __annotate_func__ contains a closure that references the class
+        # dict. We need to exclude it from pickling. Python will recreate it when
+        # __annotations__ is accessed at unpickling time.
+        clsdict.pop("__annotate_func__", None)
+
+    return (clsdict, {})
+
+
+def _enum_getstate(obj):
+    clsdict, slotstate = _class_getstate(obj)
+
+    members = {e.name: e.value for e in obj}
+    # Cleanup the clsdict that will be passed to _make_skeleton_enum:
+    # Those attributes are already handled by the metaclass.
+    for attrname in [
+        "_generate_next_value_",
+        "_member_names_",
+        "_member_map_",
+        "_member_type_",
+        "_value2member_map_",
+    ]:
+        clsdict.pop(attrname, None)
+    for member in members:
+        clsdict.pop(member)
+        # Special handling of Enum subclasses
+    return clsdict, slotstate
+
+
+# COLLECTIONS OF OBJECTS REDUCERS
+# -------------------------------
+# A reducer is a function taking a single argument (obj), and that returns a
+# tuple with all the necessary data to re-construct obj. Apart from a few
+# exceptions (list, dict, bytes, int, etc.), a reducer is necessary to
+# correctly pickle an object.
+# While many built-in objects (Exceptions objects, instances of the "object"
+# class, etc), are shipped with their own built-in reducer (invoked using
+# obj.__reduce__), some do not. The following methods were created to "fill
+# these holes".
+
+
+def _code_reduce(obj):
+    """code object reducer."""
+    # If you are not sure about the order of arguments, take a look at help
+    # of the specific type from types, for example:
+    # >>> from types import CodeType
+    # >>> help(CodeType)
+
+    # Hack to circumvent non-predictable memoization caused by string interning.
+    # See the inline comment in _class_setstate for details.
+    co_name = "".join(obj.co_name)
+
+    # Create shallow copies of these tuple to make cloudpickle payload deterministic.
+    # When creating a code object during load, copies of these four tuples are
+    # created, while in the main process, these tuples can be shared.
+    # By always creating copies, we make sure the resulting payload is deterministic.
+    co_names = tuple(name for name in obj.co_names)
+    co_varnames = tuple(name for name in obj.co_varnames)
+    co_freevars = tuple(name for name in obj.co_freevars)
+    co_cellvars = tuple(name for name in obj.co_cellvars)
+    if hasattr(obj, "co_exceptiontable"):
+        # Python 3.11 and later: there are some new attributes
+        # related to the enhanced exceptions.
+        args = (
+            obj.co_argcount,
+            obj.co_posonlyargcount,
+            obj.co_kwonlyargcount,
+            obj.co_nlocals,
+            obj.co_stacksize,
+            obj.co_flags,
+            obj.co_code,
+            obj.co_consts,
+            co_names,
+            co_varnames,
+            obj.co_filename,
+            co_name,
+            obj.co_qualname,
+            obj.co_firstlineno,
+            obj.co_linetable,
+            obj.co_exceptiontable,
+            co_freevars,
+            co_cellvars,
+        )
+    elif hasattr(obj, "co_linetable"):
+        # Python 3.10 and later: obj.co_lnotab is deprecated and constructor
+        # expects obj.co_linetable instead.
+        args = (
+            obj.co_argcount,
+            obj.co_posonlyargcount,
+            obj.co_kwonlyargcount,
+            obj.co_nlocals,
+            obj.co_stacksize,
+            obj.co_flags,
+            obj.co_code,
+            obj.co_consts,
+            co_names,
+            co_varnames,
+            obj.co_filename,
+            co_name,
+            obj.co_firstlineno,
+            obj.co_linetable,
+            co_freevars,
+            co_cellvars,
+        )
+    elif hasattr(obj, "co_nmeta"):  # pragma: no cover
+        # "nogil" Python: modified attributes from 3.9
+        args = (
+            obj.co_argcount,
+            obj.co_posonlyargcount,
+            obj.co_kwonlyargcount,
+            obj.co_nlocals,
+            obj.co_framesize,
+            obj.co_ndefaultargs,
+            obj.co_nmeta,
+            obj.co_flags,
+            obj.co_code,
+            obj.co_consts,
+            co_varnames,
+            obj.co_filename,
+            co_name,
+            obj.co_firstlineno,
+            obj.co_lnotab,
+            obj.co_exc_handlers,
+            obj.co_jump_table,
+            co_freevars,
+            co_cellvars,
+            obj.co_free2reg,
+            obj.co_cell2reg,
+        )
+    else:
+        # Backward compat for 3.8 and 3.9
+        args = (
+            obj.co_argcount,
+            obj.co_posonlyargcount,
+            obj.co_kwonlyargcount,
+            obj.co_nlocals,
+            obj.co_stacksize,
+            obj.co_flags,
+            obj.co_code,
+            obj.co_consts,
+            co_names,
+            co_varnames,
+            obj.co_filename,
+            co_name,
+            obj.co_firstlineno,
+            obj.co_lnotab,
+            co_freevars,
+            co_cellvars,
+        )
+    return types.CodeType, args
+
+
+def _cell_reduce(obj):
+    """Cell (containing values of a function's free variables) reducer."""
+    try:
+        obj.cell_contents
+    except ValueError:  # cell is empty
+        return _make_empty_cell, ()
+    else:
+        return _make_cell, (obj.cell_contents,)
+
+
+def _classmethod_reduce(obj):
+    orig_func = obj.__func__
+    return type(obj), (orig_func,)
+
+
+def _file_reduce(obj):
+    """Save a file."""
+    import io
+
+    if not hasattr(obj, "name") or not hasattr(obj, "mode"):
+        raise pickle.PicklingError(
+            "Cannot pickle files that do not map to an actual file"
+        )
+    if obj is sys.stdout:
+        return getattr, (sys, "stdout")
+    if obj is sys.stderr:
+        return getattr, (sys, "stderr")
+    if obj is sys.stdin:
+        raise pickle.PicklingError("Cannot pickle standard input")
+    if obj.closed:
+        raise pickle.PicklingError("Cannot pickle closed files")
+    if hasattr(obj, "isatty") and obj.isatty():
+        raise pickle.PicklingError("Cannot pickle files that map to tty objects")
+    if "r" not in obj.mode and "+" not in obj.mode:
+        raise pickle.PicklingError(
+            "Cannot pickle files that are not opened for reading: %s" % obj.mode
+        )
+
+    name = obj.name
+
+    retval = io.StringIO()
+
+    try:
+        # Read the whole file
+        curloc = obj.tell()
+        obj.seek(0)
+        contents = obj.read()
+        obj.seek(curloc)
+    except OSError as e:
+        raise pickle.PicklingError(
+            "Cannot pickle file %s as it cannot be read" % name
+        ) from e
+    retval.write(contents)
+    retval.seek(curloc)
+
+    retval.name = name
+    return _file_reconstructor, (retval,)
+
+
+def _getset_descriptor_reduce(obj):
+    return getattr, (obj.__objclass__, obj.__name__)
+
+
+def _mappingproxy_reduce(obj):
+    return types.MappingProxyType, (dict(obj),)
+
+
+def _memoryview_reduce(obj):
+    return bytes, (obj.tobytes(),)
+
+
+def _module_reduce(obj):
+    if _should_pickle_by_reference(obj):
+        return subimport, (obj.__name__,)
+    else:
+        # Some external libraries can populate the "__builtins__" entry of a
+        # module's `__dict__` with unpicklable objects (see #316). For that
+        # reason, we do not attempt to pickle the "__builtins__" entry, and
+        # restore a default value for it at unpickling time.
+        state = obj.__dict__.copy()
+        state.pop("__builtins__", None)
+        return dynamic_subimport, (obj.__name__, state)
+
+
+def _method_reduce(obj):
+    return (types.MethodType, (obj.__func__, obj.__self__))
+
+
+def _logger_reduce(obj):
+    return logging.getLogger, (obj.name,)
+
+
+def _root_logger_reduce(obj):
+    return logging.getLogger, ()
+
+
+def _property_reduce(obj):
+    return property, (obj.fget, obj.fset, obj.fdel, obj.__doc__)
+
+
+def _weakset_reduce(obj):
+    return weakref.WeakSet, (list(obj),)
+
+
+def _dynamic_class_reduce(obj):
+    """Save a class that can't be referenced as a module attribute.
+
+    This method is used to serialize classes that are defined inside
+    functions, or that otherwise can't be serialized as attribute lookups
+    from importable modules.
+    """
+    if Enum is not None and issubclass(obj, Enum):
+        return (
+            _make_skeleton_enum,
+            _enum_getnewargs(obj),
+            _enum_getstate(obj),
+            None,
+            None,
+            _class_setstate,
+        )
+    else:
+        return (
+            _make_skeleton_class,
+            _class_getnewargs(obj),
+            _class_getstate(obj),
+            None,
+            None,
+            _class_setstate,
+        )
+
+
+def _class_reduce(obj):
+    """Select the reducer depending on the dynamic nature of the class obj."""
+    if obj is type(None):  # noqa
+        return type, (None,)
+    elif obj is type(Ellipsis):
+        return type, (Ellipsis,)
+    elif obj is type(NotImplemented):
+        return type, (NotImplemented,)
+    elif obj in _BUILTIN_TYPE_NAMES:
+        return _builtin_type, (_BUILTIN_TYPE_NAMES[obj],)
+    elif not _should_pickle_by_reference(obj):
+        return _dynamic_class_reduce(obj)
+    return NotImplemented
+
+
+def _dict_keys_reduce(obj):
+    # Safer not to ship the full dict as sending the rest might
+    # be unintended and could potentially cause leaking of
+    # sensitive information
+    return _make_dict_keys, (list(obj),)
+
+
+def _dict_values_reduce(obj):
+    # Safer not to ship the full dict as sending the rest might
+    # be unintended and could potentially cause leaking of
+    # sensitive information
+    return _make_dict_values, (list(obj),)
+
+
+def _dict_items_reduce(obj):
+    return _make_dict_items, (dict(obj),)
+
+
+def _odict_keys_reduce(obj):
+    # Safer not to ship the full dict as sending the rest might
+    # be unintended and could potentially cause leaking of
+    # sensitive information
+    return _make_dict_keys, (list(obj), True)
+
+
+def _odict_values_reduce(obj):
+    # Safer not to ship the full dict as sending the rest might
+    # be unintended and could potentially cause leaking of
+    # sensitive information
+    return _make_dict_values, (list(obj), True)
+
+
+def _odict_items_reduce(obj):
+    return _make_dict_items, (dict(obj), True)
+
+
+def _dataclass_field_base_reduce(obj):
+    return _get_dataclass_field_type_sentinel, (obj.name,)
+
+
+# COLLECTIONS OF OBJECTS STATE SETTERS
+# ------------------------------------
+# state setters are called at unpickling time, once the object is created and
+# it has to be updated to how it was at unpickling time.
+
+
+def _function_setstate(obj, state):
+    """Update the state of a dynamic function.
+
+    As __closure__ and __globals__ are readonly attributes of a function, we
+    cannot rely on the native setstate routine of pickle.load_build, that calls
+    setattr on items of the slotstate. Instead, we have to modify them inplace.
+    """
+    state, slotstate = state
+    obj.__dict__.update(state)
+
+    obj_globals = slotstate.pop("__globals__")
+    obj_closure = slotstate.pop("__closure__")
+    # _cloudpickle_subimports is a set of submodules that must be loaded for
+    # the pickled function to work correctly at unpickling time. Now that these
+    # submodules are depickled (hence imported), they can be removed from the
+    # object's state (the object state only served as a reference holder to
+    # these submodules)
+    slotstate.pop("_cloudpickle_submodules")
+
+    obj.__globals__.update(obj_globals)
+    obj.__globals__["__builtins__"] = __builtins__
+
+    if obj_closure is not None:
+        for i, cell in enumerate(obj_closure):
+            try:
+                value = cell.cell_contents
+            except ValueError:  # cell is empty
+                continue
+            obj.__closure__[i].cell_contents = value
+
+    for k, v in slotstate.items():
+        setattr(obj, k, v)
+
+
+def _class_setstate(obj, state):
+    state, slotstate = state
+    registry = None
+    for attrname, attr in state.items():
+        if attrname == "_abc_impl":
+            registry = attr
+        else:
+            # Note: setting attribute names on a class automatically triggers their
+            # interning in CPython:
+            # https://github.com/python/cpython/blob/v3.12.0/Objects/object.c#L957
+            #
+            # This means that to get deterministic pickling for a dynamic class that
+            # was initially defined in a different Python process, the pickler
+            # needs to ensure that dynamic class and function attribute names are
+            # systematically copied into a non-interned version to avoid
+            # unpredictable pickle payloads.
+            #
+            # Indeed the Pickler's memoizer relies on physical object identity to break
+            # cycles in the reference graph of the object being serialized.
+            setattr(obj, attrname, attr)
+
+    if sys.version_info >= (3, 13) and "__firstlineno__" in state:
+        # Set the Python 3.13+ only __firstlineno__  attribute one more time, as it
+        # will be automatically deleted by the `setattr(obj, attrname, attr)` call
+        # above when `attrname` is "__firstlineno__". We assume that preserving this
+        # information might be important for some users and that it not stale in the
+        # context of cloudpickle usage, hence legitimate to propagate. Furthermore it
+        # is necessary to do so to keep deterministic chained pickling as tested in
+        # test_deterministic_str_interning_for_chained_dynamic_class_pickling.
+        obj.__firstlineno__ = state["__firstlineno__"]
+
+    if registry is not None:
+        for subclass in registry:
+            obj.register(subclass)
+
+    # PEP-649/749: During pickling, we excluded the __annotate_func__ attribute but it
+    # will be created by Python. Subsequently, annotations will be recreated when
+    # __annotations__ is accessed.
+
+    return obj
+
+
+# COLLECTION OF DATACLASS UTILITIES
+# ---------------------------------
+# There are some internal sentinel values whose identity must be preserved when
+# unpickling dataclass fields. Each sentinel value has a unique name that we can
+# use to retrieve its identity at unpickling time.
+
+
+_DATACLASSE_FIELD_TYPE_SENTINELS = {
+    dataclasses._FIELD.name: dataclasses._FIELD,
+    dataclasses._FIELD_CLASSVAR.name: dataclasses._FIELD_CLASSVAR,
+    dataclasses._FIELD_INITVAR.name: dataclasses._FIELD_INITVAR,
+}
+
+
+def _get_dataclass_field_type_sentinel(name):
+    return _DATACLASSE_FIELD_TYPE_SENTINELS[name]
+
+
+class Pickler(pickle.Pickler):
+    # set of reducers defined and used by cloudpickle (private)
+    _dispatch_table = {}
+    _dispatch_table[classmethod] = _classmethod_reduce
+    _dispatch_table[io.TextIOWrapper] = _file_reduce
+    _dispatch_table[logging.Logger] = _logger_reduce
+    _dispatch_table[logging.RootLogger] = _root_logger_reduce
+    _dispatch_table[memoryview] = _memoryview_reduce
+    _dispatch_table[property] = _property_reduce
+    _dispatch_table[staticmethod] = _classmethod_reduce
+    _dispatch_table[CellType] = _cell_reduce
+    _dispatch_table[types.CodeType] = _code_reduce
+    _dispatch_table[types.GetSetDescriptorType] = _getset_descriptor_reduce
+    _dispatch_table[types.ModuleType] = _module_reduce
+    _dispatch_table[types.MethodType] = _method_reduce
+    _dispatch_table[types.MappingProxyType] = _mappingproxy_reduce
+    _dispatch_table[weakref.WeakSet] = _weakset_reduce
+    _dispatch_table[typing.TypeVar] = _typevar_reduce
+    _dispatch_table[_collections_abc.dict_keys] = _dict_keys_reduce
+    _dispatch_table[_collections_abc.dict_values] = _dict_values_reduce
+    _dispatch_table[_collections_abc.dict_items] = _dict_items_reduce
+    _dispatch_table[type(OrderedDict().keys())] = _odict_keys_reduce
+    _dispatch_table[type(OrderedDict().values())] = _odict_values_reduce
+    _dispatch_table[type(OrderedDict().items())] = _odict_items_reduce
+    _dispatch_table[abc.abstractmethod] = _classmethod_reduce
+    _dispatch_table[abc.abstractclassmethod] = _classmethod_reduce
+    _dispatch_table[abc.abstractstaticmethod] = _classmethod_reduce
+    _dispatch_table[abc.abstractproperty] = _property_reduce
+    _dispatch_table[dataclasses._FIELD_BASE] = _dataclass_field_base_reduce
+
+    dispatch_table = ChainMap(_dispatch_table, copyreg.dispatch_table)
+
+    # function reducers are defined as instance methods of cloudpickle.Pickler
+    # objects, as they rely on a cloudpickle.Pickler attribute (globals_ref)
+    def _dynamic_function_reduce(self, func):
+        """Reduce a function that is not pickleable via attribute lookup."""
+        newargs = self._function_getnewargs(func)
+        state = _function_getstate(func)
+        return (_make_function, newargs, state, None, None, _function_setstate)
+
+    def _function_reduce(self, obj):
+        """Reducer for function objects.
+
+        If obj is a top-level attribute of a file-backed module, this reducer
+        returns NotImplemented, making the cloudpickle.Pickler fall back to
+        traditional pickle.Pickler routines to save obj. Otherwise, it reduces
+        obj using a custom cloudpickle reducer designed specifically to handle
+        dynamic functions.
+        """
+        if _should_pickle_by_reference(obj):
+            return NotImplemented
+        else:
+            return self._dynamic_function_reduce(obj)
+
+    def _function_getnewargs(self, func):
+        code = func.__code__
+
+        # base_globals represents the future global namespace of func at
+        # unpickling time. Looking it up and storing it in
+        # cloudpickle.Pickler.globals_ref allow functions sharing the same
+        # globals at pickling time to also share them once unpickled, at one
+        # condition: since globals_ref is an attribute of a cloudpickle.Pickler
+        # instance, and that a new cloudpickle.Pickler is created each time
+        # cloudpickle.dump or cloudpickle.dumps is called, functions also need
+        # to be saved within the same invocation of
+        # cloudpickle.dump/cloudpickle.dumps (for example:
+        # cloudpickle.dumps([f1, f2])). There is no such limitation when using
+        # cloudpickle.Pickler.dump, as long as the multiple invocations are
+        # bound to the same cloudpickle.Pickler instance.
+        base_globals = self.globals_ref.setdefault(id(func.__globals__), {})
+
+        if base_globals == {}:
+            # Add module attributes used to resolve relative imports
+            # instructions inside func.
+            for k in ["__package__", "__name__", "__path__", "__file__"]:
+                if k in func.__globals__:
+                    base_globals[k] = func.__globals__[k]
+
+        # Do not bind the free variables before the function is created to
+        # avoid infinite recursion.
+        if func.__closure__ is None:
+            closure = None
+        else:
+            closure = tuple(_make_empty_cell() for _ in range(len(code.co_freevars)))
+
+        return code, base_globals, None, None, closure
+
+    def dump(self, obj):
+        try:
+            return super().dump(obj)
+        except RecursionError as e:
+            msg = "Could not pickle object as excessively deep recursion required."
+            raise pickle.PicklingError(msg) from e
+
+    def __init__(self, file, protocol=None, buffer_callback=None):
+        if protocol is None:
+            protocol = DEFAULT_PROTOCOL
+        super().__init__(file, protocol=protocol, buffer_callback=buffer_callback)
+        # map functions __globals__ attribute ids, to ensure that functions
+        # sharing the same global namespace at pickling time also share
+        # their global namespace at unpickling time.
+        self.globals_ref = {}
+        self.proto = int(protocol)
+
+    if not PYPY:
+        # pickle.Pickler is the C implementation of the CPython pickler and
+        # therefore we rely on reduce_override method to customize the pickler
+        # behavior.
+
+        # `cloudpickle.Pickler.dispatch` is only left for backward
+        # compatibility - note that when using protocol 5,
+        # `cloudpickle.Pickler.dispatch` is not an extension of
+        # `pickle._Pickler.dispatch` dictionary, because `cloudpickle.Pickler`
+        # subclasses the C-implemented `pickle.Pickler`, which does not expose
+        # a `dispatch` attribute.  Earlier versions of `cloudpickle.Pickler`
+        # used `cloudpickle.Pickler.dispatch` as a class-level attribute
+        # storing all reducers implemented by cloudpickle, but the attribute
+        # name was not a great choice given because it would collide with a
+        # similarly named attribute in the pure-Python `pickle._Pickler`
+        # implementation in the standard library.
+        dispatch = dispatch_table
+
+        # Implementation of the reducer_override callback, in order to
+        # efficiently serialize dynamic functions and classes by subclassing
+        # the C-implemented `pickle.Pickler`.
+        # TODO: decorrelate reducer_override (which is tied to CPython's
+        # implementation - would it make sense to backport it to pypy? - and
+        # pickle's protocol 5 which is implementation agnostic. Currently, the
+        # availability of both notions coincide on CPython's pickle, but it may
+        # not be the case anymore when pypy implements protocol 5.
+
+        def reducer_override(self, obj):
+            """Type-agnostic reducing callback for function and classes.
+
+            For performance reasons, subclasses of the C `pickle.Pickler` class
+            cannot register custom reducers for functions and classes in the
+            dispatch_table attribute. Reducers for such types must instead
+            implemented via the special `reducer_override` method.
+
+            Note that this method will be called for any object except a few
+            builtin-types (int, lists, dicts etc.), which differs from reducers
+            in the Pickler's dispatch_table, each of them being invoked for
+            objects of a specific type only.
+
+            This property comes in handy for classes: although most classes are
+            instances of the ``type`` metaclass, some of them can be instances
+            of other custom metaclasses (such as enum.EnumMeta for example). In
+            particular, the metaclass will likely not be known in advance, and
+            thus cannot be special-cased using an entry in the dispatch_table.
+            reducer_override, among other things, allows us to register a
+            reducer that will be called for any class, independently of its
+            type.
+
+            Notes:
+
+            * reducer_override has the priority over dispatch_table-registered
+            reducers.
+            * reducer_override can be used to fix other limitations of
+              cloudpickle for other types that suffered from type-specific
+              reducers, such as Exceptions. See
+              https://github.com/cloudpipe/cloudpickle/issues/248
+            """
+            t = type(obj)
+            try:
+                is_anyclass = issubclass(t, type)
+            except TypeError:  # t is not a class (old Boost; see SF #502085)
+                is_anyclass = False
+
+            if is_anyclass:
+                return _class_reduce(obj)
+            elif isinstance(obj, types.FunctionType):
+                return self._function_reduce(obj)
+            else:
+                # fallback to save_global, including the Pickler's
+                # dispatch_table
+                return NotImplemented
+
+    else:
+        # When reducer_override is not available, hack the pure-Python
+        # Pickler's types.FunctionType and type savers. Note: the type saver
+        # must override Pickler.save_global, because pickle.py contains a
+        # hard-coded call to save_global when pickling meta-classes.
+        dispatch = pickle.Pickler.dispatch.copy()
+
+        def _save_reduce_pickle5(
+            self,
+            func,
+            args,
+            state=None,
+            listitems=None,
+            dictitems=None,
+            state_setter=None,
+            obj=None,
+        ):
+            save = self.save
+            write = self.write
+            self.save_reduce(
+                func,
+                args,
+                state=None,
+                listitems=listitems,
+                dictitems=dictitems,
+                obj=obj,
+            )
+            # backport of the Python 3.8 state_setter pickle operations
+            save(state_setter)
+            save(obj)  # simple BINGET opcode as obj is already memoized.
+            save(state)
+            write(pickle.TUPLE2)
+            # Trigger a state_setter(obj, state) function call.
+            write(pickle.REDUCE)
+            # The purpose of state_setter is to carry-out an
+            # inplace modification of obj. We do not care about what the
+            # method might return, so its output is eventually removed from
+            # the stack.
+            write(pickle.POP)
+
+        def save_global(self, obj, name=None, pack=struct.pack):
+            """Main dispatch method.
+
+            The name of this method is somewhat misleading: all types get
+            dispatched here.
+            """
+            if obj is type(None):  # noqa
+                return self.save_reduce(type, (None,), obj=obj)
+            elif obj is type(Ellipsis):
+                return self.save_reduce(type, (Ellipsis,), obj=obj)
+            elif obj is type(NotImplemented):
+                return self.save_reduce(type, (NotImplemented,), obj=obj)
+            elif obj in _BUILTIN_TYPE_NAMES:
+                return self.save_reduce(
+                    _builtin_type, (_BUILTIN_TYPE_NAMES[obj],), obj=obj
+                )
+
+            if name is not None:
+                super().save_global(obj, name=name)
+            elif not _should_pickle_by_reference(obj, name=name):
+                self._save_reduce_pickle5(*_dynamic_class_reduce(obj), obj=obj)
+            else:
+                super().save_global(obj, name=name)
+
+        dispatch[type] = save_global
+
+        def save_function(self, obj, name=None):
+            """Registered with the dispatch to handle all function types.
+
+            Determines what kind of function obj is (e.g. lambda, defined at
+            interactive prompt, etc) and handles the pickling appropriately.
+            """
+            if _should_pickle_by_reference(obj, name=name):
+                return super().save_global(obj, name=name)
+            elif PYPY and isinstance(obj.__code__, builtin_code_type):
+                return self.save_pypy_builtin_func(obj)
+            else:
+                return self._save_reduce_pickle5(
+                    *self._dynamic_function_reduce(obj), obj=obj
+                )
+
+        def save_pypy_builtin_func(self, obj):
+            """Save pypy equivalent of builtin functions.
+
+            PyPy does not have the concept of builtin-functions. Instead,
+            builtin-functions are simple function instances, but with a
+            builtin-code attribute.
+            Most of the time, builtin functions should be pickled by attribute.
+            But PyPy has flaky support for __qualname__, so some builtin
+            functions such as float.__new__ will be classified as dynamic. For
+            this reason only, we created this special routine. Because
+            builtin-functions are not expected to have closure or globals,
+            there is no additional hack (compared the one already implemented
+            in pickle) to protect ourselves from reference cycles. A simple
+            (reconstructor, newargs, obj.__dict__) tuple is save_reduced.  Note
+            also that PyPy improved their support for __qualname__ in v3.6, so
+            this routing should be removed when cloudpickle supports only PyPy
+            3.6 and later.
+            """
+            rv = (
+                types.FunctionType,
+                (obj.__code__, {}, obj.__name__, obj.__defaults__, obj.__closure__),
+                obj.__dict__,
+            )
+            self.save_reduce(*rv, obj=obj)
+
+        dispatch[types.FunctionType] = save_function
+
+
+# Shorthands similar to pickle.dump/pickle.dumps
+
+
+def dump(obj, file, protocol=None, buffer_callback=None):
+    """Serialize obj as bytes streamed into file
+
+    protocol defaults to cloudpickle.DEFAULT_PROTOCOL which is an alias to
+    pickle.HIGHEST_PROTOCOL. This setting favors maximum communication
+    speed between processes running the same Python version.
+
+    Set protocol=pickle.DEFAULT_PROTOCOL instead if you need to ensure
+    compatibility with older versions of Python (although this is not always
+    guaranteed to work because cloudpickle relies on some internal
+    implementation details that can change from one Python version to the
+    next).
+    """
+    Pickler(file, protocol=protocol, buffer_callback=buffer_callback).dump(obj)
+
+
+def dumps(obj, protocol=None, buffer_callback=None):
+    """Serialize obj as a string of bytes allocated in memory
+
+    protocol defaults to cloudpickle.DEFAULT_PROTOCOL which is an alias to
+    pickle.HIGHEST_PROTOCOL. This setting favors maximum communication
+    speed between processes running the same Python version.
+
+    Set protocol=pickle.DEFAULT_PROTOCOL instead if you need to ensure
+    compatibility with older versions of Python (although this is not always
+    guaranteed to work because cloudpickle relies on some internal
+    implementation details that can change from one Python version to the
+    next).
+    """
+    with io.BytesIO() as file:
+        cp = Pickler(file, protocol=protocol, buffer_callback=buffer_callback)
+        cp.dump(obj)
+        return file.getvalue()
+
+
+# Include pickles unloading functions in this namespace for convenience.
+load, loads = pickle.load, pickle.loads
+
+# Backward compat alias.
+CloudPickler = Pickler

lib/python3.12/site-packages/cloudpickle/cloudpickle_fast.py

@@ -0,0 +1,14 @@
+"""Compatibility module.
+
+It can be necessary to load files generated by previous versions of cloudpickle
+that rely on symbols being defined under the `cloudpickle.cloudpickle_fast`
+namespace.
+
+See: tests/test_backward_compat.py
+"""
+
+from . import cloudpickle
+
+
+def __getattr__(name):
+    return getattr(cloudpickle, name)

lib/python3.12/site-packages/ninja-1.13.0.dist-info/INSTALLER

@@ -0,0 +1 @@
+pip

lib/python3.12/site-packages/ninja-1.13.0.dist-info/METADATA

@@ -0,0 +1,102 @@
+Metadata-Version: 2.1
+Name: ninja
+Version: 1.13.0
+Summary: Ninja is a small build system with a focus on speed
+Keywords: build,c++,cross-compilation,cross-platform,fortran,ninja
+Author-Email: Jean-Christophe Fillion-Robin <scikit-build@googlegroups.com>, Henry Schreiner <henryfs@princeton.edu>
+Classifier: Development Status :: 5 - Production/Stable
+Classifier: Intended Audience :: Developers
+Classifier: License :: OSI Approved :: Apache Software License
+Classifier: License :: OSI Approved :: BSD License
+Classifier: Operating System :: OS Independent
+Classifier: Programming Language :: C
+Classifier: Programming Language :: C++
+Classifier: Programming Language :: Fortran
+Classifier: Programming Language :: Python
+Classifier: Topic :: Software Development :: Build Tools
+Classifier: Typing :: Typed
+Project-URL: Bug Tracker, https://github.com/scikit-build/ninja-python-distributions/issues
+Project-URL: Documentation, https://github.com/scikit-build/ninja-python-distributions#readme
+Project-URL: Download, https://github.com/ninja-build/ninja/releases
+Project-URL: Homepage, http://ninja-build.org/
+Project-URL: Mailing list, https://groups.google.com/forum/#!forum/scikit-build
+Project-URL: Source Code, https://github.com/scikit-build/ninja-python-distributions
+Requires-Python: >=3.8
+Description-Content-Type: text/x-rst
+
+==========================
+Ninja Python Distributions
+==========================
+
+`Ninja <http://www.ninja-build.org>`_ is a small build system with a focus on speed.
+
+The latest Ninja python wheels provide `ninja 1.13.0.gd74ef.kitware.jobserver-pipe-1 <https://ninja-build.org/manual.html>`_ executable
+and `ninja_syntax.py` for generating `.ninja` files.
+
+.. image:: https://raw.githubusercontent.com/scikit-build/ninja-python-distributions/master/ninja-python-distributions-logo.png
+
+Latest Release
+--------------
+
+.. table::
+
+  +----------------------------------------------------------------------+---------------------------------------------------------------------------+
+  | Versions                                                             | Downloads                                                                 |
+  +======================================================================+===========================================================================+
+  | .. image:: https://img.shields.io/pypi/v/ninja.svg                   | .. image:: https://img.shields.io/badge/downloads-2535k%20total-green.svg |
+  |     :target: https://pypi.python.org/pypi/ninja                      |     :target: https://pypi.python.org/pypi/ninja                           |
+  +----------------------------------------------------------------------+---------------------------------------------------------------------------+
+
+Build Status
+------------
+
+.. table::
+
+  +---------------+-------------------------------------------------------------------------------------------------------------+
+  |               | GitHub Actions (Windows, macOS, Linux)                                                                      |
+  +===============+=============================================================================================================+
+  | PyPI          | .. image:: https://github.com/scikit-build/ninja-python-distributions/actions/workflows/build.yml/badge.svg |
+  |               |     :target: https://github.com/scikit-build/ninja-python-distributions/actions/workflows/build.yml         |
+  +---------------+-------------------------------------------------------------------------------------------------------------+
+
+Maintainers
+-----------
+
+* `How to update ninja version ? <https://github.com/scikit-build/ninja-python-distributions/blob/master/docs/update_ninja_version.rst>`_
+
+* `How to make a release ? <https://github.com/scikit-build/ninja-python-distributions/blob/master/docs/make_a_release.rst>`_
+
+
+Miscellaneous
+-------------
+
+* Documentation: https://github.com/scikit-build/ninja-python-distributions#readme
+* Source code: https://github.com/scikit-build/ninja-python-distributions
+* Mailing list: https://groups.google.com/forum/#!forum/scikit-build
+
+Python Version Support
+----------------------
+
+Versions after 1.11.1.1 no longer support Python 2-3.6, and require manylinux2010+ on linux.
+Versions after 1.13 no longer support Python 3.7, and require manylinux2014+/musllinux_1_2+ on linux.
+
+License
+-------
+
+This project is maintained by Jean-Christophe Fillion-Robin from Kitware Inc.
+It is covered by the `Apache License, Version 2.0 <http://www.apache.org/licenses/LICENSE-2.0>`_.
+
+Ninja is also distributed under the `Apache License, Version 2.0 <http://www.apache.org/licenses/LICENSE-2.0>`_.
+For more information about Ninja, visit https://ninja-build.org
+
+Logo was originally created by Libby Rose from Kitware Inc.
+It is covered by `CC BY 4.0 <https://creativecommons.org/licenses/by/4.0/>`_.
+
+
+History
+-------
+
+ninja-python-distributions was initially developed in November 2016 by
+Jean-Christophe Fillion-Robin to facilitate the distribution of project using
+`scikit-build <http://scikit-build.readthedocs.io/>`_ and depending on CMake
+and Ninja.

lib/python3.12/site-packages/ninja-1.13.0.dist-info/RECORD

@@ -0,0 +1,18 @@
+../../../bin/ninja,sha256=aW-WKKednOUDFM-VVtfNGh0exSuP1Sgo9vnbFxlWW2c,372384
+ninja-1.13.0.dist-info/INSTALLER,sha256=zuuue4knoyJ-UwPPXg8fezS7VCrXJQrAP7zeNuwvFQg,4
+ninja-1.13.0.dist-info/METADATA,sha256=jXb8Tjgs7c0ivK6qdIrzmz-nkxtyRCOW0VyxgWWdCk4,5148
+ninja-1.13.0.dist-info/RECORD,,
+ninja-1.13.0.dist-info/WHEEL,sha256=zzOwTeuxXsHOT9QV_vcZk1hX9KJAUdqJD7e-2jGJxLA,150
+ninja-1.13.0.dist-info/licenses/AUTHORS.rst,sha256=bGE1t_Lhm2ir8S7n_jbLDohP84fpJ5sNCuxvDVsKNQg,142
+ninja-1.13.0.dist-info/licenses/LICENSE_Apache_20,sha256=c7p036pSC0mkAbXSFFmoUjoUbzt1GKgz7qXvqFEwv2g,10273
+ninja/__init__.py,sha256=taDHI20kpmjxOT72PiSyE42Pvz3KLXlwMfTZvvnRMKM,1533
+ninja/__main__.py,sha256=6iPLwHHAc2TMbojFVcUzERrzN0RvIsywuZc4KpJCg_4,100
+ninja/__pycache__/__init__.cpython-312.pyc,,
+ninja/__pycache__/__main__.cpython-312.pyc,,
+ninja/__pycache__/_version.cpython-312.pyc,,
+ninja/__pycache__/ninja_syntax.cpython-312.pyc,,
+ninja/_version.py,sha256=M85oP8JJdZ4yZHcp9qfGYLKUYvnN3kTyQosVcYPCPow,19
+ninja/_version.pyi,sha256=j5kbzfm6lOn8BzASXWjGIA1yT0OlHTWqlbyZ8Si_o0E,118
+ninja/ninja_syntax.py,sha256=RCXZ6Roda3lwnbhtQw6I6bJcpE98mXUZ9jocnvC4IQY,8148
+ninja/ninja_syntax.pyi,sha256=5IHH7N9CTKnMDDetFKHOsPK0SOSK_7Q4YtgzhIMNLE0,1576
+ninja/py.typed,sha256=47DEQpj8HBSa-_TImW-5JCeuQeRkm5NMpJWZG3hSuFU,0

lib/python3.12/site-packages/ninja-1.13.0.dist-info/WHEEL

@@ -0,0 +1,6 @@
+Wheel-Version: 1.0
+Generator: scikit-build-core 0.11.5
+Root-Is-Purelib: false
+Tag: py3-none-manylinux_2_17_x86_64
+Tag: py3-none-manylinux2014_x86_64
+

lib/python3.12/site-packages/ninja-1.13.0.dist-info/licenses/AUTHORS.rst

@@ -0,0 +1,5 @@
+=======
+Credits
+=======
+
+Please see the GitHub project page at https://github.com/scikit-build/ninja-python-distributions/graphs/contributors

lib/python3.12/site-packages/ninja-1.13.0.dist-info/licenses/LICENSE_Apache_20

@@ -0,0 +1,191 @@
+Apache License
+Version 2.0, January 2004
+http://www.apache.org/licenses/
+
+TERMS AND CONDITIONS FOR USE, REPRODUCTION, AND DISTRIBUTION
+
+1. Definitions.
+
+"License" shall mean the terms and conditions for use, reproduction, and
+distribution as defined by Sections 1 through 9 of this document.
+
+"Licensor" shall mean the copyright owner or entity authorized by the copyright
+owner that is granting the License.
+
+"Legal Entity" shall mean the union of the acting entity and all other entities
+that control, are controlled by, or are under common control with that entity.
+For the purposes of this definition, "control" means (i) the power, direct or
+indirect, to cause the direction or management of such entity, whether by
+contract or otherwise, or (ii) ownership of fifty percent (50%) or more of the
+outstanding shares, or (iii) beneficial ownership of such entity.
+
+"You" (or "Your") shall mean an individual or Legal Entity exercising
+permissions granted by this License.
+
+"Source" form shall mean the preferred form for making modifications, including
+but not limited to software source code, documentation source, and configuration
+files.
+
+"Object" form shall mean any form resulting from mechanical transformation or
+translation of a Source form, including but not limited to compiled object code,
+generated documentation, and conversions to other media types.
+
+"Work" shall mean the work of authorship, whether in Source or Object form, made
+available under the License, as indicated by a copyright notice that is included
+in or attached to the work (an example is provided in the Appendix below).
+
+"Derivative Works" shall mean any work, whether in Source or Object form, that
+is based on (or derived from) the Work and for which the editorial revisions,
+annotations, elaborations, or other modifications represent, as a whole, an
+original work of authorship. For the purposes of this License, Derivative Works
+shall not include works that remain separable from, or merely link (or bind by
+name) to the interfaces of, the Work and Derivative Works thereof.
+
+"Contribution" shall mean any work of authorship, including the original version
+of the Work and any modifications or additions to that Work or Derivative Works
+thereof, that is intentionally submitted to Licensor for inclusion in the Work
+by the copyright owner or by an individual or Legal Entity authorized to submit
+on behalf of the copyright owner. For the purposes of this definition,
+"submitted" means any form of electronic, verbal, or written communication sent
+to the Licensor or its representatives, including but not limited to
+communication on electronic mailing lists, source code control systems, and
+issue tracking systems that are managed by, or on behalf of, the Licensor for
+the purpose of discussing and improving the Work, but excluding communication
+that is conspicuously marked or otherwise designated in writing by the copyright
+owner as "Not a Contribution."
+
+"Contributor" shall mean Licensor and any individual or Legal Entity on behalf
+of whom a Contribution has been received by Licensor and subsequently
+incorporated within the Work.
+
+2. Grant of Copyright License.
+
+Subject to the terms and conditions of this License, each Contributor hereby
+grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free,
+irrevocable copyright license to reproduce, prepare Derivative Works of,
+publicly display, publicly perform, sublicense, and distribute the Work and such
+Derivative Works in Source or Object form.
+
+3. Grant of Patent License.
+
+Subject to the terms and conditions of this License, each Contributor hereby
+grants to You a perpetual, worldwide, non-exclusive, no-charge, royalty-free,
+irrevocable (except as stated in this section) patent license to make, have
+made, use, offer to sell, sell, import, and otherwise transfer the Work, where
+such license applies only to those patent claims licensable by such Contributor
+that are necessarily infringed by their Contribution(s) alone or by combination
+of their Contribution(s) with the Work to which such Contribution(s) was
+submitted. If You institute patent litigation against any entity (including a
+cross-claim or counterclaim in a lawsuit) alleging that the Work or a
+Contribution incorporated within the Work constitutes direct or contributory
+patent infringement, then any patent licenses granted to You under this License
+for that Work shall terminate as of the date such litigation is filed.
+
+4. Redistribution.
+
+You may reproduce and distribute copies of the Work or Derivative Works thereof
+in any medium, with or without modifications, and in Source or Object form,
+provided that You meet the following conditions:
+
+You must give any other recipients of the Work or Derivative Works a copy of
+this License; and
+You must cause any modified files to carry prominent notices stating that You
+changed the files; and
+You must retain, in the Source form of any Derivative Works that You distribute,
+all copyright, patent, trademark, and attribution notices from the Source form
+of the Work, excluding those notices that do not pertain to any part of the
+Derivative Works; and
+If the Work includes a "NOTICE" text file as part of its distribution, then any
+Derivative Works that You distribute must include a readable copy of the
+attribution notices contained within such NOTICE file, excluding those notices
+that do not pertain to any part of the Derivative Works, in at least one of the
+following places: within a NOTICE text file distributed as part of the
+Derivative Works; within the Source form or documentation, if provided along
+with the Derivative Works; or, within a display generated by the Derivative
+Works, if and wherever such third-party notices normally appear. The contents of
+the NOTICE file are for informational purposes only and do not modify the
+License. You may add Your own attribution notices within Derivative Works that
+You distribute, alongside or as an addendum to the NOTICE text from the Work,
+provided that such additional attribution notices cannot be construed as
+modifying the License.
+You may add Your own copyright statement to Your modifications and may provide
+additional or different license terms and conditions for use, reproduction, or
+distribution of Your modifications, or for any such Derivative Works as a whole,
+provided Your use, reproduction, and distribution of the Work otherwise complies
+with the conditions stated in this License.
+
+5. Submission of Contributions.
+
+Unless You explicitly state otherwise, any Contribution intentionally submitted
+for inclusion in the Work by You to the Licensor shall be under the terms and
+conditions of this License, without any additional terms or conditions.
+Notwithstanding the above, nothing herein shall supersede or modify the terms of
+any separate license agreement you may have executed with Licensor regarding
+such Contributions.
+
+6. Trademarks.
+
+This License does not grant permission to use the trade names, trademarks,
+service marks, or product names of the Licensor, except as required for
+reasonable and customary use in describing the origin of the Work and
+reproducing the content of the NOTICE file.
+
+7. Disclaimer of Warranty.
+
+Unless required by applicable law or agreed to in writing, Licensor provides the
+Work (and each Contributor provides its Contributions) on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied,
+including, without limitation, any warranties or conditions of TITLE,
+NON-INFRINGEMENT, MERCHANTABILITY, or FITNESS FOR A PARTICULAR PURPOSE. You are
+solely responsible for determining the appropriateness of using or
+redistributing the Work and assume any risks associated with Your exercise of
+permissions under this License.
+
+8. Limitation of Liability.
+
+In no event and under no legal theory, whether in tort (including negligence),
+contract, or otherwise, unless required by applicable law (such as deliberate
+and grossly negligent acts) or agreed to in writing, shall any Contributor be
+liable to You for damages, including any direct, indirect, special, incidental,
+or consequential damages of any character arising as a result of this License or
+out of the use or inability to use the Work (including but not limited to
+damages for loss of goodwill, work stoppage, computer failure or malfunction, or
+any and all other commercial damages or losses), even if such Contributor has
+been advised of the possibility of such damages.
+
+9. Accepting Warranty or Additional Liability.
+
+While redistributing the Work or Derivative Works thereof, You may choose to
+offer, and charge a fee for, acceptance of support, warranty, indemnity, or
+other liability obligations and/or rights consistent with this License. However,
+in accepting such obligations, You may act only on Your own behalf and on Your
+sole responsibility, not on behalf of any other Contributor, and only if You
+agree to indemnify, defend, and hold each Contributor harmless for any liability
+incurred by, or claims asserted against, such Contributor by reason of your
+accepting any such warranty or additional liability.
+
+END OF TERMS AND CONDITIONS
+
+APPENDIX: How to apply the Apache License to your work
+
+To apply the Apache License to your work, attach the following boilerplate
+notice, with the fields enclosed by brackets "[]" replaced with your own
+identifying information. (Don't include the brackets!) The text should be
+enclosed in the appropriate comment syntax for the file format. We also
+recommend that a file or class name and description of purpose be included on
+the same "printed page" as the copyright notice for easier identification within
+third-party archives.
+
+   Copyright [yyyy] [name of copyright owner]
+
+   Licensed under the Apache License, Version 2.0 (the "License");
+   you may not use this file except in compliance with the License.
+   You may obtain a copy of the License at
+
+     http://www.apache.org/licenses/LICENSE-2.0
+
+   Unless required by applicable law or agreed to in writing, software
+   distributed under the License is distributed on an "AS IS" BASIS,
+   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+   See the License for the specific language governing permissions and
+   limitations under the License.

lib/python3.12/site-packages/numpy/__config__.py

@@ -0,0 +1,162 @@
+# This file is generated by numpy's build process
+# It contains system_info results at the time of building this package.
+from enum import Enum
+from numpy.core._multiarray_umath import (
+    __cpu_features__,
+    __cpu_baseline__,
+    __cpu_dispatch__,
+)
+
+__all__ = ["show"]
+_built_with_meson = True
+
+
+class DisplayModes(Enum):
+    stdout = "stdout"
+    dicts = "dicts"
+
+
+def _cleanup(d):
+    """
+    Removes empty values in a `dict` recursively
+    This ensures we remove values that Meson could not provide to CONFIG
+    """
+    if isinstance(d, dict):
+        return {k: _cleanup(v) for k, v in d.items() if v and _cleanup(v)}
+    else:
+        return d
+
+
+CONFIG = _cleanup(
+    {
+        "Compilers": {
+            "c": {
+                "name": "gcc",
+                "linker": r"ld.bfd",
+                "version": "10.2.1",
+                "commands": r"cc",
+                "args": r"-fno-strict-aliasing",
+                "linker args": r"-Wl,--strip-debug, -fno-strict-aliasing",
+            },
+            "cython": {
+                "name": "cython",
+                "linker": r"cython",
+                "version": "3.0.8",
+                "commands": r"cython",
+                "args": r"",
+                "linker args": r"",
+            },
+            "c++": {
+                "name": "gcc",
+                "linker": r"ld.bfd",
+                "version": "10.2.1",
+                "commands": r"c++",
+                "args": r"",
+                "linker args": r"-Wl,--strip-debug",
+            },
+        },
+        "Machine Information": {
+            "host": {
+                "cpu": "x86_64",
+                "family": "x86_64",
+                "endian": "little",
+                "system": "linux",
+            },
+            "build": {
+                "cpu": "x86_64",
+                "family": "x86_64",
+                "endian": "little",
+                "system": "linux",
+            },
+            "cross-compiled": bool("False".lower().replace("false", "")),
+        },
+        "Build Dependencies": {
+            "blas": {
+                "name": "openblas64",
+                "found": bool("True".lower().replace("false", "")),
+                "version": "0.3.23.dev",
+                "detection method": "pkgconfig",
+                "include directory": r"/usr/local/include",
+                "lib directory": r"/usr/local/lib",
+                "openblas configuration": r"USE_64BITINT=1 DYNAMIC_ARCH=1 DYNAMIC_OLDER= NO_CBLAS= NO_LAPACK= NO_LAPACKE= NO_AFFINITY=1 USE_OPENMP= HASWELL MAX_THREADS=2",
+                "pc file directory": r"/usr/local/lib/pkgconfig",
+            },
+            "lapack": {
+                "name": "dep140551260102944",
+                "found": bool("True".lower().replace("false", "")),
+                "version": "1.26.4",
+                "detection method": "internal",
+                "include directory": r"unknown",
+                "lib directory": r"unknown",
+                "openblas configuration": r"unknown",
+                "pc file directory": r"unknown",
+            },
+        },
+        "Python Information": {
+            "path": r"/opt/python/cp312-cp312/bin/python",
+            "version": "3.12",
+        },
+        "SIMD Extensions": {
+            "baseline": __cpu_baseline__,
+            "found": [
+                feature for feature in __cpu_dispatch__ if __cpu_features__[feature]
+            ],
+            "not found": [
+                feature for feature in __cpu_dispatch__ if not __cpu_features__[feature]
+            ],
+        },
+    }
+)
+
+
+def _check_pyyaml():
+    import yaml
+
+    return yaml
+
+
+def show(mode=DisplayModes.stdout.value):
+    """
+    Show libraries and system information on which NumPy was built
+    and is being used
+
+    Parameters
+    ----------
+    mode : {`'stdout'`, `'dicts'`}, optional.
+        Indicates how to display the config information.
+        `'stdout'` prints to console, `'dicts'` returns a dictionary
+        of the configuration.
+
+    Returns
+    -------
+    out : {`dict`, `None`}
+        If mode is `'dicts'`, a dict is returned, else None
+
+    See Also
+    --------
+    get_include : Returns the directory containing NumPy C
+                  header files.
+
+    Notes
+    -----
+    1. The `'stdout'` mode will give more readable
+       output if ``pyyaml`` is installed
+
+    """
+    if mode == DisplayModes.stdout.value:
+        try:  # Non-standard library, check import
+            yaml = _check_pyyaml()
+
+            print(yaml.dump(CONFIG))
+        except ModuleNotFoundError:
+            import warnings
+            import json
+
+            warnings.warn("Install `pyyaml` for better output", stacklevel=1)
+            print(json.dumps(CONFIG, indent=2))
+    elif mode == DisplayModes.dicts.value:
+        return CONFIG
+    else:
+        raise AttributeError(
+            f"Invalid `mode`, use one of: {', '.join([e.value for e in DisplayModes])}"
+        )

lib/python3.12/site-packages/numpy/__init__.cython-30.pxd

@@ -0,0 +1,1050 @@
+# NumPy static imports for Cython >= 3.0
+#
+# If any of the PyArray_* functions are called, import_array must be
+# called first.  This is done automatically by Cython 3.0+ if a call
+# is not detected inside of the module.
+#
+# Author: Dag Sverre Seljebotn
+#
+
+from cpython.ref cimport Py_INCREF
+from cpython.object cimport PyObject, PyTypeObject, PyObject_TypeCheck
+cimport libc.stdio as stdio
+
+
+cdef extern from *:
+    # Leave a marker that the NumPy declarations came from NumPy itself and not from Cython.
+    # See https://github.com/cython/cython/issues/3573
+    """
+    /* Using NumPy API declarations from "numpy/__init__.cython-30.pxd" */
+    """
+
+
+cdef extern from "Python.h":
+    ctypedef int Py_intptr_t
+
+cdef extern from "numpy/arrayobject.h":
+    ctypedef Py_intptr_t npy_intp
+    ctypedef size_t npy_uintp
+
+    cdef enum NPY_TYPES:
+        NPY_BOOL
+        NPY_BYTE
+        NPY_UBYTE
+        NPY_SHORT
+        NPY_USHORT
+        NPY_INT
+        NPY_UINT
+        NPY_LONG
+        NPY_ULONG
+        NPY_LONGLONG
+        NPY_ULONGLONG
+        NPY_FLOAT
+        NPY_DOUBLE
+        NPY_LONGDOUBLE
+        NPY_CFLOAT
+        NPY_CDOUBLE
+        NPY_CLONGDOUBLE
+        NPY_OBJECT
+        NPY_STRING
+        NPY_UNICODE
+        NPY_VOID
+        NPY_DATETIME
+        NPY_TIMEDELTA
+        NPY_NTYPES
+        NPY_NOTYPE
+
+        NPY_INT8
+        NPY_INT16
+        NPY_INT32
+        NPY_INT64
+        NPY_INT128
+        NPY_INT256
+        NPY_UINT8
+        NPY_UINT16
+        NPY_UINT32
+        NPY_UINT64
+        NPY_UINT128
+        NPY_UINT256
+        NPY_FLOAT16
+        NPY_FLOAT32
+        NPY_FLOAT64
+        NPY_FLOAT80
+        NPY_FLOAT96
+        NPY_FLOAT128
+        NPY_FLOAT256
+        NPY_COMPLEX32
+        NPY_COMPLEX64
+        NPY_COMPLEX128
+        NPY_COMPLEX160
+        NPY_COMPLEX192
+        NPY_COMPLEX256
+        NPY_COMPLEX512
+
+        NPY_INTP
+
+    ctypedef enum NPY_ORDER:
+        NPY_ANYORDER
+        NPY_CORDER
+        NPY_FORTRANORDER
+        NPY_KEEPORDER
+
+    ctypedef enum NPY_CASTING:
+        NPY_NO_CASTING
+        NPY_EQUIV_CASTING
+        NPY_SAFE_CASTING
+        NPY_SAME_KIND_CASTING
+        NPY_UNSAFE_CASTING
+
+    ctypedef enum NPY_CLIPMODE:
+        NPY_CLIP
+        NPY_WRAP
+        NPY_RAISE
+
+    ctypedef enum NPY_SCALARKIND:
+        NPY_NOSCALAR,
+        NPY_BOOL_SCALAR,
+        NPY_INTPOS_SCALAR,
+        NPY_INTNEG_SCALAR,
+        NPY_FLOAT_SCALAR,
+        NPY_COMPLEX_SCALAR,
+        NPY_OBJECT_SCALAR
+
+    ctypedef enum NPY_SORTKIND:
+        NPY_QUICKSORT
+        NPY_HEAPSORT
+        NPY_MERGESORT
+
+    ctypedef enum NPY_SEARCHSIDE:
+        NPY_SEARCHLEFT
+        NPY_SEARCHRIGHT
+
+    enum:
+        # DEPRECATED since NumPy 1.7 ! Do not use in new code!
+        NPY_C_CONTIGUOUS
+        NPY_F_CONTIGUOUS
+        NPY_CONTIGUOUS
+        NPY_FORTRAN
+        NPY_OWNDATA
+        NPY_FORCECAST
+        NPY_ENSURECOPY
+        NPY_ENSUREARRAY
+        NPY_ELEMENTSTRIDES
+        NPY_ALIGNED
+        NPY_NOTSWAPPED
+        NPY_WRITEABLE
+        NPY_ARR_HAS_DESCR
+
+        NPY_BEHAVED
+        NPY_BEHAVED_NS
+        NPY_CARRAY
+        NPY_CARRAY_RO
+        NPY_FARRAY
+        NPY_FARRAY_RO
+        NPY_DEFAULT
+
+        NPY_IN_ARRAY
+        NPY_OUT_ARRAY
+        NPY_INOUT_ARRAY
+        NPY_IN_FARRAY
+        NPY_OUT_FARRAY
+        NPY_INOUT_FARRAY
+
+        NPY_UPDATE_ALL
+
+    enum:
+        # Added in NumPy 1.7 to replace the deprecated enums above.
+        NPY_ARRAY_C_CONTIGUOUS
+        NPY_ARRAY_F_CONTIGUOUS
+        NPY_ARRAY_OWNDATA
+        NPY_ARRAY_FORCECAST
+        NPY_ARRAY_ENSURECOPY
+        NPY_ARRAY_ENSUREARRAY
+        NPY_ARRAY_ELEMENTSTRIDES
+        NPY_ARRAY_ALIGNED
+        NPY_ARRAY_NOTSWAPPED
+        NPY_ARRAY_WRITEABLE
+        NPY_ARRAY_WRITEBACKIFCOPY
+
+        NPY_ARRAY_BEHAVED
+        NPY_ARRAY_BEHAVED_NS
+        NPY_ARRAY_CARRAY
+        NPY_ARRAY_CARRAY_RO
+        NPY_ARRAY_FARRAY
+        NPY_ARRAY_FARRAY_RO
+        NPY_ARRAY_DEFAULT
+
+        NPY_ARRAY_IN_ARRAY
+        NPY_ARRAY_OUT_ARRAY
+        NPY_ARRAY_INOUT_ARRAY
+        NPY_ARRAY_IN_FARRAY
+        NPY_ARRAY_OUT_FARRAY
+        NPY_ARRAY_INOUT_FARRAY
+
+        NPY_ARRAY_UPDATE_ALL
+
+    cdef enum:
+        NPY_MAXDIMS
+
+    npy_intp NPY_MAX_ELSIZE
+
+    ctypedef void (*PyArray_VectorUnaryFunc)(void *, void *, npy_intp, void *,  void *)
+
+    ctypedef struct PyArray_ArrayDescr:
+        # shape is a tuple, but Cython doesn't support "tuple shape"
+        # inside a non-PyObject declaration, so we have to declare it
+        # as just a PyObject*.
+        PyObject* shape
+
+    ctypedef struct PyArray_Descr:
+        pass
+
+    ctypedef class numpy.dtype [object PyArray_Descr, check_size ignore]:
+        # Use PyDataType_* macros when possible, however there are no macros
+        # for accessing some of the fields, so some are defined.
+        cdef PyTypeObject* typeobj
+        cdef char kind
+        cdef char type
+        # Numpy sometimes mutates this without warning (e.g. it'll
+        # sometimes change "|" to "<" in shared dtype objects on
+        # little-endian machines). If this matters to you, use
+        # PyArray_IsNativeByteOrder(dtype.byteorder) instead of
+        # directly accessing this field.
+        cdef char byteorder
+        cdef char flags
+        cdef int type_num
+        cdef int itemsize "elsize"
+        cdef int alignment
+        cdef object fields
+        cdef tuple names
+        # Use PyDataType_HASSUBARRAY to test whether this field is
+        # valid (the pointer can be NULL). Most users should access
+        # this field via the inline helper method PyDataType_SHAPE.
+        cdef PyArray_ArrayDescr* subarray
+
+    ctypedef class numpy.flatiter [object PyArrayIterObject, check_size ignore]:
+        # Use through macros
+        pass
+
+    ctypedef class numpy.broadcast [object PyArrayMultiIterObject, check_size ignore]:
+        # Use through macros
+        pass
+
+    ctypedef struct PyArrayObject:
+        # For use in situations where ndarray can't replace PyArrayObject*,
+        # like PyArrayObject**.
+        pass
+
+    ctypedef class numpy.ndarray [object PyArrayObject, check_size ignore]:
+        cdef __cythonbufferdefaults__ = {"mode": "strided"}
+
+        # NOTE: no field declarations since direct access is deprecated since NumPy 1.7
+        # Instead, we use properties that map to the corresponding C-API functions.
+
+        @property
+        cdef inline PyObject* base(self) nogil:
+            """Returns a borrowed reference to the object owning the data/memory.
+            """
+            return PyArray_BASE(self)
+
+        @property
+        cdef inline dtype descr(self):
+            """Returns an owned reference to the dtype of the array.
+            """
+            return <dtype>PyArray_DESCR(self)
+
+        @property
+        cdef inline int ndim(self) nogil:
+            """Returns the number of dimensions in the array.
+            """
+            return PyArray_NDIM(self)
+
+        @property
+        cdef inline npy_intp *shape(self) nogil:
+            """Returns a pointer to the dimensions/shape of the array.
+            The number of elements matches the number of dimensions of the array (ndim).
+            Can return NULL for 0-dimensional arrays.
+            """
+            return PyArray_DIMS(self)
+
+        @property
+        cdef inline npy_intp *strides(self) nogil:
+            """Returns a pointer to the strides of the array.
+            The number of elements matches the number of dimensions of the array (ndim).
+            """
+            return PyArray_STRIDES(self)
+
+        @property
+        cdef inline npy_intp size(self) nogil:
+            """Returns the total size (in number of elements) of the array.
+            """
+            return PyArray_SIZE(self)
+
+        @property
+        cdef inline char* data(self) nogil:
+            """The pointer to the data buffer as a char*.
+            This is provided for legacy reasons to avoid direct struct field access.
+            For new code that needs this access, you probably want to cast the result
+            of `PyArray_DATA()` instead, which returns a 'void*'.
+            """
+            return PyArray_BYTES(self)
+
+    ctypedef unsigned char      npy_bool
+
+    ctypedef signed char      npy_byte
+    ctypedef signed short     npy_short
+    ctypedef signed int       npy_int
+    ctypedef signed long      npy_long
+    ctypedef signed long long npy_longlong
+
+    ctypedef unsigned char      npy_ubyte
+    ctypedef unsigned short     npy_ushort
+    ctypedef unsigned int       npy_uint
+    ctypedef unsigned long      npy_ulong
+    ctypedef unsigned long long npy_ulonglong
+
+    ctypedef float        npy_float
+    ctypedef double       npy_double
+    ctypedef long double  npy_longdouble
+
+    ctypedef signed char        npy_int8
+    ctypedef signed short       npy_int16
+    ctypedef signed int         npy_int32
+    ctypedef signed long long   npy_int64
+    ctypedef signed long long   npy_int96
+    ctypedef signed long long   npy_int128
+
+    ctypedef unsigned char      npy_uint8
+    ctypedef unsigned short     npy_uint16
+    ctypedef unsigned int       npy_uint32
+    ctypedef unsigned long long npy_uint64
+    ctypedef unsigned long long npy_uint96
+    ctypedef unsigned long long npy_uint128
+
+    ctypedef float        npy_float32
+    ctypedef double       npy_float64
+    ctypedef long double  npy_float80
+    ctypedef long double  npy_float96
+    ctypedef long double  npy_float128
+
+    ctypedef struct npy_cfloat:
+        float real
+        float imag
+
+    ctypedef struct npy_cdouble:
+        double real
+        double imag
+
+    ctypedef struct npy_clongdouble:
+        long double real
+        long double imag
+
+    ctypedef struct npy_complex64:
+        float real
+        float imag
+
+    ctypedef struct npy_complex128:
+        double real
+        double imag
+
+    ctypedef struct npy_complex160:
+        long double real
+        long double imag
+
+    ctypedef struct npy_complex192:
+        long double real
+        long double imag
+
+    ctypedef struct npy_complex256:
+        long double real
+        long double imag
+
+    ctypedef struct PyArray_Dims:
+        npy_intp *ptr
+        int len
+
+    int _import_array() except -1
+    # A second definition so _import_array isn't marked as used when we use it here.
+    # Do not use - subject to change any time.
+    int __pyx_import_array "_import_array"() except -1
+
+    #
+    # Macros from ndarrayobject.h
+    #
+    bint PyArray_CHKFLAGS(ndarray m, int flags) nogil
+    bint PyArray_IS_C_CONTIGUOUS(ndarray arr) nogil
+    bint PyArray_IS_F_CONTIGUOUS(ndarray arr) nogil
+    bint PyArray_ISCONTIGUOUS(ndarray m) nogil
+    bint PyArray_ISWRITEABLE(ndarray m) nogil
+    bint PyArray_ISALIGNED(ndarray m) nogil
+
+    int PyArray_NDIM(ndarray) nogil
+    bint PyArray_ISONESEGMENT(ndarray) nogil
+    bint PyArray_ISFORTRAN(ndarray) nogil
+    int PyArray_FORTRANIF(ndarray) nogil
+
+    void* PyArray_DATA(ndarray) nogil
+    char* PyArray_BYTES(ndarray) nogil
+
+    npy_intp* PyArray_DIMS(ndarray) nogil
+    npy_intp* PyArray_STRIDES(ndarray) nogil
+    npy_intp PyArray_DIM(ndarray, size_t) nogil
+    npy_intp PyArray_STRIDE(ndarray, size_t) nogil
+
+    PyObject *PyArray_BASE(ndarray) nogil  # returns borrowed reference!
+    PyArray_Descr *PyArray_DESCR(ndarray) nogil  # returns borrowed reference to dtype!
+    PyArray_Descr *PyArray_DTYPE(ndarray) nogil  # returns borrowed reference to dtype! NP 1.7+ alias for descr.
+    int PyArray_FLAGS(ndarray) nogil
+    void PyArray_CLEARFLAGS(ndarray, int flags) nogil  # Added in NumPy 1.7
+    void PyArray_ENABLEFLAGS(ndarray, int flags) nogil  # Added in NumPy 1.7
+    npy_intp PyArray_ITEMSIZE(ndarray) nogil
+    int PyArray_TYPE(ndarray arr) nogil
+
+    object PyArray_GETITEM(ndarray arr, void *itemptr)
+    int PyArray_SETITEM(ndarray arr, void *itemptr, object obj) except -1
+
+    bint PyTypeNum_ISBOOL(int) nogil
+    bint PyTypeNum_ISUNSIGNED(int) nogil
+    bint PyTypeNum_ISSIGNED(int) nogil
+    bint PyTypeNum_ISINTEGER(int) nogil
+    bint PyTypeNum_ISFLOAT(int) nogil
+    bint PyTypeNum_ISNUMBER(int) nogil
+    bint PyTypeNum_ISSTRING(int) nogil
+    bint PyTypeNum_ISCOMPLEX(int) nogil
+    bint PyTypeNum_ISPYTHON(int) nogil
+    bint PyTypeNum_ISFLEXIBLE(int) nogil
+    bint PyTypeNum_ISUSERDEF(int) nogil
+    bint PyTypeNum_ISEXTENDED(int) nogil
+    bint PyTypeNum_ISOBJECT(int) nogil
+
+    bint PyDataType_ISBOOL(dtype) nogil
+    bint PyDataType_ISUNSIGNED(dtype) nogil
+    bint PyDataType_ISSIGNED(dtype) nogil
+    bint PyDataType_ISINTEGER(dtype) nogil
+    bint PyDataType_ISFLOAT(dtype) nogil
+    bint PyDataType_ISNUMBER(dtype) nogil
+    bint PyDataType_ISSTRING(dtype) nogil
+    bint PyDataType_ISCOMPLEX(dtype) nogil
+    bint PyDataType_ISPYTHON(dtype) nogil
+    bint PyDataType_ISFLEXIBLE(dtype) nogil
+    bint PyDataType_ISUSERDEF(dtype) nogil
+    bint PyDataType_ISEXTENDED(dtype) nogil
+    bint PyDataType_ISOBJECT(dtype) nogil
+    bint PyDataType_HASFIELDS(dtype) nogil
+    bint PyDataType_HASSUBARRAY(dtype) nogil
+
+    bint PyArray_ISBOOL(ndarray) nogil
+    bint PyArray_ISUNSIGNED(ndarray) nogil
+    bint PyArray_ISSIGNED(ndarray) nogil
+    bint PyArray_ISINTEGER(ndarray) nogil
+    bint PyArray_ISFLOAT(ndarray) nogil
+    bint PyArray_ISNUMBER(ndarray) nogil
+    bint PyArray_ISSTRING(ndarray) nogil
+    bint PyArray_ISCOMPLEX(ndarray) nogil
+    bint PyArray_ISPYTHON(ndarray) nogil
+    bint PyArray_ISFLEXIBLE(ndarray) nogil
+    bint PyArray_ISUSERDEF(ndarray) nogil
+    bint PyArray_ISEXTENDED(ndarray) nogil
+    bint PyArray_ISOBJECT(ndarray) nogil
+    bint PyArray_HASFIELDS(ndarray) nogil
+
+    bint PyArray_ISVARIABLE(ndarray) nogil
+
+    bint PyArray_SAFEALIGNEDCOPY(ndarray) nogil
+    bint PyArray_ISNBO(char) nogil              # works on ndarray.byteorder
+    bint PyArray_IsNativeByteOrder(char) nogil # works on ndarray.byteorder
+    bint PyArray_ISNOTSWAPPED(ndarray) nogil
+    bint PyArray_ISBYTESWAPPED(ndarray) nogil
+
+    bint PyArray_FLAGSWAP(ndarray, int) nogil
+
+    bint PyArray_ISCARRAY(ndarray) nogil
+    bint PyArray_ISCARRAY_RO(ndarray) nogil
+    bint PyArray_ISFARRAY(ndarray) nogil
+    bint PyArray_ISFARRAY_RO(ndarray) nogil
+    bint PyArray_ISBEHAVED(ndarray) nogil
+    bint PyArray_ISBEHAVED_RO(ndarray) nogil
+
+
+    bint PyDataType_ISNOTSWAPPED(dtype) nogil
+    bint PyDataType_ISBYTESWAPPED(dtype) nogil
+
+    bint PyArray_DescrCheck(object)
+
+    bint PyArray_Check(object)
+    bint PyArray_CheckExact(object)
+
+    # Cannot be supported due to out arg:
+    # bint PyArray_HasArrayInterfaceType(object, dtype, object, object&)
+    # bint PyArray_HasArrayInterface(op, out)
+
+
+    bint PyArray_IsZeroDim(object)
+    # Cannot be supported due to ## ## in macro:
+    # bint PyArray_IsScalar(object, verbatim work)
+    bint PyArray_CheckScalar(object)
+    bint PyArray_IsPythonNumber(object)
+    bint PyArray_IsPythonScalar(object)
+    bint PyArray_IsAnyScalar(object)
+    bint PyArray_CheckAnyScalar(object)
+
+    ndarray PyArray_GETCONTIGUOUS(ndarray)
+    bint PyArray_SAMESHAPE(ndarray, ndarray) nogil
+    npy_intp PyArray_SIZE(ndarray) nogil
+    npy_intp PyArray_NBYTES(ndarray) nogil
+
+    object PyArray_FROM_O(object)
+    object PyArray_FROM_OF(object m, int flags)
+    object PyArray_FROM_OT(object m, int type)
+    object PyArray_FROM_OTF(object m, int type, int flags)
+    object PyArray_FROMANY(object m, int type, int min, int max, int flags)
+    object PyArray_ZEROS(int nd, npy_intp* dims, int type, int fortran)
+    object PyArray_EMPTY(int nd, npy_intp* dims, int type, int fortran)
+    void PyArray_FILLWBYTE(object, int val)
+    npy_intp PyArray_REFCOUNT(object)
+    object PyArray_ContiguousFromAny(op, int, int min_depth, int max_depth)
+    unsigned char PyArray_EquivArrTypes(ndarray a1, ndarray a2)
+    bint PyArray_EquivByteorders(int b1, int b2) nogil
+    object PyArray_SimpleNew(int nd, npy_intp* dims, int typenum)
+    object PyArray_SimpleNewFromData(int nd, npy_intp* dims, int typenum, void* data)
+    #object PyArray_SimpleNewFromDescr(int nd, npy_intp* dims, dtype descr)
+    object PyArray_ToScalar(void* data, ndarray arr)
+
+    void* PyArray_GETPTR1(ndarray m, npy_intp i) nogil
+    void* PyArray_GETPTR2(ndarray m, npy_intp i, npy_intp j) nogil
+    void* PyArray_GETPTR3(ndarray m, npy_intp i, npy_intp j, npy_intp k) nogil
+    void* PyArray_GETPTR4(ndarray m, npy_intp i, npy_intp j, npy_intp k, npy_intp l) nogil
+
+    # Cannot be supported due to out arg
+    # void PyArray_DESCR_REPLACE(descr)
+
+
+    object PyArray_Copy(ndarray)
+    object PyArray_FromObject(object op, int type, int min_depth, int max_depth)
+    object PyArray_ContiguousFromObject(object op, int type, int min_depth, int max_depth)
+    object PyArray_CopyFromObject(object op, int type, int min_depth, int max_depth)
+
+    object PyArray_Cast(ndarray mp, int type_num)
+    object PyArray_Take(ndarray ap, object items, int axis)
+    object PyArray_Put(ndarray ap, object items, object values)
+
+    void PyArray_ITER_RESET(flatiter it) nogil
+    void PyArray_ITER_NEXT(flatiter it) nogil
+    void PyArray_ITER_GOTO(flatiter it, npy_intp* destination) nogil
+    void PyArray_ITER_GOTO1D(flatiter it, npy_intp ind) nogil
+    void* PyArray_ITER_DATA(flatiter it) nogil
+    bint PyArray_ITER_NOTDONE(flatiter it) nogil
+
+    void PyArray_MultiIter_RESET(broadcast multi) nogil
+    void PyArray_MultiIter_NEXT(broadcast multi) nogil
+    void PyArray_MultiIter_GOTO(broadcast multi, npy_intp dest) nogil
+    void PyArray_MultiIter_GOTO1D(broadcast multi, npy_intp ind) nogil
+    void* PyArray_MultiIter_DATA(broadcast multi, npy_intp i) nogil
+    void PyArray_MultiIter_NEXTi(broadcast multi, npy_intp i) nogil
+    bint PyArray_MultiIter_NOTDONE(broadcast multi) nogil
+
+    # Functions from __multiarray_api.h
+
+    # Functions taking dtype and returning object/ndarray are disabled
+    # for now as they steal dtype references. I'm conservative and disable
+    # more than is probably needed until it can be checked further.
+    int PyArray_SetNumericOps (object) except -1
+    object PyArray_GetNumericOps ()
+    int PyArray_INCREF (ndarray) except *  # uses PyArray_Item_INCREF...
+    int PyArray_XDECREF (ndarray) except *  # uses PyArray_Item_DECREF...
+    void PyArray_SetStringFunction (object, int)
+    dtype PyArray_DescrFromType (int)
+    object PyArray_TypeObjectFromType (int)
+    char * PyArray_Zero (ndarray)
+    char * PyArray_One (ndarray)
+    #object PyArray_CastToType (ndarray, dtype, int)
+    int PyArray_CastTo (ndarray, ndarray) except -1
+    int PyArray_CastAnyTo (ndarray, ndarray) except -1
+    int PyArray_CanCastSafely (int, int)  # writes errors
+    npy_bool PyArray_CanCastTo (dtype, dtype)  # writes errors
+    int PyArray_ObjectType (object, int) except 0
+    dtype PyArray_DescrFromObject (object, dtype)
+    #ndarray* PyArray_ConvertToCommonType (object, int *)
+    dtype PyArray_DescrFromScalar (object)
+    dtype PyArray_DescrFromTypeObject (object)
+    npy_intp PyArray_Size (object)
+    #object PyArray_Scalar (void *, dtype, object)
+    #object PyArray_FromScalar (object, dtype)
+    void PyArray_ScalarAsCtype (object, void *)
+    #int PyArray_CastScalarToCtype (object, void *, dtype)
+    #int PyArray_CastScalarDirect (object, dtype, void *, int)
+    object PyArray_ScalarFromObject (object)
+    #PyArray_VectorUnaryFunc * PyArray_GetCastFunc (dtype, int)
+    object PyArray_FromDims (int, int *, int)
+    #object PyArray_FromDimsAndDataAndDescr (int, int *, dtype, char *)
+    #object PyArray_FromAny (object, dtype, int, int, int, object)
+    object PyArray_EnsureArray (object)
+    object PyArray_EnsureAnyArray (object)
+    #object PyArray_FromFile (stdio.FILE *, dtype, npy_intp, char *)
+    #object PyArray_FromString (char *, npy_intp, dtype, npy_intp, char *)
+    #object PyArray_FromBuffer (object, dtype, npy_intp, npy_intp)
+    #object PyArray_FromIter (object, dtype, npy_intp)
+    object PyArray_Return (ndarray)
+    #object PyArray_GetField (ndarray, dtype, int)
+    #int PyArray_SetField (ndarray, dtype, int, object) except -1
+    object PyArray_Byteswap (ndarray, npy_bool)
+    object PyArray_Resize (ndarray, PyArray_Dims *, int, NPY_ORDER)
+    int PyArray_MoveInto (ndarray, ndarray) except -1
+    int PyArray_CopyInto (ndarray, ndarray) except -1
+    int PyArray_CopyAnyInto (ndarray, ndarray) except -1
+    int PyArray_CopyObject (ndarray, object) except -1
+    object PyArray_NewCopy (ndarray, NPY_ORDER)
+    object PyArray_ToList (ndarray)
+    object PyArray_ToString (ndarray, NPY_ORDER)
+    int PyArray_ToFile (ndarray, stdio.FILE *, char *, char *) except -1
+    int PyArray_Dump (object, object, int) except -1
+    object PyArray_Dumps (object, int)
+    int PyArray_ValidType (int)  # Cannot error
+    void PyArray_UpdateFlags (ndarray, int)
+    object PyArray_New (type, int, npy_intp *, int, npy_intp *, void *, int, int, object)
+    #object PyArray_NewFromDescr (type, dtype, int, npy_intp *, npy_intp *, void *, int, object)
+    #dtype PyArray_DescrNew (dtype)
+    dtype PyArray_DescrNewFromType (int)
+    double PyArray_GetPriority (object, double)  # clears errors as of 1.25
+    object PyArray_IterNew (object)
+    object PyArray_MultiIterNew (int, ...)
+
+    int PyArray_PyIntAsInt (object) except? -1
+    npy_intp PyArray_PyIntAsIntp (object)
+    int PyArray_Broadcast (broadcast) except -1
+    void PyArray_FillObjectArray (ndarray, object) except *
+    int PyArray_FillWithScalar (ndarray, object) except -1
+    npy_bool PyArray_CheckStrides (int, int, npy_intp, npy_intp, npy_intp *, npy_intp *)
+    dtype PyArray_DescrNewByteorder (dtype, char)
+    object PyArray_IterAllButAxis (object, int *)
+    #object PyArray_CheckFromAny (object, dtype, int, int, int, object)
+    #object PyArray_FromArray (ndarray, dtype, int)
+    object PyArray_FromInterface (object)
+    object PyArray_FromStructInterface (object)
+    #object PyArray_FromArrayAttr (object, dtype, object)
+    #NPY_SCALARKIND PyArray_ScalarKind (int, ndarray*)
+    int PyArray_CanCoerceScalar (int, int, NPY_SCALARKIND)
+    object PyArray_NewFlagsObject (object)
+    npy_bool PyArray_CanCastScalar (type, type)
+    #int PyArray_CompareUCS4 (npy_ucs4 *, npy_ucs4 *, register size_t)
+    int PyArray_RemoveSmallest (broadcast) except -1
+    int PyArray_ElementStrides (object)
+    void PyArray_Item_INCREF (char *, dtype) except *
+    void PyArray_Item_XDECREF (char *, dtype) except *
+    object PyArray_FieldNames (object)
+    object PyArray_Transpose (ndarray, PyArray_Dims *)
+    object PyArray_TakeFrom (ndarray, object, int, ndarray, NPY_CLIPMODE)
+    object PyArray_PutTo (ndarray, object, object, NPY_CLIPMODE)
+    object PyArray_PutMask (ndarray, object, object)
+    object PyArray_Repeat (ndarray, object, int)
+    object PyArray_Choose (ndarray, object, ndarray, NPY_CLIPMODE)
+    int PyArray_Sort (ndarray, int, NPY_SORTKIND) except -1
+    object PyArray_ArgSort (ndarray, int, NPY_SORTKIND)
+    object PyArray_SearchSorted (ndarray, object, NPY_SEARCHSIDE, PyObject *)
+    object PyArray_ArgMax (ndarray, int, ndarray)
+    object PyArray_ArgMin (ndarray, int, ndarray)
+    object PyArray_Reshape (ndarray, object)
+    object PyArray_Newshape (ndarray, PyArray_Dims *, NPY_ORDER)
+    object PyArray_Squeeze (ndarray)
+    #object PyArray_View (ndarray, dtype, type)
+    object PyArray_SwapAxes (ndarray, int, int)
+    object PyArray_Max (ndarray, int, ndarray)
+    object PyArray_Min (ndarray, int, ndarray)
+    object PyArray_Ptp (ndarray, int, ndarray)
+    object PyArray_Mean (ndarray, int, int, ndarray)
+    object PyArray_Trace (ndarray, int, int, int, int, ndarray)
+    object PyArray_Diagonal (ndarray, int, int, int)
+    object PyArray_Clip (ndarray, object, object, ndarray)
+    object PyArray_Conjugate (ndarray, ndarray)
+    object PyArray_Nonzero (ndarray)
+    object PyArray_Std (ndarray, int, int, ndarray, int)
+    object PyArray_Sum (ndarray, int, int, ndarray)
+    object PyArray_CumSum (ndarray, int, int, ndarray)
+    object PyArray_Prod (ndarray, int, int, ndarray)
+    object PyArray_CumProd (ndarray, int, int, ndarray)
+    object PyArray_All (ndarray, int, ndarray)
+    object PyArray_Any (ndarray, int, ndarray)
+    object PyArray_Compress (ndarray, object, int, ndarray)
+    object PyArray_Flatten (ndarray, NPY_ORDER)
+    object PyArray_Ravel (ndarray, NPY_ORDER)
+    npy_intp PyArray_MultiplyList (npy_intp *, int)
+    int PyArray_MultiplyIntList (int *, int)
+    void * PyArray_GetPtr (ndarray, npy_intp*)
+    int PyArray_CompareLists (npy_intp *, npy_intp *, int)
+    #int PyArray_AsCArray (object*, void *, npy_intp *, int, dtype)
+    #int PyArray_As1D (object*, char **, int *, int)
+    #int PyArray_As2D (object*, char ***, int *, int *, int)
+    int PyArray_Free (object, void *)
+    #int PyArray_Converter (object, object*)
+    int PyArray_IntpFromSequence (object, npy_intp *, int) except -1
+    object PyArray_Concatenate (object, int)
+    object PyArray_InnerProduct (object, object)
+    object PyArray_MatrixProduct (object, object)
+    object PyArray_CopyAndTranspose (object)
+    object PyArray_Correlate (object, object, int)
+    int PyArray_TypestrConvert (int, int)
+    #int PyArray_DescrConverter (object, dtype*) except 0
+    #int PyArray_DescrConverter2 (object, dtype*) except 0
+    int PyArray_IntpConverter (object, PyArray_Dims *) except 0
+    #int PyArray_BufferConverter (object, chunk) except 0
+    int PyArray_AxisConverter (object, int *) except 0
+    int PyArray_BoolConverter (object, npy_bool *) except 0
+    int PyArray_ByteorderConverter (object, char *) except 0
+    int PyArray_OrderConverter (object, NPY_ORDER *) except 0
+    unsigned char PyArray_EquivTypes (dtype, dtype)  # clears errors
+    #object PyArray_Zeros (int, npy_intp *, dtype, int)
+    #object PyArray_Empty (int, npy_intp *, dtype, int)
+    object PyArray_Where (object, object, object)
+    object PyArray_Arange (double, double, double, int)
+    #object PyArray_ArangeObj (object, object, object, dtype)
+    int PyArray_SortkindConverter (object, NPY_SORTKIND *) except 0
+    object PyArray_LexSort (object, int)
+    object PyArray_Round (ndarray, int, ndarray)
+    unsigned char PyArray_EquivTypenums (int, int)
+    int PyArray_RegisterDataType (dtype) except -1
+    int PyArray_RegisterCastFunc (dtype, int, PyArray_VectorUnaryFunc *) except -1
+    int PyArray_RegisterCanCast (dtype, int, NPY_SCALARKIND) except -1
+    #void PyArray_InitArrFuncs (PyArray_ArrFuncs *)
+    object PyArray_IntTupleFromIntp (int, npy_intp *)
+    int PyArray_TypeNumFromName (char *)
+    int PyArray_ClipmodeConverter (object, NPY_CLIPMODE *) except 0
+    #int PyArray_OutputConverter (object, ndarray*) except 0
+    object PyArray_BroadcastToShape (object, npy_intp *, int)
+    void _PyArray_SigintHandler (int)
+    void* _PyArray_GetSigintBuf ()
+    #int PyArray_DescrAlignConverter (object, dtype*) except 0
+    #int PyArray_DescrAlignConverter2 (object, dtype*) except 0
+    int PyArray_SearchsideConverter (object, void *) except 0
+    object PyArray_CheckAxis (ndarray, int *, int)
+    npy_intp PyArray_OverflowMultiplyList (npy_intp *, int)
+    int PyArray_CompareString (char *, char *, size_t)
+    int PyArray_SetBaseObject(ndarray, base) except -1 # NOTE: steals a reference to base! Use "set_array_base()" instead.
+
+
+# Typedefs that matches the runtime dtype objects in
+# the numpy module.
+
+# The ones that are commented out needs an IFDEF function
+# in Cython to enable them only on the right systems.
+
+ctypedef npy_int8       int8_t
+ctypedef npy_int16      int16_t
+ctypedef npy_int32      int32_t
+ctypedef npy_int64      int64_t
+#ctypedef npy_int96      int96_t
+#ctypedef npy_int128     int128_t
+
+ctypedef npy_uint8      uint8_t
+ctypedef npy_uint16     uint16_t
+ctypedef npy_uint32     uint32_t
+ctypedef npy_uint64     uint64_t
+#ctypedef npy_uint96     uint96_t
+#ctypedef npy_uint128    uint128_t
+
+ctypedef npy_float32    float32_t
+ctypedef npy_float64    float64_t
+#ctypedef npy_float80    float80_t
+#ctypedef npy_float128   float128_t
+
+ctypedef float complex  complex64_t
+ctypedef double complex complex128_t
+
+# The int types are mapped a bit surprising --
+# numpy.int corresponds to 'l' and numpy.long to 'q'
+ctypedef npy_long       int_t
+ctypedef npy_longlong   longlong_t
+
+ctypedef npy_ulong      uint_t
+ctypedef npy_ulonglong  ulonglong_t
+
+ctypedef npy_intp       intp_t
+ctypedef npy_uintp      uintp_t
+
+ctypedef npy_double     float_t
+ctypedef npy_double     double_t
+ctypedef npy_longdouble longdouble_t
+
+ctypedef npy_cfloat      cfloat_t
+ctypedef npy_cdouble     cdouble_t
+ctypedef npy_clongdouble clongdouble_t
+
+ctypedef npy_cdouble     complex_t
+
+cdef inline object PyArray_MultiIterNew1(a):
+    return PyArray_MultiIterNew(1, <void*>a)
+
+cdef inline object PyArray_MultiIterNew2(a, b):
+    return PyArray_MultiIterNew(2, <void*>a, <void*>b)
+
+cdef inline object PyArray_MultiIterNew3(a, b, c):
+    return PyArray_MultiIterNew(3, <void*>a, <void*>b, <void*> c)
+
+cdef inline object PyArray_MultiIterNew4(a, b, c, d):
+    return PyArray_MultiIterNew(4, <void*>a, <void*>b, <void*>c, <void*> d)
+
+cdef inline object PyArray_MultiIterNew5(a, b, c, d, e):
+    return PyArray_MultiIterNew(5, <void*>a, <void*>b, <void*>c, <void*> d, <void*> e)
+
+cdef inline tuple PyDataType_SHAPE(dtype d):
+    if PyDataType_HASSUBARRAY(d):
+        return <tuple>d.subarray.shape
+    else:
+        return ()
+
+
+cdef extern from "numpy/ndarrayobject.h":
+    PyTypeObject PyTimedeltaArrType_Type
+    PyTypeObject PyDatetimeArrType_Type
+    ctypedef int64_t npy_timedelta
+    ctypedef int64_t npy_datetime
+
+cdef extern from "numpy/ndarraytypes.h":
+    ctypedef struct PyArray_DatetimeMetaData:
+        NPY_DATETIMEUNIT base
+        int64_t num
+
+cdef extern from "numpy/arrayscalars.h":
+
+    # abstract types
+    ctypedef class numpy.generic [object PyObject]:
+        pass
+    ctypedef class numpy.number [object PyObject]:
+        pass
+    ctypedef class numpy.integer [object PyObject]:
+        pass
+    ctypedef class numpy.signedinteger [object PyObject]:
+        pass
+    ctypedef class numpy.unsignedinteger [object PyObject]:
+        pass
+    ctypedef class numpy.inexact [object PyObject]:
+        pass
+    ctypedef class numpy.floating [object PyObject]:
+        pass
+    ctypedef class numpy.complexfloating [object PyObject]:
+        pass
+    ctypedef class numpy.flexible [object PyObject]:
+        pass
+    ctypedef class numpy.character [object PyObject]:
+        pass
+
+    ctypedef struct PyDatetimeScalarObject:
+        # PyObject_HEAD
+        npy_datetime obval
+        PyArray_DatetimeMetaData obmeta
+
+    ctypedef struct PyTimedeltaScalarObject:
+        # PyObject_HEAD
+        npy_timedelta obval
+        PyArray_DatetimeMetaData obmeta
+
+    ctypedef enum NPY_DATETIMEUNIT:
+        NPY_FR_Y
+        NPY_FR_M
+        NPY_FR_W
+        NPY_FR_D
+        NPY_FR_B
+        NPY_FR_h
+        NPY_FR_m
+        NPY_FR_s
+        NPY_FR_ms
+        NPY_FR_us
+        NPY_FR_ns
+        NPY_FR_ps
+        NPY_FR_fs
+        NPY_FR_as
+        NPY_FR_GENERIC
+
+
+#
+# ufunc API
+#
+
+cdef extern from "numpy/ufuncobject.h":
+
+    ctypedef void (*PyUFuncGenericFunction) (char **, npy_intp *, npy_intp *, void *)
+
+    ctypedef class numpy.ufunc [object PyUFuncObject, check_size ignore]:
+        cdef:
+            int nin, nout, nargs
+            int identity
+            PyUFuncGenericFunction *functions
+            void **data
+            int ntypes
+            int check_return
+            char *name
+            char *types
+            char *doc
+            void *ptr
+            PyObject *obj
+            PyObject *userloops
+
+    cdef enum:
+        PyUFunc_Zero
+        PyUFunc_One
+        PyUFunc_None
+        UFUNC_ERR_IGNORE
+        UFUNC_ERR_WARN
+        UFUNC_ERR_RAISE
+        UFUNC_ERR_CALL
+        UFUNC_ERR_PRINT
+        UFUNC_ERR_LOG
+        UFUNC_MASK_DIVIDEBYZERO
+        UFUNC_MASK_OVERFLOW
+        UFUNC_MASK_UNDERFLOW
+        UFUNC_MASK_INVALID
+        UFUNC_SHIFT_DIVIDEBYZERO
+        UFUNC_SHIFT_OVERFLOW
+        UFUNC_SHIFT_UNDERFLOW
+        UFUNC_SHIFT_INVALID
+        UFUNC_FPE_DIVIDEBYZERO
+        UFUNC_FPE_OVERFLOW
+        UFUNC_FPE_UNDERFLOW
+        UFUNC_FPE_INVALID
+        UFUNC_ERR_DEFAULT
+        UFUNC_ERR_DEFAULT2
+
+    object PyUFunc_FromFuncAndData(PyUFuncGenericFunction *,
+          void **, char *, int, int, int, int, char *, char *, int)
+    int PyUFunc_RegisterLoopForType(ufunc, int,
+                                    PyUFuncGenericFunction, int *, void *) except -1
+    void PyUFunc_f_f_As_d_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_d_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_f_f \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_g_g \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_F_F_As_D_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_F_F \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_D_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_G_G \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_O_O \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_ff_f_As_dd_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_ff_f \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_dd_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_gg_g \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_FF_F_As_DD_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_DD_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_FF_F \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_GG_G \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_OO_O \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_O_O_method \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_OO_O_method \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_On_Om \
+         (char **, npy_intp *, npy_intp *, void *)
+    int PyUFunc_GetPyValues \
+        (char *, int *, int *, PyObject **)
+    int PyUFunc_checkfperr \
+           (int, PyObject *, int *)
+    void PyUFunc_clearfperr()
+    int PyUFunc_getfperr()
+    int PyUFunc_handlefperr \
+        (int, PyObject *, int, int *) except -1
+    int PyUFunc_ReplaceLoopBySignature \
+        (ufunc, PyUFuncGenericFunction, int *, PyUFuncGenericFunction *)
+    object PyUFunc_FromFuncAndDataAndSignature \
+             (PyUFuncGenericFunction *, void **, char *, int, int, int,
+              int, char *, char *, int, char *)
+
+    int _import_umath() except -1
+
+cdef inline void set_array_base(ndarray arr, object base):
+    Py_INCREF(base) # important to do this before stealing the reference below!
+    PyArray_SetBaseObject(arr, base)
+
+cdef inline object get_array_base(ndarray arr):
+    base = PyArray_BASE(arr)
+    if base is NULL:
+        return None
+    return <object>base
+
+# Versions of the import_* functions which are more suitable for
+# Cython code.
+cdef inline int import_array() except -1:
+    try:
+        __pyx_import_array()
+    except Exception:
+        raise ImportError("numpy.core.multiarray failed to import")
+
+cdef inline int import_umath() except -1:
+    try:
+        _import_umath()
+    except Exception:
+        raise ImportError("numpy.core.umath failed to import")
+
+cdef inline int import_ufunc() except -1:
+    try:
+        _import_umath()
+    except Exception:
+        raise ImportError("numpy.core.umath failed to import")
+
+
+cdef inline bint is_timedelta64_object(object obj):
+    """
+    Cython equivalent of `isinstance(obj, np.timedelta64)`
+
+    Parameters
+    ----------
+    obj : object
+
+    Returns
+    -------
+    bool
+    """
+    return PyObject_TypeCheck(obj, &PyTimedeltaArrType_Type)
+
+
+cdef inline bint is_datetime64_object(object obj):
+    """
+    Cython equivalent of `isinstance(obj, np.datetime64)`
+
+    Parameters
+    ----------
+    obj : object
+
+    Returns
+    -------
+    bool
+    """
+    return PyObject_TypeCheck(obj, &PyDatetimeArrType_Type)
+
+
+cdef inline npy_datetime get_datetime64_value(object obj) nogil:
+    """
+    returns the int64 value underlying scalar numpy datetime64 object
+
+    Note that to interpret this as a datetime, the corresponding unit is
+    also needed.  That can be found using `get_datetime64_unit`.
+    """
+    return (<PyDatetimeScalarObject*>obj).obval
+
+
+cdef inline npy_timedelta get_timedelta64_value(object obj) nogil:
+    """
+    returns the int64 value underlying scalar numpy timedelta64 object
+    """
+    return (<PyTimedeltaScalarObject*>obj).obval
+
+
+cdef inline NPY_DATETIMEUNIT get_datetime64_unit(object obj) nogil:
+    """
+    returns the unit part of the dtype for a numpy datetime64 object.
+    """
+    return <NPY_DATETIMEUNIT>(<PyDatetimeScalarObject*>obj).obmeta.base

lib/python3.12/site-packages/numpy/__init__.pxd

@@ -0,0 +1,1015 @@
+# NumPy static imports for Cython < 3.0
+#
+# If any of the PyArray_* functions are called, import_array must be
+# called first.
+#
+# Author: Dag Sverre Seljebotn
+#
+
+DEF _buffer_format_string_len = 255
+
+cimport cpython.buffer as pybuf
+from cpython.ref cimport Py_INCREF
+from cpython.mem cimport PyObject_Malloc, PyObject_Free
+from cpython.object cimport PyObject, PyTypeObject
+from cpython.buffer cimport PyObject_GetBuffer
+from cpython.type cimport type
+cimport libc.stdio as stdio
+
+cdef extern from "Python.h":
+    ctypedef int Py_intptr_t
+    bint PyObject_TypeCheck(object obj, PyTypeObject* type)
+
+cdef extern from "numpy/arrayobject.h":
+    ctypedef Py_intptr_t npy_intp
+    ctypedef size_t npy_uintp
+
+    cdef enum NPY_TYPES:
+        NPY_BOOL
+        NPY_BYTE
+        NPY_UBYTE
+        NPY_SHORT
+        NPY_USHORT
+        NPY_INT
+        NPY_UINT
+        NPY_LONG
+        NPY_ULONG
+        NPY_LONGLONG
+        NPY_ULONGLONG
+        NPY_FLOAT
+        NPY_DOUBLE
+        NPY_LONGDOUBLE
+        NPY_CFLOAT
+        NPY_CDOUBLE
+        NPY_CLONGDOUBLE
+        NPY_OBJECT
+        NPY_STRING
+        NPY_UNICODE
+        NPY_VOID
+        NPY_DATETIME
+        NPY_TIMEDELTA
+        NPY_NTYPES
+        NPY_NOTYPE
+
+        NPY_INT8
+        NPY_INT16
+        NPY_INT32
+        NPY_INT64
+        NPY_INT128
+        NPY_INT256
+        NPY_UINT8
+        NPY_UINT16
+        NPY_UINT32
+        NPY_UINT64
+        NPY_UINT128
+        NPY_UINT256
+        NPY_FLOAT16
+        NPY_FLOAT32
+        NPY_FLOAT64
+        NPY_FLOAT80
+        NPY_FLOAT96
+        NPY_FLOAT128
+        NPY_FLOAT256
+        NPY_COMPLEX32
+        NPY_COMPLEX64
+        NPY_COMPLEX128
+        NPY_COMPLEX160
+        NPY_COMPLEX192
+        NPY_COMPLEX256
+        NPY_COMPLEX512
+
+        NPY_INTP
+
+    ctypedef enum NPY_ORDER:
+        NPY_ANYORDER
+        NPY_CORDER
+        NPY_FORTRANORDER
+        NPY_KEEPORDER
+
+    ctypedef enum NPY_CASTING:
+        NPY_NO_CASTING
+        NPY_EQUIV_CASTING
+        NPY_SAFE_CASTING
+        NPY_SAME_KIND_CASTING
+        NPY_UNSAFE_CASTING
+
+    ctypedef enum NPY_CLIPMODE:
+        NPY_CLIP
+        NPY_WRAP
+        NPY_RAISE
+
+    ctypedef enum NPY_SCALARKIND:
+        NPY_NOSCALAR,
+        NPY_BOOL_SCALAR,
+        NPY_INTPOS_SCALAR,
+        NPY_INTNEG_SCALAR,
+        NPY_FLOAT_SCALAR,
+        NPY_COMPLEX_SCALAR,
+        NPY_OBJECT_SCALAR
+
+    ctypedef enum NPY_SORTKIND:
+        NPY_QUICKSORT
+        NPY_HEAPSORT
+        NPY_MERGESORT
+
+    ctypedef enum NPY_SEARCHSIDE:
+        NPY_SEARCHLEFT
+        NPY_SEARCHRIGHT
+
+    enum:
+        # DEPRECATED since NumPy 1.7 ! Do not use in new code!
+        NPY_C_CONTIGUOUS
+        NPY_F_CONTIGUOUS
+        NPY_CONTIGUOUS
+        NPY_FORTRAN
+        NPY_OWNDATA
+        NPY_FORCECAST
+        NPY_ENSURECOPY
+        NPY_ENSUREARRAY
+        NPY_ELEMENTSTRIDES
+        NPY_ALIGNED
+        NPY_NOTSWAPPED
+        NPY_WRITEABLE
+        NPY_ARR_HAS_DESCR
+
+        NPY_BEHAVED
+        NPY_BEHAVED_NS
+        NPY_CARRAY
+        NPY_CARRAY_RO
+        NPY_FARRAY
+        NPY_FARRAY_RO
+        NPY_DEFAULT
+
+        NPY_IN_ARRAY
+        NPY_OUT_ARRAY
+        NPY_INOUT_ARRAY
+        NPY_IN_FARRAY
+        NPY_OUT_FARRAY
+        NPY_INOUT_FARRAY
+
+        NPY_UPDATE_ALL
+
+    enum:
+        # Added in NumPy 1.7 to replace the deprecated enums above.
+        NPY_ARRAY_C_CONTIGUOUS
+        NPY_ARRAY_F_CONTIGUOUS
+        NPY_ARRAY_OWNDATA
+        NPY_ARRAY_FORCECAST
+        NPY_ARRAY_ENSURECOPY
+        NPY_ARRAY_ENSUREARRAY
+        NPY_ARRAY_ELEMENTSTRIDES
+        NPY_ARRAY_ALIGNED
+        NPY_ARRAY_NOTSWAPPED
+        NPY_ARRAY_WRITEABLE
+        NPY_ARRAY_WRITEBACKIFCOPY
+
+        NPY_ARRAY_BEHAVED
+        NPY_ARRAY_BEHAVED_NS
+        NPY_ARRAY_CARRAY
+        NPY_ARRAY_CARRAY_RO
+        NPY_ARRAY_FARRAY
+        NPY_ARRAY_FARRAY_RO
+        NPY_ARRAY_DEFAULT
+
+        NPY_ARRAY_IN_ARRAY
+        NPY_ARRAY_OUT_ARRAY
+        NPY_ARRAY_INOUT_ARRAY
+        NPY_ARRAY_IN_FARRAY
+        NPY_ARRAY_OUT_FARRAY
+        NPY_ARRAY_INOUT_FARRAY
+
+        NPY_ARRAY_UPDATE_ALL
+
+    cdef enum:
+        NPY_MAXDIMS
+
+    npy_intp NPY_MAX_ELSIZE
+
+    ctypedef void (*PyArray_VectorUnaryFunc)(void *, void *, npy_intp, void *,  void *)
+
+    ctypedef struct PyArray_ArrayDescr:
+        # shape is a tuple, but Cython doesn't support "tuple shape"
+        # inside a non-PyObject declaration, so we have to declare it
+        # as just a PyObject*.
+        PyObject* shape
+
+    ctypedef struct PyArray_Descr:
+        pass
+
+    ctypedef class numpy.dtype [object PyArray_Descr, check_size ignore]:
+        # Use PyDataType_* macros when possible, however there are no macros
+        # for accessing some of the fields, so some are defined.
+        cdef PyTypeObject* typeobj
+        cdef char kind
+        cdef char type
+        # Numpy sometimes mutates this without warning (e.g. it'll
+        # sometimes change "|" to "<" in shared dtype objects on
+        # little-endian machines). If this matters to you, use
+        # PyArray_IsNativeByteOrder(dtype.byteorder) instead of
+        # directly accessing this field.
+        cdef char byteorder
+        cdef char flags
+        cdef int type_num
+        cdef int itemsize "elsize"
+        cdef int alignment
+        cdef object fields
+        cdef tuple names
+        # Use PyDataType_HASSUBARRAY to test whether this field is
+        # valid (the pointer can be NULL). Most users should access
+        # this field via the inline helper method PyDataType_SHAPE.
+        cdef PyArray_ArrayDescr* subarray
+
+    ctypedef class numpy.flatiter [object PyArrayIterObject, check_size ignore]:
+        # Use through macros
+        pass
+
+    ctypedef class numpy.broadcast [object PyArrayMultiIterObject, check_size ignore]:
+        cdef int numiter
+        cdef npy_intp size, index
+        cdef int nd
+        cdef npy_intp *dimensions
+        cdef void **iters
+
+    ctypedef struct PyArrayObject:
+        # For use in situations where ndarray can't replace PyArrayObject*,
+        # like PyArrayObject**.
+        pass
+
+    ctypedef class numpy.ndarray [object PyArrayObject, check_size ignore]:
+        cdef __cythonbufferdefaults__ = {"mode": "strided"}
+
+        cdef:
+            # Only taking a few of the most commonly used and stable fields.
+            # One should use PyArray_* macros instead to access the C fields.
+            char *data
+            int ndim "nd"
+            npy_intp *shape "dimensions"
+            npy_intp *strides
+            dtype descr  # deprecated since NumPy 1.7 !
+            PyObject* base #  NOT PUBLIC, DO NOT USE !
+
+
+
+    ctypedef unsigned char      npy_bool
+
+    ctypedef signed char      npy_byte
+    ctypedef signed short     npy_short
+    ctypedef signed int       npy_int
+    ctypedef signed long      npy_long
+    ctypedef signed long long npy_longlong
+
+    ctypedef unsigned char      npy_ubyte
+    ctypedef unsigned short     npy_ushort
+    ctypedef unsigned int       npy_uint
+    ctypedef unsigned long      npy_ulong
+    ctypedef unsigned long long npy_ulonglong
+
+    ctypedef float        npy_float
+    ctypedef double       npy_double
+    ctypedef long double  npy_longdouble
+
+    ctypedef signed char        npy_int8
+    ctypedef signed short       npy_int16
+    ctypedef signed int         npy_int32
+    ctypedef signed long long   npy_int64
+    ctypedef signed long long   npy_int96
+    ctypedef signed long long   npy_int128
+
+    ctypedef unsigned char      npy_uint8
+    ctypedef unsigned short     npy_uint16
+    ctypedef unsigned int       npy_uint32
+    ctypedef unsigned long long npy_uint64
+    ctypedef unsigned long long npy_uint96
+    ctypedef unsigned long long npy_uint128
+
+    ctypedef float        npy_float32
+    ctypedef double       npy_float64
+    ctypedef long double  npy_float80
+    ctypedef long double  npy_float96
+    ctypedef long double  npy_float128
+
+    ctypedef struct npy_cfloat:
+        float real
+        float imag
+
+    ctypedef struct npy_cdouble:
+        double real
+        double imag
+
+    ctypedef struct npy_clongdouble:
+        long double real
+        long double imag
+
+    ctypedef struct npy_complex64:
+        float real
+        float imag
+
+    ctypedef struct npy_complex128:
+        double real
+        double imag
+
+    ctypedef struct npy_complex160:
+        long double real
+        long double imag
+
+    ctypedef struct npy_complex192:
+        long double real
+        long double imag
+
+    ctypedef struct npy_complex256:
+        long double real
+        long double imag
+
+    ctypedef struct PyArray_Dims:
+        npy_intp *ptr
+        int len
+
+    int _import_array() except -1
+    # A second definition so _import_array isn't marked as used when we use it here.
+    # Do not use - subject to change any time.
+    int __pyx_import_array "_import_array"() except -1
+
+    #
+    # Macros from ndarrayobject.h
+    #
+    bint PyArray_CHKFLAGS(ndarray m, int flags) nogil
+    bint PyArray_IS_C_CONTIGUOUS(ndarray arr) nogil
+    bint PyArray_IS_F_CONTIGUOUS(ndarray arr) nogil
+    bint PyArray_ISCONTIGUOUS(ndarray m) nogil
+    bint PyArray_ISWRITEABLE(ndarray m) nogil
+    bint PyArray_ISALIGNED(ndarray m) nogil
+
+    int PyArray_NDIM(ndarray) nogil
+    bint PyArray_ISONESEGMENT(ndarray) nogil
+    bint PyArray_ISFORTRAN(ndarray) nogil
+    int PyArray_FORTRANIF(ndarray) nogil
+
+    void* PyArray_DATA(ndarray) nogil
+    char* PyArray_BYTES(ndarray) nogil
+
+    npy_intp* PyArray_DIMS(ndarray) nogil
+    npy_intp* PyArray_STRIDES(ndarray) nogil
+    npy_intp PyArray_DIM(ndarray, size_t) nogil
+    npy_intp PyArray_STRIDE(ndarray, size_t) nogil
+
+    PyObject *PyArray_BASE(ndarray) nogil  # returns borrowed reference!
+    PyArray_Descr *PyArray_DESCR(ndarray) nogil  # returns borrowed reference to dtype!
+    int PyArray_FLAGS(ndarray) nogil
+    npy_intp PyArray_ITEMSIZE(ndarray) nogil
+    int PyArray_TYPE(ndarray arr) nogil
+
+    object PyArray_GETITEM(ndarray arr, void *itemptr)
+    int PyArray_SETITEM(ndarray arr, void *itemptr, object obj) except -1
+
+    bint PyTypeNum_ISBOOL(int) nogil
+    bint PyTypeNum_ISUNSIGNED(int) nogil
+    bint PyTypeNum_ISSIGNED(int) nogil
+    bint PyTypeNum_ISINTEGER(int) nogil
+    bint PyTypeNum_ISFLOAT(int) nogil
+    bint PyTypeNum_ISNUMBER(int) nogil
+    bint PyTypeNum_ISSTRING(int) nogil
+    bint PyTypeNum_ISCOMPLEX(int) nogil
+    bint PyTypeNum_ISPYTHON(int) nogil
+    bint PyTypeNum_ISFLEXIBLE(int) nogil
+    bint PyTypeNum_ISUSERDEF(int) nogil
+    bint PyTypeNum_ISEXTENDED(int) nogil
+    bint PyTypeNum_ISOBJECT(int) nogil
+
+    bint PyDataType_ISBOOL(dtype) nogil
+    bint PyDataType_ISUNSIGNED(dtype) nogil
+    bint PyDataType_ISSIGNED(dtype) nogil
+    bint PyDataType_ISINTEGER(dtype) nogil
+    bint PyDataType_ISFLOAT(dtype) nogil
+    bint PyDataType_ISNUMBER(dtype) nogil
+    bint PyDataType_ISSTRING(dtype) nogil
+    bint PyDataType_ISCOMPLEX(dtype) nogil
+    bint PyDataType_ISPYTHON(dtype) nogil
+    bint PyDataType_ISFLEXIBLE(dtype) nogil
+    bint PyDataType_ISUSERDEF(dtype) nogil
+    bint PyDataType_ISEXTENDED(dtype) nogil
+    bint PyDataType_ISOBJECT(dtype) nogil
+    bint PyDataType_HASFIELDS(dtype) nogil
+    bint PyDataType_HASSUBARRAY(dtype) nogil
+
+    bint PyArray_ISBOOL(ndarray) nogil
+    bint PyArray_ISUNSIGNED(ndarray) nogil
+    bint PyArray_ISSIGNED(ndarray) nogil
+    bint PyArray_ISINTEGER(ndarray) nogil
+    bint PyArray_ISFLOAT(ndarray) nogil
+    bint PyArray_ISNUMBER(ndarray) nogil
+    bint PyArray_ISSTRING(ndarray) nogil
+    bint PyArray_ISCOMPLEX(ndarray) nogil
+    bint PyArray_ISPYTHON(ndarray) nogil
+    bint PyArray_ISFLEXIBLE(ndarray) nogil
+    bint PyArray_ISUSERDEF(ndarray) nogil
+    bint PyArray_ISEXTENDED(ndarray) nogil
+    bint PyArray_ISOBJECT(ndarray) nogil
+    bint PyArray_HASFIELDS(ndarray) nogil
+
+    bint PyArray_ISVARIABLE(ndarray) nogil
+
+    bint PyArray_SAFEALIGNEDCOPY(ndarray) nogil
+    bint PyArray_ISNBO(char) nogil              # works on ndarray.byteorder
+    bint PyArray_IsNativeByteOrder(char) nogil # works on ndarray.byteorder
+    bint PyArray_ISNOTSWAPPED(ndarray) nogil
+    bint PyArray_ISBYTESWAPPED(ndarray) nogil
+
+    bint PyArray_FLAGSWAP(ndarray, int) nogil
+
+    bint PyArray_ISCARRAY(ndarray) nogil
+    bint PyArray_ISCARRAY_RO(ndarray) nogil
+    bint PyArray_ISFARRAY(ndarray) nogil
+    bint PyArray_ISFARRAY_RO(ndarray) nogil
+    bint PyArray_ISBEHAVED(ndarray) nogil
+    bint PyArray_ISBEHAVED_RO(ndarray) nogil
+
+
+    bint PyDataType_ISNOTSWAPPED(dtype) nogil
+    bint PyDataType_ISBYTESWAPPED(dtype) nogil
+
+    bint PyArray_DescrCheck(object)
+
+    bint PyArray_Check(object)
+    bint PyArray_CheckExact(object)
+
+    # Cannot be supported due to out arg:
+    # bint PyArray_HasArrayInterfaceType(object, dtype, object, object&)
+    # bint PyArray_HasArrayInterface(op, out)
+
+
+    bint PyArray_IsZeroDim(object)
+    # Cannot be supported due to ## ## in macro:
+    # bint PyArray_IsScalar(object, verbatim work)
+    bint PyArray_CheckScalar(object)
+    bint PyArray_IsPythonNumber(object)
+    bint PyArray_IsPythonScalar(object)
+    bint PyArray_IsAnyScalar(object)
+    bint PyArray_CheckAnyScalar(object)
+
+    ndarray PyArray_GETCONTIGUOUS(ndarray)
+    bint PyArray_SAMESHAPE(ndarray, ndarray) nogil
+    npy_intp PyArray_SIZE(ndarray) nogil
+    npy_intp PyArray_NBYTES(ndarray) nogil
+
+    object PyArray_FROM_O(object)
+    object PyArray_FROM_OF(object m, int flags)
+    object PyArray_FROM_OT(object m, int type)
+    object PyArray_FROM_OTF(object m, int type, int flags)
+    object PyArray_FROMANY(object m, int type, int min, int max, int flags)
+    object PyArray_ZEROS(int nd, npy_intp* dims, int type, int fortran)
+    object PyArray_EMPTY(int nd, npy_intp* dims, int type, int fortran)
+    void PyArray_FILLWBYTE(object, int val)
+    npy_intp PyArray_REFCOUNT(object)
+    object PyArray_ContiguousFromAny(op, int, int min_depth, int max_depth)
+    unsigned char PyArray_EquivArrTypes(ndarray a1, ndarray a2)
+    bint PyArray_EquivByteorders(int b1, int b2) nogil
+    object PyArray_SimpleNew(int nd, npy_intp* dims, int typenum)
+    object PyArray_SimpleNewFromData(int nd, npy_intp* dims, int typenum, void* data)
+    #object PyArray_SimpleNewFromDescr(int nd, npy_intp* dims, dtype descr)
+    object PyArray_ToScalar(void* data, ndarray arr)
+
+    void* PyArray_GETPTR1(ndarray m, npy_intp i) nogil
+    void* PyArray_GETPTR2(ndarray m, npy_intp i, npy_intp j) nogil
+    void* PyArray_GETPTR3(ndarray m, npy_intp i, npy_intp j, npy_intp k) nogil
+    void* PyArray_GETPTR4(ndarray m, npy_intp i, npy_intp j, npy_intp k, npy_intp l) nogil
+
+    # Cannot be supported due to out arg
+    # void PyArray_DESCR_REPLACE(descr)
+
+
+    object PyArray_Copy(ndarray)
+    object PyArray_FromObject(object op, int type, int min_depth, int max_depth)
+    object PyArray_ContiguousFromObject(object op, int type, int min_depth, int max_depth)
+    object PyArray_CopyFromObject(object op, int type, int min_depth, int max_depth)
+
+    object PyArray_Cast(ndarray mp, int type_num)
+    object PyArray_Take(ndarray ap, object items, int axis)
+    object PyArray_Put(ndarray ap, object items, object values)
+
+    void PyArray_ITER_RESET(flatiter it) nogil
+    void PyArray_ITER_NEXT(flatiter it) nogil
+    void PyArray_ITER_GOTO(flatiter it, npy_intp* destination) nogil
+    void PyArray_ITER_GOTO1D(flatiter it, npy_intp ind) nogil
+    void* PyArray_ITER_DATA(flatiter it) nogil
+    bint PyArray_ITER_NOTDONE(flatiter it) nogil
+
+    void PyArray_MultiIter_RESET(broadcast multi) nogil
+    void PyArray_MultiIter_NEXT(broadcast multi) nogil
+    void PyArray_MultiIter_GOTO(broadcast multi, npy_intp dest) nogil
+    void PyArray_MultiIter_GOTO1D(broadcast multi, npy_intp ind) nogil
+    void* PyArray_MultiIter_DATA(broadcast multi, npy_intp i) nogil
+    void PyArray_MultiIter_NEXTi(broadcast multi, npy_intp i) nogil
+    bint PyArray_MultiIter_NOTDONE(broadcast multi) nogil
+
+    # Functions from __multiarray_api.h
+
+    # Functions taking dtype and returning object/ndarray are disabled
+    # for now as they steal dtype references. I'm conservative and disable
+    # more than is probably needed until it can be checked further.
+    int PyArray_SetNumericOps (object) except -1
+    object PyArray_GetNumericOps ()
+    int PyArray_INCREF (ndarray) except *  # uses PyArray_Item_INCREF...
+    int PyArray_XDECREF (ndarray) except *  # uses PyArray_Item_DECREF...
+    void PyArray_SetStringFunction (object, int)
+    dtype PyArray_DescrFromType (int)
+    object PyArray_TypeObjectFromType (int)
+    char * PyArray_Zero (ndarray)
+    char * PyArray_One (ndarray)
+    #object PyArray_CastToType (ndarray, dtype, int)
+    int PyArray_CastTo (ndarray, ndarray) except -1
+    int PyArray_CastAnyTo (ndarray, ndarray) except -1
+    int PyArray_CanCastSafely (int, int)  # writes errors
+    npy_bool PyArray_CanCastTo (dtype, dtype)  # writes errors
+    int PyArray_ObjectType (object, int) except 0
+    dtype PyArray_DescrFromObject (object, dtype)
+    #ndarray* PyArray_ConvertToCommonType (object, int *)
+    dtype PyArray_DescrFromScalar (object)
+    dtype PyArray_DescrFromTypeObject (object)
+    npy_intp PyArray_Size (object)
+    #object PyArray_Scalar (void *, dtype, object)
+    #object PyArray_FromScalar (object, dtype)
+    void PyArray_ScalarAsCtype (object, void *)
+    #int PyArray_CastScalarToCtype (object, void *, dtype)
+    #int PyArray_CastScalarDirect (object, dtype, void *, int)
+    object PyArray_ScalarFromObject (object)
+    #PyArray_VectorUnaryFunc * PyArray_GetCastFunc (dtype, int)
+    object PyArray_FromDims (int, int *, int)
+    #object PyArray_FromDimsAndDataAndDescr (int, int *, dtype, char *)
+    #object PyArray_FromAny (object, dtype, int, int, int, object)
+    object PyArray_EnsureArray (object)
+    object PyArray_EnsureAnyArray (object)
+    #object PyArray_FromFile (stdio.FILE *, dtype, npy_intp, char *)
+    #object PyArray_FromString (char *, npy_intp, dtype, npy_intp, char *)
+    #object PyArray_FromBuffer (object, dtype, npy_intp, npy_intp)
+    #object PyArray_FromIter (object, dtype, npy_intp)
+    object PyArray_Return (ndarray)
+    #object PyArray_GetField (ndarray, dtype, int)
+    #int PyArray_SetField (ndarray, dtype, int, object) except -1
+    object PyArray_Byteswap (ndarray, npy_bool)
+    object PyArray_Resize (ndarray, PyArray_Dims *, int, NPY_ORDER)
+    int PyArray_MoveInto (ndarray, ndarray) except -1
+    int PyArray_CopyInto (ndarray, ndarray) except -1
+    int PyArray_CopyAnyInto (ndarray, ndarray) except -1
+    int PyArray_CopyObject (ndarray, object) except -1
+    object PyArray_NewCopy (ndarray, NPY_ORDER)
+    object PyArray_ToList (ndarray)
+    object PyArray_ToString (ndarray, NPY_ORDER)
+    int PyArray_ToFile (ndarray, stdio.FILE *, char *, char *) except -1
+    int PyArray_Dump (object, object, int) except -1
+    object PyArray_Dumps (object, int)
+    int PyArray_ValidType (int)  # Cannot error
+    void PyArray_UpdateFlags (ndarray, int)
+    object PyArray_New (type, int, npy_intp *, int, npy_intp *, void *, int, int, object)
+    #object PyArray_NewFromDescr (type, dtype, int, npy_intp *, npy_intp *, void *, int, object)
+    #dtype PyArray_DescrNew (dtype)
+    dtype PyArray_DescrNewFromType (int)
+    double PyArray_GetPriority (object, double)  # clears errors as of 1.25
+    object PyArray_IterNew (object)
+    object PyArray_MultiIterNew (int, ...)
+
+    int PyArray_PyIntAsInt (object) except? -1
+    npy_intp PyArray_PyIntAsIntp (object)
+    int PyArray_Broadcast (broadcast) except -1
+    void PyArray_FillObjectArray (ndarray, object) except *
+    int PyArray_FillWithScalar (ndarray, object) except -1
+    npy_bool PyArray_CheckStrides (int, int, npy_intp, npy_intp, npy_intp *, npy_intp *)
+    dtype PyArray_DescrNewByteorder (dtype, char)
+    object PyArray_IterAllButAxis (object, int *)
+    #object PyArray_CheckFromAny (object, dtype, int, int, int, object)
+    #object PyArray_FromArray (ndarray, dtype, int)
+    object PyArray_FromInterface (object)
+    object PyArray_FromStructInterface (object)
+    #object PyArray_FromArrayAttr (object, dtype, object)
+    #NPY_SCALARKIND PyArray_ScalarKind (int, ndarray*)
+    int PyArray_CanCoerceScalar (int, int, NPY_SCALARKIND)
+    object PyArray_NewFlagsObject (object)
+    npy_bool PyArray_CanCastScalar (type, type)
+    #int PyArray_CompareUCS4 (npy_ucs4 *, npy_ucs4 *, register size_t)
+    int PyArray_RemoveSmallest (broadcast) except -1
+    int PyArray_ElementStrides (object)
+    void PyArray_Item_INCREF (char *, dtype) except *
+    void PyArray_Item_XDECREF (char *, dtype) except *
+    object PyArray_FieldNames (object)
+    object PyArray_Transpose (ndarray, PyArray_Dims *)
+    object PyArray_TakeFrom (ndarray, object, int, ndarray, NPY_CLIPMODE)
+    object PyArray_PutTo (ndarray, object, object, NPY_CLIPMODE)
+    object PyArray_PutMask (ndarray, object, object)
+    object PyArray_Repeat (ndarray, object, int)
+    object PyArray_Choose (ndarray, object, ndarray, NPY_CLIPMODE)
+    int PyArray_Sort (ndarray, int, NPY_SORTKIND) except -1
+    object PyArray_ArgSort (ndarray, int, NPY_SORTKIND)
+    object PyArray_SearchSorted (ndarray, object, NPY_SEARCHSIDE, PyObject *)
+    object PyArray_ArgMax (ndarray, int, ndarray)
+    object PyArray_ArgMin (ndarray, int, ndarray)
+    object PyArray_Reshape (ndarray, object)
+    object PyArray_Newshape (ndarray, PyArray_Dims *, NPY_ORDER)
+    object PyArray_Squeeze (ndarray)
+    #object PyArray_View (ndarray, dtype, type)
+    object PyArray_SwapAxes (ndarray, int, int)
+    object PyArray_Max (ndarray, int, ndarray)
+    object PyArray_Min (ndarray, int, ndarray)
+    object PyArray_Ptp (ndarray, int, ndarray)
+    object PyArray_Mean (ndarray, int, int, ndarray)
+    object PyArray_Trace (ndarray, int, int, int, int, ndarray)
+    object PyArray_Diagonal (ndarray, int, int, int)
+    object PyArray_Clip (ndarray, object, object, ndarray)
+    object PyArray_Conjugate (ndarray, ndarray)
+    object PyArray_Nonzero (ndarray)
+    object PyArray_Std (ndarray, int, int, ndarray, int)
+    object PyArray_Sum (ndarray, int, int, ndarray)
+    object PyArray_CumSum (ndarray, int, int, ndarray)
+    object PyArray_Prod (ndarray, int, int, ndarray)
+    object PyArray_CumProd (ndarray, int, int, ndarray)
+    object PyArray_All (ndarray, int, ndarray)
+    object PyArray_Any (ndarray, int, ndarray)
+    object PyArray_Compress (ndarray, object, int, ndarray)
+    object PyArray_Flatten (ndarray, NPY_ORDER)
+    object PyArray_Ravel (ndarray, NPY_ORDER)
+    npy_intp PyArray_MultiplyList (npy_intp *, int)
+    int PyArray_MultiplyIntList (int *, int)
+    void * PyArray_GetPtr (ndarray, npy_intp*)
+    int PyArray_CompareLists (npy_intp *, npy_intp *, int)
+    #int PyArray_AsCArray (object*, void *, npy_intp *, int, dtype)
+    #int PyArray_As1D (object*, char **, int *, int)
+    #int PyArray_As2D (object*, char ***, int *, int *, int)
+    int PyArray_Free (object, void *)
+    #int PyArray_Converter (object, object*)
+    int PyArray_IntpFromSequence (object, npy_intp *, int) except -1
+    object PyArray_Concatenate (object, int)
+    object PyArray_InnerProduct (object, object)
+    object PyArray_MatrixProduct (object, object)
+    object PyArray_CopyAndTranspose (object)
+    object PyArray_Correlate (object, object, int)
+    int PyArray_TypestrConvert (int, int)
+    #int PyArray_DescrConverter (object, dtype*) except 0
+    #int PyArray_DescrConverter2 (object, dtype*) except 0
+    int PyArray_IntpConverter (object, PyArray_Dims *) except 0
+    #int PyArray_BufferConverter (object, chunk) except 0
+    int PyArray_AxisConverter (object, int *) except 0
+    int PyArray_BoolConverter (object, npy_bool *) except 0
+    int PyArray_ByteorderConverter (object, char *) except 0
+    int PyArray_OrderConverter (object, NPY_ORDER *) except 0
+    unsigned char PyArray_EquivTypes (dtype, dtype)  # clears errors
+    #object PyArray_Zeros (int, npy_intp *, dtype, int)
+    #object PyArray_Empty (int, npy_intp *, dtype, int)
+    object PyArray_Where (object, object, object)
+    object PyArray_Arange (double, double, double, int)
+    #object PyArray_ArangeObj (object, object, object, dtype)
+    int PyArray_SortkindConverter (object, NPY_SORTKIND *) except 0
+    object PyArray_LexSort (object, int)
+    object PyArray_Round (ndarray, int, ndarray)
+    unsigned char PyArray_EquivTypenums (int, int)
+    int PyArray_RegisterDataType (dtype) except -1
+    int PyArray_RegisterCastFunc (dtype, int, PyArray_VectorUnaryFunc *) except -1
+    int PyArray_RegisterCanCast (dtype, int, NPY_SCALARKIND) except -1
+    #void PyArray_InitArrFuncs (PyArray_ArrFuncs *)
+    object PyArray_IntTupleFromIntp (int, npy_intp *)
+    int PyArray_TypeNumFromName (char *)
+    int PyArray_ClipmodeConverter (object, NPY_CLIPMODE *) except 0
+    #int PyArray_OutputConverter (object, ndarray*) except 0
+    object PyArray_BroadcastToShape (object, npy_intp *, int)
+    void _PyArray_SigintHandler (int)
+    void* _PyArray_GetSigintBuf ()
+    #int PyArray_DescrAlignConverter (object, dtype*) except 0
+    #int PyArray_DescrAlignConverter2 (object, dtype*) except 0
+    int PyArray_SearchsideConverter (object, void *) except 0
+    object PyArray_CheckAxis (ndarray, int *, int)
+    npy_intp PyArray_OverflowMultiplyList (npy_intp *, int)
+    int PyArray_CompareString (char *, char *, size_t)
+    int PyArray_SetBaseObject(ndarray, base) except -1 # NOTE: steals a reference to base! Use "set_array_base()" instead.
+
+
+# Typedefs that matches the runtime dtype objects in
+# the numpy module.
+
+# The ones that are commented out needs an IFDEF function
+# in Cython to enable them only on the right systems.
+
+ctypedef npy_int8       int8_t
+ctypedef npy_int16      int16_t
+ctypedef npy_int32      int32_t
+ctypedef npy_int64      int64_t
+#ctypedef npy_int96      int96_t
+#ctypedef npy_int128     int128_t
+
+ctypedef npy_uint8      uint8_t
+ctypedef npy_uint16     uint16_t
+ctypedef npy_uint32     uint32_t
+ctypedef npy_uint64     uint64_t
+#ctypedef npy_uint96     uint96_t
+#ctypedef npy_uint128    uint128_t
+
+ctypedef npy_float32    float32_t
+ctypedef npy_float64    float64_t
+#ctypedef npy_float80    float80_t
+#ctypedef npy_float128   float128_t
+
+ctypedef float complex  complex64_t
+ctypedef double complex complex128_t
+
+# The int types are mapped a bit surprising --
+# numpy.int corresponds to 'l' and numpy.long to 'q'
+ctypedef npy_long       int_t
+ctypedef npy_longlong   longlong_t
+
+ctypedef npy_ulong      uint_t
+ctypedef npy_ulonglong  ulonglong_t
+
+ctypedef npy_intp       intp_t
+ctypedef npy_uintp      uintp_t
+
+ctypedef npy_double     float_t
+ctypedef npy_double     double_t
+ctypedef npy_longdouble longdouble_t
+
+ctypedef npy_cfloat      cfloat_t
+ctypedef npy_cdouble     cdouble_t
+ctypedef npy_clongdouble clongdouble_t
+
+ctypedef npy_cdouble     complex_t
+
+cdef inline object PyArray_MultiIterNew1(a):
+    return PyArray_MultiIterNew(1, <void*>a)
+
+cdef inline object PyArray_MultiIterNew2(a, b):
+    return PyArray_MultiIterNew(2, <void*>a, <void*>b)
+
+cdef inline object PyArray_MultiIterNew3(a, b, c):
+    return PyArray_MultiIterNew(3, <void*>a, <void*>b, <void*> c)
+
+cdef inline object PyArray_MultiIterNew4(a, b, c, d):
+    return PyArray_MultiIterNew(4, <void*>a, <void*>b, <void*>c, <void*> d)
+
+cdef inline object PyArray_MultiIterNew5(a, b, c, d, e):
+    return PyArray_MultiIterNew(5, <void*>a, <void*>b, <void*>c, <void*> d, <void*> e)
+
+cdef inline tuple PyDataType_SHAPE(dtype d):
+    if PyDataType_HASSUBARRAY(d):
+        return <tuple>d.subarray.shape
+    else:
+        return ()
+
+
+cdef extern from "numpy/ndarrayobject.h":
+    PyTypeObject PyTimedeltaArrType_Type
+    PyTypeObject PyDatetimeArrType_Type
+    ctypedef int64_t npy_timedelta
+    ctypedef int64_t npy_datetime
+
+cdef extern from "numpy/ndarraytypes.h":
+    ctypedef struct PyArray_DatetimeMetaData:
+        NPY_DATETIMEUNIT base
+        int64_t num
+
+cdef extern from "numpy/arrayscalars.h":
+
+    # abstract types
+    ctypedef class numpy.generic [object PyObject]:
+        pass
+    ctypedef class numpy.number [object PyObject]:
+        pass
+    ctypedef class numpy.integer [object PyObject]:
+        pass
+    ctypedef class numpy.signedinteger [object PyObject]:
+        pass
+    ctypedef class numpy.unsignedinteger [object PyObject]:
+        pass
+    ctypedef class numpy.inexact [object PyObject]:
+        pass
+    ctypedef class numpy.floating [object PyObject]:
+        pass
+    ctypedef class numpy.complexfloating [object PyObject]:
+        pass
+    ctypedef class numpy.flexible [object PyObject]:
+        pass
+    ctypedef class numpy.character [object PyObject]:
+        pass
+
+    ctypedef struct PyDatetimeScalarObject:
+        # PyObject_HEAD
+        npy_datetime obval
+        PyArray_DatetimeMetaData obmeta
+
+    ctypedef struct PyTimedeltaScalarObject:
+        # PyObject_HEAD
+        npy_timedelta obval
+        PyArray_DatetimeMetaData obmeta
+
+    ctypedef enum NPY_DATETIMEUNIT:
+        NPY_FR_Y
+        NPY_FR_M
+        NPY_FR_W
+        NPY_FR_D
+        NPY_FR_B
+        NPY_FR_h
+        NPY_FR_m
+        NPY_FR_s
+        NPY_FR_ms
+        NPY_FR_us
+        NPY_FR_ns
+        NPY_FR_ps
+        NPY_FR_fs
+        NPY_FR_as
+        NPY_FR_GENERIC
+
+
+#
+# ufunc API
+#
+
+cdef extern from "numpy/ufuncobject.h":
+
+    ctypedef void (*PyUFuncGenericFunction) (char **, npy_intp *, npy_intp *, void *)
+
+    ctypedef class numpy.ufunc [object PyUFuncObject, check_size ignore]:
+        cdef:
+            int nin, nout, nargs
+            int identity
+            PyUFuncGenericFunction *functions
+            void **data
+            int ntypes
+            int check_return
+            char *name
+            char *types
+            char *doc
+            void *ptr
+            PyObject *obj
+            PyObject *userloops
+
+    cdef enum:
+        PyUFunc_Zero
+        PyUFunc_One
+        PyUFunc_None
+        UFUNC_ERR_IGNORE
+        UFUNC_ERR_WARN
+        UFUNC_ERR_RAISE
+        UFUNC_ERR_CALL
+        UFUNC_ERR_PRINT
+        UFUNC_ERR_LOG
+        UFUNC_MASK_DIVIDEBYZERO
+        UFUNC_MASK_OVERFLOW
+        UFUNC_MASK_UNDERFLOW
+        UFUNC_MASK_INVALID
+        UFUNC_SHIFT_DIVIDEBYZERO
+        UFUNC_SHIFT_OVERFLOW
+        UFUNC_SHIFT_UNDERFLOW
+        UFUNC_SHIFT_INVALID
+        UFUNC_FPE_DIVIDEBYZERO
+        UFUNC_FPE_OVERFLOW
+        UFUNC_FPE_UNDERFLOW
+        UFUNC_FPE_INVALID
+        UFUNC_ERR_DEFAULT
+        UFUNC_ERR_DEFAULT2
+
+    object PyUFunc_FromFuncAndData(PyUFuncGenericFunction *,
+          void **, char *, int, int, int, int, char *, char *, int)
+    int PyUFunc_RegisterLoopForType(ufunc, int,
+                                    PyUFuncGenericFunction, int *, void *) except -1
+    void PyUFunc_f_f_As_d_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_d_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_f_f \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_g_g \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_F_F_As_D_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_F_F \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_D_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_G_G \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_O_O \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_ff_f_As_dd_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_ff_f \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_dd_d \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_gg_g \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_FF_F_As_DD_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_DD_D \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_FF_F \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_GG_G \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_OO_O \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_O_O_method \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_OO_O_method \
+         (char **, npy_intp *, npy_intp *, void *)
+    void PyUFunc_On_Om \
+         (char **, npy_intp *, npy_intp *, void *)
+    int PyUFunc_GetPyValues \
+        (char *, int *, int *, PyObject **)
+    int PyUFunc_checkfperr \
+           (int, PyObject *, int *)
+    void PyUFunc_clearfperr()
+    int PyUFunc_getfperr()
+    int PyUFunc_handlefperr \
+        (int, PyObject *, int, int *) except -1
+    int PyUFunc_ReplaceLoopBySignature \
+        (ufunc, PyUFuncGenericFunction, int *, PyUFuncGenericFunction *)
+    object PyUFunc_FromFuncAndDataAndSignature \
+             (PyUFuncGenericFunction *, void **, char *, int, int, int,
+              int, char *, char *, int, char *)
+
+    int _import_umath() except -1
+
+cdef inline void set_array_base(ndarray arr, object base):
+    Py_INCREF(base) # important to do this before stealing the reference below!
+    PyArray_SetBaseObject(arr, base)
+
+cdef inline object get_array_base(ndarray arr):
+    base = PyArray_BASE(arr)
+    if base is NULL:
+        return None
+    return <object>base
+
+# Versions of the import_* functions which are more suitable for
+# Cython code.
+cdef inline int import_array() except -1:
+    try:
+        __pyx_import_array()
+    except Exception:
+        raise ImportError("numpy.core.multiarray failed to import")
+
+cdef inline int import_umath() except -1:
+    try:
+        _import_umath()
+    except Exception:
+        raise ImportError("numpy.core.umath failed to import")
+
+cdef inline int import_ufunc() except -1:
+    try:
+        _import_umath()
+    except Exception:
+        raise ImportError("numpy.core.umath failed to import")
+
+cdef extern from *:
+    # Leave a marker that the NumPy declarations came from this file
+    # See https://github.com/cython/cython/issues/3573
+    """
+    /* NumPy API declarations from "numpy/__init__.pxd" */
+    """
+
+
+cdef inline bint is_timedelta64_object(object obj):
+    """
+    Cython equivalent of `isinstance(obj, np.timedelta64)`
+
+    Parameters
+    ----------
+    obj : object
+
+    Returns
+    -------
+    bool
+    """
+    return PyObject_TypeCheck(obj, &PyTimedeltaArrType_Type)
+
+
+cdef inline bint is_datetime64_object(object obj):
+    """
+    Cython equivalent of `isinstance(obj, np.datetime64)`
+
+    Parameters
+    ----------
+    obj : object
+
+    Returns
+    -------
+    bool
+    """
+    return PyObject_TypeCheck(obj, &PyDatetimeArrType_Type)
+
+
+cdef inline npy_datetime get_datetime64_value(object obj) nogil:
+    """
+    returns the int64 value underlying scalar numpy datetime64 object
+
+    Note that to interpret this as a datetime, the corresponding unit is
+    also needed.  That can be found using `get_datetime64_unit`.
+    """
+    return (<PyDatetimeScalarObject*>obj).obval
+
+
+cdef inline npy_timedelta get_timedelta64_value(object obj) nogil:
+    """
+    returns the int64 value underlying scalar numpy timedelta64 object
+    """
+    return (<PyTimedeltaScalarObject*>obj).obval
+
+
+cdef inline NPY_DATETIMEUNIT get_datetime64_unit(object obj) nogil:
+    """
+    returns the unit part of the dtype for a numpy datetime64 object.
+    """
+    return <NPY_DATETIMEUNIT>(<PyDatetimeScalarObject*>obj).obmeta.base

lib/python3.12/site-packages/numpy/__init__.py

@@ -0,0 +1,461 @@
+"""
+NumPy
+=====
+
+Provides
+  1. An array object of arbitrary homogeneous items
+  2. Fast mathematical operations over arrays
+  3. Linear Algebra, Fourier Transforms, Random Number Generation
+
+How to use the documentation
+----------------------------
+Documentation is available in two forms: docstrings provided
+with the code, and a loose standing reference guide, available from
+`the NumPy homepage <https://numpy.org>`_.
+
+We recommend exploring the docstrings using
+`IPython <https://ipython.org>`_, an advanced Python shell with
+TAB-completion and introspection capabilities.  See below for further
+instructions.
+
+The docstring examples assume that `numpy` has been imported as ``np``::
+
+  >>> import numpy as np
+
+Code snippets are indicated by three greater-than signs::
+
+  >>> x = 42
+  >>> x = x + 1
+
+Use the built-in ``help`` function to view a function's docstring::
+
+  >>> help(np.sort)
+  ... # doctest: +SKIP
+
+For some objects, ``np.info(obj)`` may provide additional help.  This is
+particularly true if you see the line "Help on ufunc object:" at the top
+of the help() page.  Ufuncs are implemented in C, not Python, for speed.
+The native Python help() does not know how to view their help, but our
+np.info() function does.
+
+To search for documents containing a keyword, do::
+
+  >>> np.lookfor('keyword')
+  ... # doctest: +SKIP
+
+General-purpose documents like a glossary and help on the basic concepts
+of numpy are available under the ``doc`` sub-module::
+
+  >>> from numpy import doc
+  >>> help(doc)
+  ... # doctest: +SKIP
+
+Available subpackages
+---------------------
+lib
+    Basic functions used by several sub-packages.
+random
+    Core Random Tools
+linalg
+    Core Linear Algebra Tools
+fft
+    Core FFT routines
+polynomial
+    Polynomial tools
+testing
+    NumPy testing tools
+distutils
+    Enhancements to distutils with support for
+    Fortran compilers support and more  (for Python <= 3.11).
+
+Utilities
+---------
+test
+    Run numpy unittests
+show_config
+    Show numpy build configuration
+matlib
+    Make everything matrices.
+__version__
+    NumPy version string
+
+Viewing documentation using IPython
+-----------------------------------
+
+Start IPython and import `numpy` usually under the alias ``np``: `import
+numpy as np`.  Then, directly past or use the ``%cpaste`` magic to paste
+examples into the shell.  To see which functions are available in `numpy`,
+type ``np.<TAB>`` (where ``<TAB>`` refers to the TAB key), or use
+``np.*cos*?<ENTER>`` (where ``<ENTER>`` refers to the ENTER key) to narrow
+down the list.  To view the docstring for a function, use
+``np.cos?<ENTER>`` (to view the docstring) and ``np.cos??<ENTER>`` (to view
+the source code).
+
+Copies vs. in-place operation
+-----------------------------
+Most of the functions in `numpy` return a copy of the array argument
+(e.g., `np.sort`).  In-place versions of these functions are often
+available as array methods, i.e. ``x = np.array([1,2,3]); x.sort()``.
+Exceptions to this rule are documented.
+
+"""
+import sys
+import warnings
+
+from ._globals import _NoValue, _CopyMode
+# These exceptions were moved in 1.25 and are hidden from __dir__()
+from .exceptions import (
+    ComplexWarning, ModuleDeprecationWarning, VisibleDeprecationWarning,
+    TooHardError, AxisError)
+
+
+# If a version with git hash was stored, use that instead
+from . import version
+from .version import __version__
+
+# We first need to detect if we're being called as part of the numpy setup
+# procedure itself in a reliable manner.
+try:
+    __NUMPY_SETUP__
+except NameError:
+    __NUMPY_SETUP__ = False
+
+if __NUMPY_SETUP__:
+    sys.stderr.write('Running from numpy source directory.\n')
+else:
+    # Allow distributors to run custom init code before importing numpy.core
+    from . import _distributor_init
+
+    try:
+        from numpy.__config__ import show as show_config
+    except ImportError as e:
+        msg = """Error importing numpy: you should not try to import numpy from
+        its source directory; please exit the numpy source tree, and relaunch
+        your python interpreter from there."""
+        raise ImportError(msg) from e
+
+    __all__ = [
+        'exceptions', 'ModuleDeprecationWarning', 'VisibleDeprecationWarning',
+        'ComplexWarning', 'TooHardError', 'AxisError']
+
+    # mapping of {name: (value, deprecation_msg)}
+    __deprecated_attrs__ = {}
+
+    from . import core
+    from .core import *
+    from . import compat
+    from . import exceptions
+    from . import dtypes
+    from . import lib
+    # NOTE: to be revisited following future namespace cleanup.
+    # See gh-14454 and gh-15672 for discussion.
+    from .lib import *
+
+    from . import linalg
+    from . import fft
+    from . import polynomial
+    from . import random
+    from . import ctypeslib
+    from . import ma
+    from . import matrixlib as _mat
+    from .matrixlib import *
+
+    # Deprecations introduced in NumPy 1.20.0, 2020-06-06
+    import builtins as _builtins
+
+    _msg = (
+        "module 'numpy' has no attribute '{n}'.\n"
+        "`np.{n}` was a deprecated alias for the builtin `{n}`. "
+        "To avoid this error in existing code, use `{n}` by itself. "
+        "Doing this will not modify any behavior and is safe. {extended_msg}\n"
+        "The aliases was originally deprecated in NumPy 1.20; for more "
+        "details and guidance see the original release note at:\n"
+        "    https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations")
+
+    _specific_msg = (
+        "If you specifically wanted the numpy scalar type, use `np.{}` here.")
+
+    _int_extended_msg = (
+        "When replacing `np.{}`, you may wish to use e.g. `np.int64` "
+        "or `np.int32` to specify the precision. If you wish to review "
+        "your current use, check the release note link for "
+        "additional information.")
+
+    _type_info = [
+        ("object", ""),  # The NumPy scalar only exists by name.
+        ("bool", _specific_msg.format("bool_")),
+        ("float", _specific_msg.format("float64")),
+        ("complex", _specific_msg.format("complex128")),
+        ("str", _specific_msg.format("str_")),
+        ("int", _int_extended_msg.format("int"))]
+
+    __former_attrs__ = {
+         n: _msg.format(n=n, extended_msg=extended_msg)
+         for n, extended_msg in _type_info
+     }
+
+    # Future warning introduced in NumPy 1.24.0, 2022-11-17
+    _msg = (
+        "`np.{n}` is a deprecated alias for `{an}`.  (Deprecated NumPy 1.24)")
+
+    # Some of these are awkward (since `np.str` may be preferable in the long
+    # term), but overall the names ending in 0 seem undesirable
+    _type_info = [
+        ("bool8", bool_, "np.bool_"),
+        ("int0", intp, "np.intp"),
+        ("uint0", uintp, "np.uintp"),
+        ("str0", str_, "np.str_"),
+        ("bytes0", bytes_, "np.bytes_"),
+        ("void0", void, "np.void"),
+        ("object0", object_,
+            "`np.object0` is a deprecated alias for `np.object_`. "
+            "`object` can be used instead.  (Deprecated NumPy 1.24)")]
+
+    # Some of these could be defined right away, but most were aliases to
+    # the Python objects and only removed in NumPy 1.24.  Defining them should
+    # probably wait for NumPy 1.26 or 2.0.
+    # When defined, these should possibly not be added to `__all__` to avoid
+    # import with `from numpy import *`.
+    __future_scalars__ = {"bool", "long", "ulong", "str", "bytes", "object"}
+
+    __deprecated_attrs__.update({
+        n: (alias, _msg.format(n=n, an=an)) for n, alias, an in _type_info})
+
+    import math
+
+    __deprecated_attrs__['math'] = (math,
+        "`np.math` is a deprecated alias for the standard library `math` "
+        "module (Deprecated Numpy 1.25). Replace usages of `np.math` with "
+        "`math`")
+
+    del math, _msg, _type_info
+
+    from .core import abs
+    # now that numpy modules are imported, can initialize limits
+    core.getlimits._register_known_types()
+
+    __all__.extend(['__version__', 'show_config'])
+    __all__.extend(core.__all__)
+    __all__.extend(_mat.__all__)
+    __all__.extend(lib.__all__)
+    __all__.extend(['linalg', 'fft', 'random', 'ctypeslib', 'ma'])
+
+    # Remove min and max from __all__ to avoid `from numpy import *` override
+    # the builtins min/max. Temporary fix for 1.25.x/1.26.x, see gh-24229.
+    __all__.remove('min')
+    __all__.remove('max')
+    __all__.remove('round')
+
+    # Remove one of the two occurrences of `issubdtype`, which is exposed as
+    # both `numpy.core.issubdtype` and `numpy.lib.issubdtype`.
+    __all__.remove('issubdtype')
+
+    # These are exported by np.core, but are replaced by the builtins below
+    # remove them to ensure that we don't end up with `np.long == np.int_`,
+    # which would be a breaking change.
+    del long, unicode
+    __all__.remove('long')
+    __all__.remove('unicode')
+
+    # Remove things that are in the numpy.lib but not in the numpy namespace
+    # Note that there is a test (numpy/tests/test_public_api.py:test_numpy_namespace)
+    # that prevents adding more things to the main namespace by accident.
+    # The list below will grow until the `from .lib import *` fixme above is
+    # taken care of
+    __all__.remove('Arrayterator')
+    del Arrayterator
+
+    # These names were removed in NumPy 1.20.  For at least one release,
+    # attempts to access these names in the numpy namespace will trigger
+    # a warning, and calling the function will raise an exception.
+    _financial_names = ['fv', 'ipmt', 'irr', 'mirr', 'nper', 'npv', 'pmt',
+                        'ppmt', 'pv', 'rate']
+    __expired_functions__ = {
+        name: (f'In accordance with NEP 32, the function {name} was removed '
+               'from NumPy version 1.20.  A replacement for this function '
+               'is available in the numpy_financial library: '
+               'https://pypi.org/project/numpy-financial')
+        for name in _financial_names}
+
+    # Filter out Cython harmless warnings
+    warnings.filterwarnings("ignore", message="numpy.dtype size changed")
+    warnings.filterwarnings("ignore", message="numpy.ufunc size changed")
+    warnings.filterwarnings("ignore", message="numpy.ndarray size changed")
+
+    # oldnumeric and numarray were removed in 1.9. In case some packages import
+    # but do not use them, we define them here for backward compatibility.
+    oldnumeric = 'removed'
+    numarray = 'removed'
+
+    def __getattr__(attr):
+        # Warn for expired attributes, and return a dummy function
+        # that always raises an exception.
+        import warnings
+        import math
+        try:
+            msg = __expired_functions__[attr]
+        except KeyError:
+            pass
+        else:
+            warnings.warn(msg, DeprecationWarning, stacklevel=2)
+
+            def _expired(*args, **kwds):
+                raise RuntimeError(msg)
+
+            return _expired
+
+        # Emit warnings for deprecated attributes
+        try:
+            val, msg = __deprecated_attrs__[attr]
+        except KeyError:
+            pass
+        else:
+            warnings.warn(msg, DeprecationWarning, stacklevel=2)
+            return val
+
+        if attr in __future_scalars__:
+            # And future warnings for those that will change, but also give
+            # the AttributeError
+            warnings.warn(
+                f"In the future `np.{attr}` will be defined as the "
+                "corresponding NumPy scalar.", FutureWarning, stacklevel=2)
+
+        if attr in __former_attrs__:
+            raise AttributeError(__former_attrs__[attr])
+
+        if attr == 'testing':
+            import numpy.testing as testing
+            return testing
+        elif attr == 'Tester':
+            "Removed in NumPy 1.25.0"
+            raise RuntimeError("Tester was removed in NumPy 1.25.")
+
+        raise AttributeError("module {!r} has no attribute "
+                             "{!r}".format(__name__, attr))
+
+    def __dir__():
+        public_symbols = globals().keys() | {'testing'}
+        public_symbols -= {
+            "core", "matrixlib",
+            # These were moved in 1.25 and may be deprecated eventually:
+            "ModuleDeprecationWarning", "VisibleDeprecationWarning",
+            "ComplexWarning", "TooHardError", "AxisError"
+        }
+        return list(public_symbols)
+
+    # Pytest testing
+    from numpy._pytesttester import PytestTester
+    test = PytestTester(__name__)
+    del PytestTester
+
+    def _sanity_check():
+        """
+        Quick sanity checks for common bugs caused by environment.
+        There are some cases e.g. with wrong BLAS ABI that cause wrong
+        results under specific runtime conditions that are not necessarily
+        achieved during test suite runs, and it is useful to catch those early.
+
+        See https://github.com/numpy/numpy/issues/8577 and other
+        similar bug reports.
+
+        """
+        try:
+            x = ones(2, dtype=float32)
+            if not abs(x.dot(x) - float32(2.0)) < 1e-5:
+                raise AssertionError()
+        except AssertionError:
+            msg = ("The current Numpy installation ({!r}) fails to "
+                   "pass simple sanity checks. This can be caused for example "
+                   "by incorrect BLAS library being linked in, or by mixing "
+                   "package managers (pip, conda, apt, ...). Search closed "
+                   "numpy issues for similar problems.")
+            raise RuntimeError(msg.format(__file__)) from None
+
+    _sanity_check()
+    del _sanity_check
+
+    def _mac_os_check():
+        """
+        Quick Sanity check for Mac OS look for accelerate build bugs.
+        Testing numpy polyfit calls init_dgelsd(LAPACK)
+        """
+        try:
+            c = array([3., 2., 1.])
+            x = linspace(0, 2, 5)
+            y = polyval(c, x)
+            _ = polyfit(x, y, 2, cov=True)
+        except ValueError:
+            pass
+
+    if sys.platform == "darwin":
+        from . import exceptions
+        with warnings.catch_warnings(record=True) as w:
+            _mac_os_check()
+            # Throw runtime error, if the test failed Check for warning and error_message
+            if len(w) > 0:
+                for _wn in w:
+                    if _wn.category is exceptions.RankWarning:
+                        # Ignore other warnings, they may not be relevant (see gh-25433).
+                        error_message = f"{_wn.category.__name__}: {str(_wn.message)}"
+                        msg = (
+                            "Polyfit sanity test emitted a warning, most likely due "
+                            "to using a buggy Accelerate backend."
+                            "\nIf you compiled yourself, more information is available at:"
+                            "\nhttps://numpy.org/devdocs/building/index.html"
+                            "\nOtherwise report this to the vendor "
+                            "that provided NumPy.\n\n{}\n".format(error_message))
+                        raise RuntimeError(msg)
+                del _wn
+            del w
+    del _mac_os_check
+
+    # We usually use madvise hugepages support, but on some old kernels it
+    # is slow and thus better avoided.
+    # Specifically kernel version 4.6 had a bug fix which probably fixed this:
+    # https://github.com/torvalds/linux/commit/7cf91a98e607c2f935dbcc177d70011e95b8faff
+    import os
+    use_hugepage = os.environ.get("NUMPY_MADVISE_HUGEPAGE", None)
+    if sys.platform == "linux" and use_hugepage is None:
+        # If there is an issue with parsing the kernel version,
+        # set use_hugepages to 0. Usage of LooseVersion will handle
+        # the kernel version parsing better, but avoided since it
+        # will increase the import time. See: #16679 for related discussion.
+        try:
+            use_hugepage = 1
+            kernel_version = os.uname().release.split(".")[:2]
+            kernel_version = tuple(int(v) for v in kernel_version)
+            if kernel_version < (4, 6):
+                use_hugepage = 0
+        except ValueError:
+            use_hugepages = 0
+    elif use_hugepage is None:
+        # This is not Linux, so it should not matter, just enable anyway
+        use_hugepage = 1
+    else:
+        use_hugepage = int(use_hugepage)
+
+    # Note that this will currently only make a difference on Linux
+    core.multiarray._set_madvise_hugepage(use_hugepage)
+    del use_hugepage
+
+    # Give a warning if NumPy is reloaded or imported on a sub-interpreter
+    # We do this from python, since the C-module may not be reloaded and
+    # it is tidier organized.
+    core.multiarray._multiarray_umath._reload_guard()
+
+    # default to "weak" promotion for "NumPy 2".
+    core._set_promotion_state(
+        os.environ.get("NPY_PROMOTION_STATE",
+                       "weak" if _using_numpy2_behavior() else "legacy"))
+
+    # Tell PyInstaller where to find hook-numpy.py
+    def _pyinstaller_hooks_dir():
+        from pathlib import Path
+        return [str(Path(__file__).with_name("_pyinstaller").resolve())]
+
+    # Remove symbols imported for internal use
+    del os
+
+
+# Remove symbols imported for internal use
+del sys, warnings

lib/python3.12/site-packages/numpy/_distributor_init.py

@@ -0,0 +1,15 @@
+""" Distributor init file
+
+Distributors: you can add custom code here to support particular distributions
+of numpy.
+
+For example, this is a good place to put any BLAS/LAPACK initialization code.
+
+The numpy standard source distribution will not put code in this file, so you
+can safely replace this file with your own version.
+"""
+
+try:
+    from . import _distributor_init_local
+except ImportError:
+    pass

lib/python3.12/site-packages/numpy/_globals.py

@@ -0,0 +1,95 @@
+"""
+Module defining global singleton classes.
+
+This module raises a RuntimeError if an attempt to reload it is made. In that
+way the identities of the classes defined here are fixed and will remain so
+even if numpy itself is reloaded. In particular, a function like the following
+will still work correctly after numpy is reloaded::
+
+    def foo(arg=np._NoValue):
+        if arg is np._NoValue:
+            ...
+
+That was not the case when the singleton classes were defined in the numpy
+``__init__.py`` file. See gh-7844 for a discussion of the reload problem that
+motivated this module.
+
+"""
+import enum
+
+from ._utils import set_module as _set_module
+
+__all__ = ['_NoValue', '_CopyMode']
+
+
+# Disallow reloading this module so as to preserve the identities of the
+# classes defined here.
+if '_is_loaded' in globals():
+    raise RuntimeError('Reloading numpy._globals is not allowed')
+_is_loaded = True
+
+
+class _NoValueType:
+    """Special keyword value.
+
+    The instance of this class may be used as the default value assigned to a
+    keyword if no other obvious default (e.g., `None`) is suitable,
+
+    Common reasons for using this keyword are:
+
+    - A new keyword is added to a function, and that function forwards its
+      inputs to another function or method which can be defined outside of
+      NumPy. For example, ``np.std(x)`` calls ``x.std``, so when a ``keepdims``
+      keyword was added that could only be forwarded if the user explicitly
+      specified ``keepdims``; downstream array libraries may not have added
+      the same keyword, so adding ``x.std(..., keepdims=keepdims)``
+      unconditionally could have broken previously working code.
+    - A keyword is being deprecated, and a deprecation warning must only be
+      emitted when the keyword is used.
+
+    """
+    __instance = None
+    def __new__(cls):
+        # ensure that only one instance exists
+        if not cls.__instance:
+            cls.__instance = super().__new__(cls)
+        return cls.__instance
+
+    def __repr__(self):
+        return "<no value>"
+
+
+_NoValue = _NoValueType()
+
+
+@_set_module("numpy")
+class _CopyMode(enum.Enum):
+    """
+    An enumeration for the copy modes supported
+    by numpy.copy() and numpy.array(). The following three modes are supported,
+
+    - ALWAYS: This means that a deep copy of the input
+              array will always be taken.
+    - IF_NEEDED: This means that a deep copy of the input
+                 array will be taken only if necessary.
+    - NEVER: This means that the deep copy will never be taken.
+             If a copy cannot be avoided then a `ValueError` will be
+             raised.
+
+    Note that the buffer-protocol could in theory do copies.  NumPy currently
+    assumes an object exporting the buffer protocol will never do this.
+    """
+
+    ALWAYS = True
+    IF_NEEDED = False
+    NEVER = 2
+
+    def __bool__(self):
+        # For backwards compatibility
+        if self == _CopyMode.ALWAYS:
+            return True
+
+        if self == _CopyMode.IF_NEEDED:
+            return False
+
+        raise ValueError(f"{self} is neither True nor False.")

lib/python3.12/site-packages/numpy/_pytesttester.py

@@ -0,0 +1,207 @@
+"""
+Pytest test running.
+
+This module implements the ``test()`` function for NumPy modules. The usual
+boiler plate for doing that is to put the following in the module
+``__init__.py`` file::
+
+    from numpy._pytesttester import PytestTester
+    test = PytestTester(__name__)
+    del PytestTester
+
+
+Warnings filtering and other runtime settings should be dealt with in the
+``pytest.ini`` file in the numpy repo root. The behavior of the test depends on
+whether or not that file is found as follows:
+
+* ``pytest.ini`` is present (develop mode)
+    All warnings except those explicitly filtered out are raised as error.
+* ``pytest.ini`` is absent (release mode)
+    DeprecationWarnings and PendingDeprecationWarnings are ignored, other
+    warnings are passed through.
+
+In practice, tests run from the numpy repo are run in develop mode. That
+includes the standard ``python runtests.py`` invocation.
+
+This module is imported by every numpy subpackage, so lies at the top level to
+simplify circular import issues. For the same reason, it contains no numpy
+imports at module scope, instead importing numpy within function calls.
+"""
+import sys
+import os
+
+__all__ = ['PytestTester']
+
+
+def _show_numpy_info():
+    import numpy as np
+
+    print("NumPy version %s" % np.__version__)
+    relaxed_strides = np.ones((10, 1), order="C").flags.f_contiguous
+    print("NumPy relaxed strides checking option:", relaxed_strides)
+    info = np.lib.utils._opt_info()
+    print("NumPy CPU features: ", (info if info else 'nothing enabled'))
+
+
+class PytestTester:
+    """
+    Pytest test runner.
+
+    A test function is typically added to a package's __init__.py like so::
+
+      from numpy._pytesttester import PytestTester
+      test = PytestTester(__name__).test
+      del PytestTester
+
+    Calling this test function finds and runs all tests associated with the
+    module and all its sub-modules.
+
+    Attributes
+    ----------
+    module_name : str
+        Full path to the package to test.
+
+    Parameters
+    ----------
+    module_name : module name
+        The name of the module to test.
+
+    Notes
+    -----
+    Unlike the previous ``nose``-based implementation, this class is not
+    publicly exposed as it performs some ``numpy``-specific warning
+    suppression.
+
+    """
+    def __init__(self, module_name):
+        self.module_name = module_name
+
+    def __call__(self, label='fast', verbose=1, extra_argv=None,
+                 doctests=False, coverage=False, durations=-1, tests=None):
+        """
+        Run tests for module using pytest.
+
+        Parameters
+        ----------
+        label : {'fast', 'full'}, optional
+            Identifies the tests to run. When set to 'fast', tests decorated
+            with `pytest.mark.slow` are skipped, when 'full', the slow marker
+            is ignored.
+        verbose : int, optional
+            Verbosity value for test outputs, in the range 1-3. Default is 1.
+        extra_argv : list, optional
+            List with any extra arguments to pass to pytests.
+        doctests : bool, optional
+            .. note:: Not supported
+        coverage : bool, optional
+            If True, report coverage of NumPy code. Default is False.
+            Requires installation of (pip) pytest-cov.
+        durations : int, optional
+            If < 0, do nothing, If 0, report time of all tests, if > 0,
+            report the time of the slowest `timer` tests. Default is -1.
+        tests : test or list of tests
+            Tests to be executed with pytest '--pyargs'
+
+        Returns
+        -------
+        result : bool
+            Return True on success, false otherwise.
+
+        Notes
+        -----
+        Each NumPy module exposes `test` in its namespace to run all tests for
+        it. For example, to run all tests for numpy.lib:
+
+        >>> np.lib.test() #doctest: +SKIP
+
+        Examples
+        --------
+        >>> result = np.lib.test() #doctest: +SKIP
+        ...
+        1023 passed, 2 skipped, 6 deselected, 1 xfailed in 10.39 seconds
+        >>> result
+        True
+
+        """
+        import pytest
+        import warnings
+
+        module = sys.modules[self.module_name]
+        module_path = os.path.abspath(module.__path__[0])
+
+        # setup the pytest arguments
+        pytest_args = ["-l"]
+
+        # offset verbosity. The "-q" cancels a "-v".
+        pytest_args += ["-q"]
+
+        if sys.version_info < (3, 12):
+            with warnings.catch_warnings():
+                warnings.simplefilter("always")
+                # Filter out distutils cpu warnings (could be localized to
+                # distutils tests). ASV has problems with top level import,
+                # so fetch module for suppression here.
+                from numpy.distutils import cpuinfo
+
+        with warnings.catch_warnings(record=True):
+            # Ignore the warning from importing the array_api submodule. This
+            # warning is done on import, so it would break pytest collection,
+            # but importing it early here prevents the warning from being
+            # issued when it imported again.
+            import numpy.array_api
+
+        # Filter out annoying import messages. Want these in both develop and
+        # release mode.
+        pytest_args += [
+            "-W ignore:Not importing directory",
+            "-W ignore:numpy.dtype size changed",
+            "-W ignore:numpy.ufunc size changed",
+            "-W ignore::UserWarning:cpuinfo",
+            ]
+
+        # When testing matrices, ignore their PendingDeprecationWarnings
+        pytest_args += [
+            "-W ignore:the matrix subclass is not",
+            "-W ignore:Importing from numpy.matlib is",
+            ]
+
+        if doctests:
+            pytest_args += ["--doctest-modules"]
+
+        if extra_argv:
+            pytest_args += list(extra_argv)
+
+        if verbose > 1:
+            pytest_args += ["-" + "v"*(verbose - 1)]
+
+        if coverage:
+            pytest_args += ["--cov=" + module_path]
+
+        if label == "fast":
+            # not importing at the top level to avoid circular import of module
+            from numpy.testing import IS_PYPY
+            if IS_PYPY:
+                pytest_args += ["-m", "not slow and not slow_pypy"]
+            else:
+                pytest_args += ["-m", "not slow"]
+
+        elif label != "full":
+            pytest_args += ["-m", label]
+
+        if durations >= 0:
+            pytest_args += ["--durations=%s" % durations]
+
+        if tests is None:
+            tests = [self.module_name]
+
+        pytest_args += ["--pyargs"] + list(tests)
+
+        # run tests.
+        _show_numpy_info()
+
+        try:
+            code = pytest.main(pytest_args)
+        except SystemExit as exc:
+            code = exc.code
+
+        return code == 0

lib/python3.12/site-packages/numpy/_pytesttester.pyi

@@ -0,0 +1,18 @@
+from collections.abc import Iterable
+from typing import Literal as L
+
+__all__: list[str]
+
+class PytestTester:
+    module_name: str
+    def __init__(self, module_name: str) -> None: ...
+    def __call__(
+        self,
+        label: L["fast", "full"] = ...,
+        verbose: int = ...,
+        extra_argv: None | Iterable[str] = ...,
+        doctests: L[False] = ...,
+        coverage: bool = ...,
+        durations: int = ...,
+        tests: None | Iterable[str] = ...,
+    ) -> bool: ...

lib/python3.12/site-packages/numpy/conftest.py

@@ -0,0 +1,138 @@
+"""
+Pytest configuration and fixtures for the Numpy test suite.
+"""
+import os
+import tempfile
+
+import hypothesis
+import pytest
+import numpy
+
+from numpy.core._multiarray_tests import get_fpu_mode
+
+
+_old_fpu_mode = None
+_collect_results = {}
+
+# Use a known and persistent tmpdir for hypothesis' caches, which
+# can be automatically cleared by the OS or user.
+hypothesis.configuration.set_hypothesis_home_dir(
+    os.path.join(tempfile.gettempdir(), ".hypothesis")
+)
+
+# We register two custom profiles for Numpy - for details see
+# https://hypothesis.readthedocs.io/en/latest/settings.html
+# The first is designed for our own CI runs; the latter also 
+# forces determinism and is designed for use via np.test()
+hypothesis.settings.register_profile(
+    name="numpy-profile", deadline=None, print_blob=True,
+)
+hypothesis.settings.register_profile(
+    name="np.test() profile",
+    deadline=None, print_blob=True, database=None, derandomize=True,
+    suppress_health_check=list(hypothesis.HealthCheck),
+)
+# Note that the default profile is chosen based on the presence 
+# of pytest.ini, but can be overridden by passing the 
+# --hypothesis-profile=NAME argument to pytest.
+_pytest_ini = os.path.join(os.path.dirname(__file__), "..", "pytest.ini")
+hypothesis.settings.load_profile(
+    "numpy-profile" if os.path.isfile(_pytest_ini) else "np.test() profile"
+)
+
+# The experimentalAPI is used in _umath_tests
+os.environ["NUMPY_EXPERIMENTAL_DTYPE_API"] = "1"
+
+def pytest_configure(config):
+    config.addinivalue_line("markers",
+        "valgrind_error: Tests that are known to error under valgrind.")
+    config.addinivalue_line("markers",
+        "leaks_references: Tests that are known to leak references.")
+    config.addinivalue_line("markers",
+        "slow: Tests that are very slow.")
+    config.addinivalue_line("markers",
+        "slow_pypy: Tests that are very slow on pypy.")
+
+
+def pytest_addoption(parser):
+    parser.addoption("--available-memory", action="store", default=None,
+                     help=("Set amount of memory available for running the "
+                           "test suite. This can result to tests requiring "
+                           "especially large amounts of memory to be skipped. "
+                           "Equivalent to setting environment variable "
+                           "NPY_AVAILABLE_MEM. Default: determined"
+                           "automatically."))
+
+
+def pytest_sessionstart(session):
+    available_mem = session.config.getoption('available_memory')
+    if available_mem is not None:
+        os.environ['NPY_AVAILABLE_MEM'] = available_mem
+
+
+#FIXME when yield tests are gone.
+@pytest.hookimpl()
+def pytest_itemcollected(item):
+    """
+    Check FPU precision mode was not changed during test collection.
+
+    The clumsy way we do it here is mainly necessary because numpy
+    still uses yield tests, which can execute code at test collection
+    time.
+    """
+    global _old_fpu_mode
+
+    mode = get_fpu_mode()
+
+    if _old_fpu_mode is None:
+        _old_fpu_mode = mode
+    elif mode != _old_fpu_mode:
+        _collect_results[item] = (_old_fpu_mode, mode)
+        _old_fpu_mode = mode
+
+
+@pytest.fixture(scope="function", autouse=True)
+def check_fpu_mode(request):
+    """
+    Check FPU precision mode was not changed during the test.
+    """
+    old_mode = get_fpu_mode()
+    yield
+    new_mode = get_fpu_mode()
+
+    if old_mode != new_mode:
+        raise AssertionError("FPU precision mode changed from {0:#x} to {1:#x}"
+                             " during the test".format(old_mode, new_mode))
+
+    collect_result = _collect_results.get(request.node)
+    if collect_result is not None:
+        old_mode, new_mode = collect_result
+        raise AssertionError("FPU precision mode changed from {0:#x} to {1:#x}"
+                             " when collecting the test".format(old_mode,
+                                                                new_mode))
+
+
+@pytest.fixture(autouse=True)
+def add_np(doctest_namespace):
+    doctest_namespace['np'] = numpy
+
+@pytest.fixture(autouse=True)
+def env_setup(monkeypatch):
+    monkeypatch.setenv('PYTHONHASHSEED', '0')
+
+
+@pytest.fixture(params=[True, False])
+def weak_promotion(request):
+    """
+    Fixture to ensure "legacy" promotion state or change it to use the new
+    weak promotion (plus warning).  `old_promotion` should be used as a
+    parameter in the function.
+    """
+    state = numpy._get_promotion_state()
+    if request.param:
+        numpy._set_promotion_state("weak_and_warn")
+    else:
+        numpy._set_promotion_state("legacy")
+
+    yield request.param
+    numpy._set_promotion_state(state)

lib/python3.12/site-packages/numpy/ctypeslib.py

@@ -0,0 +1,545 @@
+"""
+============================
+``ctypes`` Utility Functions
+============================
+
+See Also
+--------
+load_library : Load a C library.
+ndpointer : Array restype/argtype with verification.
+as_ctypes : Create a ctypes array from an ndarray.
+as_array : Create an ndarray from a ctypes array.
+
+References
+----------
+.. [1] "SciPy Cookbook: ctypes", https://scipy-cookbook.readthedocs.io/items/Ctypes.html
+
+Examples
+--------
+Load the C library:
+
+>>> _lib = np.ctypeslib.load_library('libmystuff', '.')     #doctest: +SKIP
+
+Our result type, an ndarray that must be of type double, be 1-dimensional
+and is C-contiguous in memory:
+
+>>> array_1d_double = np.ctypeslib.ndpointer(
+...                          dtype=np.double,
+...                          ndim=1, flags='CONTIGUOUS')    #doctest: +SKIP
+
+Our C-function typically takes an array and updates its values
+in-place.  For example::
+
+    void foo_func(double* x, int length)
+    {
+        int i;
+        for (i = 0; i < length; i++) {
+            x[i] = i*i;
+        }
+    }
+
+We wrap it using:
+
+>>> _lib.foo_func.restype = None                      #doctest: +SKIP
+>>> _lib.foo_func.argtypes = [array_1d_double, c_int] #doctest: +SKIP
+
+Then, we're ready to call ``foo_func``:
+
+>>> out = np.empty(15, dtype=np.double)
+>>> _lib.foo_func(out, len(out))                #doctest: +SKIP
+
+"""
+__all__ = ['load_library', 'ndpointer', 'c_intp', 'as_ctypes', 'as_array',
+           'as_ctypes_type']
+
+import os
+from numpy import (
+    integer, ndarray, dtype as _dtype, asarray, frombuffer
+)
+from numpy.core.multiarray import _flagdict, flagsobj
+
+try:
+    import ctypes
+except ImportError:
+    ctypes = None
+
+if ctypes is None:
+    def _dummy(*args, **kwds):
+        """
+        Dummy object that raises an ImportError if ctypes is not available.
+
+        Raises
+        ------
+        ImportError
+            If ctypes is not available.
+
+        """
+        raise ImportError("ctypes is not available.")
+    load_library = _dummy
+    as_ctypes = _dummy
+    as_array = _dummy
+    from numpy import intp as c_intp
+    _ndptr_base = object
+else:
+    import numpy.core._internal as nic
+    c_intp = nic._getintp_ctype()
+    del nic
+    _ndptr_base = ctypes.c_void_p
+
+    # Adapted from Albert Strasheim
+    def load_library(libname, loader_path):
+        """
+        It is possible to load a library using
+
+        >>> lib = ctypes.cdll[<full_path_name>] # doctest: +SKIP
+
+        But there are cross-platform considerations, such as library file extensions,
+        plus the fact Windows will just load the first library it finds with that name.
+        NumPy supplies the load_library function as a convenience.
+
+        .. versionchanged:: 1.20.0
+            Allow libname and loader_path to take any
+            :term:`python:path-like object`.
+
+        Parameters
+        ----------
+        libname : path-like
+            Name of the library, which can have 'lib' as a prefix,
+            but without an extension.
+        loader_path : path-like
+            Where the library can be found.
+
+        Returns
+        -------
+        ctypes.cdll[libpath] : library object
+           A ctypes library object
+
+        Raises
+        ------
+        OSError
+            If there is no library with the expected extension, or the
+            library is defective and cannot be loaded.
+        """
+        # Convert path-like objects into strings
+        libname = os.fsdecode(libname)
+        loader_path = os.fsdecode(loader_path)
+
+        ext = os.path.splitext(libname)[1]
+        if not ext:
+            import sys
+            import sysconfig
+            # Try to load library with platform-specific name, otherwise
+            # default to libname.[so|dll|dylib].  Sometimes, these files are
+            # built erroneously on non-linux platforms.
+            base_ext = ".so"
+            if sys.platform.startswith("darwin"):
+                base_ext = ".dylib"
+            elif sys.platform.startswith("win"):
+                base_ext = ".dll"
+            libname_ext = [libname + base_ext]
+            so_ext = sysconfig.get_config_var("EXT_SUFFIX")
+            if not so_ext == base_ext:
+                libname_ext.insert(0, libname + so_ext)
+        else:
+            libname_ext = [libname]
+
+        loader_path = os.path.abspath(loader_path)
+        if not os.path.isdir(loader_path):
+            libdir = os.path.dirname(loader_path)
+        else:
+            libdir = loader_path
+
+        for ln in libname_ext:
+            libpath = os.path.join(libdir, ln)
+            if os.path.exists(libpath):
+                try:
+                    return ctypes.cdll[libpath]
+                except OSError:
+                    ## defective lib file
+                    raise
+        ## if no successful return in the libname_ext loop:
+        raise OSError("no file with expected extension")
+
+
+def _num_fromflags(flaglist):
+    num = 0
+    for val in flaglist:
+        num += _flagdict[val]
+    return num
+
+_flagnames = ['C_CONTIGUOUS', 'F_CONTIGUOUS', 'ALIGNED', 'WRITEABLE',
+              'OWNDATA', 'WRITEBACKIFCOPY']
+def _flags_fromnum(num):
+    res = []
+    for key in _flagnames:
+        value = _flagdict[key]
+        if (num & value):
+            res.append(key)
+    return res
+
+
+class _ndptr(_ndptr_base):
+    @classmethod
+    def from_param(cls, obj):
+        if not isinstance(obj, ndarray):
+            raise TypeError("argument must be an ndarray")
+        if cls._dtype_ is not None \
+               and obj.dtype != cls._dtype_:
+            raise TypeError("array must have data type %s" % cls._dtype_)
+        if cls._ndim_ is not None \
+               and obj.ndim != cls._ndim_:
+            raise TypeError("array must have %d dimension(s)" % cls._ndim_)
+        if cls._shape_ is not None \
+               and obj.shape != cls._shape_:
+            raise TypeError("array must have shape %s" % str(cls._shape_))
+        if cls._flags_ is not None \
+               and ((obj.flags.num & cls._flags_) != cls._flags_):
+            raise TypeError("array must have flags %s" %
+                    _flags_fromnum(cls._flags_))
+        return obj.ctypes
+
+
+class _concrete_ndptr(_ndptr):
+    """
+    Like _ndptr, but with `_shape_` and `_dtype_` specified.
+
+    Notably, this means the pointer has enough information to reconstruct
+    the array, which is not generally true.
+    """
+    def _check_retval_(self):
+        """
+        This method is called when this class is used as the .restype
+        attribute for a shared-library function, to automatically wrap the
+        pointer into an array.
+        """
+        return self.contents
+
+    @property
+    def contents(self):
+        """
+        Get an ndarray viewing the data pointed to by this pointer.
+
+        This mirrors the `contents` attribute of a normal ctypes pointer
+        """
+        full_dtype = _dtype((self._dtype_, self._shape_))
+        full_ctype = ctypes.c_char * full_dtype.itemsize
+        buffer = ctypes.cast(self, ctypes.POINTER(full_ctype)).contents
+        return frombuffer(buffer, dtype=full_dtype).squeeze(axis=0)
+
+
+# Factory for an array-checking class with from_param defined for
+#  use with ctypes argtypes mechanism
+_pointer_type_cache = {}
+def ndpointer(dtype=None, ndim=None, shape=None, flags=None):
+    """
+    Array-checking restype/argtypes.
+
+    An ndpointer instance is used to describe an ndarray in restypes
+    and argtypes specifications.  This approach is more flexible than
+    using, for example, ``POINTER(c_double)``, since several restrictions
+    can be specified, which are verified upon calling the ctypes function.
+    These include data type, number of dimensions, shape and flags.  If a
+    given array does not satisfy the specified restrictions,
+    a ``TypeError`` is raised.
+
+    Parameters
+    ----------
+    dtype : data-type, optional
+        Array data-type.
+    ndim : int, optional
+        Number of array dimensions.
+    shape : tuple of ints, optional
+        Array shape.
+    flags : str or tuple of str
+        Array flags; may be one or more of:
+
+          - C_CONTIGUOUS / C / CONTIGUOUS
+          - F_CONTIGUOUS / F / FORTRAN
+          - OWNDATA / O
+          - WRITEABLE / W
+          - ALIGNED / A
+          - WRITEBACKIFCOPY / X
+
+    Returns
+    -------
+    klass : ndpointer type object
+        A type object, which is an ``_ndtpr`` instance containing
+        dtype, ndim, shape and flags information.
+
+    Raises
+    ------
+    TypeError
+        If a given array does not satisfy the specified restrictions.
+
+    Examples
+    --------
+    >>> clib.somefunc.argtypes = [np.ctypeslib.ndpointer(dtype=np.float64,
+    ...                                                  ndim=1,
+    ...                                                  flags='C_CONTIGUOUS')]
+    ... #doctest: +SKIP
+    >>> clib.somefunc(np.array([1, 2, 3], dtype=np.float64))
+    ... #doctest: +SKIP
+
+    """
+
+    # normalize dtype to an Optional[dtype]
+    if dtype is not None:
+        dtype = _dtype(dtype)
+
+    # normalize flags to an Optional[int]
+    num = None
+    if flags is not None:
+        if isinstance(flags, str):
+            flags = flags.split(',')
+        elif isinstance(flags, (int, integer)):
+            num = flags
+            flags = _flags_fromnum(num)
+        elif isinstance(flags, flagsobj):
+            num = flags.num
+            flags = _flags_fromnum(num)
+        if num is None:
+            try:
+                flags = [x.strip().upper() for x in flags]
+            except Exception as e:
+                raise TypeError("invalid flags specification") from e
+            num = _num_fromflags(flags)
+
+    # normalize shape to an Optional[tuple]
+    if shape is not None:
+        try:
+            shape = tuple(shape)
+        except TypeError:
+            # single integer -> 1-tuple
+            shape = (shape,)
+
+    cache_key = (dtype, ndim, shape, num)
+
+    try:
+        return _pointer_type_cache[cache_key]
+    except KeyError:
+        pass
+
+    # produce a name for the new type
+    if dtype is None:
+        name = 'any'
+    elif dtype.names is not None:
+        name = str(id(dtype))
+    else:
+        name = dtype.str
+    if ndim is not None:
+        name += "_%dd" % ndim
+    if shape is not None:
+        name += "_"+"x".join(str(x) for x in shape)
+    if flags is not None:
+        name += "_"+"_".join(flags)
+
+    if dtype is not None and shape is not None:
+        base = _concrete_ndptr
+    else:
+        base = _ndptr
+
+    klass = type("ndpointer_%s"%name, (base,),
+                 {"_dtype_": dtype,
+                  "_shape_" : shape,
+                  "_ndim_" : ndim,
+                  "_flags_" : num})
+    _pointer_type_cache[cache_key] = klass
+    return klass
+
+
+if ctypes is not None:
+    def _ctype_ndarray(element_type, shape):
+        """ Create an ndarray of the given element type and shape """
+        for dim in shape[::-1]:
+            element_type = dim * element_type
+            # prevent the type name include np.ctypeslib
+            element_type.__module__ = None
+        return element_type
+
+
+    def _get_scalar_type_map():
+        """
+        Return a dictionary mapping native endian scalar dtype to ctypes types
+        """
+        ct = ctypes
+        simple_types = [
+            ct.c_byte, ct.c_short, ct.c_int, ct.c_long, ct.c_longlong,
+            ct.c_ubyte, ct.c_ushort, ct.c_uint, ct.c_ulong, ct.c_ulonglong,
+            ct.c_float, ct.c_double,
+            ct.c_bool,
+        ]
+        return {_dtype(ctype): ctype for ctype in simple_types}
+
+
+    _scalar_type_map = _get_scalar_type_map()
+
+
+    def _ctype_from_dtype_scalar(dtype):
+        # swapping twice ensure that `=` is promoted to <, >, or |
+        dtype_with_endian = dtype.newbyteorder('S').newbyteorder('S')
+        dtype_native = dtype.newbyteorder('=')
+        try:
+            ctype = _scalar_type_map[dtype_native]
+        except KeyError as e:
+            raise NotImplementedError(
+                "Converting {!r} to a ctypes type".format(dtype)
+            ) from None
+
+        if dtype_with_endian.byteorder == '>':
+            ctype = ctype.__ctype_be__
+        elif dtype_with_endian.byteorder == '<':
+            ctype = ctype.__ctype_le__
+
+        return ctype
+
+
+    def _ctype_from_dtype_subarray(dtype):
+        element_dtype, shape = dtype.subdtype
+        ctype = _ctype_from_dtype(element_dtype)
+        return _ctype_ndarray(ctype, shape)
+
+
+    def _ctype_from_dtype_structured(dtype):
+        # extract offsets of each field
+        field_data = []
+        for name in dtype.names:
+            field_dtype, offset = dtype.fields[name][:2]
+            field_data.append((offset, name, _ctype_from_dtype(field_dtype)))
+
+        # ctypes doesn't care about field order
+        field_data = sorted(field_data, key=lambda f: f[0])
+
+        if len(field_data) > 1 and all(offset == 0 for offset, name, ctype in field_data):
+            # union, if multiple fields all at address 0
+            size = 0
+            _fields_ = []
+            for offset, name, ctype in field_data:
+                _fields_.append((name, ctype))
+                size = max(size, ctypes.sizeof(ctype))
+
+            # pad to the right size
+            if dtype.itemsize != size:
+                _fields_.append(('', ctypes.c_char * dtype.itemsize))
+
+            # we inserted manual padding, so always `_pack_`
+            return type('union', (ctypes.Union,), dict(
+                _fields_=_fields_,
+                _pack_=1,
+                __module__=None,
+            ))
+        else:
+            last_offset = 0
+            _fields_ = []
+            for offset, name, ctype in field_data:
+                padding = offset - last_offset
+                if padding < 0:
+                    raise NotImplementedError("Overlapping fields")
+                if padding > 0:
+                    _fields_.append(('', ctypes.c_char * padding))
+
+                _fields_.append((name, ctype))
+                last_offset = offset + ctypes.sizeof(ctype)
+
+
+            padding = dtype.itemsize - last_offset
+            if padding > 0:
+                _fields_.append(('', ctypes.c_char * padding))
+
+            # we inserted manual padding, so always `_pack_`
+            return type('struct', (ctypes.Structure,), dict(
+                _fields_=_fields_,
+                _pack_=1,
+                __module__=None,
+            ))
+
+
+    def _ctype_from_dtype(dtype):
+        if dtype.fields is not None:
+            return _ctype_from_dtype_structured(dtype)
+        elif dtype.subdtype is not None:
+            return _ctype_from_dtype_subarray(dtype)
+        else:
+            return _ctype_from_dtype_scalar(dtype)
+
+
+    def as_ctypes_type(dtype):
+        r"""
+        Convert a dtype into a ctypes type.
+
+        Parameters
+        ----------
+        dtype : dtype
+            The dtype to convert
+
+        Returns
+        -------
+        ctype
+            A ctype scalar, union, array, or struct
+
+        Raises
+        ------
+        NotImplementedError
+            If the conversion is not possible
+
+        Notes
+        -----
+        This function does not losslessly round-trip in either direction.
+
+        ``np.dtype(as_ctypes_type(dt))`` will:
+
+         - insert padding fields
+         - reorder fields to be sorted by offset
+         - discard field titles
+
+        ``as_ctypes_type(np.dtype(ctype))`` will:
+
+         - discard the class names of `ctypes.Structure`\ s and
+           `ctypes.Union`\ s
+         - convert single-element `ctypes.Union`\ s into single-element
+           `ctypes.Structure`\ s
+         - insert padding fields
+
+        """
+        return _ctype_from_dtype(_dtype(dtype))
+
+
+    def as_array(obj, shape=None):
+        """
+        Create a numpy array from a ctypes array or POINTER.
+
+        The numpy array shares the memory with the ctypes object.
+
+        The shape parameter must be given if converting from a ctypes POINTER.
+        The shape parameter is ignored if converting from a ctypes array
+        """
+        if isinstance(obj, ctypes._Pointer):
+            # convert pointers to an array of the desired shape
+            if shape is None:
+                raise TypeError(
+                    'as_array() requires a shape argument when called on a '
+                    'pointer')
+            p_arr_type = ctypes.POINTER(_ctype_ndarray(obj._type_, shape))
+            obj = ctypes.cast(obj, p_arr_type).contents
+
+        return asarray(obj)
+
+
+    def as_ctypes(obj):
+        """Create and return a ctypes object from a numpy array.  Actually
+        anything that exposes the __array_interface__ is accepted."""
+        ai = obj.__array_interface__
+        if ai["strides"]:
+            raise TypeError("strided arrays not supported")
+        if ai["version"] != 3:
+            raise TypeError("only __array_interface__ version 3 supported")
+        addr, readonly = ai["data"]
+        if readonly:
+            raise TypeError("readonly arrays unsupported")
+
+        # can't use `_dtype((ai["typestr"], ai["shape"]))` here, as it overflows
+        # dtype.itemsize (gh-14214)
+        ctype_scalar = as_ctypes_type(ai["typestr"])
+        result_type = _ctype_ndarray(ctype_scalar, ai["shape"])
+        result = result_type.from_address(addr)
+        result.__keep = obj
+        return result

lib/python3.12/site-packages/numpy/ctypeslib.pyi

@@ -0,0 +1,251 @@
+# NOTE: Numpy's mypy plugin is used for importing the correct
+# platform-specific `ctypes._SimpleCData[int]` sub-type
+from ctypes import c_int64 as _c_intp
+
+import os
+import sys
+import ctypes
+from collections.abc import Iterable, Sequence
+from typing import (
+    Literal as L,
+    Any,
+    Union,
+    TypeVar,
+    Generic,
+    overload,
+    ClassVar,
+)
+
+from numpy import (
+    ndarray,
+    dtype,
+    generic,
+    bool_,
+    byte,
+    short,
+    intc,
+    int_,
+    longlong,
+    ubyte,
+    ushort,
+    uintc,
+    uint,
+    ulonglong,
+    single,
+    double,
+    longdouble,
+    void,
+)
+from numpy.core._internal import _ctypes
+from numpy.core.multiarray import flagsobj
+from numpy._typing import (
+    # Arrays
+    NDArray,
+    _ArrayLike,
+
+    # Shapes
+    _ShapeLike,
+
+    # DTypes
+    DTypeLike,
+    _DTypeLike,
+    _VoidDTypeLike,
+    _BoolCodes,
+    _UByteCodes,
+    _UShortCodes,
+    _UIntCCodes,
+    _UIntCodes,
+    _ULongLongCodes,
+    _ByteCodes,
+    _ShortCodes,
+    _IntCCodes,
+    _IntCodes,
+    _LongLongCodes,
+    _SingleCodes,
+    _DoubleCodes,
+    _LongDoubleCodes,
+)
+
+# TODO: Add a proper `_Shape` bound once we've got variadic typevars
+_DType = TypeVar("_DType", bound=dtype[Any])
+_DTypeOptional = TypeVar("_DTypeOptional", bound=None | dtype[Any])
+_SCT = TypeVar("_SCT", bound=generic)
+
+_FlagsKind = L[
+    'C_CONTIGUOUS', 'CONTIGUOUS', 'C',
+    'F_CONTIGUOUS', 'FORTRAN', 'F',
+    'ALIGNED', 'A',
+    'WRITEABLE', 'W',
+    'OWNDATA', 'O',
+    'WRITEBACKIFCOPY', 'X',
+]
+
+# TODO: Add a shape typevar once we have variadic typevars (PEP 646)
+class _ndptr(ctypes.c_void_p, Generic[_DTypeOptional]):
+    # In practice these 4 classvars are defined in the dynamic class
+    # returned by `ndpointer`
+    _dtype_: ClassVar[_DTypeOptional]
+    _shape_: ClassVar[None]
+    _ndim_: ClassVar[None | int]
+    _flags_: ClassVar[None | list[_FlagsKind]]
+
+    @overload
+    @classmethod
+    def from_param(cls: type[_ndptr[None]], obj: ndarray[Any, Any]) -> _ctypes[Any]: ...
+    @overload
+    @classmethod
+    def from_param(cls: type[_ndptr[_DType]], obj: ndarray[Any, _DType]) -> _ctypes[Any]: ...
+
+class _concrete_ndptr(_ndptr[_DType]):
+    _dtype_: ClassVar[_DType]
+    _shape_: ClassVar[tuple[int, ...]]
+    @property
+    def contents(self) -> ndarray[Any, _DType]: ...
+
+def load_library(
+    libname: str | bytes | os.PathLike[str] | os.PathLike[bytes],
+    loader_path: str | bytes | os.PathLike[str] | os.PathLike[bytes],
+) -> ctypes.CDLL: ...
+
+__all__: list[str]
+
+c_intp = _c_intp
+
+@overload
+def ndpointer(
+    dtype: None = ...,
+    ndim: int = ...,
+    shape: None | _ShapeLike = ...,
+    flags: None | _FlagsKind | Iterable[_FlagsKind] | int | flagsobj = ...,
+) -> type[_ndptr[None]]: ...
+@overload
+def ndpointer(
+    dtype: _DTypeLike[_SCT],
+    ndim: int = ...,
+    *,
+    shape: _ShapeLike,
+    flags: None | _FlagsKind | Iterable[_FlagsKind] | int | flagsobj = ...,
+) -> type[_concrete_ndptr[dtype[_SCT]]]: ...
+@overload
+def ndpointer(
+    dtype: DTypeLike,
+    ndim: int = ...,
+    *,
+    shape: _ShapeLike,
+    flags: None | _FlagsKind | Iterable[_FlagsKind] | int | flagsobj = ...,
+) -> type[_concrete_ndptr[dtype[Any]]]: ...
+@overload
+def ndpointer(
+    dtype: _DTypeLike[_SCT],
+    ndim: int = ...,
+    shape: None = ...,
+    flags: None | _FlagsKind | Iterable[_FlagsKind] | int | flagsobj = ...,
+) -> type[_ndptr[dtype[_SCT]]]: ...
+@overload
+def ndpointer(
+    dtype: DTypeLike,
+    ndim: int = ...,
+    shape: None = ...,
+    flags: None | _FlagsKind | Iterable[_FlagsKind] | int | flagsobj = ...,
+) -> type[_ndptr[dtype[Any]]]: ...
+
+@overload
+def as_ctypes_type(dtype: _BoolCodes | _DTypeLike[bool_] | type[ctypes.c_bool]) -> type[ctypes.c_bool]: ...
+@overload
+def as_ctypes_type(dtype: _ByteCodes | _DTypeLike[byte] | type[ctypes.c_byte]) -> type[ctypes.c_byte]: ...
+@overload
+def as_ctypes_type(dtype: _ShortCodes | _DTypeLike[short] | type[ctypes.c_short]) -> type[ctypes.c_short]: ...
+@overload
+def as_ctypes_type(dtype: _IntCCodes | _DTypeLike[intc] | type[ctypes.c_int]) -> type[ctypes.c_int]: ...
+@overload
+def as_ctypes_type(dtype: _IntCodes | _DTypeLike[int_] | type[int | ctypes.c_long]) -> type[ctypes.c_long]: ...
+@overload
+def as_ctypes_type(dtype: _LongLongCodes | _DTypeLike[longlong] | type[ctypes.c_longlong]) -> type[ctypes.c_longlong]: ...
+@overload
+def as_ctypes_type(dtype: _UByteCodes | _DTypeLike[ubyte] | type[ctypes.c_ubyte]) -> type[ctypes.c_ubyte]: ...
+@overload
+def as_ctypes_type(dtype: _UShortCodes | _DTypeLike[ushort] | type[ctypes.c_ushort]) -> type[ctypes.c_ushort]: ...
+@overload
+def as_ctypes_type(dtype: _UIntCCodes | _DTypeLike[uintc] | type[ctypes.c_uint]) -> type[ctypes.c_uint]: ...
+@overload
+def as_ctypes_type(dtype: _UIntCodes | _DTypeLike[uint] | type[ctypes.c_ulong]) -> type[ctypes.c_ulong]: ...
+@overload
+def as_ctypes_type(dtype: _ULongLongCodes | _DTypeLike[ulonglong] | type[ctypes.c_ulonglong]) -> type[ctypes.c_ulonglong]: ...
+@overload
+def as_ctypes_type(dtype: _SingleCodes | _DTypeLike[single] | type[ctypes.c_float]) -> type[ctypes.c_float]: ...
+@overload
+def as_ctypes_type(dtype: _DoubleCodes | _DTypeLike[double] | type[float | ctypes.c_double]) -> type[ctypes.c_double]: ...
+@overload
+def as_ctypes_type(dtype: _LongDoubleCodes | _DTypeLike[longdouble] | type[ctypes.c_longdouble]) -> type[ctypes.c_longdouble]: ...
+@overload
+def as_ctypes_type(dtype: _VoidDTypeLike) -> type[Any]: ...  # `ctypes.Union` or `ctypes.Structure`
+@overload
+def as_ctypes_type(dtype: str) -> type[Any]: ...
+
+@overload
+def as_array(obj: ctypes._PointerLike, shape: Sequence[int]) -> NDArray[Any]: ...
+@overload
+def as_array(obj: _ArrayLike[_SCT], shape: None | _ShapeLike = ...) -> NDArray[_SCT]: ...
+@overload
+def as_array(obj: object, shape: None | _ShapeLike = ...) -> NDArray[Any]: ...
+
+@overload
+def as_ctypes(obj: bool_) -> ctypes.c_bool: ...
+@overload
+def as_ctypes(obj: byte) -> ctypes.c_byte: ...
+@overload
+def as_ctypes(obj: short) -> ctypes.c_short: ...
+@overload
+def as_ctypes(obj: intc) -> ctypes.c_int: ...
+@overload
+def as_ctypes(obj: int_) -> ctypes.c_long: ...
+@overload
+def as_ctypes(obj: longlong) -> ctypes.c_longlong: ...
+@overload
+def as_ctypes(obj: ubyte) -> ctypes.c_ubyte: ...
+@overload
+def as_ctypes(obj: ushort) -> ctypes.c_ushort: ...
+@overload
+def as_ctypes(obj: uintc) -> ctypes.c_uint: ...
+@overload
+def as_ctypes(obj: uint) -> ctypes.c_ulong: ...
+@overload
+def as_ctypes(obj: ulonglong) -> ctypes.c_ulonglong: ...
+@overload
+def as_ctypes(obj: single) -> ctypes.c_float: ...
+@overload
+def as_ctypes(obj: double) -> ctypes.c_double: ...
+@overload
+def as_ctypes(obj: longdouble) -> ctypes.c_longdouble: ...
+@overload
+def as_ctypes(obj: void) -> Any: ...  # `ctypes.Union` or `ctypes.Structure`
+@overload
+def as_ctypes(obj: NDArray[bool_]) -> ctypes.Array[ctypes.c_bool]: ...
+@overload
+def as_ctypes(obj: NDArray[byte]) -> ctypes.Array[ctypes.c_byte]: ...
+@overload
+def as_ctypes(obj: NDArray[short]) -> ctypes.Array[ctypes.c_short]: ...
+@overload
+def as_ctypes(obj: NDArray[intc]) -> ctypes.Array[ctypes.c_int]: ...
+@overload
+def as_ctypes(obj: NDArray[int_]) -> ctypes.Array[ctypes.c_long]: ...
+@overload
+def as_ctypes(obj: NDArray[longlong]) -> ctypes.Array[ctypes.c_longlong]: ...
+@overload
+def as_ctypes(obj: NDArray[ubyte]) -> ctypes.Array[ctypes.c_ubyte]: ...
+@overload
+def as_ctypes(obj: NDArray[ushort]) -> ctypes.Array[ctypes.c_ushort]: ...
+@overload
+def as_ctypes(obj: NDArray[uintc]) -> ctypes.Array[ctypes.c_uint]: ...
+@overload
+def as_ctypes(obj: NDArray[uint]) -> ctypes.Array[ctypes.c_ulong]: ...
+@overload
+def as_ctypes(obj: NDArray[ulonglong]) -> ctypes.Array[ctypes.c_ulonglong]: ...
+@overload
+def as_ctypes(obj: NDArray[single]) -> ctypes.Array[ctypes.c_float]: ...
+@overload
+def as_ctypes(obj: NDArray[double]) -> ctypes.Array[ctypes.c_double]: ...
+@overload
+def as_ctypes(obj: NDArray[longdouble]) -> ctypes.Array[ctypes.c_longdouble]: ...
+@overload
+def as_ctypes(obj: NDArray[void]) -> ctypes.Array[Any]: ...  # `ctypes.Union` or `ctypes.Structure`

lib/python3.12/site-packages/numpy/dtypes.py

@@ -0,0 +1,77 @@
+"""
+DType classes and utility (:mod:`numpy.dtypes`)
+===============================================
+
+This module is home to specific dtypes related functionality and their classes.
+For more general information about dtypes, also see `numpy.dtype` and
+:ref:`arrays.dtypes`.
+
+Similar to the builtin ``types`` module, this submodule defines types (classes)
+that are not widely used directly.
+
+.. versionadded:: NumPy 1.25
+
+    The dtypes module is new in NumPy 1.25.  Previously DType classes were
+    only accessible indirectly.
+
+
+DType classes
+-------------
+
+The following are the classes of the corresponding NumPy dtype instances and
+NumPy scalar types.  The classes can be used in ``isinstance`` checks and can
+also be instantiated or used directly.  Direct use of these classes is not
+typical, since their scalar counterparts (e.g. ``np.float64``) or strings
+like ``"float64"`` can be used.
+
+.. list-table::
+    :header-rows: 1
+
+    * - Group
+      - DType class
+
+    * - Boolean
+      - ``BoolDType``
+
+    * - Bit-sized integers
+      - ``Int8DType``, ``UInt8DType``, ``Int16DType``, ``UInt16DType``,
+        ``Int32DType``, ``UInt32DType``, ``Int64DType``, ``UInt64DType``
+
+    * - C-named integers (may be aliases)
+      - ``ByteDType``, ``UByteDType``, ``ShortDType``, ``UShortDType``,
+        ``IntDType``, ``UIntDType``, ``LongDType``, ``ULongDType``,
+        ``LongLongDType``, ``ULongLongDType``
+
+    * - Floating point
+      - ``Float16DType``, ``Float32DType``, ``Float64DType``,
+        ``LongDoubleDType``
+
+    * - Complex
+      - ``Complex64DType``, ``Complex128DType``, ``CLongDoubleDType``
+
+    * - Strings
+      - ``BytesDType``, ``BytesDType``
+
+    * - Times
+      - ``DateTime64DType``, ``TimeDelta64DType``
+
+    * - Others
+      - ``ObjectDType``, ``VoidDType``
+
+"""
+
+__all__ = []
+
+
+def _add_dtype_helper(DType, alias):
+    # Function to add DTypes a bit more conveniently without channeling them
+    # through `numpy.core._multiarray_umath` namespace or similar.
+    from numpy import dtypes
+
+    setattr(dtypes, DType.__name__, DType)
+    __all__.append(DType.__name__)
+
+    if alias:
+        alias = alias.removeprefix("numpy.dtypes.")
+        setattr(dtypes, alias, DType)
+        __all__.append(alias)

lib/python3.12/site-packages/numpy/dtypes.pyi

@@ -0,0 +1,43 @@
+import numpy as np
+
+
+__all__: list[str]
+
+# Boolean:
+BoolDType = np.dtype[np.bool_]
+# Sized integers:
+Int8DType = np.dtype[np.int8]
+UInt8DType = np.dtype[np.uint8]
+Int16DType = np.dtype[np.int16]
+UInt16DType = np.dtype[np.uint16]
+Int32DType = np.dtype[np.int32]
+UInt32DType = np.dtype[np.uint32]
+Int64DType = np.dtype[np.int64]
+UInt64DType = np.dtype[np.uint64]
+# Standard C-named version/alias:
+ByteDType = np.dtype[np.byte]
+UByteDType = np.dtype[np.ubyte]
+ShortDType = np.dtype[np.short]
+UShortDType = np.dtype[np.ushort]
+IntDType = np.dtype[np.intc]
+UIntDType = np.dtype[np.uintc]
+LongDType = np.dtype[np.int_]  # Unfortunately, the correct scalar
+ULongDType = np.dtype[np.uint]  # Unfortunately, the correct scalar
+LongLongDType = np.dtype[np.longlong]
+ULongLongDType = np.dtype[np.ulonglong]
+# Floats
+Float16DType = np.dtype[np.float16]
+Float32DType = np.dtype[np.float32]
+Float64DType = np.dtype[np.float64]
+LongDoubleDType = np.dtype[np.longdouble]
+# Complex:
+Complex64DType = np.dtype[np.complex64]
+Complex128DType = np.dtype[np.complex128]
+CLongDoubleDType = np.dtype[np.clongdouble]
+# Others:
+ObjectDType = np.dtype[np.object_]
+BytesDType = np.dtype[np.bytes_]
+StrDType = np.dtype[np.str_]
+VoidDType = np.dtype[np.void]
+DateTime64DType = np.dtype[np.datetime64]
+TimeDelta64DType = np.dtype[np.timedelta64]

lib/python3.12/site-packages/numpy/exceptions.py

@@ -0,0 +1,231 @@
+"""
+Exceptions and Warnings (:mod:`numpy.exceptions`)
+=================================================
+
+General exceptions used by NumPy.  Note that some exceptions may be module
+specific, such as linear algebra errors.
+
+.. versionadded:: NumPy 1.25
+
+    The exceptions module is new in NumPy 1.25.  Older exceptions remain
+    available through the main NumPy namespace for compatibility.
+
+.. currentmodule:: numpy.exceptions
+
+Warnings
+--------
+.. autosummary::
+   :toctree: generated/
+
+   ComplexWarning             Given when converting complex to real.
+   VisibleDeprecationWarning  Same as a DeprecationWarning, but more visible.
+
+Exceptions
+----------
+.. autosummary::
+   :toctree: generated/
+
+    AxisError          Given when an axis was invalid.
+    DTypePromotionError   Given when no common dtype could be found.
+    TooHardError       Error specific to `numpy.shares_memory`.
+
+"""
+
+
+__all__ = [
+    "ComplexWarning", "VisibleDeprecationWarning", "ModuleDeprecationWarning",
+    "TooHardError", "AxisError", "DTypePromotionError"]
+
+
+# Disallow reloading this module so as to preserve the identities of the
+# classes defined here.
+if '_is_loaded' in globals():
+    raise RuntimeError('Reloading numpy._globals is not allowed')
+_is_loaded = True
+
+
+class ComplexWarning(RuntimeWarning):
+    """
+    The warning raised when casting a complex dtype to a real dtype.
+
+    As implemented, casting a complex number to a real discards its imaginary
+    part, but this behavior may not be what the user actually wants.
+
+    """
+    pass
+
+
+class ModuleDeprecationWarning(DeprecationWarning):
+    """Module deprecation warning.
+
+    .. warning::
+
+        This warning should not be used, since nose testing is not relevant
+        anymore.
+
+    The nose tester turns ordinary Deprecation warnings into test failures.
+    That makes it hard to deprecate whole modules, because they get
+    imported by default. So this is a special Deprecation warning that the
+    nose tester will let pass without making tests fail.
+
+    """
+
+
+class VisibleDeprecationWarning(UserWarning):
+    """Visible deprecation warning.
+
+    By default, python will not show deprecation warnings, so this class
+    can be used when a very visible warning is helpful, for example because
+    the usage is most likely a user bug.
+
+    """
+
+
+# Exception used in shares_memory()
+class TooHardError(RuntimeError):
+    """max_work was exceeded.
+
+    This is raised whenever the maximum number of candidate solutions
+    to consider specified by the ``max_work`` parameter is exceeded.
+    Assigning a finite number to max_work may have caused the operation
+    to fail.
+
+    """
+
+    pass
+
+
+class AxisError(ValueError, IndexError):
+    """Axis supplied was invalid.
+
+    This is raised whenever an ``axis`` parameter is specified that is larger
+    than the number of array dimensions.
+    For compatibility with code written against older numpy versions, which
+    raised a mixture of `ValueError` and `IndexError` for this situation, this
+    exception subclasses both to ensure that ``except ValueError`` and
+    ``except IndexError`` statements continue to catch `AxisError`.
+
+    .. versionadded:: 1.13
+
+    Parameters
+    ----------
+    axis : int or str
+        The out of bounds axis or a custom exception message.
+        If an axis is provided, then `ndim` should be specified as well.
+    ndim : int, optional
+        The number of array dimensions.
+    msg_prefix : str, optional
+        A prefix for the exception message.
+
+    Attributes
+    ----------
+    axis : int, optional
+        The out of bounds axis or ``None`` if a custom exception
+        message was provided. This should be the axis as passed by
+        the user, before any normalization to resolve negative indices.
+
+        .. versionadded:: 1.22
+    ndim : int, optional
+        The number of array dimensions or ``None`` if a custom exception
+        message was provided.
+
+        .. versionadded:: 1.22
+
+
+    Examples
+    --------
+    >>> array_1d = np.arange(10)
+    >>> np.cumsum(array_1d, axis=1)
+    Traceback (most recent call last):
+      ...
+    numpy.exceptions.AxisError: axis 1 is out of bounds for array of dimension 1
+
+    Negative axes are preserved:
+
+    >>> np.cumsum(array_1d, axis=-2)
+    Traceback (most recent call last):
+      ...
+    numpy.exceptions.AxisError: axis -2 is out of bounds for array of dimension 1
+
+    The class constructor generally takes the axis and arrays'
+    dimensionality as arguments:
+
+    >>> print(np.AxisError(2, 1, msg_prefix='error'))
+    error: axis 2 is out of bounds for array of dimension 1
+
+    Alternatively, a custom exception message can be passed:
+
+    >>> print(np.AxisError('Custom error message'))
+    Custom error message
+
+    """
+
+    __slots__ = ("axis", "ndim", "_msg")
+
+    def __init__(self, axis, ndim=None, msg_prefix=None):
+        if ndim is msg_prefix is None:
+            # single-argument form: directly set the error message
+            self._msg = axis
+            self.axis = None
+            self.ndim = None
+        else:
+            self._msg = msg_prefix
+            self.axis = axis
+            self.ndim = ndim
+
+    def __str__(self):
+        axis = self.axis
+        ndim = self.ndim
+
+        if axis is ndim is None:
+            return self._msg
+        else:
+            msg = f"axis {axis} is out of bounds for array of dimension {ndim}"
+            if self._msg is not None:
+                msg = f"{self._msg}: {msg}"
+            return msg
+
+
+class DTypePromotionError(TypeError):
+    """Multiple DTypes could not be converted to a common one.
+
+    This exception derives from ``TypeError`` and is raised whenever dtypes
+    cannot be converted to a single common one.  This can be because they
+    are of a different category/class or incompatible instances of the same
+    one (see Examples).
+
+    Notes
+    -----
+    Many functions will use promotion to find the correct result and
+    implementation.  For these functions the error will typically be chained
+    with a more specific error indicating that no implementation was found
+    for the input dtypes.
+
+    Typically promotion should be considered "invalid" between the dtypes of
+    two arrays when `arr1 == arr2` can safely return all ``False`` because the
+    dtypes are fundamentally different.
+
+    Examples
+    --------
+    Datetimes and complex numbers are incompatible classes and cannot be
+    promoted:
+
+    >>> np.result_type(np.dtype("M8[s]"), np.complex128)
+    DTypePromotionError: The DType <class 'numpy.dtype[datetime64]'> could not
+    be promoted by <class 'numpy.dtype[complex128]'>. This means that no common
+    DType exists for the given inputs. For example they cannot be stored in a
+    single array unless the dtype is `object`. The full list of DTypes is:
+    (<class 'numpy.dtype[datetime64]'>, <class 'numpy.dtype[complex128]'>)
+
+    For example for structured dtypes, the structure can mismatch and the
+    same ``DTypePromotionError`` is given when two structured dtypes with
+    a mismatch in their number of fields is given:
+
+    >>> dtype1 = np.dtype([("field1", np.float64), ("field2", np.int64)])
+    >>> dtype2 = np.dtype([("field1", np.float64)])
+    >>> np.promote_types(dtype1, dtype2)
+    DTypePromotionError: field names `('field1', 'field2')` and `('field1',)`
+    mismatch.
+
+    """
+    pass

lib/python3.12/site-packages/numpy/exceptions.pyi

@@ -0,0 +1,18 @@
+from typing import overload
+
+__all__: list[str]
+
+class ComplexWarning(RuntimeWarning): ...
+class ModuleDeprecationWarning(DeprecationWarning): ...
+class VisibleDeprecationWarning(UserWarning): ...
+class TooHardError(RuntimeError): ...
+class DTypePromotionError(TypeError): ...
+
+class AxisError(ValueError, IndexError):
+    axis: None | int
+    ndim: None | int
+    @overload
+    def __init__(self, axis: str, ndim: None = ..., msg_prefix: None = ...) -> None: ...
+    @overload
+    def __init__(self, axis: int, ndim: int, msg_prefix: None | str = ...) -> None: ...
+    def __str__(self) -> str: ...

lib/python3.12/site-packages/numpy/matlib.py

@@ -0,0 +1,378 @@
+import warnings
+
+# 2018-05-29, PendingDeprecationWarning added to matrix.__new__
+# 2020-01-23, numpy 1.19.0 PendingDeprecatonWarning
+warnings.warn("Importing from numpy.matlib is deprecated since 1.19.0. "
+              "The matrix subclass is not the recommended way to represent "
+              "matrices or deal with linear algebra (see "
+              "https://docs.scipy.org/doc/numpy/user/numpy-for-matlab-users.html). "
+              "Please adjust your code to use regular ndarray. ",
+              PendingDeprecationWarning, stacklevel=2)
+
+import numpy as np
+from numpy.matrixlib.defmatrix import matrix, asmatrix
+# Matlib.py contains all functions in the numpy namespace with a few
+# replacements. See doc/source/reference/routines.matlib.rst for details.
+# Need * as we're copying the numpy namespace.
+from numpy import *  # noqa: F403
+
+__version__ = np.__version__
+
+__all__ = np.__all__[:] # copy numpy namespace
+__all__ += ['rand', 'randn', 'repmat']
+
+def empty(shape, dtype=None, order='C'):
+    """Return a new matrix of given shape and type, without initializing entries.
+
+    Parameters
+    ----------
+    shape : int or tuple of int
+        Shape of the empty matrix.
+    dtype : data-type, optional
+        Desired output data-type.
+    order : {'C', 'F'}, optional
+        Whether to store multi-dimensional data in row-major
+        (C-style) or column-major (Fortran-style) order in
+        memory.
+
+    See Also
+    --------
+    empty_like, zeros
+
+    Notes
+    -----
+    `empty`, unlike `zeros`, does not set the matrix values to zero,
+    and may therefore be marginally faster.  On the other hand, it requires
+    the user to manually set all the values in the array, and should be
+    used with caution.
+
+    Examples
+    --------
+    >>> import numpy.matlib
+    >>> np.matlib.empty((2, 2))    # filled with random data
+    matrix([[  6.76425276e-320,   9.79033856e-307], # random
+            [  7.39337286e-309,   3.22135945e-309]])
+    >>> np.matlib.empty((2, 2), dtype=int)
+    matrix([[ 6600475,        0], # random
+            [ 6586976, 22740995]])
+
+    """
+    return ndarray.__new__(matrix, shape, dtype, order=order)
+
+def ones(shape, dtype=None, order='C'):
+    """
+    Matrix of ones.
+
+    Return a matrix of given shape and type, filled with ones.
+
+    Parameters
+    ----------
+    shape : {sequence of ints, int}
+        Shape of the matrix
+    dtype : data-type, optional
+        The desired data-type for the matrix, default is np.float64.
+    order : {'C', 'F'}, optional
+        Whether to store matrix in C- or Fortran-contiguous order,
+        default is 'C'.
+
+    Returns
+    -------
+    out : matrix
+        Matrix of ones of given shape, dtype, and order.
+
+    See Also
+    --------
+    ones : Array of ones.
+    matlib.zeros : Zero matrix.
+
+    Notes
+    -----
+    If `shape` has length one i.e. ``(N,)``, or is a scalar ``N``,
+    `out` becomes a single row matrix of shape ``(1,N)``.
+
+    Examples
+    --------
+    >>> np.matlib.ones((2,3))
+    matrix([[1.,  1.,  1.],
+            [1.,  1.,  1.]])
+
+    >>> np.matlib.ones(2)
+    matrix([[1.,  1.]])
+
+    """
+    a = ndarray.__new__(matrix, shape, dtype, order=order)
+    a.fill(1)
+    return a
+
+def zeros(shape, dtype=None, order='C'):
+    """
+    Return a matrix of given shape and type, filled with zeros.
+
+    Parameters
+    ----------
+    shape : int or sequence of ints
+        Shape of the matrix
+    dtype : data-type, optional
+        The desired data-type for the matrix, default is float.
+    order : {'C', 'F'}, optional
+        Whether to store the result in C- or Fortran-contiguous order,
+        default is 'C'.
+
+    Returns
+    -------
+    out : matrix
+        Zero matrix of given shape, dtype, and order.
+
+    See Also
+    --------
+    numpy.zeros : Equivalent array function.
+    matlib.ones : Return a matrix of ones.
+
+    Notes
+    -----
+    If `shape` has length one i.e. ``(N,)``, or is a scalar ``N``,
+    `out` becomes a single row matrix of shape ``(1,N)``.
+
+    Examples
+    --------
+    >>> import numpy.matlib
+    >>> np.matlib.zeros((2, 3))
+    matrix([[0.,  0.,  0.],
+            [0.,  0.,  0.]])
+
+    >>> np.matlib.zeros(2)
+    matrix([[0.,  0.]])
+
+    """
+    a = ndarray.__new__(matrix, shape, dtype, order=order)
+    a.fill(0)
+    return a
+
+def identity(n,dtype=None):
+    """
+    Returns the square identity matrix of given size.
+
+    Parameters
+    ----------
+    n : int
+        Size of the returned identity matrix.
+    dtype : data-type, optional
+        Data-type of the output. Defaults to ``float``.
+
+    Returns
+    -------
+    out : matrix
+        `n` x `n` matrix with its main diagonal set to one,
+        and all other elements zero.
+
+    See Also
+    --------
+    numpy.identity : Equivalent array function.
+    matlib.eye : More general matrix identity function.
+
+    Examples
+    --------
+    >>> import numpy.matlib
+    >>> np.matlib.identity(3, dtype=int)
+    matrix([[1, 0, 0],
+            [0, 1, 0],
+            [0, 0, 1]])
+
+    """
+    a = array([1]+n*[0], dtype=dtype)
+    b = empty((n, n), dtype=dtype)
+    b.flat = a
+    return b
+
+def eye(n,M=None, k=0, dtype=float, order='C'):
+    """
+    Return a matrix with ones on the diagonal and zeros elsewhere.
+
+    Parameters
+    ----------
+    n : int
+        Number of rows in the output.
+    M : int, optional
+        Number of columns in the output, defaults to `n`.
+    k : int, optional
+        Index of the diagonal: 0 refers to the main diagonal,
+        a positive value refers to an upper diagonal,
+        and a negative value to a lower diagonal.
+    dtype : dtype, optional
+        Data-type of the returned matrix.
+    order : {'C', 'F'}, optional
+        Whether the output should be stored in row-major (C-style) or
+        column-major (Fortran-style) order in memory.
+
+        .. versionadded:: 1.14.0
+
+    Returns
+    -------
+    I : matrix
+        A `n` x `M` matrix where all elements are equal to zero,
+        except for the `k`-th diagonal, whose values are equal to one.
+
+    See Also
+    --------
+    numpy.eye : Equivalent array function.
+    identity : Square identity matrix.
+
+    Examples
+    --------
+    >>> import numpy.matlib
+    >>> np.matlib.eye(3, k=1, dtype=float)
+    matrix([[0.,  1.,  0.],
+            [0.,  0.,  1.],
+            [0.,  0.,  0.]])
+
+    """
+    return asmatrix(np.eye(n, M=M, k=k, dtype=dtype, order=order))
+
+def rand(*args):
+    """
+    Return a matrix of random values with given shape.
+
+    Create a matrix of the given shape and propagate it with
+    random samples from a uniform distribution over ``[0, 1)``.
+
+    Parameters
+    ----------
+    \\*args : Arguments
+        Shape of the output.
+        If given as N integers, each integer specifies the size of one
+        dimension.
+        If given as a tuple, this tuple gives the complete shape.
+
+    Returns
+    -------
+    out : ndarray
+        The matrix of random values with shape given by `\\*args`.
+
+    See Also
+    --------
+    randn, numpy.random.RandomState.rand
+
+    Examples
+    --------
+    >>> np.random.seed(123)
+    >>> import numpy.matlib
+    >>> np.matlib.rand(2, 3)
+    matrix([[0.69646919, 0.28613933, 0.22685145],
+            [0.55131477, 0.71946897, 0.42310646]])
+    >>> np.matlib.rand((2, 3))
+    matrix([[0.9807642 , 0.68482974, 0.4809319 ],
+            [0.39211752, 0.34317802, 0.72904971]])
+
+    If the first argument is a tuple, other arguments are ignored:
+
+    >>> np.matlib.rand((2, 3), 4)
+    matrix([[0.43857224, 0.0596779 , 0.39804426],
+            [0.73799541, 0.18249173, 0.17545176]])
+
+    """
+    if isinstance(args[0], tuple):
+        args = args[0]
+    return asmatrix(np.random.rand(*args))
+
+def randn(*args):
+    """
+    Return a random matrix with data from the "standard normal" distribution.
+
+    `randn` generates a matrix filled with random floats sampled from a
+    univariate "normal" (Gaussian) distribution of mean 0 and variance 1.
+
+    Parameters
+    ----------
+    \\*args : Arguments
+        Shape of the output.
+        If given as N integers, each integer specifies the size of one
+        dimension. If given as a tuple, this tuple gives the complete shape.
+
+    Returns
+    -------
+    Z : matrix of floats
+        A matrix of floating-point samples drawn from the standard normal
+        distribution.
+
+    See Also
+    --------
+    rand, numpy.random.RandomState.randn
+
+    Notes
+    -----
+    For random samples from the normal distribution with mean ``mu`` and
+    standard deviation ``sigma``, use::
+
+        sigma * np.matlib.randn(...) + mu
+
+    Examples
+    --------
+    >>> np.random.seed(123)
+    >>> import numpy.matlib
+    >>> np.matlib.randn(1)
+    matrix([[-1.0856306]])
+    >>> np.matlib.randn(1, 2, 3)
+    matrix([[ 0.99734545,  0.2829785 , -1.50629471],
+            [-0.57860025,  1.65143654, -2.42667924]])
+
+    Two-by-four matrix of samples from the normal distribution with
+    mean 3 and standard deviation 2.5:
+
+    >>> 2.5 * np.matlib.randn((2, 4)) + 3
+    matrix([[1.92771843, 6.16484065, 0.83314899, 1.30278462],
+            [2.76322758, 6.72847407, 1.40274501, 1.8900451 ]])
+
+    """
+    if isinstance(args[0], tuple):
+        args = args[0]
+    return asmatrix(np.random.randn(*args))
+
+def repmat(a, m, n):
+    """
+    Repeat a 0-D to 2-D array or matrix MxN times.
+
+    Parameters
+    ----------
+    a : array_like
+        The array or matrix to be repeated.
+    m, n : int
+        The number of times `a` is repeated along the first and second axes.
+
+    Returns
+    -------
+    out : ndarray
+        The result of repeating `a`.
+
+    Examples
+    --------
+    >>> import numpy.matlib
+    >>> a0 = np.array(1)
+    >>> np.matlib.repmat(a0, 2, 3)
+    array([[1, 1, 1],
+           [1, 1, 1]])
+
+    >>> a1 = np.arange(4)
+    >>> np.matlib.repmat(a1, 2, 2)
+    array([[0, 1, 2, 3, 0, 1, 2, 3],
+           [0, 1, 2, 3, 0, 1, 2, 3]])
+
+    >>> a2 = np.asmatrix(np.arange(6).reshape(2, 3))
+    >>> np.matlib.repmat(a2, 2, 3)
+    matrix([[0, 1, 2, 0, 1, 2, 0, 1, 2],
+            [3, 4, 5, 3, 4, 5, 3, 4, 5],
+            [0, 1, 2, 0, 1, 2, 0, 1, 2],
+            [3, 4, 5, 3, 4, 5, 3, 4, 5]])
+
+    """
+    a = asanyarray(a)
+    ndim = a.ndim
+    if ndim == 0:
+        origrows, origcols = (1, 1)
+    elif ndim == 1:
+        origrows, origcols = (1, a.shape[0])
+    else:
+        origrows, origcols = a.shape
+    rows = origrows * m
+    cols = origcols * n
+    c = a.reshape(1, a.size).repeat(m, 0).reshape(rows, origcols).repeat(n, 0)
+    return c.reshape(rows, cols)

lib/python3.12/site-packages/numpy/polynomial/__init__.py

@@ -0,0 +1,185 @@
+"""
+A sub-package for efficiently dealing with polynomials.
+
+Within the documentation for this sub-package, a "finite power series,"
+i.e., a polynomial (also referred to simply as a "series") is represented
+by a 1-D numpy array of the polynomial's coefficients, ordered from lowest
+order term to highest.  For example, array([1,2,3]) represents
+``P_0 + 2*P_1 + 3*P_2``, where P_n is the n-th order basis polynomial
+applicable to the specific module in question, e.g., `polynomial` (which
+"wraps" the "standard" basis) or `chebyshev`.  For optimal performance,
+all operations on polynomials, including evaluation at an argument, are
+implemented as operations on the coefficients.  Additional (module-specific)
+information can be found in the docstring for the module of interest.
+
+This package provides *convenience classes* for each of six different kinds
+of polynomials:
+
+         ========================    ================
+         **Name**                    **Provides**
+         ========================    ================
+         `~polynomial.Polynomial`    Power series
+         `~chebyshev.Chebyshev`      Chebyshev series
+         `~legendre.Legendre`        Legendre series
+         `~laguerre.Laguerre`        Laguerre series
+         `~hermite.Hermite`          Hermite series
+         `~hermite_e.HermiteE`       HermiteE series
+         ========================    ================
+
+These *convenience classes* provide a consistent interface for creating,
+manipulating, and fitting data with polynomials of different bases.
+The convenience classes are the preferred interface for the `~numpy.polynomial`
+package, and are available from the ``numpy.polynomial`` namespace.
+This eliminates the need to navigate to the corresponding submodules, e.g.
+``np.polynomial.Polynomial`` or ``np.polynomial.Chebyshev`` instead of
+``np.polynomial.polynomial.Polynomial`` or
+``np.polynomial.chebyshev.Chebyshev``, respectively.
+The classes provide a more consistent and concise interface than the
+type-specific functions defined in the submodules for each type of polynomial.
+For example, to fit a Chebyshev polynomial with degree ``1`` to data given
+by arrays ``xdata`` and ``ydata``, the
+`~chebyshev.Chebyshev.fit` class method::
+
+    >>> from numpy.polynomial import Chebyshev
+    >>> c = Chebyshev.fit(xdata, ydata, deg=1)
+
+is preferred over the `chebyshev.chebfit` function from the
+``np.polynomial.chebyshev`` module::
+
+    >>> from numpy.polynomial.chebyshev import chebfit
+    >>> c = chebfit(xdata, ydata, deg=1)
+
+See :doc:`routines.polynomials.classes` for more details.
+
+Convenience Classes
+===================
+
+The following lists the various constants and methods common to all of
+the classes representing the various kinds of polynomials. In the following,
+the term ``Poly`` represents any one of the convenience classes (e.g.
+`~polynomial.Polynomial`, `~chebyshev.Chebyshev`, `~hermite.Hermite`, etc.)
+while the lowercase ``p`` represents an **instance** of a polynomial class.
+
+Constants
+---------
+
+- ``Poly.domain``     -- Default domain
+- ``Poly.window``     -- Default window
+- ``Poly.basis_name`` -- String used to represent the basis
+- ``Poly.maxpower``   -- Maximum value ``n`` such that ``p**n`` is allowed
+- ``Poly.nickname``   -- String used in printing
+
+Creation
+--------
+
+Methods for creating polynomial instances.
+
+- ``Poly.basis(degree)``    -- Basis polynomial of given degree
+- ``Poly.identity()``       -- ``p`` where ``p(x) = x`` for all ``x``
+- ``Poly.fit(x, y, deg)``   -- ``p`` of degree ``deg`` with coefficients
+  determined by the least-squares fit to the data ``x``, ``y``
+- ``Poly.fromroots(roots)`` -- ``p`` with specified roots
+- ``p.copy()``              -- Create a copy of ``p``
+
+Conversion
+----------
+
+Methods for converting a polynomial instance of one kind to another.
+
+- ``p.cast(Poly)``    -- Convert ``p`` to instance of kind ``Poly``
+- ``p.convert(Poly)`` -- Convert ``p`` to instance of kind ``Poly`` or map
+  between ``domain`` and ``window``
+
+Calculus
+--------
+- ``p.deriv()`` -- Take the derivative of ``p``
+- ``p.integ()`` -- Integrate ``p``
+
+Validation
+----------
+- ``Poly.has_samecoef(p1, p2)``   -- Check if coefficients match
+- ``Poly.has_samedomain(p1, p2)`` -- Check if domains match
+- ``Poly.has_sametype(p1, p2)``   -- Check if types match
+- ``Poly.has_samewindow(p1, p2)`` -- Check if windows match
+
+Misc
+----
+- ``p.linspace()`` -- Return ``x, p(x)`` at equally-spaced points in ``domain``
+- ``p.mapparms()`` -- Return the parameters for the linear mapping between
+  ``domain`` and ``window``.
+- ``p.roots()``    -- Return the roots of `p`.
+- ``p.trim()``     -- Remove trailing coefficients.
+- ``p.cutdeg(degree)`` -- Truncate p to given degree
+- ``p.truncate(size)`` -- Truncate p to given size
+
+"""
+from .polynomial import Polynomial
+from .chebyshev import Chebyshev
+from .legendre import Legendre
+from .hermite import Hermite
+from .hermite_e import HermiteE
+from .laguerre import Laguerre
+
+__all__ = [
+    "set_default_printstyle",
+    "polynomial", "Polynomial",
+    "chebyshev", "Chebyshev",
+    "legendre", "Legendre",
+    "hermite", "Hermite",
+    "hermite_e", "HermiteE",
+    "laguerre", "Laguerre",
+]
+
+
+def set_default_printstyle(style):
+    """
+    Set the default format for the string representation of polynomials.
+
+    Values for ``style`` must be valid inputs to ``__format__``, i.e. 'ascii'
+    or 'unicode'.
+
+    Parameters
+    ----------
+    style : str
+        Format string for default printing style. Must be either 'ascii' or
+        'unicode'.
+
+    Notes
+    -----
+    The default format depends on the platform: 'unicode' is used on
+    Unix-based systems and 'ascii' on Windows. This determination is based on
+    default font support for the unicode superscript and subscript ranges.
+
+    Examples
+    --------
+    >>> p = np.polynomial.Polynomial([1, 2, 3])
+    >>> c = np.polynomial.Chebyshev([1, 2, 3])
+    >>> np.polynomial.set_default_printstyle('unicode')
+    >>> print(p)
+    1.0 + 2.0·x + 3.0·x²
+    >>> print(c)
+    1.0 + 2.0·T₁(x) + 3.0·T₂(x)
+    >>> np.polynomial.set_default_printstyle('ascii')
+    >>> print(p)
+    1.0 + 2.0 x + 3.0 x**2
+    >>> print(c)
+    1.0 + 2.0 T_1(x) + 3.0 T_2(x)
+    >>> # Formatting supersedes all class/package-level defaults
+    >>> print(f"{p:unicode}")
+    1.0 + 2.0·x + 3.0·x²
+    """
+    if style not in ('unicode', 'ascii'):
+        raise ValueError(
+            f"Unsupported format string '{style}'. Valid options are 'ascii' "
+            f"and 'unicode'"
+        )
+    _use_unicode = True
+    if style == 'ascii':
+        _use_unicode = False
+    from ._polybase import ABCPolyBase
+    ABCPolyBase._use_unicode = _use_unicode
+
+
+from numpy._pytesttester import PytestTester
+test = PytestTester(__name__)
+del PytestTester

lib/python3.12/site-packages/numpy/polynomial/__init__.pyi

@@ -0,0 +1,22 @@
+from numpy._pytesttester import PytestTester
+
+from numpy.polynomial import (
+    chebyshev as chebyshev,
+    hermite as hermite,
+    hermite_e as hermite_e,
+    laguerre as laguerre,
+    legendre as legendre,
+    polynomial as polynomial,
+)
+from numpy.polynomial.chebyshev import Chebyshev as Chebyshev
+from numpy.polynomial.hermite import Hermite as Hermite
+from numpy.polynomial.hermite_e import HermiteE as HermiteE
+from numpy.polynomial.laguerre import Laguerre as Laguerre
+from numpy.polynomial.legendre import Legendre as Legendre
+from numpy.polynomial.polynomial import Polynomial as Polynomial
+
+__all__: list[str]
+__path__: list[str]
+test: PytestTester
+
+def set_default_printstyle(style): ...

lib/python3.12/site-packages/numpy/polynomial/_polybase.py

@@ -0,0 +1,1206 @@
+"""
+Abstract base class for the various polynomial Classes.
+
+The ABCPolyBase class provides the methods needed to implement the common API
+for the various polynomial classes. It operates as a mixin, but uses the
+abc module from the stdlib, hence it is only available for Python >= 2.6.
+
+"""
+import os
+import abc
+import numbers
+
+import numpy as np
+from . import polyutils as pu
+
+__all__ = ['ABCPolyBase']
+
+class ABCPolyBase(abc.ABC):
+    """An abstract base class for immutable series classes.
+
+    ABCPolyBase provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' along with the
+    methods listed below.
+
+    .. versionadded:: 1.9.0
+
+    Parameters
+    ----------
+    coef : array_like
+        Series coefficients in order of increasing degree, i.e.,
+        ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``, where
+        ``P_i`` is the basis polynomials of degree ``i``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is the derived class domain.
+    window : (2,) array_like, optional
+        Window, see domain for its use. The default value is the
+        derived class window.
+    symbol : str, optional
+        Symbol used to represent the independent variable in string 
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    Attributes
+    ----------
+    coef : (N,) ndarray
+        Series coefficients in order of increasing degree.
+    domain : (2,) ndarray
+        Domain that is mapped to window.
+    window : (2,) ndarray
+        Window that domain is mapped to.
+    symbol : str
+        Symbol representing the independent variable.
+
+    Class Attributes
+    ----------------
+    maxpower : int
+        Maximum power allowed, i.e., the largest number ``n`` such that
+        ``p(x)**n`` is allowed. This is to limit runaway polynomial size.
+    domain : (2,) ndarray
+        Default domain of the class.
+    window : (2,) ndarray
+        Default window of the class.
+
+    """
+
+    # Not hashable
+    __hash__ = None
+
+    # Opt out of numpy ufuncs and Python ops with ndarray subclasses.
+    __array_ufunc__ = None
+
+    # Limit runaway size. T_n^m has degree n*m
+    maxpower = 100
+
+    # Unicode character mappings for improved __str__
+    _superscript_mapping = str.maketrans({
+        "0": "⁰",
+        "1": "¹",
+        "2": "²",
+        "3": "³",
+        "4": "⁴",
+        "5": "⁵",
+        "6": "⁶",
+        "7": "⁷",
+        "8": "⁸",
+        "9": "⁹"
+    })
+    _subscript_mapping = str.maketrans({
+        "0": "₀",
+        "1": "₁",
+        "2": "₂",
+        "3": "₃",
+        "4": "₄",
+        "5": "₅",
+        "6": "₆",
+        "7": "₇",
+        "8": "₈",
+        "9": "₉"
+    })
+    # Some fonts don't support full unicode character ranges necessary for
+    # the full set of superscripts and subscripts, including common/default
+    # fonts in Windows shells/terminals. Therefore, default to ascii-only
+    # printing on windows.
+    _use_unicode = not os.name == 'nt'
+
+    @property
+    def symbol(self):
+        return self._symbol
+
+    @property
+    @abc.abstractmethod
+    def domain(self):
+        pass
+
+    @property
+    @abc.abstractmethod
+    def window(self):
+        pass
+
+    @property
+    @abc.abstractmethod
+    def basis_name(self):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _add(c1, c2):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _sub(c1, c2):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _mul(c1, c2):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _div(c1, c2):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _pow(c, pow, maxpower=None):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _val(x, c):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _int(c, m, k, lbnd, scl):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _der(c, m, scl):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _fit(x, y, deg, rcond, full):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _line(off, scl):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _roots(c):
+        pass
+
+    @staticmethod
+    @abc.abstractmethod
+    def _fromroots(r):
+        pass
+
+    def has_samecoef(self, other):
+        """Check if coefficients match.
+
+        .. versionadded:: 1.6.0
+
+        Parameters
+        ----------
+        other : class instance
+            The other class must have the ``coef`` attribute.
+
+        Returns
+        -------
+        bool : boolean
+            True if the coefficients are the same, False otherwise.
+
+        """
+        if len(self.coef) != len(other.coef):
+            return False
+        elif not np.all(self.coef == other.coef):
+            return False
+        else:
+            return True
+
+    def has_samedomain(self, other):
+        """Check if domains match.
+
+        .. versionadded:: 1.6.0
+
+        Parameters
+        ----------
+        other : class instance
+            The other class must have the ``domain`` attribute.
+
+        Returns
+        -------
+        bool : boolean
+            True if the domains are the same, False otherwise.
+
+        """
+        return np.all(self.domain == other.domain)
+
+    def has_samewindow(self, other):
+        """Check if windows match.
+
+        .. versionadded:: 1.6.0
+
+        Parameters
+        ----------
+        other : class instance
+            The other class must have the ``window`` attribute.
+
+        Returns
+        -------
+        bool : boolean
+            True if the windows are the same, False otherwise.
+
+        """
+        return np.all(self.window == other.window)
+
+    def has_sametype(self, other):
+        """Check if types match.
+
+        .. versionadded:: 1.7.0
+
+        Parameters
+        ----------
+        other : object
+            Class instance.
+
+        Returns
+        -------
+        bool : boolean
+            True if other is same class as self
+
+        """
+        return isinstance(other, self.__class__)
+
+    def _get_coefficients(self, other):
+        """Interpret other as polynomial coefficients.
+
+        The `other` argument is checked to see if it is of the same
+        class as self with identical domain and window. If so,
+        return its coefficients, otherwise return `other`.
+
+        .. versionadded:: 1.9.0
+
+        Parameters
+        ----------
+        other : anything
+            Object to be checked.
+
+        Returns
+        -------
+        coef
+            The coefficients of`other` if it is a compatible instance,
+            of ABCPolyBase, otherwise `other`.
+
+        Raises
+        ------
+        TypeError
+            When `other` is an incompatible instance of ABCPolyBase.
+
+        """
+        if isinstance(other, ABCPolyBase):
+            if not isinstance(other, self.__class__):
+                raise TypeError("Polynomial types differ")
+            elif not np.all(self.domain == other.domain):
+                raise TypeError("Domains differ")
+            elif not np.all(self.window == other.window):
+                raise TypeError("Windows differ")
+            elif self.symbol != other.symbol:
+                raise ValueError("Polynomial symbols differ")
+            return other.coef
+        return other
+
+    def __init__(self, coef, domain=None, window=None, symbol='x'):
+        [coef] = pu.as_series([coef], trim=False)
+        self.coef = coef
+
+        if domain is not None:
+            [domain] = pu.as_series([domain], trim=False)
+            if len(domain) != 2:
+                raise ValueError("Domain has wrong number of elements.")
+            self.domain = domain
+
+        if window is not None:
+            [window] = pu.as_series([window], trim=False)
+            if len(window) != 2:
+                raise ValueError("Window has wrong number of elements.")
+            self.window = window
+
+        # Validation for symbol
+        try:
+            if not symbol.isidentifier():
+                raise ValueError(
+                    "Symbol string must be a valid Python identifier"
+                )
+        # If a user passes in something other than a string, the above
+        # results in an AttributeError. Catch this and raise a more
+        # informative exception
+        except AttributeError:
+            raise TypeError("Symbol must be a non-empty string")
+
+        self._symbol = symbol
+
+    def __repr__(self):
+        coef = repr(self.coef)[6:-1]
+        domain = repr(self.domain)[6:-1]
+        window = repr(self.window)[6:-1]
+        name = self.__class__.__name__
+        return (f"{name}({coef}, domain={domain}, window={window}, "
+                f"symbol='{self.symbol}')")
+
+    def __format__(self, fmt_str):
+        if fmt_str == '':
+            return self.__str__()
+        if fmt_str not in ('ascii', 'unicode'):
+            raise ValueError(
+                f"Unsupported format string '{fmt_str}' passed to "
+                f"{self.__class__}.__format__. Valid options are "
+                f"'ascii' and 'unicode'"
+            )
+        if fmt_str == 'ascii':
+            return self._generate_string(self._str_term_ascii)
+        return self._generate_string(self._str_term_unicode)
+
+    def __str__(self):
+        if self._use_unicode:
+            return self._generate_string(self._str_term_unicode)
+        return self._generate_string(self._str_term_ascii)
+
+    def _generate_string(self, term_method):
+        """
+        Generate the full string representation of the polynomial, using
+        ``term_method`` to generate each polynomial term.
+        """
+        # Get configuration for line breaks
+        linewidth = np.get_printoptions().get('linewidth', 75)
+        if linewidth < 1:
+            linewidth = 1
+        out = pu.format_float(self.coef[0])
+        for i, coef in enumerate(self.coef[1:]):
+            out += " "
+            power = str(i + 1)
+            # Polynomial coefficient
+            # The coefficient array can be an object array with elements that
+            # will raise a TypeError with >= 0 (e.g. strings or Python
+            # complex). In this case, represent the coefficient as-is.
+            try:
+                if coef >= 0:
+                    next_term = f"+ " + pu.format_float(coef, parens=True)
+                else:
+                    next_term = f"- " + pu.format_float(-coef, parens=True)
+            except TypeError:
+                next_term = f"+ {coef}"
+            # Polynomial term
+            next_term += term_method(power, self.symbol)
+            # Length of the current line with next term added
+            line_len = len(out.split('\n')[-1]) + len(next_term)
+            # If not the last term in the polynomial, it will be two
+            # characters longer due to the +/- with the next term
+            if i < len(self.coef[1:]) - 1:
+                line_len += 2
+            # Handle linebreaking
+            if line_len >= linewidth:
+                next_term = next_term.replace(" ", "\n", 1)
+            out += next_term
+        return out
+
+    @classmethod
+    def _str_term_unicode(cls, i, arg_str):
+        """
+        String representation of single polynomial term using unicode
+        characters for superscripts and subscripts.
+        """
+        if cls.basis_name is None:
+            raise NotImplementedError(
+                "Subclasses must define either a basis_name, or override "
+                "_str_term_unicode(cls, i, arg_str)"
+            )
+        return (f"·{cls.basis_name}{i.translate(cls._subscript_mapping)}"
+                f"({arg_str})")
+
+    @classmethod
+    def _str_term_ascii(cls, i, arg_str):
+        """
+        String representation of a single polynomial term using ** and _ to
+        represent superscripts and subscripts, respectively.
+        """
+        if cls.basis_name is None:
+            raise NotImplementedError(
+                "Subclasses must define either a basis_name, or override "
+                "_str_term_ascii(cls, i, arg_str)"
+            )
+        return f" {cls.basis_name}_{i}({arg_str})"
+
+    @classmethod
+    def _repr_latex_term(cls, i, arg_str, needs_parens):
+        if cls.basis_name is None:
+            raise NotImplementedError(
+                "Subclasses must define either a basis name, or override "
+                "_repr_latex_term(i, arg_str, needs_parens)")
+        # since we always add parens, we don't care if the expression needs them
+        return f"{{{cls.basis_name}}}_{{{i}}}({arg_str})"
+
+    @staticmethod
+    def _repr_latex_scalar(x, parens=False):
+        # TODO: we're stuck with disabling math formatting until we handle
+        # exponents in this function
+        return r'\text{{{}}}'.format(pu.format_float(x, parens=parens))
+
+    def _repr_latex_(self):
+        # get the scaled argument string to the basis functions
+        off, scale = self.mapparms()
+        if off == 0 and scale == 1:
+            term = self.symbol
+            needs_parens = False
+        elif scale == 1:
+            term = f"{self._repr_latex_scalar(off)} + {self.symbol}"
+            needs_parens = True
+        elif off == 0:
+            term = f"{self._repr_latex_scalar(scale)}{self.symbol}"
+            needs_parens = True
+        else:
+            term = (
+                f"{self._repr_latex_scalar(off)} + "
+                f"{self._repr_latex_scalar(scale)}{self.symbol}"
+            )
+            needs_parens = True
+
+        mute = r"\color{{LightGray}}{{{}}}".format
+
+        parts = []
+        for i, c in enumerate(self.coef):
+            # prevent duplication of + and - signs
+            if i == 0:
+                coef_str = f"{self._repr_latex_scalar(c)}"
+            elif not isinstance(c, numbers.Real):
+                coef_str = f" + ({self._repr_latex_scalar(c)})"
+            elif not np.signbit(c):
+                coef_str = f" + {self._repr_latex_scalar(c, parens=True)}"
+            else:
+                coef_str = f" - {self._repr_latex_scalar(-c, parens=True)}"
+
+            # produce the string for the term
+            term_str = self._repr_latex_term(i, term, needs_parens)
+            if term_str == '1':
+                part = coef_str
+            else:
+                part = rf"{coef_str}\,{term_str}"
+
+            if c == 0:
+                part = mute(part)
+
+            parts.append(part)
+
+        if parts:
+            body = ''.join(parts)
+        else:
+            # in case somehow there are no coefficients at all
+            body = '0'
+
+        return rf"${self.symbol} \mapsto {body}$"
+
+
+
+    # Pickle and copy
+
+    def __getstate__(self):
+        ret = self.__dict__.copy()
+        ret['coef'] = self.coef.copy()
+        ret['domain'] = self.domain.copy()
+        ret['window'] = self.window.copy()
+        ret['symbol'] = self.symbol
+        return ret
+
+    def __setstate__(self, dict):
+        self.__dict__ = dict
+
+    # Call
+
+    def __call__(self, arg):
+        off, scl = pu.mapparms(self.domain, self.window)
+        arg = off + scl*arg
+        return self._val(arg, self.coef)
+
+    def __iter__(self):
+        return iter(self.coef)
+
+    def __len__(self):
+        return len(self.coef)
+
+    # Numeric properties.
+
+    def __neg__(self):
+        return self.__class__(
+            -self.coef, self.domain, self.window, self.symbol
+        )
+
+    def __pos__(self):
+        return self
+
+    def __add__(self, other):
+        othercoef = self._get_coefficients(other)
+        try:
+            coef = self._add(self.coef, othercoef)
+        except Exception:
+            return NotImplemented
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def __sub__(self, other):
+        othercoef = self._get_coefficients(other)
+        try:
+            coef = self._sub(self.coef, othercoef)
+        except Exception:
+            return NotImplemented
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def __mul__(self, other):
+        othercoef = self._get_coefficients(other)
+        try:
+            coef = self._mul(self.coef, othercoef)
+        except Exception:
+            return NotImplemented
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def __truediv__(self, other):
+        # there is no true divide if the rhs is not a Number, although it
+        # could return the first n elements of an infinite series.
+        # It is hard to see where n would come from, though.
+        if not isinstance(other, numbers.Number) or isinstance(other, bool):
+            raise TypeError(
+                f"unsupported types for true division: "
+                f"'{type(self)}', '{type(other)}'"
+            )
+        return self.__floordiv__(other)
+
+    def __floordiv__(self, other):
+        res = self.__divmod__(other)
+        if res is NotImplemented:
+            return res
+        return res[0]
+
+    def __mod__(self, other):
+        res = self.__divmod__(other)
+        if res is NotImplemented:
+            return res
+        return res[1]
+
+    def __divmod__(self, other):
+        othercoef = self._get_coefficients(other)
+        try:
+            quo, rem = self._div(self.coef, othercoef)
+        except ZeroDivisionError:
+            raise
+        except Exception:
+            return NotImplemented
+        quo = self.__class__(quo, self.domain, self.window, self.symbol)
+        rem = self.__class__(rem, self.domain, self.window, self.symbol)
+        return quo, rem
+
+    def __pow__(self, other):
+        coef = self._pow(self.coef, other, maxpower=self.maxpower)
+        res = self.__class__(coef, self.domain, self.window, self.symbol)
+        return res
+
+    def __radd__(self, other):
+        try:
+            coef = self._add(other, self.coef)
+        except Exception:
+            return NotImplemented
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def __rsub__(self, other):
+        try:
+            coef = self._sub(other, self.coef)
+        except Exception:
+            return NotImplemented
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def __rmul__(self, other):
+        try:
+            coef = self._mul(other, self.coef)
+        except Exception:
+            return NotImplemented
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def __rdiv__(self, other):
+        # set to __floordiv__ /.
+        return self.__rfloordiv__(other)
+
+    def __rtruediv__(self, other):
+        # An instance of ABCPolyBase is not considered a
+        # Number.
+        return NotImplemented
+
+    def __rfloordiv__(self, other):
+        res = self.__rdivmod__(other)
+        if res is NotImplemented:
+            return res
+        return res[0]
+
+    def __rmod__(self, other):
+        res = self.__rdivmod__(other)
+        if res is NotImplemented:
+            return res
+        return res[1]
+
+    def __rdivmod__(self, other):
+        try:
+            quo, rem = self._div(other, self.coef)
+        except ZeroDivisionError:
+            raise
+        except Exception:
+            return NotImplemented
+        quo = self.__class__(quo, self.domain, self.window, self.symbol)
+        rem = self.__class__(rem, self.domain, self.window, self.symbol)
+        return quo, rem
+
+    def __eq__(self, other):
+        res = (isinstance(other, self.__class__) and
+               np.all(self.domain == other.domain) and
+               np.all(self.window == other.window) and
+               (self.coef.shape == other.coef.shape) and
+               np.all(self.coef == other.coef) and
+               (self.symbol == other.symbol))
+        return res
+
+    def __ne__(self, other):
+        return not self.__eq__(other)
+
+    #
+    # Extra methods.
+    #
+
+    def copy(self):
+        """Return a copy.
+
+        Returns
+        -------
+        new_series : series
+            Copy of self.
+
+        """
+        return self.__class__(self.coef, self.domain, self.window, self.symbol)
+
+    def degree(self):
+        """The degree of the series.
+
+        .. versionadded:: 1.5.0
+
+        Returns
+        -------
+        degree : int
+            Degree of the series, one less than the number of coefficients.
+
+        Examples
+        --------
+
+        Create a polynomial object for ``1 + 7*x + 4*x**2``:
+
+        >>> poly = np.polynomial.Polynomial([1, 7, 4])
+        >>> print(poly)
+        1.0 + 7.0·x + 4.0·x²
+        >>> poly.degree()
+        2
+
+        Note that this method does not check for non-zero coefficients.
+        You must trim the polynomial to remove any trailing zeroes:
+
+        >>> poly = np.polynomial.Polynomial([1, 7, 0])
+        >>> print(poly)
+        1.0 + 7.0·x + 0.0·x²
+        >>> poly.degree()
+        2
+        >>> poly.trim().degree()
+        1
+
+        """
+        return len(self) - 1
+
+    def cutdeg(self, deg):
+        """Truncate series to the given degree.
+
+        Reduce the degree of the series to `deg` by discarding the
+        high order terms. If `deg` is greater than the current degree a
+        copy of the current series is returned. This can be useful in least
+        squares where the coefficients of the high degree terms may be very
+        small.
+
+        .. versionadded:: 1.5.0
+
+        Parameters
+        ----------
+        deg : non-negative int
+            The series is reduced to degree `deg` by discarding the high
+            order terms. The value of `deg` must be a non-negative integer.
+
+        Returns
+        -------
+        new_series : series
+            New instance of series with reduced degree.
+
+        """
+        return self.truncate(deg + 1)
+
+    def trim(self, tol=0):
+        """Remove trailing coefficients
+
+        Remove trailing coefficients until a coefficient is reached whose
+        absolute value greater than `tol` or the beginning of the series is
+        reached. If all the coefficients would be removed the series is set
+        to ``[0]``. A new series instance is returned with the new
+        coefficients.  The current instance remains unchanged.
+
+        Parameters
+        ----------
+        tol : non-negative number.
+            All trailing coefficients less than `tol` will be removed.
+
+        Returns
+        -------
+        new_series : series
+            New instance of series with trimmed coefficients.
+
+        """
+        coef = pu.trimcoef(self.coef, tol)
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def truncate(self, size):
+        """Truncate series to length `size`.
+
+        Reduce the series to length `size` by discarding the high
+        degree terms. The value of `size` must be a positive integer. This
+        can be useful in least squares where the coefficients of the
+        high degree terms may be very small.
+
+        Parameters
+        ----------
+        size : positive int
+            The series is reduced to length `size` by discarding the high
+            degree terms. The value of `size` must be a positive integer.
+
+        Returns
+        -------
+        new_series : series
+            New instance of series with truncated coefficients.
+
+        """
+        isize = int(size)
+        if isize != size or isize < 1:
+            raise ValueError("size must be a positive integer")
+        if isize >= len(self.coef):
+            coef = self.coef
+        else:
+            coef = self.coef[:isize]
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def convert(self, domain=None, kind=None, window=None):
+        """Convert series to a different kind and/or domain and/or window.
+
+        Parameters
+        ----------
+        domain : array_like, optional
+            The domain of the converted series. If the value is None,
+            the default domain of `kind` is used.
+        kind : class, optional
+            The polynomial series type class to which the current instance
+            should be converted. If kind is None, then the class of the
+            current instance is used.
+        window : array_like, optional
+            The window of the converted series. If the value is None,
+            the default window of `kind` is used.
+
+        Returns
+        -------
+        new_series : series
+            The returned class can be of different type than the current
+            instance and/or have a different domain and/or different
+            window.
+
+        Notes
+        -----
+        Conversion between domains and class types can result in
+        numerically ill defined series.
+
+        """
+        if kind is None:
+            kind = self.__class__
+        if domain is None:
+            domain = kind.domain
+        if window is None:
+            window = kind.window
+        return self(kind.identity(domain, window=window, symbol=self.symbol))
+
+    def mapparms(self):
+        """Return the mapping parameters.
+
+        The returned values define a linear map ``off + scl*x`` that is
+        applied to the input arguments before the series is evaluated. The
+        map depends on the ``domain`` and ``window``; if the current
+        ``domain`` is equal to the ``window`` the resulting map is the
+        identity.  If the coefficients of the series instance are to be
+        used by themselves outside this class, then the linear function
+        must be substituted for the ``x`` in the standard representation of
+        the base polynomials.
+
+        Returns
+        -------
+        off, scl : float or complex
+            The mapping function is defined by ``off + scl*x``.
+
+        Notes
+        -----
+        If the current domain is the interval ``[l1, r1]`` and the window
+        is ``[l2, r2]``, then the linear mapping function ``L`` is
+        defined by the equations::
+
+            L(l1) = l2
+            L(r1) = r2
+
+        """
+        return pu.mapparms(self.domain, self.window)
+
+    def integ(self, m=1, k=[], lbnd=None):
+        """Integrate.
+
+        Return a series instance that is the definite integral of the
+        current series.
+
+        Parameters
+        ----------
+        m : non-negative int
+            The number of integrations to perform.
+        k : array_like
+            Integration constants. The first constant is applied to the
+            first integration, the second to the second, and so on. The
+            list of values must less than or equal to `m` in length and any
+            missing values are set to zero.
+        lbnd : Scalar
+            The lower bound of the definite integral.
+
+        Returns
+        -------
+        new_series : series
+            A new series representing the integral. The domain is the same
+            as the domain of the integrated series.
+
+        """
+        off, scl = self.mapparms()
+        if lbnd is None:
+            lbnd = 0
+        else:
+            lbnd = off + scl*lbnd
+        coef = self._int(self.coef, m, k, lbnd, 1./scl)
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def deriv(self, m=1):
+        """Differentiate.
+
+        Return a series instance of that is the derivative of the current
+        series.
+
+        Parameters
+        ----------
+        m : non-negative int
+            Find the derivative of order `m`.
+
+        Returns
+        -------
+        new_series : series
+            A new series representing the derivative. The domain is the same
+            as the domain of the differentiated series.
+
+        """
+        off, scl = self.mapparms()
+        coef = self._der(self.coef, m, scl)
+        return self.__class__(coef, self.domain, self.window, self.symbol)
+
+    def roots(self):
+        """Return the roots of the series polynomial.
+
+        Compute the roots for the series. Note that the accuracy of the
+        roots decreases the further outside the `domain` they lie.
+
+        Returns
+        -------
+        roots : ndarray
+            Array containing the roots of the series.
+
+        """
+        roots = self._roots(self.coef)
+        return pu.mapdomain(roots, self.window, self.domain)
+
+    def linspace(self, n=100, domain=None):
+        """Return x, y values at equally spaced points in domain.
+
+        Returns the x, y values at `n` linearly spaced points across the
+        domain.  Here y is the value of the polynomial at the points x. By
+        default the domain is the same as that of the series instance.
+        This method is intended mostly as a plotting aid.
+
+        .. versionadded:: 1.5.0
+
+        Parameters
+        ----------
+        n : int, optional
+            Number of point pairs to return. The default value is 100.
+        domain : {None, array_like}, optional
+            If not None, the specified domain is used instead of that of
+            the calling instance. It should be of the form ``[beg,end]``.
+            The default is None which case the class domain is used.
+
+        Returns
+        -------
+        x, y : ndarray
+            x is equal to linspace(self.domain[0], self.domain[1], n) and
+            y is the series evaluated at element of x.
+
+        """
+        if domain is None:
+            domain = self.domain
+        x = np.linspace(domain[0], domain[1], n)
+        y = self(x)
+        return x, y
+
+    @classmethod
+    def fit(cls, x, y, deg, domain=None, rcond=None, full=False, w=None,
+        window=None, symbol='x'):
+        """Least squares fit to data.
+
+        Return a series instance that is the least squares fit to the data
+        `y` sampled at `x`. The domain of the returned instance can be
+        specified and this will often result in a superior fit with less
+        chance of ill conditioning.
+
+        Parameters
+        ----------
+        x : array_like, shape (M,)
+            x-coordinates of the M sample points ``(x[i], y[i])``.
+        y : array_like, shape (M,)
+            y-coordinates of the M sample points ``(x[i], y[i])``.
+        deg : int or 1-D array_like
+            Degree(s) of the fitting polynomials. If `deg` is a single integer
+            all terms up to and including the `deg`'th term are included in the
+            fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+            degrees of the terms to include may be used instead.
+        domain : {None, [beg, end], []}, optional
+            Domain to use for the returned series. If ``None``,
+            then a minimal domain that covers the points `x` is chosen.  If
+            ``[]`` the class domain is used. The default value was the
+            class domain in NumPy 1.4 and ``None`` in later versions.
+            The ``[]`` option was added in numpy 1.5.0.
+        rcond : float, optional
+            Relative condition number of the fit. Singular values smaller
+            than this relative to the largest singular value will be
+            ignored. The default value is len(x)*eps, where eps is the
+            relative precision of the float type, about 2e-16 in most
+            cases.
+        full : bool, optional
+            Switch determining nature of return value. When it is False
+            (the default) just the coefficients are returned, when True
+            diagnostic information from the singular value decomposition is
+            also returned.
+        w : array_like, shape (M,), optional
+            Weights. If not None, the weight ``w[i]`` applies to the unsquared
+            residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+            chosen so that the errors of the products ``w[i]*y[i]`` all have
+            the same variance.  When using inverse-variance weighting, use
+            ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+            .. versionadded:: 1.5.0
+        window : {[beg, end]}, optional
+            Window to use for the returned series. The default
+            value is the default class domain
+
+            .. versionadded:: 1.6.0
+        symbol : str, optional
+            Symbol representing the independent variable. Default is 'x'.
+
+        Returns
+        -------
+        new_series : series
+            A series that represents the least squares fit to the data and
+            has the domain and window specified in the call. If the
+            coefficients for the unscaled and unshifted basis polynomials are
+            of interest, do ``new_series.convert().coef``.
+
+        [resid, rank, sv, rcond] : list
+            These values are only returned if ``full == True``
+
+            - resid -- sum of squared residuals of the least squares fit
+            - rank -- the numerical rank of the scaled Vandermonde matrix
+            - sv -- singular values of the scaled Vandermonde matrix
+            - rcond -- value of `rcond`.
+
+            For more details, see `linalg.lstsq`.
+
+        """
+        if domain is None:
+            domain = pu.getdomain(x)
+        elif type(domain) is list and len(domain) == 0:
+            domain = cls.domain
+
+        if window is None:
+            window = cls.window
+
+        xnew = pu.mapdomain(x, domain, window)
+        res = cls._fit(xnew, y, deg, w=w, rcond=rcond, full=full)
+        if full:
+            [coef, status] = res
+            return (
+                cls(coef, domain=domain, window=window, symbol=symbol), status
+            )
+        else:
+            coef = res
+            return cls(coef, domain=domain, window=window, symbol=symbol)
+
+    @classmethod
+    def fromroots(cls, roots, domain=[], window=None, symbol='x'):
+        """Return series instance that has the specified roots.
+
+        Returns a series representing the product
+        ``(x - r[0])*(x - r[1])*...*(x - r[n-1])``, where ``r`` is a
+        list of roots.
+
+        Parameters
+        ----------
+        roots : array_like
+            List of roots.
+        domain : {[], None, array_like}, optional
+            Domain for the resulting series. If None the domain is the
+            interval from the smallest root to the largest. If [] the
+            domain is the class domain. The default is [].
+        window : {None, array_like}, optional
+            Window for the returned series. If None the class window is
+            used. The default is None.
+        symbol : str, optional
+            Symbol representing the independent variable. Default is 'x'.
+
+        Returns
+        -------
+        new_series : series
+            Series with the specified roots.
+
+        """
+        [roots] = pu.as_series([roots], trim=False)
+        if domain is None:
+            domain = pu.getdomain(roots)
+        elif type(domain) is list and len(domain) == 0:
+            domain = cls.domain
+
+        if window is None:
+            window = cls.window
+
+        deg = len(roots)
+        off, scl = pu.mapparms(domain, window)
+        rnew = off + scl*roots
+        coef = cls._fromroots(rnew) / scl**deg
+        return cls(coef, domain=domain, window=window, symbol=symbol)
+
+    @classmethod
+    def identity(cls, domain=None, window=None, symbol='x'):
+        """Identity function.
+
+        If ``p`` is the returned series, then ``p(x) == x`` for all
+        values of x.
+
+        Parameters
+        ----------
+        domain : {None, array_like}, optional
+            If given, the array must be of the form ``[beg, end]``, where
+            ``beg`` and ``end`` are the endpoints of the domain. If None is
+            given then the class domain is used. The default is None.
+        window : {None, array_like}, optional
+            If given, the resulting array must be if the form
+            ``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
+            the window. If None is given then the class window is used. The
+            default is None.
+        symbol : str, optional
+            Symbol representing the independent variable. Default is 'x'.
+
+        Returns
+        -------
+        new_series : series
+             Series of representing the identity.
+
+        """
+        if domain is None:
+            domain = cls.domain
+        if window is None:
+            window = cls.window
+        off, scl = pu.mapparms(window, domain)
+        coef = cls._line(off, scl)
+        return cls(coef, domain, window, symbol)
+
+    @classmethod
+    def basis(cls, deg, domain=None, window=None, symbol='x'):
+        """Series basis polynomial of degree `deg`.
+
+        Returns the series representing the basis polynomial of degree `deg`.
+
+        .. versionadded:: 1.7.0
+
+        Parameters
+        ----------
+        deg : int
+            Degree of the basis polynomial for the series. Must be >= 0.
+        domain : {None, array_like}, optional
+            If given, the array must be of the form ``[beg, end]``, where
+            ``beg`` and ``end`` are the endpoints of the domain. If None is
+            given then the class domain is used. The default is None.
+        window : {None, array_like}, optional
+            If given, the resulting array must be if the form
+            ``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
+            the window. If None is given then the class window is used. The
+            default is None.
+        symbol : str, optional
+            Symbol representing the independent variable. Default is 'x'.
+
+        Returns
+        -------
+        new_series : series
+            A series with the coefficient of the `deg` term set to one and
+            all others zero.
+
+        """
+        if domain is None:
+            domain = cls.domain
+        if window is None:
+            window = cls.window
+        ideg = int(deg)
+
+        if ideg != deg or ideg < 0:
+            raise ValueError("deg must be non-negative integer")
+        return cls([0]*ideg + [1], domain, window, symbol)
+
+    @classmethod
+    def cast(cls, series, domain=None, window=None):
+        """Convert series to series of this class.
+
+        The `series` is expected to be an instance of some polynomial
+        series of one of the types supported by by the numpy.polynomial
+        module, but could be some other class that supports the convert
+        method.
+
+        .. versionadded:: 1.7.0
+
+        Parameters
+        ----------
+        series : series
+            The series instance to be converted.
+        domain : {None, array_like}, optional
+            If given, the array must be of the form ``[beg, end]``, where
+            ``beg`` and ``end`` are the endpoints of the domain. If None is
+            given then the class domain is used. The default is None.
+        window : {None, array_like}, optional
+            If given, the resulting array must be if the form
+            ``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
+            the window. If None is given then the class window is used. The
+            default is None.
+
+        Returns
+        -------
+        new_series : series
+            A series of the same kind as the calling class and equal to
+            `series` when evaluated.
+
+        See Also
+        --------
+        convert : similar instance method
+
+        """
+        if domain is None:
+            domain = cls.domain
+        if window is None:
+            window = cls.window
+        return series.convert(domain, cls, window)

lib/python3.12/site-packages/numpy/polynomial/_polybase.pyi

@@ -0,0 +1,71 @@
+import abc
+from typing import Any, ClassVar
+
+__all__: list[str]
+
+class ABCPolyBase(abc.ABC):
+    __hash__: ClassVar[None]  # type: ignore[assignment]
+    __array_ufunc__: ClassVar[None]
+    maxpower: ClassVar[int]
+    coef: Any
+    @property
+    def symbol(self) -> str: ...
+    @property
+    @abc.abstractmethod
+    def domain(self): ...
+    @property
+    @abc.abstractmethod
+    def window(self): ...
+    @property
+    @abc.abstractmethod
+    def basis_name(self): ...
+    def has_samecoef(self, other): ...
+    def has_samedomain(self, other): ...
+    def has_samewindow(self, other): ...
+    def has_sametype(self, other): ...
+    def __init__(self, coef, domain=..., window=..., symbol: str = ...) -> None: ...
+    def __format__(self, fmt_str): ...
+    def __call__(self, arg): ...
+    def __iter__(self): ...
+    def __len__(self): ...
+    def __neg__(self): ...
+    def __pos__(self): ...
+    def __add__(self, other): ...
+    def __sub__(self, other): ...
+    def __mul__(self, other): ...
+    def __truediv__(self, other): ...
+    def __floordiv__(self, other): ...
+    def __mod__(self, other): ...
+    def __divmod__(self, other): ...
+    def __pow__(self, other): ...
+    def __radd__(self, other): ...
+    def __rsub__(self, other): ...
+    def __rmul__(self, other): ...
+    def __rdiv__(self, other): ...
+    def __rtruediv__(self, other): ...
+    def __rfloordiv__(self, other): ...
+    def __rmod__(self, other): ...
+    def __rdivmod__(self, other): ...
+    def __eq__(self, other): ...
+    def __ne__(self, other): ...
+    def copy(self): ...
+    def degree(self): ...
+    def cutdeg(self, deg): ...
+    def trim(self, tol=...): ...
+    def truncate(self, size): ...
+    def convert(self, domain=..., kind=..., window=...): ...
+    def mapparms(self): ...
+    def integ(self, m=..., k = ..., lbnd=...): ...
+    def deriv(self, m=...): ...
+    def roots(self): ...
+    def linspace(self, n=..., domain=...): ...
+    @classmethod
+    def fit(cls, x, y, deg, domain=..., rcond=..., full=..., w=..., window=...): ...
+    @classmethod
+    def fromroots(cls, roots, domain = ..., window=...): ...
+    @classmethod
+    def identity(cls, domain=..., window=...): ...
+    @classmethod
+    def basis(cls, deg, domain=..., window=...): ...
+    @classmethod
+    def cast(cls, series, domain=..., window=...): ...

lib/python3.12/site-packages/numpy/polynomial/chebyshev.py

@@ -0,0 +1,2082 @@
+"""
+====================================================
+Chebyshev Series (:mod:`numpy.polynomial.chebyshev`)
+====================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Chebyshev series, including a `Chebyshev` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+
+.. autosummary::
+   :toctree: generated/
+
+   Chebyshev
+
+
+Constants
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebdomain
+   chebzero
+   chebone
+   chebx
+
+Arithmetic
+----------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebadd
+   chebsub
+   chebmulx
+   chebmul
+   chebdiv
+   chebpow
+   chebval
+   chebval2d
+   chebval3d
+   chebgrid2d
+   chebgrid3d
+
+Calculus
+--------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebder
+   chebint
+
+Misc Functions
+--------------
+
+.. autosummary::
+   :toctree: generated/
+
+   chebfromroots
+   chebroots
+   chebvander
+   chebvander2d
+   chebvander3d
+   chebgauss
+   chebweight
+   chebcompanion
+   chebfit
+   chebpts1
+   chebpts2
+   chebtrim
+   chebline
+   cheb2poly
+   poly2cheb
+   chebinterpolate
+
+See also
+--------
+`numpy.polynomial`
+
+Notes
+-----
+The implementations of multiplication, division, integration, and
+differentiation use the algebraic identities [1]_:
+
+.. math::
+    T_n(x) = \\frac{z^n + z^{-n}}{2} \\\\
+    z\\frac{dx}{dz} = \\frac{z - z^{-1}}{2}.
+
+where
+
+.. math:: x = \\frac{z + z^{-1}}{2}.
+
+These identities allow a Chebyshev series to be expressed as a finite,
+symmetric Laurent series.  In this module, this sort of Laurent series
+is referred to as a "z-series."
+
+References
+----------
+.. [1] A. T. Benjamin, et al., "Combinatorial Trigonometry with Chebyshev
+  Polynomials," *Journal of Statistical Planning and Inference 14*, 2008
+  (https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)
+
+"""
+import numpy as np
+import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd',
+    'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval',
+    'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots',
+    'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1',
+    'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d',
+    'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion',
+    'chebgauss', 'chebweight', 'chebinterpolate']
+
+chebtrim = pu.trimcoef
+
+#
+# A collection of functions for manipulating z-series. These are private
+# functions and do minimal error checking.
+#
+
+def _cseries_to_zseries(c):
+    """Convert Chebyshev series to z-series.
+
+    Convert a Chebyshev series to the equivalent z-series. The result is
+    never an empty array. The dtype of the return is the same as that of
+    the input. No checks are run on the arguments as this routine is for
+    internal use.
+
+    Parameters
+    ----------
+    c : 1-D ndarray
+        Chebyshev coefficients, ordered from low to high
+
+    Returns
+    -------
+    zs : 1-D ndarray
+        Odd length symmetric z-series, ordered from  low to high.
+
+    """
+    n = c.size
+    zs = np.zeros(2*n-1, dtype=c.dtype)
+    zs[n-1:] = c/2
+    return zs + zs[::-1]
+
+
+def _zseries_to_cseries(zs):
+    """Convert z-series to a Chebyshev series.
+
+    Convert a z series to the equivalent Chebyshev series. The result is
+    never an empty array. The dtype of the return is the same as that of
+    the input. No checks are run on the arguments as this routine is for
+    internal use.
+
+    Parameters
+    ----------
+    zs : 1-D ndarray
+        Odd length symmetric z-series, ordered from  low to high.
+
+    Returns
+    -------
+    c : 1-D ndarray
+        Chebyshev coefficients, ordered from  low to high.
+
+    """
+    n = (zs.size + 1)//2
+    c = zs[n-1:].copy()
+    c[1:n] *= 2
+    return c
+
+
+def _zseries_mul(z1, z2):
+    """Multiply two z-series.
+
+    Multiply two z-series to produce a z-series.
+
+    Parameters
+    ----------
+    z1, z2 : 1-D ndarray
+        The arrays must be 1-D but this is not checked.
+
+    Returns
+    -------
+    product : 1-D ndarray
+        The product z-series.
+
+    Notes
+    -----
+    This is simply convolution. If symmetric/anti-symmetric z-series are
+    denoted by S/A then the following rules apply:
+
+    S*S, A*A -> S
+    S*A, A*S -> A
+
+    """
+    return np.convolve(z1, z2)
+
+
+def _zseries_div(z1, z2):
+    """Divide the first z-series by the second.
+
+    Divide `z1` by `z2` and return the quotient and remainder as z-series.
+    Warning: this implementation only applies when both z1 and z2 have the
+    same symmetry, which is sufficient for present purposes.
+
+    Parameters
+    ----------
+    z1, z2 : 1-D ndarray
+        The arrays must be 1-D and have the same symmetry, but this is not
+        checked.
+
+    Returns
+    -------
+
+    (quotient, remainder) : 1-D ndarrays
+        Quotient and remainder as z-series.
+
+    Notes
+    -----
+    This is not the same as polynomial division on account of the desired form
+    of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
+    then the following rules apply:
+
+    S/S -> S,S
+    A/A -> S,A
+
+    The restriction to types of the same symmetry could be fixed but seems like
+    unneeded generality. There is no natural form for the remainder in the case
+    where there is no symmetry.
+
+    """
+    z1 = z1.copy()
+    z2 = z2.copy()
+    lc1 = len(z1)
+    lc2 = len(z2)
+    if lc2 == 1:
+        z1 /= z2
+        return z1, z1[:1]*0
+    elif lc1 < lc2:
+        return z1[:1]*0, z1
+    else:
+        dlen = lc1 - lc2
+        scl = z2[0]
+        z2 /= scl
+        quo = np.empty(dlen + 1, dtype=z1.dtype)
+        i = 0
+        j = dlen
+        while i < j:
+            r = z1[i]
+            quo[i] = z1[i]
+            quo[dlen - i] = r
+            tmp = r*z2
+            z1[i:i+lc2] -= tmp
+            z1[j:j+lc2] -= tmp
+            i += 1
+            j -= 1
+        r = z1[i]
+        quo[i] = r
+        tmp = r*z2
+        z1[i:i+lc2] -= tmp
+        quo /= scl
+        rem = z1[i+1:i-1+lc2].copy()
+        return quo, rem
+
+
+def _zseries_der(zs):
+    """Differentiate a z-series.
+
+    The derivative is with respect to x, not z. This is achieved using the
+    chain rule and the value of dx/dz given in the module notes.
+
+    Parameters
+    ----------
+    zs : z-series
+        The z-series to differentiate.
+
+    Returns
+    -------
+    derivative : z-series
+        The derivative
+
+    Notes
+    -----
+    The zseries for x (ns) has been multiplied by two in order to avoid
+    using floats that are incompatible with Decimal and likely other
+    specialized scalar types. This scaling has been compensated by
+    multiplying the value of zs by two also so that the two cancels in the
+    division.
+
+    """
+    n = len(zs)//2
+    ns = np.array([-1, 0, 1], dtype=zs.dtype)
+    zs *= np.arange(-n, n+1)*2
+    d, r = _zseries_div(zs, ns)
+    return d
+
+
+def _zseries_int(zs):
+    """Integrate a z-series.
+
+    The integral is with respect to x, not z. This is achieved by a change
+    of variable using dx/dz given in the module notes.
+
+    Parameters
+    ----------
+    zs : z-series
+        The z-series to integrate
+
+    Returns
+    -------
+    integral : z-series
+        The indefinite integral
+
+    Notes
+    -----
+    The zseries for x (ns) has been multiplied by two in order to avoid
+    using floats that are incompatible with Decimal and likely other
+    specialized scalar types. This scaling has been compensated by
+    dividing the resulting zs by two.
+
+    """
+    n = 1 + len(zs)//2
+    ns = np.array([-1, 0, 1], dtype=zs.dtype)
+    zs = _zseries_mul(zs, ns)
+    div = np.arange(-n, n+1)*2
+    zs[:n] /= div[:n]
+    zs[n+1:] /= div[n+1:]
+    zs[n] = 0
+    return zs
+
+#
+# Chebyshev series functions
+#
+
+
+def poly2cheb(pol):
+    """
+    Convert a polynomial to a Chebyshev series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Chebyshev series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Chebyshev
+        series.
+
+    See Also
+    --------
+    cheb2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy import polynomial as P
+    >>> p = P.Polynomial(range(4))
+    >>> p
+    Polynomial([0., 1., 2., 3.], domain=[-1,  1], window=[-1,  1])
+    >>> c = p.convert(kind=P.Chebyshev)
+    >>> c
+    Chebyshev([1.  , 3.25, 1.  , 0.75], domain=[-1.,  1.], window=[-1.,  1.])
+    >>> P.chebyshev.poly2cheb(range(4))
+    array([1.  , 3.25, 1.  , 0.75])
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = chebadd(chebmulx(res), pol[i])
+    return res
+
+
+def cheb2poly(c):
+    """
+    Convert a Chebyshev series to a polynomial.
+
+    Convert an array representing the coefficients of a Chebyshev series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Chebyshev series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2cheb
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy import polynomial as P
+    >>> c = P.Chebyshev(range(4))
+    >>> c
+    Chebyshev([0., 1., 2., 3.], domain=[-1,  1], window=[-1,  1])
+    >>> p = c.convert(kind=P.Polynomial)
+    >>> p
+    Polynomial([-2., -8.,  4., 12.], domain=[-1.,  1.], window=[-1.,  1.])
+    >>> P.chebyshev.cheb2poly(range(4))
+    array([-2.,  -8.,   4.,  12.])
+
+    """
+    from .polynomial import polyadd, polysub, polymulx
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n < 3:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], c1)
+            c1 = polyadd(tmp, polymulx(c1)*2)
+        return polyadd(c0, polymulx(c1))
+
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Chebyshev default domain.
+chebdomain = np.array([-1, 1])
+
+# Chebyshev coefficients representing zero.
+chebzero = np.array([0])
+
+# Chebyshev coefficients representing one.
+chebone = np.array([1])
+
+# Chebyshev coefficients representing the identity x.
+chebx = np.array([0, 1])
+
+
+def chebline(off, scl):
+    """
+    Chebyshev series whose graph is a straight line.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Chebyshev series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as C
+    >>> C.chebline(3,2)
+    array([3, 2])
+    >>> C.chebval(-3, C.chebline(3,2)) # should be -3
+    -3.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def chebfromroots(roots):
+    """
+    Generate a Chebyshev series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Chebyshev form, where the `r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * T_1(x) + ... +  c_n * T_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Chebyshev form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as C
+    >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
+    array([ 0.  , -0.25,  0.  ,  0.25])
+    >>> j = complex(0,1)
+    >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
+    array([1.5+0.j, 0. +0.j, 0.5+0.j])
+
+    """
+    return pu._fromroots(chebline, chebmul, roots)
+
+
+def chebadd(c1, c2):
+    """
+    Add one Chebyshev series to another.
+
+    Returns the sum of two Chebyshev series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Chebyshev series of their sum.
+
+    See Also
+    --------
+    chebsub, chebmulx, chebmul, chebdiv, chebpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Chebyshev series
+    is a Chebyshev series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebadd(c1,c2)
+    array([4., 4., 4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def chebsub(c1, c2):
+    """
+    Subtract one Chebyshev series from another.
+
+    Returns the difference of two Chebyshev series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Chebyshev series coefficients representing their difference.
+
+    See Also
+    --------
+    chebadd, chebmulx, chebmul, chebdiv, chebpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Chebyshev
+    series is a Chebyshev series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebsub(c1,c2)
+    array([-2.,  0.,  2.])
+    >>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
+    array([ 2.,  0., -2.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def chebmulx(c):
+    """Multiply a Chebyshev series by x.
+
+    Multiply the polynomial `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.5.0
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> C.chebmulx([1,2,3])
+    array([1. , 2.5, 1. , 1.5])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0]*0
+    prd[1] = c[0]
+    if len(c) > 1:
+        tmp = c[1:]/2
+        prd[2:] = tmp
+        prd[0:-2] += tmp
+    return prd
+
+
+def chebmul(c1, c2):
+    """
+    Multiply one Chebyshev series by another.
+
+    Returns the product of two Chebyshev series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Chebyshev series coefficients representing their product.
+
+    See Also
+    --------
+    chebadd, chebsub, chebmulx, chebdiv, chebpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Chebyshev polynomial basis set.  Thus, to express
+    the product as a C-series, it is typically necessary to "reproject"
+    the product onto said basis set, which typically produces
+    "unintuitive live" (but correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebmul(c1,c2) # multiplication requires "reprojection"
+    array([  6.5,  12. ,  12. ,   4. ,   1.5])
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    z1 = _cseries_to_zseries(c1)
+    z2 = _cseries_to_zseries(c2)
+    prd = _zseries_mul(z1, z2)
+    ret = _zseries_to_cseries(prd)
+    return pu.trimseq(ret)
+
+
+def chebdiv(c1, c2):
+    """
+    Divide one Chebyshev series by another.
+
+    Returns the quotient-with-remainder of two Chebyshev series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``T_0 + 2*T_1 + 3*T_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Chebyshev series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Chebyshev series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    chebadd, chebsub, chebmulx, chebmul, chebpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one C-series by another
+    results in quotient and remainder terms that are not in the Chebyshev
+    polynomial basis set.  Thus, to express these results as C-series, it
+    is typically necessary to "reproject" the results onto said basis
+    set, which typically produces "unintuitive" (but correct) results;
+    see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
+    (array([3.]), array([-8., -4.]))
+    >>> c2 = (0,1,2,3)
+    >>> C.chebdiv(c2,c1) # neither "intuitive"
+    (array([0., 2.]), array([-2., -4.]))
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    if c2[-1] == 0:
+        raise ZeroDivisionError()
+
+    # note: this is more efficient than `pu._div(chebmul, c1, c2)`
+    lc1 = len(c1)
+    lc2 = len(c2)
+    if lc1 < lc2:
+        return c1[:1]*0, c1
+    elif lc2 == 1:
+        return c1/c2[-1], c1[:1]*0
+    else:
+        z1 = _cseries_to_zseries(c1)
+        z2 = _cseries_to_zseries(c2)
+        quo, rem = _zseries_div(z1, z2)
+        quo = pu.trimseq(_zseries_to_cseries(quo))
+        rem = pu.trimseq(_zseries_to_cseries(rem))
+        return quo, rem
+
+
+def chebpow(c, pow, maxpower=16):
+    """Raise a Chebyshev series to a power.
+
+    Returns the Chebyshev series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``T_0 + 2*T_1 + 3*T_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Chebyshev series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Chebyshev series of power.
+
+    See Also
+    --------
+    chebadd, chebsub, chebmulx, chebmul, chebdiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> C.chebpow([1, 2, 3, 4], 2)
+    array([15.5, 22. , 16. , ..., 12.5, 12. ,  8. ])
+
+    """
+    # note: this is more efficient than `pu._pow(chebmul, c1, c2)`, as it
+    # avoids converting between z and c series repeatedly
+
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    power = int(pow)
+    if power != pow or power < 0:
+        raise ValueError("Power must be a non-negative integer.")
+    elif maxpower is not None and power > maxpower:
+        raise ValueError("Power is too large")
+    elif power == 0:
+        return np.array([1], dtype=c.dtype)
+    elif power == 1:
+        return c
+    else:
+        # This can be made more efficient by using powers of two
+        # in the usual way.
+        zs = _cseries_to_zseries(c)
+        prd = zs
+        for i in range(2, power + 1):
+            prd = np.convolve(prd, zs)
+        return _zseries_to_cseries(prd)
+
+
+def chebder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Chebyshev series.
+
+    Returns the Chebyshev series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2``
+    while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) +
+    2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Chebyshev series coefficients. If c is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    der : ndarray
+        Chebyshev series of the derivative.
+
+    See Also
+    --------
+    chebint
+
+    Notes
+    -----
+    In general, the result of differentiating a C-series needs to be
+    "reprojected" onto the C-series basis set. Thus, typically, the
+    result of this function is "unintuitive," albeit correct; see Examples
+    section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c = (1,2,3,4)
+    >>> C.chebder(c)
+    array([14., 12., 24.])
+    >>> C.chebder(c,3)
+    array([96.])
+    >>> C.chebder(c,scl=-1)
+    array([-14., -12., -24.])
+    >>> C.chebder(c,2,-1)
+    array([12.,  96.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._deprecate_as_int(m, "the order of derivation")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1]*0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 2, -1):
+                der[j - 1] = (2*j)*c[j]
+                c[j - 2] += (j*c[j])/(j - 2)
+            if n > 1:
+                der[1] = 4*c[2]
+            der[0] = c[1]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Chebyshev series.
+
+    Returns the Chebyshev series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]]
+    represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) +
+    2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Chebyshev series coefficients. If c is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at zero
+        is the first value in the list, the value of the second integral
+        at zero is the second value, etc.  If ``k == []`` (the default),
+        all constants are set to zero.  If ``m == 1``, a single scalar can
+        be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    S : ndarray
+        C-series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    chebder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a`- perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import chebyshev as C
+    >>> c = (1,2,3)
+    >>> C.chebint(c)
+    array([ 0.5, -0.5,  0.5,  0.5])
+    >>> C.chebint(c,3)
+    array([ 0.03125   , -0.1875    ,  0.04166667, -0.05208333,  0.01041667, # may vary
+        0.00625   ])
+    >>> C.chebint(c, k=3)
+    array([ 3.5, -0.5,  0.5,  0.5])
+    >>> C.chebint(c,lbnd=-2)
+    array([ 8.5, -0.5,  0.5,  0.5])
+    >>> C.chebint(c,scl=-2)
+    array([-1.,  1., -1., -1.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._deprecate_as_int(m, "the order of integration")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0]*(cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0]*0
+            tmp[1] = c[0]
+            if n > 1:
+                tmp[2] = c[1]/4
+            for j in range(2, n):
+                tmp[j + 1] = c[j]/(2*(j + 1))
+                tmp[j - 1] -= c[j]/(2*(j - 1))
+            tmp[0] += k[i] - chebval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def chebval(x, c, tensor=True):
+    """
+    Evaluate a Chebyshev series at points x.
+
+    If `c` is of length `n + 1`, this function returns the value:
+
+    .. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    chebval2d, chebgrid2d, chebval3d, chebgrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,)*x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        x2 = 2*x
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            c0 = c[-i] - c1
+            c1 = tmp + c1*x2
+    return c0 + c1*x
+
+
+def chebval2d(x, y, c):
+    """
+    Evaluate a 2-D Chebyshev series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points `(x, y)`,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than 2 the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points formed
+        from pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    chebval, chebgrid2d, chebval3d, chebgrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(chebval, c, x, y)
+
+
+def chebgrid2d(x, y, c):
+    """
+    Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_j(b),
+
+    where the points `(a, b)` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points in the
+        Cartesian product of `x` and `y`.
+
+    See Also
+    --------
+    chebval, chebval2d, chebval3d, chebgrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(chebval, c, x, y)
+
+
+def chebval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Chebyshev series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    chebval, chebval2d, chebgrid2d, chebgrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(chebval, c, x, y, z)
+
+
+def chebgrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c)
+
+    where the points `(a, b, c)` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    chebval, chebval2d, chebgrid2d, chebval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(chebval, c, x, y, z)
+
+
+def chebvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = T_i(x),
+
+    where `0 <= i <= deg`. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Chebyshev polynomial.
+
+    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
+    matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
+    ``chebval(x, c)`` are the same up to roundoff.  This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Chebyshev series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Chebyshev polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=False, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    # Use forward recursion to generate the entries.
+    v[0] = x*0 + 1
+    if ideg > 0:
+        x2 = 2*x
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = v[i-1]*x2 - v[i-2]
+    return np.moveaxis(v, 0, -1)
+
+
+def chebvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y)`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y),
+
+    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
+    `V` index the points `(x, y)` and the last index encodes the degrees of
+    the Chebyshev polynomials.
+
+    If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Chebyshev
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    chebvander, chebvander3d, chebval2d, chebval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((chebvander, chebvander), (x, y), deg)
+
+
+def chebvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z),
+
+    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
+    indices of `V` index the points `(x, y, z)` and the last index encodes
+    the degrees of the Chebyshev polynomials.
+
+    If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Chebyshev
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    chebvander, chebvander3d, chebval2d, chebval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((chebvander, chebvander, chebvander), (x, y, z), deg)
+
+
+def chebfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Chebyshev series to data.
+
+    Return the coefficients of a Chebyshev series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer,
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+        .. versionadded:: 1.5.0
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Chebyshev coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column k  of `y` are in column
+        `k`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    chebval : Evaluates a Chebyshev series.
+    chebvander : Vandermonde matrix of Chebyshev series.
+    chebweight : Chebyshev weight function.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Chebyshev series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where :math:`w_j` are the weights. This problem is solved by setting up
+    as the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `RankWarning` will be issued. This means that the
+    coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Chebyshev series are usually better conditioned than fits
+    using power series, but much can depend on the distribution of the
+    sample points and the smoothness of the data. If the quality of the fit
+    is inadequate splines may be a good alternative.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+
+    """
+    return pu._fit(chebvander, x, y, deg, rcond, full, w)
+
+
+def chebcompanion(c):
+    """Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is a Chebyshev basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Chebyshev series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0]/c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = np.array([1.] + [np.sqrt(.5)]*(n-1))
+    top = mat.reshape(-1)[1::n+1]
+    bot = mat.reshape(-1)[n::n+1]
+    top[0] = np.sqrt(.5)
+    top[1:] = 1/2
+    bot[...] = top
+    mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5
+    return mat
+
+
+def chebroots(c):
+    """
+    Compute the roots of a Chebyshev series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * T_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The Chebyshev series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as cheb
+    >>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
+    array([ -5.00000000e-01,   2.60860684e-17,   1.00000000e+00]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0]/c[1]])
+
+    # rotated companion matrix reduces error
+    m = chebcompanion(c)[::-1,::-1]
+    r = la.eigvals(m)
+    r.sort()
+    return r
+
+
+def chebinterpolate(func, deg, args=()):
+    """Interpolate a function at the Chebyshev points of the first kind.
+
+    Returns the Chebyshev series that interpolates `func` at the Chebyshev
+    points of the first kind in the interval [-1, 1]. The interpolating
+    series tends to a minmax approximation to `func` with increasing `deg`
+    if the function is continuous in the interval.
+
+    .. versionadded:: 1.14.0
+
+    Parameters
+    ----------
+    func : function
+        The function to be approximated. It must be a function of a single
+        variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
+        extra arguments passed in the `args` parameter.
+    deg : int
+        Degree of the interpolating polynomial
+    args : tuple, optional
+        Extra arguments to be used in the function call. Default is no extra
+        arguments.
+
+    Returns
+    -------
+    coef : ndarray, shape (deg + 1,)
+        Chebyshev coefficients of the interpolating series ordered from low to
+        high.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.chebyshev as C
+    >>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8)
+    array([  5.00000000e-01,   8.11675684e-01,  -9.86864911e-17,
+            -5.42457905e-02,  -2.71387850e-16,   4.51658839e-03,
+             2.46716228e-17,  -3.79694221e-04,  -3.26899002e-16])
+
+    Notes
+    -----
+
+    The Chebyshev polynomials used in the interpolation are orthogonal when
+    sampled at the Chebyshev points of the first kind. If it is desired to
+    constrain some of the coefficients they can simply be set to the desired
+    value after the interpolation, no new interpolation or fit is needed. This
+    is especially useful if it is known apriori that some of coefficients are
+    zero. For instance, if the function is even then the coefficients of the
+    terms of odd degree in the result can be set to zero.
+
+    """
+    deg = np.asarray(deg)
+
+    # check arguments.
+    if deg.ndim > 0 or deg.dtype.kind not in 'iu' or deg.size == 0:
+        raise TypeError("deg must be an int")
+    if deg < 0:
+        raise ValueError("expected deg >= 0")
+
+    order = deg + 1
+    xcheb = chebpts1(order)
+    yfunc = func(xcheb, *args)
+    m = chebvander(xcheb, deg)
+    c = np.dot(m.T, yfunc)
+    c[0] /= order
+    c[1:] /= 0.5*order
+
+    return c
+
+
+def chebgauss(deg):
+    """
+    Gauss-Chebyshev quadrature.
+
+    Computes the sample points and weights for Gauss-Chebyshev quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
+    the weight function :math:`f(x) = 1/\\sqrt{1 - x^2}`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. For Gauss-Chebyshev there are closed form solutions for
+    the sample points and weights. If n = `deg`, then
+
+    .. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n))
+
+    .. math:: w_i = \\pi / n
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    x = np.cos(np.pi * np.arange(1, 2*ideg, 2) / (2.0*ideg))
+    w = np.ones(ideg)*(np.pi/ideg)
+
+    return x, w
+
+
+def chebweight(x):
+    """
+    The weight function of the Chebyshev polynomials.
+
+    The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of
+    integration is :math:`[-1, 1]`. The Chebyshev polynomials are
+    orthogonal, but not normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    w = 1./(np.sqrt(1. + x) * np.sqrt(1. - x))
+    return w
+
+
+def chebpts1(npts):
+    """
+    Chebyshev points of the first kind.
+
+    The Chebyshev points of the first kind are the points ``cos(x)``,
+    where ``x = [pi*(k + .5)/npts for k in range(npts)]``.
+
+    Parameters
+    ----------
+    npts : int
+        Number of sample points desired.
+
+    Returns
+    -------
+    pts : ndarray
+        The Chebyshev points of the first kind.
+
+    See Also
+    --------
+    chebpts2
+
+    Notes
+    -----
+
+    .. versionadded:: 1.5.0
+
+    """
+    _npts = int(npts)
+    if _npts != npts:
+        raise ValueError("npts must be integer")
+    if _npts < 1:
+        raise ValueError("npts must be >= 1")
+
+    x = 0.5 * np.pi / _npts * np.arange(-_npts+1, _npts+1, 2)
+    return np.sin(x)
+
+
+def chebpts2(npts):
+    """
+    Chebyshev points of the second kind.
+
+    The Chebyshev points of the second kind are the points ``cos(x)``,
+    where ``x = [pi*k/(npts - 1) for k in range(npts)]`` sorted in ascending
+    order.
+
+    Parameters
+    ----------
+    npts : int
+        Number of sample points desired.
+
+    Returns
+    -------
+    pts : ndarray
+        The Chebyshev points of the second kind.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.5.0
+
+    """
+    _npts = int(npts)
+    if _npts != npts:
+        raise ValueError("npts must be integer")
+    if _npts < 2:
+        raise ValueError("npts must be >= 2")
+
+    x = np.linspace(-np.pi, 0, _npts)
+    return np.cos(x)
+
+
+#
+# Chebyshev series class
+#
+
+class Chebyshev(ABCPolyBase):
+    """A Chebyshev series class.
+
+    The Chebyshev class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    methods listed below.
+
+    Parameters
+    ----------
+    coef : array_like
+        Chebyshev coefficients in order of increasing degree, i.e.,
+        ``(1, 2, 3)`` gives ``1*T_0(x) + 2*T_1(x) + 3*T_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1, 1].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1, 1].
+
+        .. versionadded:: 1.6.0
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(chebadd)
+    _sub = staticmethod(chebsub)
+    _mul = staticmethod(chebmul)
+    _div = staticmethod(chebdiv)
+    _pow = staticmethod(chebpow)
+    _val = staticmethod(chebval)
+    _int = staticmethod(chebint)
+    _der = staticmethod(chebder)
+    _fit = staticmethod(chebfit)
+    _line = staticmethod(chebline)
+    _roots = staticmethod(chebroots)
+    _fromroots = staticmethod(chebfromroots)
+
+    @classmethod
+    def interpolate(cls, func, deg, domain=None, args=()):
+        """Interpolate a function at the Chebyshev points of the first kind.
+
+        Returns the series that interpolates `func` at the Chebyshev points of
+        the first kind scaled and shifted to the `domain`. The resulting series
+        tends to a minmax approximation of `func` when the function is
+        continuous in the domain.
+
+        .. versionadded:: 1.14.0
+
+        Parameters
+        ----------
+        func : function
+            The function to be interpolated. It must be a function of a single
+            variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
+            extra arguments passed in the `args` parameter.
+        deg : int
+            Degree of the interpolating polynomial.
+        domain : {None, [beg, end]}, optional
+            Domain over which `func` is interpolated. The default is None, in
+            which case the domain is [-1, 1].
+        args : tuple, optional
+            Extra arguments to be used in the function call. Default is no
+            extra arguments.
+
+        Returns
+        -------
+        polynomial : Chebyshev instance
+            Interpolating Chebyshev instance.
+
+        Notes
+        -----
+        See `numpy.polynomial.chebfromfunction` for more details.
+
+        """
+        if domain is None:
+            domain = cls.domain
+        xfunc = lambda x: func(pu.mapdomain(x, cls.window, domain), *args)
+        coef = chebinterpolate(xfunc, deg)
+        return cls(coef, domain=domain)
+
+    # Virtual properties
+    domain = np.array(chebdomain)
+    window = np.array(chebdomain)
+    basis_name = 'T'

lib/python3.12/site-packages/numpy/polynomial/chebyshev.pyi

@@ -0,0 +1,51 @@
+from typing import Any
+
+from numpy import ndarray, dtype, int_
+from numpy.polynomial._polybase import ABCPolyBase
+from numpy.polynomial.polyutils import trimcoef
+
+__all__: list[str]
+
+chebtrim = trimcoef
+
+def poly2cheb(pol): ...
+def cheb2poly(c): ...
+
+chebdomain: ndarray[Any, dtype[int_]]
+chebzero: ndarray[Any, dtype[int_]]
+chebone: ndarray[Any, dtype[int_]]
+chebx: ndarray[Any, dtype[int_]]
+
+def chebline(off, scl): ...
+def chebfromroots(roots): ...
+def chebadd(c1, c2): ...
+def chebsub(c1, c2): ...
+def chebmulx(c): ...
+def chebmul(c1, c2): ...
+def chebdiv(c1, c2): ...
+def chebpow(c, pow, maxpower=...): ...
+def chebder(c, m=..., scl=..., axis=...): ...
+def chebint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ...
+def chebval(x, c, tensor=...): ...
+def chebval2d(x, y, c): ...
+def chebgrid2d(x, y, c): ...
+def chebval3d(x, y, z, c): ...
+def chebgrid3d(x, y, z, c): ...
+def chebvander(x, deg): ...
+def chebvander2d(x, y, deg): ...
+def chebvander3d(x, y, z, deg): ...
+def chebfit(x, y, deg, rcond=..., full=..., w=...): ...
+def chebcompanion(c): ...
+def chebroots(c): ...
+def chebinterpolate(func, deg, args = ...): ...
+def chebgauss(deg): ...
+def chebweight(x): ...
+def chebpts1(npts): ...
+def chebpts2(npts): ...
+
+class Chebyshev(ABCPolyBase):
+    @classmethod
+    def interpolate(cls, func, deg, domain=..., args = ...): ...
+    domain: Any
+    window: Any
+    basis_name: Any

lib/python3.12/site-packages/numpy/polynomial/hermite.py

@@ -0,0 +1,1703 @@
+"""
+==============================================================
+Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`)
+==============================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Hermite series, including a `Hermite` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   Hermite
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   hermdomain
+   hermzero
+   hermone
+   hermx
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   hermadd
+   hermsub
+   hermmulx
+   hermmul
+   hermdiv
+   hermpow
+   hermval
+   hermval2d
+   hermval3d
+   hermgrid2d
+   hermgrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   hermder
+   hermint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   hermfromroots
+   hermroots
+   hermvander
+   hermvander2d
+   hermvander3d
+   hermgauss
+   hermweight
+   hermcompanion
+   hermfit
+   hermtrim
+   hermline
+   herm2poly
+   poly2herm
+
+See also
+--------
+`numpy.polynomial`
+
+"""
+import numpy as np
+import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd',
+    'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval',
+    'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots',
+    'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite',
+    'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d',
+    'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight']
+
+hermtrim = pu.trimcoef
+
+
+def poly2herm(pol):
+    """
+    poly2herm(pol)
+
+    Convert a polynomial to a Hermite series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Hermite series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Hermite
+        series.
+
+    See Also
+    --------
+    herm2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import poly2herm
+    >>> poly2herm(np.arange(4))
+    array([1.   ,  2.75 ,  0.5  ,  0.375])
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = hermadd(hermmulx(res), pol[i])
+    return res
+
+
+def herm2poly(c):
+    """
+    Convert a Hermite series to a polynomial.
+
+    Convert an array representing the coefficients of a Hermite series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Hermite series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2herm
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import herm2poly
+    >>> herm2poly([ 1.   ,  2.75 ,  0.5  ,  0.375])
+    array([0., 1., 2., 3.])
+
+    """
+    from .polynomial import polyadd, polysub, polymulx
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n == 1:
+        return c
+    if n == 2:
+        c[1] *= 2
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], c1*(2*(i - 1)))
+            c1 = polyadd(tmp, polymulx(c1)*2)
+        return polyadd(c0, polymulx(c1)*2)
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Hermite
+hermdomain = np.array([-1, 1])
+
+# Hermite coefficients representing zero.
+hermzero = np.array([0])
+
+# Hermite coefficients representing one.
+hermone = np.array([1])
+
+# Hermite coefficients representing the identity x.
+hermx = np.array([0, 1/2])
+
+
+def hermline(off, scl):
+    """
+    Hermite series whose graph is a straight line.
+
+
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Hermite series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermline, hermval
+    >>> hermval(0,hermline(3, 2))
+    3.0
+    >>> hermval(1,hermline(3, 2))
+    5.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl/2])
+    else:
+        return np.array([off])
+
+
+def hermfromroots(roots):
+    """
+    Generate a Hermite series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Hermite form, where the `r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * H_1(x) + ... +  c_n * H_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Hermite form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermfromroots, hermval
+    >>> coef = hermfromroots((-1, 0, 1))
+    >>> hermval((-1, 0, 1), coef)
+    array([0.,  0.,  0.])
+    >>> coef = hermfromroots((-1j, 1j))
+    >>> hermval((-1j, 1j), coef)
+    array([0.+0.j, 0.+0.j])
+
+    """
+    return pu._fromroots(hermline, hermmul, roots)
+
+
+def hermadd(c1, c2):
+    """
+    Add one Hermite series to another.
+
+    Returns the sum of two Hermite series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Hermite series of their sum.
+
+    See Also
+    --------
+    hermsub, hermmulx, hermmul, hermdiv, hermpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Hermite series
+    is a Hermite series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermadd
+    >>> hermadd([1, 2, 3], [1, 2, 3, 4])
+    array([2., 4., 6., 4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def hermsub(c1, c2):
+    """
+    Subtract one Hermite series from another.
+
+    Returns the difference of two Hermite series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their difference.
+
+    See Also
+    --------
+    hermadd, hermmulx, hermmul, hermdiv, hermpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Hermite
+    series is a Hermite series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermsub
+    >>> hermsub([1, 2, 3, 4], [1, 2, 3])
+    array([0.,  0.,  0.,  4.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def hermmulx(c):
+    """Multiply a Hermite series by x.
+
+    Multiply the Hermite series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmul, hermdiv, hermpow
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Hermite
+    polynomials in the form
+
+    .. math::
+
+        xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x))
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermmulx
+    >>> hermmulx([1, 2, 3])
+    array([2. , 6.5, 1. , 1.5])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0]*0
+    prd[1] = c[0]/2
+    for i in range(1, len(c)):
+        prd[i + 1] = c[i]/2
+        prd[i - 1] += c[i]*i
+    return prd
+
+
+def hermmul(c1, c2):
+    """
+    Multiply one Hermite series by another.
+
+    Returns the product of two Hermite series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their product.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmulx, hermdiv, hermpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Hermite polynomial basis set.  Thus, to express
+    the product as a Hermite series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermmul
+    >>> hermmul([1, 2, 3], [0, 1, 2])
+    array([52.,  29.,  52.,   7.,   6.])
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0]*xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]*xs
+        c1 = c[1]*xs
+    else:
+        nd = len(c)
+        c0 = c[-2]*xs
+        c1 = c[-1]*xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
+            c1 = hermadd(tmp, hermmulx(c1)*2)
+    return hermadd(c0, hermmulx(c1)*2)
+
+
+def hermdiv(c1, c2):
+    """
+    Divide one Hermite series by another.
+
+    Returns the quotient-with-remainder of two Hermite series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Hermite series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmulx, hermmul, hermpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Hermite series by another
+    results in quotient and remainder terms that are not in the Hermite
+    polynomial basis set.  Thus, to express these results as a Hermite
+    series, it is necessary to "reproject" the results onto the Hermite
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermdiv
+    >>> hermdiv([ 52.,  29.,  52.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([0.]))
+    >>> hermdiv([ 54.,  31.,  52.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([2., 2.]))
+    >>> hermdiv([ 53.,  30.,  52.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([1., 1.]))
+
+    """
+    return pu._div(hermmul, c1, c2)
+
+
+def hermpow(c, pow, maxpower=16):
+    """Raise a Hermite series to a power.
+
+    Returns the Hermite series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Hermite series of power.
+
+    See Also
+    --------
+    hermadd, hermsub, hermmulx, hermmul, hermdiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermpow
+    >>> hermpow([1, 2, 3], 2)
+    array([81.,  52.,  82.,  12.,   9.])
+
+    """
+    return pu._pow(hermmul, c, pow, maxpower)
+
+
+def hermder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Hermite series.
+
+    Returns the Hermite series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2``
+    while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) +
+    2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite series coefficients. If `c` is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    der : ndarray
+        Hermite series of the derivative.
+
+    See Also
+    --------
+    hermint
+
+    Notes
+    -----
+    In general, the result of differentiating a Hermite series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermder
+    >>> hermder([ 1. ,  0.5,  0.5,  0.5])
+    array([1., 2., 3.])
+    >>> hermder([-0.5,  1./2.,  1./8.,  1./12.,  1./16.], m=2)
+    array([1., 2., 3.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._deprecate_as_int(m, "the order of derivation")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1]*0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 0, -1):
+                der[j - 1] = (2*j)*c[j]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Hermite series.
+
+    Returns the Hermite series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
+    represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
+    2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite series coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    S : ndarray
+        Hermite series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    hermder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermint
+    >>> hermint([1,2,3]) # integrate once, value 0 at 0.
+    array([1. , 0.5, 0.5, 0.5])
+    >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0
+    array([-0.5       ,  0.5       ,  0.125     ,  0.08333333,  0.0625    ]) # may vary
+    >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0.
+    array([2. , 0.5, 0.5, 0.5])
+    >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1
+    array([-2. ,  0.5,  0.5,  0.5])
+    >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1)
+    array([ 1.66666667, -0.5       ,  0.125     ,  0.08333333,  0.0625    ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._deprecate_as_int(m, "the order of integration")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0]*(cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0]*0
+            tmp[1] = c[0]/2
+            for j in range(1, n):
+                tmp[j + 1] = c[j]/(2*(j + 1))
+            tmp[0] += k[i] - hermval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermval(x, c, tensor=True):
+    """
+    Evaluate an Hermite series at points x.
+
+    If `c` is of length `n + 1`, this function returns the value:
+
+    .. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    hermval2d, hermgrid2d, hermval3d, hermgrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermval
+    >>> coef = [1,2,3]
+    >>> hermval(1, coef)
+    11.0
+    >>> hermval([[1,2],[3,4]], coef)
+    array([[ 11.,   51.],
+           [115.,  203.]])
+
+    """
+    c = np.array(c, ndmin=1, copy=False)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,)*x.ndim)
+
+    x2 = x*2
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - c1*(2*(nd - 1))
+            c1 = tmp + c1*x2
+    return c0 + c1*x2
+
+
+def hermval2d(x, y, c):
+    """
+    Evaluate a 2-D Hermite series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points `(x, y)`,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    hermval, hermgrid2d, hermval3d, hermgrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(hermval, c, x, y)
+
+
+def hermgrid2d(x, y, c):
+    """
+    Evaluate a 2-D Hermite series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b)
+
+    where the points `(a, b)` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermval, hermval2d, hermval3d, hermgrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(hermval, c, x, y)
+
+
+def hermval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Hermite series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermval, hermval2d, hermgrid2d, hermgrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(hermval, c, x, y, z)
+
+
+def hermgrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * H_i(a) * H_j(b) * H_k(c)
+
+    where the points `(a, b, c)` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermval, hermval2d, hermgrid2d, hermval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(hermval, c, x, y, z)
+
+
+def hermvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = H_i(x),
+
+    where `0 <= i <= deg`. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Hermite polynomial.
+
+    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
+    array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and
+    ``hermval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Hermite series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Hermite polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermvander
+    >>> x = np.array([-1, 0, 1])
+    >>> hermvander(x, 3)
+    array([[ 1., -2.,  2.,  4.],
+           [ 1.,  0., -2., -0.],
+           [ 1.,  2.,  2., -4.]])
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=False, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x*0 + 1
+    if ideg > 0:
+        x2 = x*2
+        v[1] = x2
+        for i in range(2, ideg + 1):
+            v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1)))
+    return np.moveaxis(v, 0, -1)
+
+
+def hermvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y)`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y),
+
+    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
+    `V` index the points `(x, y)` and the last index encodes the degrees of
+    the Hermite polynomials.
+
+    If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Hermite
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    hermvander, hermvander3d, hermval2d, hermval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((hermvander, hermvander), (x, y), deg)
+
+
+def hermvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = H_i(x)*H_j(y)*H_k(z),
+
+    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
+    indices of `V` index the points `(x, y, z)` and the last index encodes
+    the degrees of the Hermite polynomials.
+
+    If ``V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``hermval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Hermite
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermvander, hermvander3d, hermval2d, hermval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((hermvander, hermvander, hermvander), (x, y, z), deg)
+
+
+def hermfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Hermite series to data.
+
+    Return the coefficients of a Hermite series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Hermite coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column k  of `y` are in column
+        `k`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.hermite_e.hermefit
+    hermval : Evaluates a Hermite series.
+    hermvander : Vandermonde matrix of Hermite series.
+    hermweight : Hermite weight function
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Hermite series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `RankWarning` will be issued. This means that the
+    coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Hermite series are probably most useful when the data can be
+    approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Hermite
+    weight. In that case the weight ``sqrt(w(x[i]))`` should be used
+    together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
+    available as `hermweight`.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermfit, hermval
+    >>> x = np.linspace(-10, 10)
+    >>> err = np.random.randn(len(x))/10
+    >>> y = hermval(x, [1, 2, 3]) + err
+    >>> hermfit(x, y, 2)
+    array([1.0218, 1.9986, 2.9999]) # may vary
+
+    """
+    return pu._fit(hermvander, x, y, deg, rcond, full, w)
+
+
+def hermcompanion(c):
+    """Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is an Hermite basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-.5*c[0]/c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
+    scl = np.multiply.accumulate(scl)[::-1]
+    top = mat.reshape(-1)[1::n+1]
+    bot = mat.reshape(-1)[n::n+1]
+    top[...] = np.sqrt(.5*np.arange(1, n))
+    bot[...] = top
+    mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
+    return mat
+
+
+def hermroots(c):
+    """
+    Compute the roots of a Hermite series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * H_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The Hermite series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite import hermroots, hermfromroots
+    >>> coef = hermfromroots([-1, 0, 1])
+    >>> coef
+    array([0.   ,  0.25 ,  0.   ,  0.125])
+    >>> hermroots(coef)
+    array([-1.00000000e+00, -1.38777878e-17,  1.00000000e+00])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) <= 1:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-.5*c[0]/c[1]])
+
+    # rotated companion matrix reduces error
+    m = hermcompanion(c)[::-1,::-1]
+    r = la.eigvals(m)
+    r.sort()
+    return r
+
+
+def _normed_hermite_n(x, n):
+    """
+    Evaluate a normalized Hermite polynomial.
+
+    Compute the value of the normalized Hermite polynomial of degree ``n``
+    at the points ``x``.
+
+
+    Parameters
+    ----------
+    x : ndarray of double.
+        Points at which to evaluate the function
+    n : int
+        Degree of the normalized Hermite function to be evaluated.
+
+    Returns
+    -------
+    values : ndarray
+        The shape of the return value is described above.
+
+    Notes
+    -----
+    .. versionadded:: 1.10.0
+
+    This function is needed for finding the Gauss points and integration
+    weights for high degrees. The values of the standard Hermite functions
+    overflow when n >= 207.
+
+    """
+    if n == 0:
+        return np.full(x.shape, 1/np.sqrt(np.sqrt(np.pi)))
+
+    c0 = 0.
+    c1 = 1./np.sqrt(np.sqrt(np.pi))
+    nd = float(n)
+    for i in range(n - 1):
+        tmp = c0
+        c0 = -c1*np.sqrt((nd - 1.)/nd)
+        c1 = tmp + c1*x*np.sqrt(2./nd)
+        nd = nd - 1.0
+    return c0 + c1*x*np.sqrt(2)
+
+
+def hermgauss(deg):
+    """
+    Gauss-Hermite quadrature.
+
+    Computes the sample points and weights for Gauss-Hermite quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]`
+    with the weight function :math:`f(x) = \\exp(-x^2)`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`H_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0]*deg + [1], dtype=np.float64)
+    m = hermcompanion(c)
+    x = la.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = _normed_hermite_n(x, ideg)
+    df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg)
+    x -= dy/df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = _normed_hermite_n(x, ideg - 1)
+    fm /= np.abs(fm).max()
+    w = 1/(fm * fm)
+
+    # for Hermite we can also symmetrize
+    w = (w + w[::-1])/2
+    x = (x - x[::-1])/2
+
+    # scale w to get the right value
+    w *= np.sqrt(np.pi) / w.sum()
+
+    return x, w
+
+
+def hermweight(x):
+    """
+    Weight function of the Hermite polynomials.
+
+    The weight function is :math:`\\exp(-x^2)` and the interval of
+    integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are
+    orthogonal, but not normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    w = np.exp(-x**2)
+    return w
+
+
+#
+# Hermite series class
+#
+
+class Hermite(ABCPolyBase):
+    """An Hermite series class.
+
+    The Hermite class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed in the `ABCPolyBase` documentation.
+
+    Parameters
+    ----------
+    coef : array_like
+        Hermite coefficients in order of increasing degree, i.e,
+        ``(1, 2, 3)`` gives ``1*H_0(x) + 2*H_1(X) + 3*H_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1, 1].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1, 1].
+
+        .. versionadded:: 1.6.0
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(hermadd)
+    _sub = staticmethod(hermsub)
+    _mul = staticmethod(hermmul)
+    _div = staticmethod(hermdiv)
+    _pow = staticmethod(hermpow)
+    _val = staticmethod(hermval)
+    _int = staticmethod(hermint)
+    _der = staticmethod(hermder)
+    _fit = staticmethod(hermfit)
+    _line = staticmethod(hermline)
+    _roots = staticmethod(hermroots)
+    _fromroots = staticmethod(hermfromroots)
+
+    # Virtual properties
+    domain = np.array(hermdomain)
+    window = np.array(hermdomain)
+    basis_name = 'H'

lib/python3.12/site-packages/numpy/polynomial/hermite.pyi

@@ -0,0 +1,46 @@
+from typing import Any
+
+from numpy import ndarray, dtype, int_, float_
+from numpy.polynomial._polybase import ABCPolyBase
+from numpy.polynomial.polyutils import trimcoef
+
+__all__: list[str]
+
+hermtrim = trimcoef
+
+def poly2herm(pol): ...
+def herm2poly(c): ...
+
+hermdomain: ndarray[Any, dtype[int_]]
+hermzero: ndarray[Any, dtype[int_]]
+hermone: ndarray[Any, dtype[int_]]
+hermx: ndarray[Any, dtype[float_]]
+
+def hermline(off, scl): ...
+def hermfromroots(roots): ...
+def hermadd(c1, c2): ...
+def hermsub(c1, c2): ...
+def hermmulx(c): ...
+def hermmul(c1, c2): ...
+def hermdiv(c1, c2): ...
+def hermpow(c, pow, maxpower=...): ...
+def hermder(c, m=..., scl=..., axis=...): ...
+def hermint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ...
+def hermval(x, c, tensor=...): ...
+def hermval2d(x, y, c): ...
+def hermgrid2d(x, y, c): ...
+def hermval3d(x, y, z, c): ...
+def hermgrid3d(x, y, z, c): ...
+def hermvander(x, deg): ...
+def hermvander2d(x, y, deg): ...
+def hermvander3d(x, y, z, deg): ...
+def hermfit(x, y, deg, rcond=..., full=..., w=...): ...
+def hermcompanion(c): ...
+def hermroots(c): ...
+def hermgauss(deg): ...
+def hermweight(x): ...
+
+class Hermite(ABCPolyBase):
+    domain: Any
+    window: Any
+    basis_name: Any

lib/python3.12/site-packages/numpy/polynomial/hermite_e.py

@@ -0,0 +1,1695 @@
+"""
+===================================================================
+HermiteE Series, "Probabilists" (:mod:`numpy.polynomial.hermite_e`)
+===================================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Hermite_e series, including a `HermiteE` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   HermiteE
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   hermedomain
+   hermezero
+   hermeone
+   hermex
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   hermeadd
+   hermesub
+   hermemulx
+   hermemul
+   hermediv
+   hermepow
+   hermeval
+   hermeval2d
+   hermeval3d
+   hermegrid2d
+   hermegrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   hermeder
+   hermeint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   hermefromroots
+   hermeroots
+   hermevander
+   hermevander2d
+   hermevander3d
+   hermegauss
+   hermeweight
+   hermecompanion
+   hermefit
+   hermetrim
+   hermeline
+   herme2poly
+   poly2herme
+
+See also
+--------
+`numpy.polynomial`
+
+"""
+import numpy as np
+import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline',
+    'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv',
+    'hermepow', 'hermeval', 'hermeder', 'hermeint', 'herme2poly',
+    'poly2herme', 'hermefromroots', 'hermevander', 'hermefit', 'hermetrim',
+    'hermeroots', 'HermiteE', 'hermeval2d', 'hermeval3d', 'hermegrid2d',
+    'hermegrid3d', 'hermevander2d', 'hermevander3d', 'hermecompanion',
+    'hermegauss', 'hermeweight']
+
+hermetrim = pu.trimcoef
+
+
+def poly2herme(pol):
+    """
+    poly2herme(pol)
+
+    Convert a polynomial to a Hermite series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Hermite series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Hermite
+        series.
+
+    See Also
+    --------
+    herme2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import poly2herme
+    >>> poly2herme(np.arange(4))
+    array([  2.,  10.,   2.,   3.])
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = hermeadd(hermemulx(res), pol[i])
+    return res
+
+
+def herme2poly(c):
+    """
+    Convert a Hermite series to a polynomial.
+
+    Convert an array representing the coefficients of a Hermite series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Hermite series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2herme
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import herme2poly
+    >>> herme2poly([  2.,  10.,   2.,   3.])
+    array([0.,  1.,  2.,  3.])
+
+    """
+    from .polynomial import polyadd, polysub, polymulx
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n == 1:
+        return c
+    if n == 2:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], c1*(i - 1))
+            c1 = polyadd(tmp, polymulx(c1))
+        return polyadd(c0, polymulx(c1))
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Hermite
+hermedomain = np.array([-1, 1])
+
+# Hermite coefficients representing zero.
+hermezero = np.array([0])
+
+# Hermite coefficients representing one.
+hermeone = np.array([1])
+
+# Hermite coefficients representing the identity x.
+hermex = np.array([0, 1])
+
+
+def hermeline(off, scl):
+    """
+    Hermite series whose graph is a straight line.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Hermite series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeline
+    >>> from numpy.polynomial.hermite_e import hermeline, hermeval
+    >>> hermeval(0,hermeline(3, 2))
+    3.0
+    >>> hermeval(1,hermeline(3, 2))
+    5.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def hermefromroots(roots):
+    """
+    Generate a HermiteE series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in HermiteE form, where the `r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * He_1(x) + ... +  c_n * He_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in HermiteE form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermefromroots, hermeval
+    >>> coef = hermefromroots((-1, 0, 1))
+    >>> hermeval((-1, 0, 1), coef)
+    array([0., 0., 0.])
+    >>> coef = hermefromroots((-1j, 1j))
+    >>> hermeval((-1j, 1j), coef)
+    array([0.+0.j, 0.+0.j])
+
+    """
+    return pu._fromroots(hermeline, hermemul, roots)
+
+
+def hermeadd(c1, c2):
+    """
+    Add one Hermite series to another.
+
+    Returns the sum of two Hermite series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Hermite series of their sum.
+
+    See Also
+    --------
+    hermesub, hermemulx, hermemul, hermediv, hermepow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Hermite series
+    is a Hermite series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeadd
+    >>> hermeadd([1, 2, 3], [1, 2, 3, 4])
+    array([2.,  4.,  6.,  4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def hermesub(c1, c2):
+    """
+    Subtract one Hermite series from another.
+
+    Returns the difference of two Hermite series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their difference.
+
+    See Also
+    --------
+    hermeadd, hermemulx, hermemul, hermediv, hermepow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Hermite
+    series is a Hermite series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermesub
+    >>> hermesub([1, 2, 3, 4], [1, 2, 3])
+    array([0., 0., 0., 4.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def hermemulx(c):
+    """Multiply a Hermite series by x.
+
+    Multiply the Hermite series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Hermite
+    polynomials in the form
+
+    .. math::
+
+        xP_i(x) = (P_{i + 1}(x) + iP_{i - 1}(x)))
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermemulx
+    >>> hermemulx([1, 2, 3])
+    array([2.,  7.,  2.,  3.])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0]*0
+    prd[1] = c[0]
+    for i in range(1, len(c)):
+        prd[i + 1] = c[i]
+        prd[i - 1] += c[i]*i
+    return prd
+
+
+def hermemul(c1, c2):
+    """
+    Multiply one Hermite series by another.
+
+    Returns the product of two Hermite series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Hermite series coefficients representing their product.
+
+    See Also
+    --------
+    hermeadd, hermesub, hermemulx, hermediv, hermepow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Hermite polynomial basis set.  Thus, to express
+    the product as a Hermite series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermemul
+    >>> hermemul([1, 2, 3], [0, 1, 2])
+    array([14.,  15.,  28.,   7.,   6.])
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0]*xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]*xs
+        c1 = c[1]*xs
+    else:
+        nd = len(c)
+        c0 = c[-2]*xs
+        c1 = c[-1]*xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = hermesub(c[-i]*xs, c1*(nd - 1))
+            c1 = hermeadd(tmp, hermemulx(c1))
+    return hermeadd(c0, hermemulx(c1))
+
+
+def hermediv(c1, c2):
+    """
+    Divide one Hermite series by another.
+
+    Returns the quotient-with-remainder of two Hermite series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Hermite series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Hermite series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    hermeadd, hermesub, hermemulx, hermemul, hermepow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Hermite series by another
+    results in quotient and remainder terms that are not in the Hermite
+    polynomial basis set.  Thus, to express these results as a Hermite
+    series, it is necessary to "reproject" the results onto the Hermite
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermediv
+    >>> hermediv([ 14.,  15.,  28.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([0.]))
+    >>> hermediv([ 15.,  17.,  28.,   7.,   6.], [0, 1, 2])
+    (array([1., 2., 3.]), array([1., 2.]))
+
+    """
+    return pu._div(hermemul, c1, c2)
+
+
+def hermepow(c, pow, maxpower=16):
+    """Raise a Hermite series to a power.
+
+    Returns the Hermite series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Hermite series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Hermite series of power.
+
+    See Also
+    --------
+    hermeadd, hermesub, hermemulx, hermemul, hermediv
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermepow
+    >>> hermepow([1, 2, 3], 2)
+    array([23.,  28.,  46.,  12.,   9.])
+
+    """
+    return pu._pow(hermemul, c, pow, maxpower)
+
+
+def hermeder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Hermite_e series.
+
+    Returns the series coefficients `c` differentiated `m` times along
+    `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*He_0 + 2*He_1 + 3*He_2``
+    while [[1,2],[1,2]] represents ``1*He_0(x)*He_0(y) + 1*He_1(x)*He_0(y)
+    + 2*He_0(x)*He_1(y) + 2*He_1(x)*He_1(y)`` if axis=0 is ``x`` and axis=1
+    is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite_e series coefficients. If `c` is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    der : ndarray
+        Hermite series of the derivative.
+
+    See Also
+    --------
+    hermeint
+
+    Notes
+    -----
+    In general, the result of differentiating a Hermite series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeder
+    >>> hermeder([ 1.,  1.,  1.,  1.])
+    array([1.,  2.,  3.])
+    >>> hermeder([-0.25,  1.,  1./2.,  1./3.,  1./4 ], m=2)
+    array([1.,  2.,  3.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._deprecate_as_int(m, "the order of derivation")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        return c[:1]*0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 0, -1):
+                der[j - 1] = j*c[j]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Hermite_e series.
+
+    Returns the Hermite_e series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
+    represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
+    2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Hermite_e series coefficients. If c is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    S : ndarray
+        Hermite_e series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    hermeder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeint
+    >>> hermeint([1, 2, 3]) # integrate once, value 0 at 0.
+    array([1., 1., 1., 1.])
+    >>> hermeint([1, 2, 3], m=2) # integrate twice, value & deriv 0 at 0
+    array([-0.25      ,  1.        ,  0.5       ,  0.33333333,  0.25      ]) # may vary
+    >>> hermeint([1, 2, 3], k=1) # integrate once, value 1 at 0.
+    array([2., 1., 1., 1.])
+    >>> hermeint([1, 2, 3], lbnd=-1) # integrate once, value 0 at -1
+    array([-1.,  1.,  1.,  1.])
+    >>> hermeint([1, 2, 3], m=2, k=[1, 2], lbnd=-1)
+    array([ 1.83333333,  0.        ,  0.5       ,  0.33333333,  0.25      ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._deprecate_as_int(m, "the order of integration")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0]*(cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0]*0
+            tmp[1] = c[0]
+            for j in range(1, n):
+                tmp[j + 1] = c[j]/(j + 1)
+            tmp[0] += k[i] - hermeval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def hermeval(x, c, tensor=True):
+    """
+    Evaluate an HermiteE series at points x.
+
+    If `c` is of length `n + 1`, this function returns the value:
+
+    .. math:: p(x) = c_0 * He_0(x) + c_1 * He_1(x) + ... + c_n * He_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        with themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    hermeval2d, hermegrid2d, hermeval3d, hermegrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeval
+    >>> coef = [1,2,3]
+    >>> hermeval(1, coef)
+    3.0
+    >>> hermeval([[1,2],[3,4]], coef)
+    array([[ 3., 14.],
+           [31., 54.]])
+
+    """
+    c = np.array(c, ndmin=1, copy=False)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,)*x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - c1*(nd - 1)
+            c1 = tmp + c1*x
+    return c0 + c1*x
+
+
+def hermeval2d(x, y, c):
+    """
+    Evaluate a 2-D HermiteE series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * He_i(x) * He_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points `(x, y)`,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    hermeval, hermegrid2d, hermeval3d, hermegrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(hermeval, c, x, y)
+
+
+def hermegrid2d(x, y, c):
+    """
+    Evaluate a 2-D HermiteE series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b)
+
+    where the points `(a, b)` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermeval, hermeval2d, hermeval3d, hermegrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(hermeval, c, x, y)
+
+
+def hermeval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Hermite_e series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * He_i(x) * He_j(y) * He_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermeval, hermeval2d, hermegrid2d, hermegrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(hermeval, c, x, y, z)
+
+
+def hermegrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D HermiteE series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * He_i(a) * He_j(b) * He_k(c)
+
+    where the points `(a, b, c)` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    hermeval, hermeval2d, hermegrid2d, hermeval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(hermeval, c, x, y, z)
+
+
+def hermevander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = He_i(x),
+
+    where `0 <= i <= deg`. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the HermiteE polynomial.
+
+    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
+    array ``V = hermevander(x, n)``, then ``np.dot(V, c)`` and
+    ``hermeval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of HermiteE series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding HermiteE polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermevander
+    >>> x = np.array([-1, 0, 1])
+    >>> hermevander(x, 3)
+    array([[ 1., -1.,  0.,  2.],
+           [ 1.,  0., -1., -0.],
+           [ 1.,  1.,  0., -2.]])
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=False, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x*0 + 1
+    if ideg > 0:
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = (v[i-1]*x - v[i-2]*(i - 1))
+    return np.moveaxis(v, 0, -1)
+
+
+def hermevander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y)`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = He_i(x) * He_j(y),
+
+    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
+    `V` index the points `(x, y)` and the last index encodes the degrees of
+    the HermiteE polynomials.
+
+    If ``V = hermevander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``hermeval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D HermiteE
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    hermevander, hermevander3d, hermeval2d, hermeval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((hermevander, hermevander), (x, y), deg)
+
+
+def hermevander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
+    then Hehe pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = He_i(x)*He_j(y)*He_k(z),
+
+    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
+    indices of `V` index the points `(x, y, z)` and the last index encodes
+    the degrees of the HermiteE polynomials.
+
+    If ``V = hermevander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``hermeval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D HermiteE
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    hermevander, hermevander3d, hermeval2d, hermeval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((hermevander, hermevander, hermevander), (x, y, z), deg)
+
+
+def hermefit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Hermite series to data.
+
+    Return the coefficients of a HermiteE series of degree `deg` that is
+    the least squares fit to the data values `y` given at points `x`. If
+    `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D
+    multiple fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * He_1(x) + ... + c_n * He_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Hermite coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column k  of `y` are in column
+        `k`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full = False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.laguerre.lagfit
+    hermeval : Evaluates a Hermite series.
+    hermevander : pseudo Vandermonde matrix of Hermite series.
+    hermeweight : HermiteE weight function.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the HermiteE series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the pseudo Vandermonde matrix of `x`, the elements of `c`
+    are the coefficients to be solved for, and the elements of `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `RankWarning` will be issued. This means that the
+    coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using HermiteE series are probably most useful when the data can
+    be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the HermiteE
+    weight. In that case the weight ``sqrt(w(x[i]))`` should be used
+    together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
+    available as `hermeweight`.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermefit, hermeval
+    >>> x = np.linspace(-10, 10)
+    >>> np.random.seed(123)
+    >>> err = np.random.randn(len(x))/10
+    >>> y = hermeval(x, [1, 2, 3]) + err
+    >>> hermefit(x, y, 2)
+    array([ 1.01690445,  1.99951418,  2.99948696]) # may vary
+
+    """
+    return pu._fit(hermevander, x, y, deg, rcond, full, w)
+
+
+def hermecompanion(c):
+    """
+    Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is an HermiteE basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of HermiteE series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0]/c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = np.hstack((1., 1./np.sqrt(np.arange(n - 1, 0, -1))))
+    scl = np.multiply.accumulate(scl)[::-1]
+    top = mat.reshape(-1)[1::n+1]
+    bot = mat.reshape(-1)[n::n+1]
+    top[...] = np.sqrt(np.arange(1, n))
+    bot[...] = top
+    mat[:, -1] -= scl*c[:-1]/c[-1]
+    return mat
+
+
+def hermeroots(c):
+    """
+    Compute the roots of a HermiteE series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * He_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.chebyshev.chebroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The HermiteE series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.hermite_e import hermeroots, hermefromroots
+    >>> coef = hermefromroots([-1, 0, 1])
+    >>> coef
+    array([0., 2., 0., 1.])
+    >>> hermeroots(coef)
+    array([-1.,  0.,  1.]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) <= 1:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0]/c[1]])
+
+    # rotated companion matrix reduces error
+    m = hermecompanion(c)[::-1,::-1]
+    r = la.eigvals(m)
+    r.sort()
+    return r
+
+
+def _normed_hermite_e_n(x, n):
+    """
+    Evaluate a normalized HermiteE polynomial.
+
+    Compute the value of the normalized HermiteE polynomial of degree ``n``
+    at the points ``x``.
+
+
+    Parameters
+    ----------
+    x : ndarray of double.
+        Points at which to evaluate the function
+    n : int
+        Degree of the normalized HermiteE function to be evaluated.
+
+    Returns
+    -------
+    values : ndarray
+        The shape of the return value is described above.
+
+    Notes
+    -----
+    .. versionadded:: 1.10.0
+
+    This function is needed for finding the Gauss points and integration
+    weights for high degrees. The values of the standard HermiteE functions
+    overflow when n >= 207.
+
+    """
+    if n == 0:
+        return np.full(x.shape, 1/np.sqrt(np.sqrt(2*np.pi)))
+
+    c0 = 0.
+    c1 = 1./np.sqrt(np.sqrt(2*np.pi))
+    nd = float(n)
+    for i in range(n - 1):
+        tmp = c0
+        c0 = -c1*np.sqrt((nd - 1.)/nd)
+        c1 = tmp + c1*x*np.sqrt(1./nd)
+        nd = nd - 1.0
+    return c0 + c1*x
+
+
+def hermegauss(deg):
+    """
+    Gauss-HermiteE quadrature.
+
+    Computes the sample points and weights for Gauss-HermiteE quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]`
+    with the weight function :math:`f(x) = \\exp(-x^2/2)`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (He'_n(x_k) * He_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`He_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0]*deg + [1])
+    m = hermecompanion(c)
+    x = la.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = _normed_hermite_e_n(x, ideg)
+    df = _normed_hermite_e_n(x, ideg - 1) * np.sqrt(ideg)
+    x -= dy/df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = _normed_hermite_e_n(x, ideg - 1)
+    fm /= np.abs(fm).max()
+    w = 1/(fm * fm)
+
+    # for Hermite_e we can also symmetrize
+    w = (w + w[::-1])/2
+    x = (x - x[::-1])/2
+
+    # scale w to get the right value
+    w *= np.sqrt(2*np.pi) / w.sum()
+
+    return x, w
+
+
+def hermeweight(x):
+    """Weight function of the Hermite_e polynomials.
+
+    The weight function is :math:`\\exp(-x^2/2)` and the interval of
+    integration is :math:`[-\\inf, \\inf]`. the HermiteE polynomials are
+    orthogonal, but not normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    w = np.exp(-.5*x**2)
+    return w
+
+
+#
+# HermiteE series class
+#
+
+class HermiteE(ABCPolyBase):
+    """An HermiteE series class.
+
+    The HermiteE class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed in the `ABCPolyBase` documentation.
+
+    Parameters
+    ----------
+    coef : array_like
+        HermiteE coefficients in order of increasing degree, i.e,
+        ``(1, 2, 3)`` gives ``1*He_0(x) + 2*He_1(X) + 3*He_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1, 1].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1, 1].
+
+        .. versionadded:: 1.6.0
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(hermeadd)
+    _sub = staticmethod(hermesub)
+    _mul = staticmethod(hermemul)
+    _div = staticmethod(hermediv)
+    _pow = staticmethod(hermepow)
+    _val = staticmethod(hermeval)
+    _int = staticmethod(hermeint)
+    _der = staticmethod(hermeder)
+    _fit = staticmethod(hermefit)
+    _line = staticmethod(hermeline)
+    _roots = staticmethod(hermeroots)
+    _fromroots = staticmethod(hermefromroots)
+
+    # Virtual properties
+    domain = np.array(hermedomain)
+    window = np.array(hermedomain)
+    basis_name = 'He'

lib/python3.12/site-packages/numpy/polynomial/hermite_e.pyi

@@ -0,0 +1,46 @@
+from typing import Any
+
+from numpy import ndarray, dtype, int_
+from numpy.polynomial._polybase import ABCPolyBase
+from numpy.polynomial.polyutils import trimcoef
+
+__all__: list[str]
+
+hermetrim = trimcoef
+
+def poly2herme(pol): ...
+def herme2poly(c): ...
+
+hermedomain: ndarray[Any, dtype[int_]]
+hermezero: ndarray[Any, dtype[int_]]
+hermeone: ndarray[Any, dtype[int_]]
+hermex: ndarray[Any, dtype[int_]]
+
+def hermeline(off, scl): ...
+def hermefromroots(roots): ...
+def hermeadd(c1, c2): ...
+def hermesub(c1, c2): ...
+def hermemulx(c): ...
+def hermemul(c1, c2): ...
+def hermediv(c1, c2): ...
+def hermepow(c, pow, maxpower=...): ...
+def hermeder(c, m=..., scl=..., axis=...): ...
+def hermeint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ...
+def hermeval(x, c, tensor=...): ...
+def hermeval2d(x, y, c): ...
+def hermegrid2d(x, y, c): ...
+def hermeval3d(x, y, z, c): ...
+def hermegrid3d(x, y, z, c): ...
+def hermevander(x, deg): ...
+def hermevander2d(x, y, deg): ...
+def hermevander3d(x, y, z, deg): ...
+def hermefit(x, y, deg, rcond=..., full=..., w=...): ...
+def hermecompanion(c): ...
+def hermeroots(c): ...
+def hermegauss(deg): ...
+def hermeweight(x): ...
+
+class HermiteE(ABCPolyBase):
+    domain: Any
+    window: Any
+    basis_name: Any

lib/python3.12/site-packages/numpy/polynomial/laguerre.py

@@ -0,0 +1,1651 @@
+"""
+==================================================
+Laguerre Series (:mod:`numpy.polynomial.laguerre`)
+==================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Laguerre series, including a `Laguerre` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   Laguerre
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   lagdomain
+   lagzero
+   lagone
+   lagx
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   lagadd
+   lagsub
+   lagmulx
+   lagmul
+   lagdiv
+   lagpow
+   lagval
+   lagval2d
+   lagval3d
+   laggrid2d
+   laggrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   lagder
+   lagint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   lagfromroots
+   lagroots
+   lagvander
+   lagvander2d
+   lagvander3d
+   laggauss
+   lagweight
+   lagcompanion
+   lagfit
+   lagtrim
+   lagline
+   lag2poly
+   poly2lag
+
+See also
+--------
+`numpy.polynomial`
+
+"""
+import numpy as np
+import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd',
+    'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', 'lagval', 'lagder',
+    'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', 'lagvander',
+    'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', 'lagval3d',
+    'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d', 'lagcompanion',
+    'laggauss', 'lagweight']
+
+lagtrim = pu.trimcoef
+
+
+def poly2lag(pol):
+    """
+    poly2lag(pol)
+
+    Convert a polynomial to a Laguerre series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Laguerre series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Laguerre
+        series.
+
+    See Also
+    --------
+    lag2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import poly2lag
+    >>> poly2lag(np.arange(4))
+    array([ 23., -63.,  58., -18.])
+
+    """
+    [pol] = pu.as_series([pol])
+    res = 0
+    for p in pol[::-1]:
+        res = lagadd(lagmulx(res), p)
+    return res
+
+
+def lag2poly(c):
+    """
+    Convert a Laguerre series to a polynomial.
+
+    Convert an array representing the coefficients of a Laguerre series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Laguerre series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2lag
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lag2poly
+    >>> lag2poly([ 23., -63.,  58., -18.])
+    array([0., 1., 2., 3.])
+
+    """
+    from .polynomial import polyadd, polysub, polymulx
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n == 1:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], (c1*(i - 1))/i)
+            c1 = polyadd(tmp, polysub((2*i - 1)*c1, polymulx(c1))/i)
+        return polyadd(c0, polysub(c1, polymulx(c1)))
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Laguerre
+lagdomain = np.array([0, 1])
+
+# Laguerre coefficients representing zero.
+lagzero = np.array([0])
+
+# Laguerre coefficients representing one.
+lagone = np.array([1])
+
+# Laguerre coefficients representing the identity x.
+lagx = np.array([1, -1])
+
+
+def lagline(off, scl):
+    """
+    Laguerre series whose graph is a straight line.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Laguerre series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagline, lagval
+    >>> lagval(0,lagline(3, 2))
+    3.0
+    >>> lagval(1,lagline(3, 2))
+    5.0
+
+    """
+    if scl != 0:
+        return np.array([off + scl, -scl])
+    else:
+        return np.array([off])
+
+
+def lagfromroots(roots):
+    """
+    Generate a Laguerre series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Laguerre form, where the `r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * L_1(x) + ... +  c_n * L_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Laguerre form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagfromroots, lagval
+    >>> coef = lagfromroots((-1, 0, 1))
+    >>> lagval((-1, 0, 1), coef)
+    array([0.,  0.,  0.])
+    >>> coef = lagfromroots((-1j, 1j))
+    >>> lagval((-1j, 1j), coef)
+    array([0.+0.j, 0.+0.j])
+
+    """
+    return pu._fromroots(lagline, lagmul, roots)
+
+
+def lagadd(c1, c2):
+    """
+    Add one Laguerre series to another.
+
+    Returns the sum of two Laguerre series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Laguerre series of their sum.
+
+    See Also
+    --------
+    lagsub, lagmulx, lagmul, lagdiv, lagpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Laguerre series
+    is a Laguerre series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagadd
+    >>> lagadd([1, 2, 3], [1, 2, 3, 4])
+    array([2.,  4.,  6.,  4.])
+
+
+    """
+    return pu._add(c1, c2)
+
+
+def lagsub(c1, c2):
+    """
+    Subtract one Laguerre series from another.
+
+    Returns the difference of two Laguerre series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Laguerre series coefficients representing their difference.
+
+    See Also
+    --------
+    lagadd, lagmulx, lagmul, lagdiv, lagpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Laguerre
+    series is a Laguerre series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagsub
+    >>> lagsub([1, 2, 3, 4], [1, 2, 3])
+    array([0.,  0.,  0.,  4.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def lagmulx(c):
+    """Multiply a Laguerre series by x.
+
+    Multiply the Laguerre series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmul, lagdiv, lagpow
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Laguerre
+    polynomials in the form
+
+    .. math::
+
+        xP_i(x) = (-(i + 1)*P_{i + 1}(x) + (2i + 1)P_{i}(x) - iP_{i - 1}(x))
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagmulx
+    >>> lagmulx([1, 2, 3])
+    array([-1.,  -1.,  11.,  -9.])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0]
+    prd[1] = -c[0]
+    for i in range(1, len(c)):
+        prd[i + 1] = -c[i]*(i + 1)
+        prd[i] += c[i]*(2*i + 1)
+        prd[i - 1] -= c[i]*i
+    return prd
+
+
+def lagmul(c1, c2):
+    """
+    Multiply one Laguerre series by another.
+
+    Returns the product of two Laguerre series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Laguerre series coefficients representing their product.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmulx, lagdiv, lagpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Laguerre polynomial basis set.  Thus, to express
+    the product as a Laguerre series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagmul
+    >>> lagmul([1, 2, 3], [0, 1, 2])
+    array([  8., -13.,  38., -51.,  36.])
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0]*xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]*xs
+        c1 = c[1]*xs
+    else:
+        nd = len(c)
+        c0 = c[-2]*xs
+        c1 = c[-1]*xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = lagsub(c[-i]*xs, (c1*(nd - 1))/nd)
+            c1 = lagadd(tmp, lagsub((2*nd - 1)*c1, lagmulx(c1))/nd)
+    return lagadd(c0, lagsub(c1, lagmulx(c1)))
+
+
+def lagdiv(c1, c2):
+    """
+    Divide one Laguerre series by another.
+
+    Returns the quotient-with-remainder of two Laguerre series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Laguerre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of Laguerre series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmulx, lagmul, lagpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Laguerre series by another
+    results in quotient and remainder terms that are not in the Laguerre
+    polynomial basis set.  Thus, to express these results as a Laguerre
+    series, it is necessary to "reproject" the results onto the Laguerre
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagdiv
+    >>> lagdiv([  8., -13.,  38., -51.,  36.], [0, 1, 2])
+    (array([1., 2., 3.]), array([0.]))
+    >>> lagdiv([  9., -12.,  38., -51.,  36.], [0, 1, 2])
+    (array([1., 2., 3.]), array([1., 1.]))
+
+    """
+    return pu._div(lagmul, c1, c2)
+
+
+def lagpow(c, pow, maxpower=16):
+    """Raise a Laguerre series to a power.
+
+    Returns the Laguerre series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Laguerre series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Laguerre series of power.
+
+    See Also
+    --------
+    lagadd, lagsub, lagmulx, lagmul, lagdiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagpow
+    >>> lagpow([1, 2, 3], 2)
+    array([ 14., -16.,  56., -72.,  54.])
+
+    """
+    return pu._pow(lagmul, c, pow, maxpower)
+
+
+def lagder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Laguerre series.
+
+    Returns the Laguerre series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
+    while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
+    2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Laguerre series coefficients. If `c` is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    der : ndarray
+        Laguerre series of the derivative.
+
+    See Also
+    --------
+    lagint
+
+    Notes
+    -----
+    In general, the result of differentiating a Laguerre series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagder
+    >>> lagder([ 1.,  1.,  1., -3.])
+    array([1.,  2.,  3.])
+    >>> lagder([ 1.,  0.,  0., -4.,  3.], m=2)
+    array([1.,  2.,  3.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+
+    cnt = pu._deprecate_as_int(m, "the order of derivation")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1]*0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 1, -1):
+                der[j - 1] = -c[j]
+                c[j - 1] += c[j]
+            der[0] = -c[1]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Laguerre series.
+
+    Returns the Laguerre series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
+    represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
+    2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Laguerre series coefficients. If `c` is multidimensional
+        the different axis correspond to different variables with the
+        degree in each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    S : ndarray
+        Laguerre series coefficients of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    lagder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagint
+    >>> lagint([1,2,3])
+    array([ 1.,  1.,  1., -3.])
+    >>> lagint([1,2,3], m=2)
+    array([ 1.,  0.,  0., -4.,  3.])
+    >>> lagint([1,2,3], k=1)
+    array([ 2.,  1.,  1., -3.])
+    >>> lagint([1,2,3], lbnd=-1)
+    array([11.5,  1. ,  1. , -3. ])
+    >>> lagint([1,2], m=2, k=[1,2], lbnd=-1)
+    array([ 11.16666667,  -5.        ,  -3.        ,   2.        ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._deprecate_as_int(m, "the order of integration")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0]*(cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0]
+            tmp[1] = -c[0]
+            for j in range(1, n):
+                tmp[j] += c[j]
+                tmp[j + 1] = -c[j]
+            tmp[0] += k[i] - lagval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def lagval(x, c, tensor=True):
+    """
+    Evaluate a Laguerre series at points x.
+
+    If `c` is of length `n + 1`, this function returns the value:
+
+    .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    lagval2d, laggrid2d, lagval3d, laggrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagval
+    >>> coef = [1,2,3]
+    >>> lagval(1, coef)
+    -0.5
+    >>> lagval([[1,2],[3,4]], coef)
+    array([[-0.5, -4. ],
+           [-4.5, -2. ]])
+
+    """
+    c = np.array(c, ndmin=1, copy=False)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,)*x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - (c1*(nd - 1))/nd
+            c1 = tmp + (c1*((2*nd - 1) - x))/nd
+    return c0 + c1*(1 - x)
+
+
+def lagval2d(x, y, c):
+    """
+    Evaluate a 2-D Laguerre series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points `(x, y)`,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    lagval, laggrid2d, lagval3d, laggrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(lagval, c, x, y)
+
+
+def laggrid2d(x, y, c):
+    """
+    Evaluate a 2-D Laguerre series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
+
+    where the points `(a, b)` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points in the
+        Cartesian product of `x` and `y`.
+
+    See Also
+    --------
+    lagval, lagval2d, lagval3d, laggrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(lagval, c, x, y)
+
+
+def lagval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Laguerre series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    lagval, lagval2d, laggrid2d, laggrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(lagval, c, x, y, z)
+
+
+def laggrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Laguerre series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
+
+    where the points `(a, b, c)` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    lagval, lagval2d, laggrid2d, lagval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(lagval, c, x, y, z)
+
+
+def lagvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = L_i(x)
+
+    where `0 <= i <= deg`. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Laguerre polynomial.
+
+    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
+    array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and
+    ``lagval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Laguerre series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Laguerre polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagvander
+    >>> x = np.array([0, 1, 2])
+    >>> lagvander(x, 3)
+    array([[ 1.        ,  1.        ,  1.        ,  1.        ],
+           [ 1.        ,  0.        , -0.5       , -0.66666667],
+           [ 1.        , -1.        , -1.        , -0.33333333]])
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=False, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x*0 + 1
+    if ideg > 0:
+        v[1] = 1 - x
+        for i in range(2, ideg + 1):
+            v[i] = (v[i-1]*(2*i - 1 - x) - v[i-2]*(i - 1))/i
+    return np.moveaxis(v, 0, -1)
+
+
+def lagvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y)`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
+
+    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
+    `V` index the points `(x, y)` and the last index encodes the degrees of
+    the Laguerre polynomials.
+
+    If ``V = lagvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``lagval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Laguerre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    lagvander, lagvander3d, lagval2d, lagval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((lagvander, lagvander), (x, y), deg)
+
+
+def lagvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
+
+    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
+    indices of `V` index the points `(x, y, z)` and the last index encodes
+    the degrees of the Laguerre polynomials.
+
+    If ``V = lagvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``lagval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Laguerre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    lagvander, lagvander3d, lagval2d, lagval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((lagvander, lagvander, lagvander), (x, y, z), deg)
+
+
+def lagfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Laguerre series to data.
+
+    Return the coefficients of a Laguerre series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
+
+    where ``n`` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Laguerre coefficients ordered from low to high. If `y` was 2-D,
+        the coefficients for the data in column *k*  of `y` are in column
+        *k*.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    lagval : Evaluates a Laguerre series.
+    lagvander : pseudo Vandermonde matrix of Laguerre series.
+    lagweight : Laguerre weight function.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Laguerre series ``p`` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up as the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where ``V`` is the weighted pseudo Vandermonde matrix of `x`, ``c`` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of ``V``.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `RankWarning` will be issued. This means that the
+    coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Laguerre series are probably most useful when the data can
+    be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Laguerre
+    weight. In that case the weight ``sqrt(w(x[i]))`` should be used
+    together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
+    available as `lagweight`.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagfit, lagval
+    >>> x = np.linspace(0, 10)
+    >>> err = np.random.randn(len(x))/10
+    >>> y = lagval(x, [1, 2, 3]) + err
+    >>> lagfit(x, y, 2)
+    array([ 0.96971004,  2.00193749,  3.00288744]) # may vary
+
+    """
+    return pu._fit(lagvander, x, y, deg, rcond, full, w)
+
+
+def lagcompanion(c):
+    """
+    Return the companion matrix of c.
+
+    The usual companion matrix of the Laguerre polynomials is already
+    symmetric when `c` is a basis Laguerre polynomial, so no scaling is
+    applied.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Laguerre series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Companion matrix of dimensions (deg, deg).
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[1 + c[0]/c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    top = mat.reshape(-1)[1::n+1]
+    mid = mat.reshape(-1)[0::n+1]
+    bot = mat.reshape(-1)[n::n+1]
+    top[...] = -np.arange(1, n)
+    mid[...] = 2.*np.arange(n) + 1.
+    bot[...] = top
+    mat[:, -1] += (c[:-1]/c[-1])*n
+    return mat
+
+
+def lagroots(c):
+    """
+    Compute the roots of a Laguerre series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * L_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    The Laguerre series basis polynomials aren't powers of `x` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.laguerre import lagroots, lagfromroots
+    >>> coef = lagfromroots([0, 1, 2])
+    >>> coef
+    array([  2.,  -8.,  12.,  -6.])
+    >>> lagroots(coef)
+    array([-4.4408921e-16,  1.0000000e+00,  2.0000000e+00])
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) <= 1:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([1 + c[0]/c[1]])
+
+    # rotated companion matrix reduces error
+    m = lagcompanion(c)[::-1,::-1]
+    r = la.eigvals(m)
+    r.sort()
+    return r
+
+
+def laggauss(deg):
+    """
+    Gauss-Laguerre quadrature.
+
+    Computes the sample points and weights for Gauss-Laguerre quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[0, \\inf]`
+    with the weight function :math:`f(x) = \\exp(-x)`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    The results have only been tested up to degree 100 higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`L_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0]*deg + [1])
+    m = lagcompanion(c)
+    x = la.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = lagval(x, c)
+    df = lagval(x, lagder(c))
+    x -= dy/df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = lagval(x, c[1:])
+    fm /= np.abs(fm).max()
+    df /= np.abs(df).max()
+    w = 1/(fm * df)
+
+    # scale w to get the right value, 1 in this case
+    w /= w.sum()
+
+    return x, w
+
+
+def lagweight(x):
+    """Weight function of the Laguerre polynomials.
+
+    The weight function is :math:`exp(-x)` and the interval of integration
+    is :math:`[0, \\inf]`. The Laguerre polynomials are orthogonal, but not
+    normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    w = np.exp(-x)
+    return w
+
+#
+# Laguerre series class
+#
+
+class Laguerre(ABCPolyBase):
+    """A Laguerre series class.
+
+    The Laguerre class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed in the `ABCPolyBase` documentation.
+
+    Parameters
+    ----------
+    coef : array_like
+        Laguerre coefficients in order of increasing degree, i.e,
+        ``(1, 2, 3)`` gives ``1*L_0(x) + 2*L_1(X) + 3*L_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [0, 1].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [0, 1].
+
+        .. versionadded:: 1.6.0
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(lagadd)
+    _sub = staticmethod(lagsub)
+    _mul = staticmethod(lagmul)
+    _div = staticmethod(lagdiv)
+    _pow = staticmethod(lagpow)
+    _val = staticmethod(lagval)
+    _int = staticmethod(lagint)
+    _der = staticmethod(lagder)
+    _fit = staticmethod(lagfit)
+    _line = staticmethod(lagline)
+    _roots = staticmethod(lagroots)
+    _fromroots = staticmethod(lagfromroots)
+
+    # Virtual properties
+    domain = np.array(lagdomain)
+    window = np.array(lagdomain)
+    basis_name = 'L'

lib/python3.12/site-packages/numpy/polynomial/laguerre.pyi

@@ -0,0 +1,46 @@
+from typing import Any
+
+from numpy import ndarray, dtype, int_
+from numpy.polynomial._polybase import ABCPolyBase
+from numpy.polynomial.polyutils import trimcoef
+
+__all__: list[str]
+
+lagtrim = trimcoef
+
+def poly2lag(pol): ...
+def lag2poly(c): ...
+
+lagdomain: ndarray[Any, dtype[int_]]
+lagzero: ndarray[Any, dtype[int_]]
+lagone: ndarray[Any, dtype[int_]]
+lagx: ndarray[Any, dtype[int_]]
+
+def lagline(off, scl): ...
+def lagfromroots(roots): ...
+def lagadd(c1, c2): ...
+def lagsub(c1, c2): ...
+def lagmulx(c): ...
+def lagmul(c1, c2): ...
+def lagdiv(c1, c2): ...
+def lagpow(c, pow, maxpower=...): ...
+def lagder(c, m=..., scl=..., axis=...): ...
+def lagint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ...
+def lagval(x, c, tensor=...): ...
+def lagval2d(x, y, c): ...
+def laggrid2d(x, y, c): ...
+def lagval3d(x, y, z, c): ...
+def laggrid3d(x, y, z, c): ...
+def lagvander(x, deg): ...
+def lagvander2d(x, y, deg): ...
+def lagvander3d(x, y, z, deg): ...
+def lagfit(x, y, deg, rcond=..., full=..., w=...): ...
+def lagcompanion(c): ...
+def lagroots(c): ...
+def laggauss(deg): ...
+def lagweight(x): ...
+
+class Laguerre(ABCPolyBase):
+    domain: Any
+    window: Any
+    basis_name: Any

lib/python3.12/site-packages/numpy/polynomial/legendre.py

@@ -0,0 +1,1664 @@
+"""
+==================================================
+Legendre Series (:mod:`numpy.polynomial.legendre`)
+==================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with Legendre series, including a `Legendre` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with such polynomials is in the
+docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+    Legendre
+
+Constants
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   legdomain
+   legzero
+   legone
+   legx
+
+Arithmetic
+----------
+
+.. autosummary::
+   :toctree: generated/
+
+   legadd
+   legsub
+   legmulx
+   legmul
+   legdiv
+   legpow
+   legval
+   legval2d
+   legval3d
+   leggrid2d
+   leggrid3d
+
+Calculus
+--------
+
+.. autosummary::
+   :toctree: generated/
+
+   legder
+   legint
+
+Misc Functions
+--------------
+
+.. autosummary::
+   :toctree: generated/
+
+   legfromroots
+   legroots
+   legvander
+   legvander2d
+   legvander3d
+   leggauss
+   legweight
+   legcompanion
+   legfit
+   legtrim
+   legline
+   leg2poly
+   poly2leg
+
+See also
+--------
+numpy.polynomial
+
+"""
+import numpy as np
+import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+__all__ = [
+    'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd',
+    'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder',
+    'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander',
+    'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d',
+    'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion',
+    'leggauss', 'legweight']
+
+legtrim = pu.trimcoef
+
+
+def poly2leg(pol):
+    """
+    Convert a polynomial to a Legendre series.
+
+    Convert an array representing the coefficients of a polynomial (relative
+    to the "standard" basis) ordered from lowest degree to highest, to an
+    array of the coefficients of the equivalent Legendre series, ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    pol : array_like
+        1-D array containing the polynomial coefficients
+
+    Returns
+    -------
+    c : ndarray
+        1-D array containing the coefficients of the equivalent Legendre
+        series.
+
+    See Also
+    --------
+    leg2poly
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy import polynomial as P
+    >>> p = P.Polynomial(np.arange(4))
+    >>> p
+    Polynomial([0.,  1.,  2.,  3.], domain=[-1,  1], window=[-1,  1])
+    >>> c = P.Legendre(P.legendre.poly2leg(p.coef))
+    >>> c
+    Legendre([ 1.  ,  3.25,  1.  ,  0.75], domain=[-1,  1], window=[-1,  1]) # may vary
+
+    """
+    [pol] = pu.as_series([pol])
+    deg = len(pol) - 1
+    res = 0
+    for i in range(deg, -1, -1):
+        res = legadd(legmulx(res), pol[i])
+    return res
+
+
+def leg2poly(c):
+    """
+    Convert a Legendre series to a polynomial.
+
+    Convert an array representing the coefficients of a Legendre series,
+    ordered from lowest degree to highest, to an array of the coefficients
+    of the equivalent polynomial (relative to the "standard" basis) ordered
+    from lowest to highest degree.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array containing the Legendre series coefficients, ordered
+        from lowest order term to highest.
+
+    Returns
+    -------
+    pol : ndarray
+        1-D array containing the coefficients of the equivalent polynomial
+        (relative to the "standard" basis) ordered from lowest order term
+        to highest.
+
+    See Also
+    --------
+    poly2leg
+
+    Notes
+    -----
+    The easy way to do conversions between polynomial basis sets
+    is to use the convert method of a class instance.
+
+    Examples
+    --------
+    >>> from numpy import polynomial as P
+    >>> c = P.Legendre(range(4))
+    >>> c
+    Legendre([0., 1., 2., 3.], domain=[-1,  1], window=[-1,  1])
+    >>> p = c.convert(kind=P.Polynomial)
+    >>> p
+    Polynomial([-1. , -3.5,  3. ,  7.5], domain=[-1.,  1.], window=[-1.,  1.])
+    >>> P.legendre.leg2poly(range(4))
+    array([-1. , -3.5,  3. ,  7.5])
+
+
+    """
+    from .polynomial import polyadd, polysub, polymulx
+
+    [c] = pu.as_series([c])
+    n = len(c)
+    if n < 3:
+        return c
+    else:
+        c0 = c[-2]
+        c1 = c[-1]
+        # i is the current degree of c1
+        for i in range(n - 1, 1, -1):
+            tmp = c0
+            c0 = polysub(c[i - 2], (c1*(i - 1))/i)
+            c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i)
+        return polyadd(c0, polymulx(c1))
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Legendre
+legdomain = np.array([-1, 1])
+
+# Legendre coefficients representing zero.
+legzero = np.array([0])
+
+# Legendre coefficients representing one.
+legone = np.array([1])
+
+# Legendre coefficients representing the identity x.
+legx = np.array([0, 1])
+
+
+def legline(off, scl):
+    """
+    Legendre series whose graph is a straight line.
+
+
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The specified line is given by ``off + scl*x``.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the Legendre series for
+        ``off + scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyline
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> import numpy.polynomial.legendre as L
+    >>> L.legline(3,2)
+    array([3, 2])
+    >>> L.legval(-3, L.legline(3,2)) # should be -3
+    -3.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def legfromroots(roots):
+    """
+    Generate a Legendre series with given roots.
+
+    The function returns the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    in Legendre form, where the `r_n` are the roots specified in `roots`.
+    If a zero has multiplicity n, then it must appear in `roots` n times.
+    For instance, if 2 is a root of multiplicity three and 3 is a root of
+    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
+    roots can appear in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * L_1(x) + ... +  c_n * L_n(x)
+
+    The coefficient of the last term is not generally 1 for monic
+    polynomials in Legendre form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of coefficients.  If all roots are real then `out` is a
+        real array, if some of the roots are complex, then `out` is complex
+        even if all the coefficients in the result are real (see Examples
+        below).
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfromroots
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Examples
+    --------
+    >>> import numpy.polynomial.legendre as L
+    >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis
+    array([ 0. , -0.4,  0. ,  0.4])
+    >>> j = complex(0,1)
+    >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis
+    array([ 1.33333333+0.j,  0.00000000+0.j,  0.66666667+0.j]) # may vary
+
+    """
+    return pu._fromroots(legline, legmul, roots)
+
+
+def legadd(c1, c2):
+    """
+    Add one Legendre series to another.
+
+    Returns the sum of two Legendre series `c1` + `c2`.  The arguments
+    are sequences of coefficients ordered from lowest order term to
+    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the Legendre series of their sum.
+
+    See Also
+    --------
+    legsub, legmulx, legmul, legdiv, legpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the sum of two Legendre series
+    is a Legendre series (without having to "reproject" the result onto
+    the basis set) so addition, just like that of "standard" polynomials,
+    is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> L.legadd(c1,c2)
+    array([4.,  4.,  4.])
+
+    """
+    return pu._add(c1, c2)
+
+
+def legsub(c1, c2):
+    """
+    Subtract one Legendre series from another.
+
+    Returns the difference of two Legendre series `c1` - `c2`.  The
+    sequences of coefficients are from lowest order term to highest, i.e.,
+    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Legendre series coefficients representing their difference.
+
+    See Also
+    --------
+    legadd, legmulx, legmul, legdiv, legpow
+
+    Notes
+    -----
+    Unlike multiplication, division, etc., the difference of two Legendre
+    series is a Legendre series (without having to "reproject" the result
+    onto the basis set) so subtraction, just like that of "standard"
+    polynomials, is simply "component-wise."
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> L.legsub(c1,c2)
+    array([-2.,  0.,  2.])
+    >>> L.legsub(c2,c1) # -C.legsub(c1,c2)
+    array([ 2.,  0., -2.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def legmulx(c):
+    """Multiply a Legendre series by x.
+
+    Multiply the Legendre series `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    legadd, legmul, legdiv, legpow
+
+    Notes
+    -----
+    The multiplication uses the recursion relationship for Legendre
+    polynomials in the form
+
+    .. math::
+
+      xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> L.legmulx([1,2,3])
+    array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0]*0
+    prd[1] = c[0]
+    for i in range(1, len(c)):
+        j = i + 1
+        k = i - 1
+        s = i + j
+        prd[j] = (c[i]*j)/s
+        prd[k] += (c[i]*i)/s
+    return prd
+
+
+def legmul(c1, c2):
+    """
+    Multiply one Legendre series by another.
+
+    Returns the product of two Legendre series `c1` * `c2`.  The arguments
+    are sequences of coefficients, from lowest order "term" to highest,
+    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of Legendre series coefficients representing their product.
+
+    See Also
+    --------
+    legadd, legsub, legmulx, legdiv, legpow
+
+    Notes
+    -----
+    In general, the (polynomial) product of two C-series results in terms
+    that are not in the Legendre polynomial basis set.  Thus, to express
+    the product as a Legendre series, it is necessary to "reproject" the
+    product onto said basis set, which may produce "unintuitive" (but
+    correct) results; see Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2)
+    >>> L.legmul(c1,c2) # multiplication requires "reprojection"
+    array([  4.33333333,  10.4       ,  11.66666667,   3.6       ]) # may vary
+
+    """
+    # s1, s2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+
+    if len(c1) > len(c2):
+        c = c2
+        xs = c1
+    else:
+        c = c1
+        xs = c2
+
+    if len(c) == 1:
+        c0 = c[0]*xs
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]*xs
+        c1 = c[1]*xs
+    else:
+        nd = len(c)
+        c0 = c[-2]*xs
+        c1 = c[-1]*xs
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd)
+            c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
+    return legadd(c0, legmulx(c1))
+
+
+def legdiv(c1, c2):
+    """
+    Divide one Legendre series by another.
+
+    Returns the quotient-with-remainder of two Legendre series
+    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
+    order "term" to highest, e.g., [1,2,3] represents the series
+    ``P_0 + 2*P_1 + 3*P_2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of Legendre series coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    quo, rem : ndarrays
+        Of Legendre series coefficients representing the quotient and
+        remainder.
+
+    See Also
+    --------
+    legadd, legsub, legmulx, legmul, legpow
+
+    Notes
+    -----
+    In general, the (polynomial) division of one Legendre series by another
+    results in quotient and remainder terms that are not in the Legendre
+    polynomial basis set.  Thus, to express these results as a Legendre
+    series, it is necessary to "reproject" the results onto the Legendre
+    basis set, which may produce "unintuitive" (but correct) results; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
+    (array([3.]), array([-8., -4.]))
+    >>> c2 = (0,1,2,3)
+    >>> L.legdiv(c2,c1) # neither "intuitive"
+    (array([-0.07407407,  1.66666667]), array([-1.03703704, -2.51851852])) # may vary
+
+    """
+    return pu._div(legmul, c1, c2)
+
+
+def legpow(c, pow, maxpower=16):
+    """Raise a Legendre series to a power.
+
+    Returns the Legendre series `c` raised to the power `pow`. The
+    argument `c` is a sequence of coefficients ordered from low to high.
+    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Legendre series coefficients ordered from low to
+        high.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Legendre series of power.
+
+    See Also
+    --------
+    legadd, legsub, legmulx, legmul, legdiv
+
+    """
+    return pu._pow(legmul, c, pow, maxpower)
+
+
+def legder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a Legendre series.
+
+    Returns the Legendre series coefficients `c` differentiated `m` times
+    along `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable). The argument
+    `c` is an array of coefficients from low to high degree along each
+    axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
+    while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
+    2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Legendre series coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change of
+        variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    der : ndarray
+        Legendre series of the derivative.
+
+    See Also
+    --------
+    legint
+
+    Notes
+    -----
+    In general, the result of differentiating a Legendre series does not
+    resemble the same operation on a power series. Thus the result of this
+    function may be "unintuitive," albeit correct; see Examples section
+    below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c = (1,2,3,4)
+    >>> L.legder(c)
+    array([  6.,   9.,  20.])
+    >>> L.legder(c, 3)
+    array([60.])
+    >>> L.legder(c, scl=-1)
+    array([ -6.,  -9., -20.])
+    >>> L.legder(c, 2,-1)
+    array([  9.,  60.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    cnt = pu._deprecate_as_int(m, "the order of derivation")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1]*0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
+            for j in range(n, 2, -1):
+                der[j - 1] = (2*j - 1)*c[j]
+                c[j - 2] += c[j]
+            if n > 1:
+                der[1] = 3*c[2]
+            der[0] = c[1]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a Legendre series.
+
+    Returns the Legendre series coefficients `c` integrated `m` times from
+    `lbnd` along `axis`. At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.)  The argument `c` is an array of
+    coefficients from low to high degree along each axis, e.g., [1,2,3]
+    represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
+    represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
+    2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of Legendre series coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree in
+        each axis given by the corresponding index.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at
+        ``lbnd`` is the first value in the list, the value of the second
+        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
+        default), all constants are set to zero.  If ``m == 1``, a single
+        scalar can be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    S : ndarray
+        Legendre series coefficient array of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    legder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.
+    Why is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`.  Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Also note that, in general, the result of integrating a C-series needs
+    to be "reprojected" onto the C-series basis set.  Thus, typically,
+    the result of this function is "unintuitive," albeit correct; see
+    Examples section below.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import legendre as L
+    >>> c = (1,2,3)
+    >>> L.legint(c)
+    array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
+    >>> L.legint(c, 3)
+    array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02, # may vary
+             -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])
+    >>> L.legint(c, k=3)
+     array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
+    >>> L.legint(c, lbnd=-2)
+    array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary
+    >>> L.legint(c, scl=2)
+    array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ]) # may vary
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._deprecate_as_int(m, "the order of integration")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    k = list(k) + [0]*(cnt - len(k))
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
+            tmp[0] = c[0]*0
+            tmp[1] = c[0]
+            if n > 1:
+                tmp[2] = c[1]/3
+            for j in range(2, n):
+                t = c[j]/(2*j + 1)
+                tmp[j + 1] = t
+                tmp[j - 1] -= t
+            tmp[0] += k[i] - legval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def legval(x, c, tensor=True):
+    """
+    Evaluate a Legendre series at points x.
+
+    If `c` is of length `n + 1`, this function returns the value:
+
+    .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    values : ndarray, algebra_like
+        The shape of the return value is described above.
+
+    See Also
+    --------
+    legval2d, leggrid2d, legval3d, leggrid3d
+
+    Notes
+    -----
+    The evaluation uses Clenshaw recursion, aka synthetic division.
+
+    """
+    c = np.array(c, ndmin=1, copy=False)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        c = c.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,)*x.ndim)
+
+    if len(c) == 1:
+        c0 = c[0]
+        c1 = 0
+    elif len(c) == 2:
+        c0 = c[0]
+        c1 = c[1]
+    else:
+        nd = len(c)
+        c0 = c[-2]
+        c1 = c[-1]
+        for i in range(3, len(c) + 1):
+            tmp = c0
+            nd = nd - 1
+            c0 = c[-i] - (c1*(nd - 1))/nd
+            c1 = tmp + (c1*x*(2*nd - 1))/nd
+    return c0 + c1*x
+
+
+def legval2d(x, y, c):
+    """
+    Evaluate a 2-D Legendre series at points (x, y).
+
+    This function returns the values:
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` is a 1-D array a one is implicitly appended to its shape to make
+    it 2-D. The shape of the result will be c.shape[2:] + x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points `(x, y)`,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and if it isn't an ndarray it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in ``c[i,j]``. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Legendre series at points formed
+        from pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    legval, leggrid2d, legval3d, leggrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(legval, c, x, y)
+
+
+def leggrid2d(x, y, c):
+    """
+    Evaluate a 2-D Legendre series on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
+
+    where the points `(a, b)` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional Chebyshev series at points in the
+        Cartesian product of `x` and `y`.
+
+    See Also
+    --------
+    legval, legval2d, legval3d, leggrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(legval, c, x, y)
+
+
+def legval3d(x, y, z, c):
+    """
+    Evaluate a 3-D Legendre series at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    legval, legval2d, leggrid2d, leggrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(legval, c, x, y, z)
+
+
+def leggrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
+
+    where the points `(a, b, c)` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    legval, legval2d, leggrid2d, legval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(legval, c, x, y, z)
+
+
+def legvander(x, deg):
+    """Pseudo-Vandermonde matrix of given degree.
+
+    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
+    `x`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = L_i(x)
+
+    where `0 <= i <= deg`. The leading indices of `V` index the elements of
+    `x` and the last index is the degree of the Legendre polynomial.
+
+    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
+    array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
+    ``legval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of Legendre series of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray
+        The pseudo-Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where The last index is the degree of the
+        corresponding Legendre polynomial.  The dtype will be the same as
+        the converted `x`.
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=False, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    # Use forward recursion to generate the entries. This is not as accurate
+    # as reverse recursion in this application but it is more efficient.
+    v[0] = x*0 + 1
+    if ideg > 0:
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
+    return np.moveaxis(v, 0, -1)
+
+
+def legvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y)`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
+
+    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
+    `V` index the points `(x, y)` and the last index encodes the degrees of
+    the Legendre polynomials.
+
+    If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D Legendre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    legvander, legvander3d, legval2d, legval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((legvander, legvander), (x, y), deg)
+
+
+def legvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
+
+    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
+    indices of `V` index the points `(x, y, z)` and the last index encodes
+    the degrees of the Legendre polynomials.
+
+    If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D Legendre
+    series of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    legvander, legvander3d, legval2d, legval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg)
+
+
+def legfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least squares fit of Legendre series to data.
+
+    Return the coefficients of a Legendre series of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (M,)
+        x-coordinates of the M sample points ``(x[i], y[i])``.
+    y : array_like, shape (M,) or (M, K)
+        y-coordinates of the sample points. Several data sets of sample
+        points sharing the same x-coordinates can be fitted at once by
+        passing in a 2D-array that contains one dataset per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit. Singular values smaller than
+        this relative to the largest singular value will be ignored. The
+        default value is len(x)*eps, where eps is the relative precision of
+        the float type, about 2e-16 in most cases.
+    full : bool, optional
+        Switch determining nature of return value. When it is False (the
+        default) just the coefficients are returned, when True diagnostic
+        information from the singular value decomposition is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+        .. versionadded:: 1.5.0
+
+    Returns
+    -------
+    coef : ndarray, shape (M,) or (M, K)
+        Legendre coefficients ordered from low to high. If `y` was
+        2-D, the coefficients for the data in column k of `y` are in
+        column `k`. If `deg` is specified as a list, coefficients for
+        terms not included in the fit are set equal to zero in the
+        returned `coef`.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Warns
+    -----
+    RankWarning
+        The rank of the coefficient matrix in the least-squares fit is
+        deficient. The warning is only raised if ``full == False``.  The
+        warnings can be turned off by
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyfit
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    legval : Evaluates a Legendre series.
+    legvander : Vandermonde matrix of Legendre series.
+    legweight : Legendre weight function (= 1).
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the Legendre series `p` that
+    minimizes the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where :math:`w_j` are the weights. This problem is solved by setting up
+    as the (typically) overdetermined matrix equation
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected, then a `RankWarning` will be issued. This means that the
+    coefficient values may be poorly determined. Using a lower order fit
+    will usually get rid of the warning.  The `rcond` parameter can also be
+    set to a value smaller than its default, but the resulting fit may be
+    spurious and have large contributions from roundoff error.
+
+    Fits using Legendre series are usually better conditioned than fits
+    using power series, but much can depend on the distribution of the
+    sample points and the smoothness of the data. If the quality of the fit
+    is inadequate splines may be a good alternative.
+
+    References
+    ----------
+    .. [1] Wikipedia, "Curve fitting",
+           https://en.wikipedia.org/wiki/Curve_fitting
+
+    Examples
+    --------
+
+    """
+    return pu._fit(legvander, x, y, deg, rcond, full, w)
+
+
+def legcompanion(c):
+    """Return the scaled companion matrix of c.
+
+    The basis polynomials are scaled so that the companion matrix is
+    symmetric when `c` is an Legendre basis polynomial. This provides
+    better eigenvalue estimates than the unscaled case and for basis
+    polynomials the eigenvalues are guaranteed to be real if
+    `numpy.linalg.eigvalsh` is used to obtain them.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of Legendre series coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Scaled companion matrix of dimensions (deg, deg).
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0]/c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    scl = 1./np.sqrt(2*np.arange(n) + 1)
+    top = mat.reshape(-1)[1::n+1]
+    bot = mat.reshape(-1)[n::n+1]
+    top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
+    bot[...] = top
+    mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
+    return mat
+
+
+def legroots(c):
+    """
+    Compute the roots of a Legendre series.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * L_i(x).
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the series. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.polynomial.polyroots
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the series for such values.
+    Roots with multiplicity greater than 1 will also show larger errors as
+    the value of the series near such points is relatively insensitive to
+    errors in the roots. Isolated roots near the origin can be improved by
+    a few iterations of Newton's method.
+
+    The Legendre series basis polynomials aren't powers of ``x`` so the
+    results of this function may seem unintuitive.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.legendre as leg
+    >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
+    array([-0.85099543, -0.11407192,  0.51506735]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0]/c[1]])
+
+    # rotated companion matrix reduces error
+    m = legcompanion(c)[::-1,::-1]
+    r = la.eigvals(m)
+    r.sort()
+    return r
+
+
+def leggauss(deg):
+    """
+    Gauss-Legendre quadrature.
+
+    Computes the sample points and weights for Gauss-Legendre quadrature.
+    These sample points and weights will correctly integrate polynomials of
+    degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
+    the weight function :math:`f(x) = 1`.
+
+    Parameters
+    ----------
+    deg : int
+        Number of sample points and weights. It must be >= 1.
+
+    Returns
+    -------
+    x : ndarray
+        1-D ndarray containing the sample points.
+    y : ndarray
+        1-D ndarray containing the weights.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    The results have only been tested up to degree 100, higher degrees may
+    be problematic. The weights are determined by using the fact that
+
+    .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
+
+    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
+    is the k'th root of :math:`L_n`, and then scaling the results to get
+    the right value when integrating 1.
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg <= 0:
+        raise ValueError("deg must be a positive integer")
+
+    # first approximation of roots. We use the fact that the companion
+    # matrix is symmetric in this case in order to obtain better zeros.
+    c = np.array([0]*deg + [1])
+    m = legcompanion(c)
+    x = la.eigvalsh(m)
+
+    # improve roots by one application of Newton
+    dy = legval(x, c)
+    df = legval(x, legder(c))
+    x -= dy/df
+
+    # compute the weights. We scale the factor to avoid possible numerical
+    # overflow.
+    fm = legval(x, c[1:])
+    fm /= np.abs(fm).max()
+    df /= np.abs(df).max()
+    w = 1/(fm * df)
+
+    # for Legendre we can also symmetrize
+    w = (w + w[::-1])/2
+    x = (x - x[::-1])/2
+
+    # scale w to get the right value
+    w *= 2. / w.sum()
+
+    return x, w
+
+
+def legweight(x):
+    """
+    Weight function of the Legendre polynomials.
+
+    The weight function is :math:`1` and the interval of integration is
+    :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not
+    normalized, with respect to this weight function.
+
+    Parameters
+    ----------
+    x : array_like
+       Values at which the weight function will be computed.
+
+    Returns
+    -------
+    w : ndarray
+       The weight function at `x`.
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    w = x*0.0 + 1.0
+    return w
+
+#
+# Legendre series class
+#
+
+class Legendre(ABCPolyBase):
+    """A Legendre series class.
+
+    The Legendre class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed in the `ABCPolyBase` documentation.
+
+    Parameters
+    ----------
+    coef : array_like
+        Legendre coefficients in order of increasing degree, i.e.,
+        ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1, 1].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1, 1].
+
+        .. versionadded:: 1.6.0
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(legadd)
+    _sub = staticmethod(legsub)
+    _mul = staticmethod(legmul)
+    _div = staticmethod(legdiv)
+    _pow = staticmethod(legpow)
+    _val = staticmethod(legval)
+    _int = staticmethod(legint)
+    _der = staticmethod(legder)
+    _fit = staticmethod(legfit)
+    _line = staticmethod(legline)
+    _roots = staticmethod(legroots)
+    _fromroots = staticmethod(legfromroots)
+
+    # Virtual properties
+    domain = np.array(legdomain)
+    window = np.array(legdomain)
+    basis_name = 'P'

lib/python3.12/site-packages/numpy/polynomial/legendre.pyi

@@ -0,0 +1,46 @@
+from typing import Any
+
+from numpy import ndarray, dtype, int_
+from numpy.polynomial._polybase import ABCPolyBase
+from numpy.polynomial.polyutils import trimcoef
+
+__all__: list[str]
+
+legtrim = trimcoef
+
+def poly2leg(pol): ...
+def leg2poly(c): ...
+
+legdomain: ndarray[Any, dtype[int_]]
+legzero: ndarray[Any, dtype[int_]]
+legone: ndarray[Any, dtype[int_]]
+legx: ndarray[Any, dtype[int_]]
+
+def legline(off, scl): ...
+def legfromroots(roots): ...
+def legadd(c1, c2): ...
+def legsub(c1, c2): ...
+def legmulx(c): ...
+def legmul(c1, c2): ...
+def legdiv(c1, c2): ...
+def legpow(c, pow, maxpower=...): ...
+def legder(c, m=..., scl=..., axis=...): ...
+def legint(c, m=..., k = ..., lbnd=..., scl=..., axis=...): ...
+def legval(x, c, tensor=...): ...
+def legval2d(x, y, c): ...
+def leggrid2d(x, y, c): ...
+def legval3d(x, y, z, c): ...
+def leggrid3d(x, y, z, c): ...
+def legvander(x, deg): ...
+def legvander2d(x, y, deg): ...
+def legvander3d(x, y, z, deg): ...
+def legfit(x, y, deg, rcond=..., full=..., w=...): ...
+def legcompanion(c): ...
+def legroots(c): ...
+def leggauss(deg): ...
+def legweight(x): ...
+
+class Legendre(ABCPolyBase):
+    domain: Any
+    window: Any
+    basis_name: Any

lib/python3.12/site-packages/numpy/polynomial/polynomial.py

@@ -0,0 +1,1542 @@
+"""
+=================================================
+Power Series (:mod:`numpy.polynomial.polynomial`)
+=================================================
+
+This module provides a number of objects (mostly functions) useful for
+dealing with polynomials, including a `Polynomial` class that
+encapsulates the usual arithmetic operations.  (General information
+on how this module represents and works with polynomial objects is in
+the docstring for its "parent" sub-package, `numpy.polynomial`).
+
+Classes
+-------
+.. autosummary::
+   :toctree: generated/
+
+   Polynomial
+
+Constants
+---------
+.. autosummary::
+   :toctree: generated/
+
+   polydomain
+   polyzero
+   polyone
+   polyx
+
+Arithmetic
+----------
+.. autosummary::
+   :toctree: generated/
+
+   polyadd
+   polysub
+   polymulx
+   polymul
+   polydiv
+   polypow
+   polyval
+   polyval2d
+   polyval3d
+   polygrid2d
+   polygrid3d
+
+Calculus
+--------
+.. autosummary::
+   :toctree: generated/
+
+   polyder
+   polyint
+
+Misc Functions
+--------------
+.. autosummary::
+   :toctree: generated/
+
+   polyfromroots
+   polyroots
+   polyvalfromroots
+   polyvander
+   polyvander2d
+   polyvander3d
+   polycompanion
+   polyfit
+   polytrim
+   polyline
+
+See Also
+--------
+`numpy.polynomial`
+
+"""
+__all__ = [
+    'polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd',
+    'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', 'polyval',
+    'polyvalfromroots', 'polyder', 'polyint', 'polyfromroots', 'polyvander',
+    'polyfit', 'polytrim', 'polyroots', 'Polynomial', 'polyval2d', 'polyval3d',
+    'polygrid2d', 'polygrid3d', 'polyvander2d', 'polyvander3d']
+
+import numpy as np
+import numpy.linalg as la
+from numpy.core.multiarray import normalize_axis_index
+
+from . import polyutils as pu
+from ._polybase import ABCPolyBase
+
+polytrim = pu.trimcoef
+
+#
+# These are constant arrays are of integer type so as to be compatible
+# with the widest range of other types, such as Decimal.
+#
+
+# Polynomial default domain.
+polydomain = np.array([-1, 1])
+
+# Polynomial coefficients representing zero.
+polyzero = np.array([0])
+
+# Polynomial coefficients representing one.
+polyone = np.array([1])
+
+# Polynomial coefficients representing the identity x.
+polyx = np.array([0, 1])
+
+#
+# Polynomial series functions
+#
+
+
+def polyline(off, scl):
+    """
+    Returns an array representing a linear polynomial.
+
+    Parameters
+    ----------
+    off, scl : scalars
+        The "y-intercept" and "slope" of the line, respectively.
+
+    Returns
+    -------
+    y : ndarray
+        This module's representation of the linear polynomial ``off +
+        scl*x``.
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebline
+    numpy.polynomial.legendre.legline
+    numpy.polynomial.laguerre.lagline
+    numpy.polynomial.hermite.hermline
+    numpy.polynomial.hermite_e.hermeline
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> P.polyline(1,-1)
+    array([ 1, -1])
+    >>> P.polyval(1, P.polyline(1,-1)) # should be 0
+    0.0
+
+    """
+    if scl != 0:
+        return np.array([off, scl])
+    else:
+        return np.array([off])
+
+
+def polyfromroots(roots):
+    """
+    Generate a monic polynomial with given roots.
+
+    Return the coefficients of the polynomial
+
+    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
+
+    where the ``r_n`` are the roots specified in `roots`.  If a zero has
+    multiplicity n, then it must appear in `roots` n times. For instance,
+    if 2 is a root of multiplicity three and 3 is a root of multiplicity 2,
+    then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear
+    in any order.
+
+    If the returned coefficients are `c`, then
+
+    .. math:: p(x) = c_0 + c_1 * x + ... +  x^n
+
+    The coefficient of the last term is 1 for monic polynomials in this
+    form.
+
+    Parameters
+    ----------
+    roots : array_like
+        Sequence containing the roots.
+
+    Returns
+    -------
+    out : ndarray
+        1-D array of the polynomial's coefficients If all the roots are
+        real, then `out` is also real, otherwise it is complex.  (see
+        Examples below).
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfromroots
+    numpy.polynomial.legendre.legfromroots
+    numpy.polynomial.laguerre.lagfromroots
+    numpy.polynomial.hermite.hermfromroots
+    numpy.polynomial.hermite_e.hermefromroots
+
+    Notes
+    -----
+    The coefficients are determined by multiplying together linear factors
+    of the form ``(x - r_i)``, i.e.
+
+    .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n)
+
+    where ``n == len(roots) - 1``; note that this implies that ``1`` is always
+    returned for :math:`a_n`.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
+    array([ 0., -1.,  0.,  1.])
+    >>> j = complex(0,1)
+    >>> P.polyfromroots((-j,j)) # complex returned, though values are real
+    array([1.+0.j,  0.+0.j,  1.+0.j])
+
+    """
+    return pu._fromroots(polyline, polymul, roots)
+
+
+def polyadd(c1, c2):
+    """
+    Add one polynomial to another.
+
+    Returns the sum of two polynomials `c1` + `c2`.  The arguments are
+    sequences of coefficients from lowest order term to highest, i.e.,
+    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of polynomial coefficients ordered from low to high.
+
+    Returns
+    -------
+    out : ndarray
+        The coefficient array representing their sum.
+
+    See Also
+    --------
+    polysub, polymulx, polymul, polydiv, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> sum = P.polyadd(c1,c2); sum
+    array([4.,  4.,  4.])
+    >>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
+    28.0
+
+    """
+    return pu._add(c1, c2)
+
+
+def polysub(c1, c2):
+    """
+    Subtract one polynomial from another.
+
+    Returns the difference of two polynomials `c1` - `c2`.  The arguments
+    are sequences of coefficients from lowest order term to highest, i.e.,
+    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of polynomial coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Of coefficients representing their difference.
+
+    See Also
+    --------
+    polyadd, polymulx, polymul, polydiv, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> P.polysub(c1,c2)
+    array([-2.,  0.,  2.])
+    >>> P.polysub(c2,c1) # -P.polysub(c1,c2)
+    array([ 2.,  0., -2.])
+
+    """
+    return pu._sub(c1, c2)
+
+
+def polymulx(c):
+    """Multiply a polynomial by x.
+
+    Multiply the polynomial `c` by x, where x is the independent
+    variable.
+
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of polynomial coefficients ordered from low to
+        high.
+
+    Returns
+    -------
+    out : ndarray
+        Array representing the result of the multiplication.
+
+    See Also
+    --------
+    polyadd, polysub, polymul, polydiv, polypow
+
+    Notes
+    -----
+
+    .. versionadded:: 1.5.0
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    # The zero series needs special treatment
+    if len(c) == 1 and c[0] == 0:
+        return c
+
+    prd = np.empty(len(c) + 1, dtype=c.dtype)
+    prd[0] = c[0]*0
+    prd[1:] = c
+    return prd
+
+
+def polymul(c1, c2):
+    """
+    Multiply one polynomial by another.
+
+    Returns the product of two polynomials `c1` * `c2`.  The arguments are
+    sequences of coefficients, from lowest order term to highest, e.g.,
+    [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of coefficients representing a polynomial, relative to the
+        "standard" basis, and ordered from lowest order term to highest.
+
+    Returns
+    -------
+    out : ndarray
+        Of the coefficients of their product.
+
+    See Also
+    --------
+    polyadd, polysub, polymulx, polydiv, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> P.polymul(c1,c2)
+    array([  3.,   8.,  14.,   8.,   3.])
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    ret = np.convolve(c1, c2)
+    return pu.trimseq(ret)
+
+
+def polydiv(c1, c2):
+    """
+    Divide one polynomial by another.
+
+    Returns the quotient-with-remainder of two polynomials `c1` / `c2`.
+    The arguments are sequences of coefficients, from lowest order term
+    to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.
+
+    Parameters
+    ----------
+    c1, c2 : array_like
+        1-D arrays of polynomial coefficients ordered from low to high.
+
+    Returns
+    -------
+    [quo, rem] : ndarrays
+        Of coefficient series representing the quotient and remainder.
+
+    See Also
+    --------
+    polyadd, polysub, polymulx, polymul, polypow
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c1 = (1,2,3)
+    >>> c2 = (3,2,1)
+    >>> P.polydiv(c1,c2)
+    (array([3.]), array([-8., -4.]))
+    >>> P.polydiv(c2,c1)
+    (array([ 0.33333333]), array([ 2.66666667,  1.33333333])) # may vary
+
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = pu.as_series([c1, c2])
+    if c2[-1] == 0:
+        raise ZeroDivisionError()
+
+    # note: this is more efficient than `pu._div(polymul, c1, c2)`
+    lc1 = len(c1)
+    lc2 = len(c2)
+    if lc1 < lc2:
+        return c1[:1]*0, c1
+    elif lc2 == 1:
+        return c1/c2[-1], c1[:1]*0
+    else:
+        dlen = lc1 - lc2
+        scl = c2[-1]
+        c2 = c2[:-1]/scl
+        i = dlen
+        j = lc1 - 1
+        while i >= 0:
+            c1[i:j] -= c2*c1[j]
+            i -= 1
+            j -= 1
+        return c1[j+1:]/scl, pu.trimseq(c1[:j+1])
+
+
+def polypow(c, pow, maxpower=None):
+    """Raise a polynomial to a power.
+
+    Returns the polynomial `c` raised to the power `pow`. The argument
+    `c` is a sequence of coefficients ordered from low to high. i.e.,
+    [1,2,3] is the series  ``1 + 2*x + 3*x**2.``
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of array of series coefficients ordered from low to
+        high degree.
+    pow : integer
+        Power to which the series will be raised
+    maxpower : integer, optional
+        Maximum power allowed. This is mainly to limit growth of the series
+        to unmanageable size. Default is 16
+
+    Returns
+    -------
+    coef : ndarray
+        Power series of power.
+
+    See Also
+    --------
+    polyadd, polysub, polymulx, polymul, polydiv
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> P.polypow([1,2,3], 2)
+    array([ 1., 4., 10., 12., 9.])
+
+    """
+    # note: this is more efficient than `pu._pow(polymul, c1, c2)`, as it
+    # avoids calling `as_series` repeatedly
+    return pu._pow(np.convolve, c, pow, maxpower)
+
+
+def polyder(c, m=1, scl=1, axis=0):
+    """
+    Differentiate a polynomial.
+
+    Returns the polynomial coefficients `c` differentiated `m` times along
+    `axis`.  At each iteration the result is multiplied by `scl` (the
+    scaling factor is for use in a linear change of variable).  The
+    argument `c` is an array of coefficients from low to high degree along
+    each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``
+    while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is
+    ``x`` and axis=1 is ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        Array of polynomial coefficients. If c is multidimensional the
+        different axis correspond to different variables with the degree
+        in each axis given by the corresponding index.
+    m : int, optional
+        Number of derivatives taken, must be non-negative. (Default: 1)
+    scl : scalar, optional
+        Each differentiation is multiplied by `scl`.  The end result is
+        multiplication by ``scl**m``.  This is for use in a linear change
+        of variable. (Default: 1)
+    axis : int, optional
+        Axis over which the derivative is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    der : ndarray
+        Polynomial coefficients of the derivative.
+
+    See Also
+    --------
+    polyint
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3
+    >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2
+    array([  2.,   6.,  12.])
+    >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24
+    array([24.])
+    >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2
+    array([ -2.,  -6., -12.])
+    >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x
+    array([  6.,  24.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        # astype fails with NA
+        c = c + 0.0
+    cdt = c.dtype
+    cnt = pu._deprecate_as_int(m, "the order of derivation")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of derivation must be non-negative")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    c = np.moveaxis(c, iaxis, 0)
+    n = len(c)
+    if cnt >= n:
+        c = c[:1]*0
+    else:
+        for i in range(cnt):
+            n = n - 1
+            c *= scl
+            der = np.empty((n,) + c.shape[1:], dtype=cdt)
+            for j in range(n, 0, -1):
+                der[j - 1] = j*c[j]
+            c = der
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
+    """
+    Integrate a polynomial.
+
+    Returns the polynomial coefficients `c` integrated `m` times from
+    `lbnd` along `axis`.  At each iteration the resulting series is
+    **multiplied** by `scl` and an integration constant, `k`, is added.
+    The scaling factor is for use in a linear change of variable.  ("Buyer
+    beware": note that, depending on what one is doing, one may want `scl`
+    to be the reciprocal of what one might expect; for more information,
+    see the Notes section below.) The argument `c` is an array of
+    coefficients, from low to high degree along each axis, e.g., [1,2,3]
+    represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]]
+    represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is
+    ``y``.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of polynomial coefficients, ordered from low to high.
+    m : int, optional
+        Order of integration, must be positive. (Default: 1)
+    k : {[], list, scalar}, optional
+        Integration constant(s).  The value of the first integral at zero
+        is the first value in the list, the value of the second integral
+        at zero is the second value, etc.  If ``k == []`` (the default),
+        all constants are set to zero.  If ``m == 1``, a single scalar can
+        be given instead of a list.
+    lbnd : scalar, optional
+        The lower bound of the integral. (Default: 0)
+    scl : scalar, optional
+        Following each integration the result is *multiplied* by `scl`
+        before the integration constant is added. (Default: 1)
+    axis : int, optional
+        Axis over which the integral is taken. (Default: 0).
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    S : ndarray
+        Coefficient array of the integral.
+
+    Raises
+    ------
+    ValueError
+        If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
+        ``np.ndim(scl) != 0``.
+
+    See Also
+    --------
+    polyder
+
+    Notes
+    -----
+    Note that the result of each integration is *multiplied* by `scl`.  Why
+    is this important to note?  Say one is making a linear change of
+    variable :math:`u = ax + b` in an integral relative to `x`. Then
+    :math:`dx = du/a`, so one will need to set `scl` equal to
+    :math:`1/a` - perhaps not what one would have first thought.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polynomial as P
+    >>> c = (1,2,3)
+    >>> P.polyint(c) # should return array([0, 1, 1, 1])
+    array([0.,  1.,  1.,  1.])
+    >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20])
+     array([ 0.        ,  0.        ,  0.        ,  0.16666667,  0.08333333, # may vary
+             0.05      ])
+    >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1])
+    array([3.,  1.,  1.,  1.])
+    >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1])
+    array([6.,  1.,  1.,  1.])
+    >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2])
+    array([ 0., -2., -2., -2.])
+
+    """
+    c = np.array(c, ndmin=1, copy=True)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        # astype doesn't preserve mask attribute.
+        c = c + 0.0
+    cdt = c.dtype
+    if not np.iterable(k):
+        k = [k]
+    cnt = pu._deprecate_as_int(m, "the order of integration")
+    iaxis = pu._deprecate_as_int(axis, "the axis")
+    if cnt < 0:
+        raise ValueError("The order of integration must be non-negative")
+    if len(k) > cnt:
+        raise ValueError("Too many integration constants")
+    if np.ndim(lbnd) != 0:
+        raise ValueError("lbnd must be a scalar.")
+    if np.ndim(scl) != 0:
+        raise ValueError("scl must be a scalar.")
+    iaxis = normalize_axis_index(iaxis, c.ndim)
+
+    if cnt == 0:
+        return c
+
+    k = list(k) + [0]*(cnt - len(k))
+    c = np.moveaxis(c, iaxis, 0)
+    for i in range(cnt):
+        n = len(c)
+        c *= scl
+        if n == 1 and np.all(c[0] == 0):
+            c[0] += k[i]
+        else:
+            tmp = np.empty((n + 1,) + c.shape[1:], dtype=cdt)
+            tmp[0] = c[0]*0
+            tmp[1] = c[0]
+            for j in range(1, n):
+                tmp[j + 1] = c[j]/(j + 1)
+            tmp[0] += k[i] - polyval(lbnd, tmp)
+            c = tmp
+    c = np.moveaxis(c, 0, iaxis)
+    return c
+
+
+def polyval(x, c, tensor=True):
+    """
+    Evaluate a polynomial at points x.
+
+    If `c` is of length `n + 1`, this function returns the value
+
+    .. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `c`.
+
+    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If
+    `c` is multidimensional, then the shape of the result depends on the
+    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
+    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
+    scalars have shape (,).
+
+    Trailing zeros in the coefficients will be used in the evaluation, so
+    they should be avoided if efficiency is a concern.
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        with themselves and with the elements of `c`.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree n are contained in c[n]. If `c` is multidimensional the
+        remaining indices enumerate multiple polynomials. In the two
+        dimensional case the coefficients may be thought of as stored in
+        the columns of `c`.
+    tensor : boolean, optional
+        If True, the shape of the coefficient array is extended with ones
+        on the right, one for each dimension of `x`. Scalars have dimension 0
+        for this action. The result is that every column of coefficients in
+        `c` is evaluated for every element of `x`. If False, `x` is broadcast
+        over the columns of `c` for the evaluation.  This keyword is useful
+        when `c` is multidimensional. The default value is True.
+
+        .. versionadded:: 1.7.0
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The shape of the returned array is described above.
+
+    See Also
+    --------
+    polyval2d, polygrid2d, polyval3d, polygrid3d
+
+    Notes
+    -----
+    The evaluation uses Horner's method.
+
+    Examples
+    --------
+    >>> from numpy.polynomial.polynomial import polyval
+    >>> polyval(1, [1,2,3])
+    6.0
+    >>> a = np.arange(4).reshape(2,2)
+    >>> a
+    array([[0, 1],
+           [2, 3]])
+    >>> polyval(a, [1,2,3])
+    array([[ 1.,   6.],
+           [17.,  34.]])
+    >>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
+    >>> coef
+    array([[0, 1],
+           [2, 3]])
+    >>> polyval([1,2], coef, tensor=True)
+    array([[2.,  4.],
+           [4.,  7.]])
+    >>> polyval([1,2], coef, tensor=False)
+    array([2.,  7.])
+
+    """
+    c = np.array(c, ndmin=1, copy=False)
+    if c.dtype.char in '?bBhHiIlLqQpP':
+        # astype fails with NA
+        c = c + 0.0
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray) and tensor:
+        c = c.reshape(c.shape + (1,)*x.ndim)
+
+    c0 = c[-1] + x*0
+    for i in range(2, len(c) + 1):
+        c0 = c[-i] + c0*x
+    return c0
+
+
+def polyvalfromroots(x, r, tensor=True):
+    """
+    Evaluate a polynomial specified by its roots at points x.
+
+    If `r` is of length `N`, this function returns the value
+
+    .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n)
+
+    The parameter `x` is converted to an array only if it is a tuple or a
+    list, otherwise it is treated as a scalar. In either case, either `x`
+    or its elements must support multiplication and addition both with
+    themselves and with the elements of `r`.
+
+    If `r` is a 1-D array, then `p(x)` will have the same shape as `x`.  If `r`
+    is multidimensional, then the shape of the result depends on the value of
+    `tensor`. If `tensor` is ``True`` the shape will be r.shape[1:] + x.shape;
+    that is, each polynomial is evaluated at every value of `x`. If `tensor` is
+    ``False``, the shape will be r.shape[1:]; that is, each polynomial is
+    evaluated only for the corresponding broadcast value of `x`. Note that
+    scalars have shape (,).
+
+    .. versionadded:: 1.12
+
+    Parameters
+    ----------
+    x : array_like, compatible object
+        If `x` is a list or tuple, it is converted to an ndarray, otherwise
+        it is left unchanged and treated as a scalar. In either case, `x`
+        or its elements must support addition and multiplication with
+        with themselves and with the elements of `r`.
+    r : array_like
+        Array of roots. If `r` is multidimensional the first index is the
+        root index, while the remaining indices enumerate multiple
+        polynomials. For instance, in the two dimensional case the roots
+        of each polynomial may be thought of as stored in the columns of `r`.
+    tensor : boolean, optional
+        If True, the shape of the roots array is extended with ones on the
+        right, one for each dimension of `x`. Scalars have dimension 0 for this
+        action. The result is that every column of coefficients in `r` is
+        evaluated for every element of `x`. If False, `x` is broadcast over the
+        columns of `r` for the evaluation.  This keyword is useful when `r` is
+        multidimensional. The default value is True.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The shape of the returned array is described above.
+
+    See Also
+    --------
+    polyroots, polyfromroots, polyval
+
+    Examples
+    --------
+    >>> from numpy.polynomial.polynomial import polyvalfromroots
+    >>> polyvalfromroots(1, [1,2,3])
+    0.0
+    >>> a = np.arange(4).reshape(2,2)
+    >>> a
+    array([[0, 1],
+           [2, 3]])
+    >>> polyvalfromroots(a, [-1, 0, 1])
+    array([[-0.,   0.],
+           [ 6.,  24.]])
+    >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients
+    >>> r # each column of r defines one polynomial
+    array([[-2, -1],
+           [ 0,  1]])
+    >>> b = [-2, 1]
+    >>> polyvalfromroots(b, r, tensor=True)
+    array([[-0.,  3.],
+           [ 3., 0.]])
+    >>> polyvalfromroots(b, r, tensor=False)
+    array([-0.,  0.])
+    """
+    r = np.array(r, ndmin=1, copy=False)
+    if r.dtype.char in '?bBhHiIlLqQpP':
+        r = r.astype(np.double)
+    if isinstance(x, (tuple, list)):
+        x = np.asarray(x)
+    if isinstance(x, np.ndarray):
+        if tensor:
+            r = r.reshape(r.shape + (1,)*x.ndim)
+        elif x.ndim >= r.ndim:
+            raise ValueError("x.ndim must be < r.ndim when tensor == False")
+    return np.prod(x - r, axis=0)
+
+
+def polyval2d(x, y, c):
+    """
+    Evaluate a 2-D polynomial at points (x, y).
+
+    This function returns the value
+
+    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars and they
+    must have the same shape after conversion. In either case, either `x`
+    and `y` or their elements must support multiplication and addition both
+    with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points `(x, y)`,
+        where `x` and `y` must have the same shape. If `x` or `y` is a list
+        or tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term
+        of multi-degree i,j is contained in `c[i,j]`. If `c` has
+        dimension greater than two the remaining indices enumerate multiple
+        sets of coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points formed with
+        pairs of corresponding values from `x` and `y`.
+
+    See Also
+    --------
+    polyval, polygrid2d, polyval3d, polygrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(polyval, c, x, y)
+
+
+def polygrid2d(x, y, c):
+    """
+    Evaluate a 2-D polynomial on the Cartesian product of x and y.
+
+    This function returns the values:
+
+    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j
+
+    where the points `(a, b)` consist of all pairs formed by taking
+    `a` from `x` and `b` from `y`. The resulting points form a grid with
+    `x` in the first dimension and `y` in the second.
+
+    The parameters `x` and `y` are converted to arrays only if they are
+    tuples or a lists, otherwise they are treated as a scalars. In either
+    case, either `x` and `y` or their elements must support multiplication
+    and addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than two dimensions, ones are implicitly appended to
+    its shape to make it 2-D. The shape of the result will be c.shape[2:] +
+    x.shape + y.shape.
+
+    Parameters
+    ----------
+    x, y : array_like, compatible objects
+        The two dimensional series is evaluated at the points in the
+        Cartesian product of `x` and `y`.  If `x` or `y` is a list or
+        tuple, it is first converted to an ndarray, otherwise it is left
+        unchanged and, if it isn't an ndarray, it is treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    polyval, polyval2d, polyval3d, polygrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(polyval, c, x, y)
+
+
+def polyval3d(x, y, z, c):
+    """
+    Evaluate a 3-D polynomial at points (x, y, z).
+
+    This function returns the values:
+
+    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if
+    they are tuples or a lists, otherwise they are treated as a scalars and
+    they must have the same shape after conversion. In either case, either
+    `x`, `y`, and `z` or their elements must support multiplication and
+    addition both with themselves and with the elements of `c`.
+
+    If `c` has fewer than 3 dimensions, ones are implicitly appended to its
+    shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible object
+        The three dimensional series is evaluated at the points
+        `(x, y, z)`, where `x`, `y`, and `z` must have the same shape.  If
+        any of `x`, `y`, or `z` is a list or tuple, it is first converted
+        to an ndarray, otherwise it is left unchanged and if it isn't an
+        ndarray it is  treated as a scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficient of the term of
+        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
+        greater than 3 the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the multidimensional polynomial on points formed with
+        triples of corresponding values from `x`, `y`, and `z`.
+
+    See Also
+    --------
+    polyval, polyval2d, polygrid2d, polygrid3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._valnd(polyval, c, x, y, z)
+
+
+def polygrid3d(x, y, z, c):
+    """
+    Evaluate a 3-D polynomial on the Cartesian product of x, y and z.
+
+    This function returns the values:
+
+    .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k
+
+    where the points `(a, b, c)` consist of all triples formed by taking
+    `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
+    a grid with `x` in the first dimension, `y` in the second, and `z` in
+    the third.
+
+    The parameters `x`, `y`, and `z` are converted to arrays only if they
+    are tuples or a lists, otherwise they are treated as a scalars. In
+    either case, either `x`, `y`, and `z` or their elements must support
+    multiplication and addition both with themselves and with the elements
+    of `c`.
+
+    If `c` has fewer than three dimensions, ones are implicitly appended to
+    its shape to make it 3-D. The shape of the result will be c.shape[3:] +
+    x.shape + y.shape + z.shape.
+
+    Parameters
+    ----------
+    x, y, z : array_like, compatible objects
+        The three dimensional series is evaluated at the points in the
+        Cartesian product of `x`, `y`, and `z`.  If `x`,`y`, or `z` is a
+        list or tuple, it is first converted to an ndarray, otherwise it is
+        left unchanged and, if it isn't an ndarray, it is treated as a
+        scalar.
+    c : array_like
+        Array of coefficients ordered so that the coefficients for terms of
+        degree i,j are contained in ``c[i,j]``. If `c` has dimension
+        greater than two the remaining indices enumerate multiple sets of
+        coefficients.
+
+    Returns
+    -------
+    values : ndarray, compatible object
+        The values of the two dimensional polynomial at points in the Cartesian
+        product of `x` and `y`.
+
+    See Also
+    --------
+    polyval, polyval2d, polygrid2d, polyval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._gridnd(polyval, c, x, y, z)
+
+
+def polyvander(x, deg):
+    """Vandermonde matrix of given degree.
+
+    Returns the Vandermonde matrix of degree `deg` and sample points
+    `x`. The Vandermonde matrix is defined by
+
+    .. math:: V[..., i] = x^i,
+
+    where `0 <= i <= deg`. The leading indices of `V` index the elements of
+    `x` and the last index is the power of `x`.
+
+    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
+    matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and
+    ``polyval(x, c)`` are the same up to roundoff. This equivalence is
+    useful both for least squares fitting and for the evaluation of a large
+    number of polynomials of the same degree and sample points.
+
+    Parameters
+    ----------
+    x : array_like
+        Array of points. The dtype is converted to float64 or complex128
+        depending on whether any of the elements are complex. If `x` is
+        scalar it is converted to a 1-D array.
+    deg : int
+        Degree of the resulting matrix.
+
+    Returns
+    -------
+    vander : ndarray.
+        The Vandermonde matrix. The shape of the returned matrix is
+        ``x.shape + (deg + 1,)``, where the last index is the power of `x`.
+        The dtype will be the same as the converted `x`.
+
+    See Also
+    --------
+    polyvander2d, polyvander3d
+
+    """
+    ideg = pu._deprecate_as_int(deg, "deg")
+    if ideg < 0:
+        raise ValueError("deg must be non-negative")
+
+    x = np.array(x, copy=False, ndmin=1) + 0.0
+    dims = (ideg + 1,) + x.shape
+    dtyp = x.dtype
+    v = np.empty(dims, dtype=dtyp)
+    v[0] = x*0 + 1
+    if ideg > 0:
+        v[1] = x
+        for i in range(2, ideg + 1):
+            v[i] = v[i-1]*x
+    return np.moveaxis(v, 0, -1)
+
+
+def polyvander2d(x, y, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y)`. The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j,
+
+    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
+    `V` index the points `(x, y)` and the last index encodes the powers of
+    `x` and `y`.
+
+    If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
+    correspond to the elements of a 2-D coefficient array `c` of shape
+    (xdeg + 1, ydeg + 1) in the order
+
+    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
+
+    and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same
+    up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 2-D polynomials
+    of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg].
+
+    Returns
+    -------
+    vander2d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg([1]+1)`.  The dtype will be the same
+        as the converted `x` and `y`.
+
+    See Also
+    --------
+    polyvander, polyvander3d, polyval2d, polyval3d
+
+    """
+    return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg)
+
+
+def polyvander3d(x, y, z, deg):
+    """Pseudo-Vandermonde matrix of given degrees.
+
+    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
+    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
+    then The pseudo-Vandermonde matrix is defined by
+
+    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k,
+
+    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading
+    indices of `V` index the points `(x, y, z)` and the last index encodes
+    the powers of `x`, `y`, and `z`.
+
+    If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
+    of `V` correspond to the elements of a 3-D coefficient array `c` of
+    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
+
+    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
+
+    and  ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the
+    same up to roundoff. This equivalence is useful both for least squares
+    fitting and for the evaluation of a large number of 3-D polynomials
+    of the same degrees and sample points.
+
+    Parameters
+    ----------
+    x, y, z : array_like
+        Arrays of point coordinates, all of the same shape. The dtypes will
+        be converted to either float64 or complex128 depending on whether
+        any of the elements are complex. Scalars are converted to 1-D
+        arrays.
+    deg : list of ints
+        List of maximum degrees of the form [x_deg, y_deg, z_deg].
+
+    Returns
+    -------
+    vander3d : ndarray
+        The shape of the returned matrix is ``x.shape + (order,)``, where
+        :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`.  The dtype will
+        be the same as the converted `x`, `y`, and `z`.
+
+    See Also
+    --------
+    polyvander, polyvander3d, polyval2d, polyval3d
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg)
+
+
+def polyfit(x, y, deg, rcond=None, full=False, w=None):
+    """
+    Least-squares fit of a polynomial to data.
+
+    Return the coefficients of a polynomial of degree `deg` that is the
+    least squares fit to the data values `y` given at points `x`. If `y` is
+    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
+    fits are done, one for each column of `y`, and the resulting
+    coefficients are stored in the corresponding columns of a 2-D return.
+    The fitted polynomial(s) are in the form
+
+    .. math::  p(x) = c_0 + c_1 * x + ... + c_n * x^n,
+
+    where `n` is `deg`.
+
+    Parameters
+    ----------
+    x : array_like, shape (`M`,)
+        x-coordinates of the `M` sample (data) points ``(x[i], y[i])``.
+    y : array_like, shape (`M`,) or (`M`, `K`)
+        y-coordinates of the sample points.  Several sets of sample points
+        sharing the same x-coordinates can be (independently) fit with one
+        call to `polyfit` by passing in for `y` a 2-D array that contains
+        one data set per column.
+    deg : int or 1-D array_like
+        Degree(s) of the fitting polynomials. If `deg` is a single integer
+        all terms up to and including the `deg`'th term are included in the
+        fit. For NumPy versions >= 1.11.0 a list of integers specifying the
+        degrees of the terms to include may be used instead.
+    rcond : float, optional
+        Relative condition number of the fit.  Singular values smaller
+        than `rcond`, relative to the largest singular value, will be
+        ignored.  The default value is ``len(x)*eps``, where `eps` is the
+        relative precision of the platform's float type, about 2e-16 in
+        most cases.
+    full : bool, optional
+        Switch determining the nature of the return value.  When ``False``
+        (the default) just the coefficients are returned; when ``True``,
+        diagnostic information from the singular value decomposition (used
+        to solve the fit's matrix equation) is also returned.
+    w : array_like, shape (`M`,), optional
+        Weights. If not None, the weight ``w[i]`` applies to the unsquared
+        residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
+        chosen so that the errors of the products ``w[i]*y[i]`` all have the
+        same variance.  When using inverse-variance weighting, use
+        ``w[i] = 1/sigma(y[i])``.  The default value is None.
+
+        .. versionadded:: 1.5.0
+
+    Returns
+    -------
+    coef : ndarray, shape (`deg` + 1,) or (`deg` + 1, `K`)
+        Polynomial coefficients ordered from low to high.  If `y` was 2-D,
+        the coefficients in column `k` of `coef` represent the polynomial
+        fit to the data in `y`'s `k`-th column.
+
+    [residuals, rank, singular_values, rcond] : list
+        These values are only returned if ``full == True``
+
+        - residuals -- sum of squared residuals of the least squares fit
+        - rank -- the numerical rank of the scaled Vandermonde matrix
+        - singular_values -- singular values of the scaled Vandermonde matrix
+        - rcond -- value of `rcond`.
+
+        For more details, see `numpy.linalg.lstsq`.
+
+    Raises
+    ------
+    RankWarning
+        Raised if the matrix in the least-squares fit is rank deficient.
+        The warning is only raised if ``full == False``.  The warnings can
+        be turned off by:
+
+        >>> import warnings
+        >>> warnings.simplefilter('ignore', np.RankWarning)
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebfit
+    numpy.polynomial.legendre.legfit
+    numpy.polynomial.laguerre.lagfit
+    numpy.polynomial.hermite.hermfit
+    numpy.polynomial.hermite_e.hermefit
+    polyval : Evaluates a polynomial.
+    polyvander : Vandermonde matrix for powers.
+    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
+    scipy.interpolate.UnivariateSpline : Computes spline fits.
+
+    Notes
+    -----
+    The solution is the coefficients of the polynomial `p` that minimizes
+    the sum of the weighted squared errors
+
+    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
+
+    where the :math:`w_j` are the weights. This problem is solved by
+    setting up the (typically) over-determined matrix equation:
+
+    .. math:: V(x) * c = w * y,
+
+    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
+    coefficients to be solved for, `w` are the weights, and `y` are the
+    observed values.  This equation is then solved using the singular value
+    decomposition of `V`.
+
+    If some of the singular values of `V` are so small that they are
+    neglected (and `full` == ``False``), a `RankWarning` will be raised.
+    This means that the coefficient values may be poorly determined.
+    Fitting to a lower order polynomial will usually get rid of the warning
+    (but may not be what you want, of course; if you have independent
+    reason(s) for choosing the degree which isn't working, you may have to:
+    a) reconsider those reasons, and/or b) reconsider the quality of your
+    data).  The `rcond` parameter can also be set to a value smaller than
+    its default, but the resulting fit may be spurious and have large
+    contributions from roundoff error.
+
+    Polynomial fits using double precision tend to "fail" at about
+    (polynomial) degree 20. Fits using Chebyshev or Legendre series are
+    generally better conditioned, but much can still depend on the
+    distribution of the sample points and the smoothness of the data.  If
+    the quality of the fit is inadequate, splines may be a good
+    alternative.
+
+    Examples
+    --------
+    >>> np.random.seed(123)
+    >>> from numpy.polynomial import polynomial as P
+    >>> x = np.linspace(-1,1,51) # x "data": [-1, -0.96, ..., 0.96, 1]
+    >>> y = x**3 - x + np.random.randn(len(x))  # x^3 - x + Gaussian noise
+    >>> c, stats = P.polyfit(x,y,3,full=True)
+    >>> np.random.seed(123)
+    >>> c # c[0], c[2] should be approx. 0, c[1] approx. -1, c[3] approx. 1
+    array([ 0.01909725, -1.30598256, -0.00577963,  1.02644286]) # may vary
+    >>> stats # note the large SSR, explaining the rather poor results
+     [array([ 38.06116253]), 4, array([ 1.38446749,  1.32119158,  0.50443316, # may vary
+              0.28853036]), 1.1324274851176597e-014]
+
+    Same thing without the added noise
+
+    >>> y = x**3 - x
+    >>> c, stats = P.polyfit(x,y,3,full=True)
+    >>> c # c[0], c[2] should be "very close to 0", c[1] ~= -1, c[3] ~= 1
+    array([-6.36925336e-18, -1.00000000e+00, -4.08053781e-16,  1.00000000e+00])
+    >>> stats # note the minuscule SSR
+    [array([  7.46346754e-31]), 4, array([ 1.38446749,  1.32119158, # may vary
+               0.50443316,  0.28853036]), 1.1324274851176597e-014]
+
+    """
+    return pu._fit(polyvander, x, y, deg, rcond, full, w)
+
+
+def polycompanion(c):
+    """
+    Return the companion matrix of c.
+
+    The companion matrix for power series cannot be made symmetric by
+    scaling the basis, so this function differs from those for the
+    orthogonal polynomials.
+
+    Parameters
+    ----------
+    c : array_like
+        1-D array of polynomial coefficients ordered from low to high
+        degree.
+
+    Returns
+    -------
+    mat : ndarray
+        Companion matrix of dimensions (deg, deg).
+
+    Notes
+    -----
+
+    .. versionadded:: 1.7.0
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        raise ValueError('Series must have maximum degree of at least 1.')
+    if len(c) == 2:
+        return np.array([[-c[0]/c[1]]])
+
+    n = len(c) - 1
+    mat = np.zeros((n, n), dtype=c.dtype)
+    bot = mat.reshape(-1)[n::n+1]
+    bot[...] = 1
+    mat[:, -1] -= c[:-1]/c[-1]
+    return mat
+
+
+def polyroots(c):
+    """
+    Compute the roots of a polynomial.
+
+    Return the roots (a.k.a. "zeros") of the polynomial
+
+    .. math:: p(x) = \\sum_i c[i] * x^i.
+
+    Parameters
+    ----------
+    c : 1-D array_like
+        1-D array of polynomial coefficients.
+
+    Returns
+    -------
+    out : ndarray
+        Array of the roots of the polynomial. If all the roots are real,
+        then `out` is also real, otherwise it is complex.
+
+    See Also
+    --------
+    numpy.polynomial.chebyshev.chebroots
+    numpy.polynomial.legendre.legroots
+    numpy.polynomial.laguerre.lagroots
+    numpy.polynomial.hermite.hermroots
+    numpy.polynomial.hermite_e.hermeroots
+
+    Notes
+    -----
+    The root estimates are obtained as the eigenvalues of the companion
+    matrix, Roots far from the origin of the complex plane may have large
+    errors due to the numerical instability of the power series for such
+    values. Roots with multiplicity greater than 1 will also show larger
+    errors as the value of the series near such points is relatively
+    insensitive to errors in the roots. Isolated roots near the origin can
+    be improved by a few iterations of Newton's method.
+
+    Examples
+    --------
+    >>> import numpy.polynomial.polynomial as poly
+    >>> poly.polyroots(poly.polyfromroots((-1,0,1)))
+    array([-1.,  0.,  1.])
+    >>> poly.polyroots(poly.polyfromroots((-1,0,1))).dtype
+    dtype('float64')
+    >>> j = complex(0,1)
+    >>> poly.polyroots(poly.polyfromroots((-j,0,j)))
+    array([  0.00000000e+00+0.j,   0.00000000e+00+1.j,   2.77555756e-17-1.j]) # may vary
+
+    """
+    # c is a trimmed copy
+    [c] = pu.as_series([c])
+    if len(c) < 2:
+        return np.array([], dtype=c.dtype)
+    if len(c) == 2:
+        return np.array([-c[0]/c[1]])
+
+    # rotated companion matrix reduces error
+    m = polycompanion(c)[::-1,::-1]
+    r = la.eigvals(m)
+    r.sort()
+    return r
+
+
+#
+# polynomial class
+#
+
+class Polynomial(ABCPolyBase):
+    """A power series class.
+
+    The Polynomial class provides the standard Python numerical methods
+    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
+    attributes and methods listed in the `ABCPolyBase` documentation.
+
+    Parameters
+    ----------
+    coef : array_like
+        Polynomial coefficients in order of increasing degree, i.e.,
+        ``(1, 2, 3)`` give ``1 + 2*x + 3*x**2``.
+    domain : (2,) array_like, optional
+        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
+        to the interval ``[window[0], window[1]]`` by shifting and scaling.
+        The default value is [-1, 1].
+    window : (2,) array_like, optional
+        Window, see `domain` for its use. The default value is [-1, 1].
+
+        .. versionadded:: 1.6.0
+    symbol : str, optional
+        Symbol used to represent the independent variable in string
+        representations of the polynomial expression, e.g. for printing.
+        The symbol must be a valid Python identifier. Default value is 'x'.
+
+        .. versionadded:: 1.24
+
+    """
+    # Virtual Functions
+    _add = staticmethod(polyadd)
+    _sub = staticmethod(polysub)
+    _mul = staticmethod(polymul)
+    _div = staticmethod(polydiv)
+    _pow = staticmethod(polypow)
+    _val = staticmethod(polyval)
+    _int = staticmethod(polyint)
+    _der = staticmethod(polyder)
+    _fit = staticmethod(polyfit)
+    _line = staticmethod(polyline)
+    _roots = staticmethod(polyroots)
+    _fromroots = staticmethod(polyfromroots)
+
+    # Virtual properties
+    domain = np.array(polydomain)
+    window = np.array(polydomain)
+    basis_name = None
+
+    @classmethod
+    def _str_term_unicode(cls, i, arg_str):
+        if i == '1':
+            return f"·{arg_str}"
+        else:
+            return f"·{arg_str}{i.translate(cls._superscript_mapping)}"
+
+    @staticmethod
+    def _str_term_ascii(i, arg_str):
+        if i == '1':
+            return f" {arg_str}"
+        else:
+            return f" {arg_str}**{i}"
+
+    @staticmethod
+    def _repr_latex_term(i, arg_str, needs_parens):
+        if needs_parens:
+            arg_str = rf"\left({arg_str}\right)"
+        if i == 0:
+            return '1'
+        elif i == 1:
+            return arg_str
+        else:
+            return f"{arg_str}^{{{i}}}"

lib/python3.12/site-packages/numpy/polynomial/polynomial.pyi

@@ -0,0 +1,41 @@
+from typing import Any
+
+from numpy import ndarray, dtype, int_
+from numpy.polynomial._polybase import ABCPolyBase
+from numpy.polynomial.polyutils import trimcoef
+
+__all__: list[str]
+
+polytrim = trimcoef
+
+polydomain: ndarray[Any, dtype[int_]]
+polyzero: ndarray[Any, dtype[int_]]
+polyone: ndarray[Any, dtype[int_]]
+polyx: ndarray[Any, dtype[int_]]
+
+def polyline(off, scl): ...
+def polyfromroots(roots): ...
+def polyadd(c1, c2): ...
+def polysub(c1, c2): ...
+def polymulx(c): ...
+def polymul(c1, c2): ...
+def polydiv(c1, c2): ...
+def polypow(c, pow, maxpower=...): ...
+def polyder(c, m=..., scl=..., axis=...): ...
+def polyint(c, m=..., k=..., lbnd=..., scl=..., axis=...): ...
+def polyval(x, c, tensor=...): ...
+def polyvalfromroots(x, r, tensor=...): ...
+def polyval2d(x, y, c): ...
+def polygrid2d(x, y, c): ...
+def polyval3d(x, y, z, c): ...
+def polygrid3d(x, y, z, c): ...
+def polyvander(x, deg): ...
+def polyvander2d(x, y, deg): ...
+def polyvander3d(x, y, z, deg): ...
+def polyfit(x, y, deg, rcond=..., full=..., w=...): ...
+def polyroots(c): ...
+
+class Polynomial(ABCPolyBase):
+    domain: Any
+    window: Any
+    basis_name: Any

lib/python3.12/site-packages/numpy/polynomial/polyutils.py

@@ -0,0 +1,789 @@
+"""
+Utility classes and functions for the polynomial modules.
+
+This module provides: error and warning objects; a polynomial base class;
+and some routines used in both the `polynomial` and `chebyshev` modules.
+
+Warning objects
+---------------
+
+.. autosummary::
+   :toctree: generated/
+
+   RankWarning  raised in least-squares fit for rank-deficient matrix.
+
+Functions
+---------
+
+.. autosummary::
+   :toctree: generated/
+
+   as_series    convert list of array_likes into 1-D arrays of common type.
+   trimseq      remove trailing zeros.
+   trimcoef     remove small trailing coefficients.
+   getdomain    return the domain appropriate for a given set of abscissae.
+   mapdomain    maps points between domains.
+   mapparms     parameters of the linear map between domains.
+
+"""
+import operator
+import functools
+import warnings
+
+import numpy as np
+
+from numpy.core.multiarray import dragon4_positional, dragon4_scientific
+from numpy.core.umath import absolute
+
+__all__ = [
+    'RankWarning', 'as_series', 'trimseq',
+    'trimcoef', 'getdomain', 'mapdomain', 'mapparms',
+    'format_float']
+
+#
+# Warnings and Exceptions
+#
+
+class RankWarning(UserWarning):
+    """Issued by chebfit when the design matrix is rank deficient."""
+    pass
+
+#
+# Helper functions to convert inputs to 1-D arrays
+#
+def trimseq(seq):
+    """Remove small Poly series coefficients.
+
+    Parameters
+    ----------
+    seq : sequence
+        Sequence of Poly series coefficients. This routine fails for
+        empty sequences.
+
+    Returns
+    -------
+    series : sequence
+        Subsequence with trailing zeros removed. If the resulting sequence
+        would be empty, return the first element. The returned sequence may
+        or may not be a view.
+
+    Notes
+    -----
+    Do not lose the type info if the sequence contains unknown objects.
+
+    """
+    if len(seq) == 0:
+        return seq
+    else:
+        for i in range(len(seq) - 1, -1, -1):
+            if seq[i] != 0:
+                break
+        return seq[:i+1]
+
+
+def as_series(alist, trim=True):
+    """
+    Return argument as a list of 1-d arrays.
+
+    The returned list contains array(s) of dtype double, complex double, or
+    object.  A 1-d argument of shape ``(N,)`` is parsed into ``N`` arrays of
+    size one; a 2-d argument of shape ``(M,N)`` is parsed into ``M`` arrays
+    of size ``N`` (i.e., is "parsed by row"); and a higher dimensional array
+    raises a Value Error if it is not first reshaped into either a 1-d or 2-d
+    array.
+
+    Parameters
+    ----------
+    alist : array_like
+        A 1- or 2-d array_like
+    trim : boolean, optional
+        When True, trailing zeros are removed from the inputs.
+        When False, the inputs are passed through intact.
+
+    Returns
+    -------
+    [a1, a2,...] : list of 1-D arrays
+        A copy of the input data as a list of 1-d arrays.
+
+    Raises
+    ------
+    ValueError
+        Raised when `as_series` cannot convert its input to 1-d arrays, or at
+        least one of the resulting arrays is empty.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polyutils as pu
+    >>> a = np.arange(4)
+    >>> pu.as_series(a)
+    [array([0.]), array([1.]), array([2.]), array([3.])]
+    >>> b = np.arange(6).reshape((2,3))
+    >>> pu.as_series(b)
+    [array([0., 1., 2.]), array([3., 4., 5.])]
+
+    >>> pu.as_series((1, np.arange(3), np.arange(2, dtype=np.float16)))
+    [array([1.]), array([0., 1., 2.]), array([0., 1.])]
+
+    >>> pu.as_series([2, [1.1, 0.]])
+    [array([2.]), array([1.1])]
+
+    >>> pu.as_series([2, [1.1, 0.]], trim=False)
+    [array([2.]), array([1.1, 0. ])]
+
+    """
+    arrays = [np.array(a, ndmin=1, copy=False) for a in alist]
+    if min([a.size for a in arrays]) == 0:
+        raise ValueError("Coefficient array is empty")
+    if any(a.ndim != 1 for a in arrays):
+        raise ValueError("Coefficient array is not 1-d")
+    if trim:
+        arrays = [trimseq(a) for a in arrays]
+
+    if any(a.dtype == np.dtype(object) for a in arrays):
+        ret = []
+        for a in arrays:
+            if a.dtype != np.dtype(object):
+                tmp = np.empty(len(a), dtype=np.dtype(object))
+                tmp[:] = a[:]
+                ret.append(tmp)
+            else:
+                ret.append(a.copy())
+    else:
+        try:
+            dtype = np.common_type(*arrays)
+        except Exception as e:
+            raise ValueError("Coefficient arrays have no common type") from e
+        ret = [np.array(a, copy=True, dtype=dtype) for a in arrays]
+    return ret
+
+
+def trimcoef(c, tol=0):
+    """
+    Remove "small" "trailing" coefficients from a polynomial.
+
+    "Small" means "small in absolute value" and is controlled by the
+    parameter `tol`; "trailing" means highest order coefficient(s), e.g., in
+    ``[0, 1, 1, 0, 0]`` (which represents ``0 + x + x**2 + 0*x**3 + 0*x**4``)
+    both the 3-rd and 4-th order coefficients would be "trimmed."
+
+    Parameters
+    ----------
+    c : array_like
+        1-d array of coefficients, ordered from lowest order to highest.
+    tol : number, optional
+        Trailing (i.e., highest order) elements with absolute value less
+        than or equal to `tol` (default value is zero) are removed.
+
+    Returns
+    -------
+    trimmed : ndarray
+        1-d array with trailing zeros removed.  If the resulting series
+        would be empty, a series containing a single zero is returned.
+
+    Raises
+    ------
+    ValueError
+        If `tol` < 0
+
+    See Also
+    --------
+    trimseq
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polyutils as pu
+    >>> pu.trimcoef((0,0,3,0,5,0,0))
+    array([0.,  0.,  3.,  0.,  5.])
+    >>> pu.trimcoef((0,0,1e-3,0,1e-5,0,0),1e-3) # item == tol is trimmed
+    array([0.])
+    >>> i = complex(0,1) # works for complex
+    >>> pu.trimcoef((3e-4,1e-3*(1-i),5e-4,2e-5*(1+i)), 1e-3)
+    array([0.0003+0.j   , 0.001 -0.001j])
+
+    """
+    if tol < 0:
+        raise ValueError("tol must be non-negative")
+
+    [c] = as_series([c])
+    [ind] = np.nonzero(np.abs(c) > tol)
+    if len(ind) == 0:
+        return c[:1]*0
+    else:
+        return c[:ind[-1] + 1].copy()
+
+def getdomain(x):
+    """
+    Return a domain suitable for given abscissae.
+
+    Find a domain suitable for a polynomial or Chebyshev series
+    defined at the values supplied.
+
+    Parameters
+    ----------
+    x : array_like
+        1-d array of abscissae whose domain will be determined.
+
+    Returns
+    -------
+    domain : ndarray
+        1-d array containing two values.  If the inputs are complex, then
+        the two returned points are the lower left and upper right corners
+        of the smallest rectangle (aligned with the axes) in the complex
+        plane containing the points `x`. If the inputs are real, then the
+        two points are the ends of the smallest interval containing the
+        points `x`.
+
+    See Also
+    --------
+    mapparms, mapdomain
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polyutils as pu
+    >>> points = np.arange(4)**2 - 5; points
+    array([-5, -4, -1,  4])
+    >>> pu.getdomain(points)
+    array([-5.,  4.])
+    >>> c = np.exp(complex(0,1)*np.pi*np.arange(12)/6) # unit circle
+    >>> pu.getdomain(c)
+    array([-1.-1.j,  1.+1.j])
+
+    """
+    [x] = as_series([x], trim=False)
+    if x.dtype.char in np.typecodes['Complex']:
+        rmin, rmax = x.real.min(), x.real.max()
+        imin, imax = x.imag.min(), x.imag.max()
+        return np.array((complex(rmin, imin), complex(rmax, imax)))
+    else:
+        return np.array((x.min(), x.max()))
+
+def mapparms(old, new):
+    """
+    Linear map parameters between domains.
+
+    Return the parameters of the linear map ``offset + scale*x`` that maps
+    `old` to `new` such that ``old[i] -> new[i]``, ``i = 0, 1``.
+
+    Parameters
+    ----------
+    old, new : array_like
+        Domains. Each domain must (successfully) convert to a 1-d array
+        containing precisely two values.
+
+    Returns
+    -------
+    offset, scale : scalars
+        The map ``L(x) = offset + scale*x`` maps the first domain to the
+        second.
+
+    See Also
+    --------
+    getdomain, mapdomain
+
+    Notes
+    -----
+    Also works for complex numbers, and thus can be used to calculate the
+    parameters required to map any line in the complex plane to any other
+    line therein.
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polyutils as pu
+    >>> pu.mapparms((-1,1),(-1,1))
+    (0.0, 1.0)
+    >>> pu.mapparms((1,-1),(-1,1))
+    (-0.0, -1.0)
+    >>> i = complex(0,1)
+    >>> pu.mapparms((-i,-1),(1,i))
+    ((1+1j), (1-0j))
+
+    """
+    oldlen = old[1] - old[0]
+    newlen = new[1] - new[0]
+    off = (old[1]*new[0] - old[0]*new[1])/oldlen
+    scl = newlen/oldlen
+    return off, scl
+
+def mapdomain(x, old, new):
+    """
+    Apply linear map to input points.
+
+    The linear map ``offset + scale*x`` that maps the domain `old` to
+    the domain `new` is applied to the points `x`.
+
+    Parameters
+    ----------
+    x : array_like
+        Points to be mapped. If `x` is a subtype of ndarray the subtype
+        will be preserved.
+    old, new : array_like
+        The two domains that determine the map.  Each must (successfully)
+        convert to 1-d arrays containing precisely two values.
+
+    Returns
+    -------
+    x_out : ndarray
+        Array of points of the same shape as `x`, after application of the
+        linear map between the two domains.
+
+    See Also
+    --------
+    getdomain, mapparms
+
+    Notes
+    -----
+    Effectively, this implements:
+
+    .. math::
+        x\\_out = new[0] + m(x - old[0])
+
+    where
+
+    .. math::
+        m = \\frac{new[1]-new[0]}{old[1]-old[0]}
+
+    Examples
+    --------
+    >>> from numpy.polynomial import polyutils as pu
+    >>> old_domain = (-1,1)
+    >>> new_domain = (0,2*np.pi)
+    >>> x = np.linspace(-1,1,6); x
+    array([-1. , -0.6, -0.2,  0.2,  0.6,  1. ])
+    >>> x_out = pu.mapdomain(x, old_domain, new_domain); x_out
+    array([ 0.        ,  1.25663706,  2.51327412,  3.76991118,  5.02654825, # may vary
+            6.28318531])
+    >>> x - pu.mapdomain(x_out, new_domain, old_domain)
+    array([0., 0., 0., 0., 0., 0.])
+
+    Also works for complex numbers (and thus can be used to map any line in
+    the complex plane to any other line therein).
+
+    >>> i = complex(0,1)
+    >>> old = (-1 - i, 1 + i)
+    >>> new = (-1 + i, 1 - i)
+    >>> z = np.linspace(old[0], old[1], 6); z
+    array([-1. -1.j , -0.6-0.6j, -0.2-0.2j,  0.2+0.2j,  0.6+0.6j,  1. +1.j ])
+    >>> new_z = pu.mapdomain(z, old, new); new_z
+    array([-1.0+1.j , -0.6+0.6j, -0.2+0.2j,  0.2-0.2j,  0.6-0.6j,  1.0-1.j ]) # may vary
+
+    """
+    x = np.asanyarray(x)
+    off, scl = mapparms(old, new)
+    return off + scl*x
+
+
+def _nth_slice(i, ndim):
+    sl = [np.newaxis] * ndim
+    sl[i] = slice(None)
+    return tuple(sl)
+
+
+def _vander_nd(vander_fs, points, degrees):
+    r"""
+    A generalization of the Vandermonde matrix for N dimensions
+
+    The result is built by combining the results of 1d Vandermonde matrices,
+
+    .. math::
+        W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{V_k(x_k)[i_0, \ldots, i_M, j_k]}
+
+    where
+
+    .. math::
+        N &= \texttt{len(points)} = \texttt{len(degrees)} = \texttt{len(vander\_fs)} \\
+        M &= \texttt{points[k].ndim} \\
+        V_k &= \texttt{vander\_fs[k]} \\
+        x_k &= \texttt{points[k]} \\
+        0 \le j_k &\le \texttt{degrees[k]}
+
+    Expanding the one-dimensional :math:`V_k` functions gives:
+
+    .. math::
+        W[i_0, \ldots, i_M, j_0, \ldots, j_N] = \prod_{k=0}^N{B_{k, j_k}(x_k[i_0, \ldots, i_M])}
+
+    where :math:`B_{k,m}` is the m'th basis of the polynomial construction used along
+    dimension :math:`k`. For a regular polynomial, :math:`B_{k, m}(x) = P_m(x) = x^m`.
+
+    Parameters
+    ----------
+    vander_fs : Sequence[function(array_like, int) -> ndarray]
+        The 1d vander function to use for each axis, such as ``polyvander``
+    points : Sequence[array_like]
+        Arrays of point coordinates, all of the same shape. The dtypes
+        will be converted to either float64 or complex128 depending on
+        whether any of the elements are complex. Scalars are converted to
+        1-D arrays.
+        This must be the same length as `vander_fs`.
+    degrees : Sequence[int]
+        The maximum degree (inclusive) to use for each axis.
+        This must be the same length as `vander_fs`.
+
+    Returns
+    -------
+    vander_nd : ndarray
+        An array of shape ``points[0].shape + tuple(d + 1 for d in degrees)``.
+    """
+    n_dims = len(vander_fs)
+    if n_dims != len(points):
+        raise ValueError(
+            f"Expected {n_dims} dimensions of sample points, got {len(points)}")
+    if n_dims != len(degrees):
+        raise ValueError(
+            f"Expected {n_dims} dimensions of degrees, got {len(degrees)}")
+    if n_dims == 0:
+        raise ValueError("Unable to guess a dtype or shape when no points are given")
+
+    # convert to the same shape and type
+    points = tuple(np.array(tuple(points), copy=False) + 0.0)
+
+    # produce the vandermonde matrix for each dimension, placing the last
+    # axis of each in an independent trailing axis of the output
+    vander_arrays = (
+        vander_fs[i](points[i], degrees[i])[(...,) + _nth_slice(i, n_dims)]
+        for i in range(n_dims)
+    )
+
+    # we checked this wasn't empty already, so no `initial` needed
+    return functools.reduce(operator.mul, vander_arrays)
+
+
+def _vander_nd_flat(vander_fs, points, degrees):
+    """
+    Like `_vander_nd`, but flattens the last ``len(degrees)`` axes into a single axis
+
+    Used to implement the public ``<type>vander<n>d`` functions.
+    """
+    v = _vander_nd(vander_fs, points, degrees)
+    return v.reshape(v.shape[:-len(degrees)] + (-1,))
+
+
+def _fromroots(line_f, mul_f, roots):
+    """
+    Helper function used to implement the ``<type>fromroots`` functions.
+
+    Parameters
+    ----------
+    line_f : function(float, float) -> ndarray
+        The ``<type>line`` function, such as ``polyline``
+    mul_f : function(array_like, array_like) -> ndarray
+        The ``<type>mul`` function, such as ``polymul``
+    roots
+        See the ``<type>fromroots`` functions for more detail
+    """
+    if len(roots) == 0:
+        return np.ones(1)
+    else:
+        [roots] = as_series([roots], trim=False)
+        roots.sort()
+        p = [line_f(-r, 1) for r in roots]
+        n = len(p)
+        while n > 1:
+            m, r = divmod(n, 2)
+            tmp = [mul_f(p[i], p[i+m]) for i in range(m)]
+            if r:
+                tmp[0] = mul_f(tmp[0], p[-1])
+            p = tmp
+            n = m
+        return p[0]
+
+
+def _valnd(val_f, c, *args):
+    """
+    Helper function used to implement the ``<type>val<n>d`` functions.
+
+    Parameters
+    ----------
+    val_f : function(array_like, array_like, tensor: bool) -> array_like
+        The ``<type>val`` function, such as ``polyval``
+    c, args
+        See the ``<type>val<n>d`` functions for more detail
+    """
+    args = [np.asanyarray(a) for a in args]
+    shape0 = args[0].shape
+    if not all((a.shape == shape0 for a in args[1:])):
+        if len(args) == 3:
+            raise ValueError('x, y, z are incompatible')
+        elif len(args) == 2:
+            raise ValueError('x, y are incompatible')
+        else:
+            raise ValueError('ordinates are incompatible')
+    it = iter(args)
+    x0 = next(it)
+
+    # use tensor on only the first
+    c = val_f(x0, c)
+    for xi in it:
+        c = val_f(xi, c, tensor=False)
+    return c
+
+
+def _gridnd(val_f, c, *args):
+    """
+    Helper function used to implement the ``<type>grid<n>d`` functions.
+
+    Parameters
+    ----------
+    val_f : function(array_like, array_like, tensor: bool) -> array_like
+        The ``<type>val`` function, such as ``polyval``
+    c, args
+        See the ``<type>grid<n>d`` functions for more detail
+    """
+    for xi in args:
+        c = val_f(xi, c)
+    return c
+
+
+def _div(mul_f, c1, c2):
+    """
+    Helper function used to implement the ``<type>div`` functions.
+
+    Implementation uses repeated subtraction of c2 multiplied by the nth basis.
+    For some polynomial types, a more efficient approach may be possible.
+
+    Parameters
+    ----------
+    mul_f : function(array_like, array_like) -> array_like
+        The ``<type>mul`` function, such as ``polymul``
+    c1, c2
+        See the ``<type>div`` functions for more detail
+    """
+    # c1, c2 are trimmed copies
+    [c1, c2] = as_series([c1, c2])
+    if c2[-1] == 0:
+        raise ZeroDivisionError()
+
+    lc1 = len(c1)
+    lc2 = len(c2)
+    if lc1 < lc2:
+        return c1[:1]*0, c1
+    elif lc2 == 1:
+        return c1/c2[-1], c1[:1]*0
+    else:
+        quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
+        rem = c1
+        for i in range(lc1 - lc2, - 1, -1):
+            p = mul_f([0]*i + [1], c2)
+            q = rem[-1]/p[-1]
+            rem = rem[:-1] - q*p[:-1]
+            quo[i] = q
+        return quo, trimseq(rem)
+
+
+def _add(c1, c2):
+    """ Helper function used to implement the ``<type>add`` functions. """
+    # c1, c2 are trimmed copies
+    [c1, c2] = as_series([c1, c2])
+    if len(c1) > len(c2):
+        c1[:c2.size] += c2
+        ret = c1
+    else:
+        c2[:c1.size] += c1
+        ret = c2
+    return trimseq(ret)
+
+
+def _sub(c1, c2):
+    """ Helper function used to implement the ``<type>sub`` functions. """
+    # c1, c2 are trimmed copies
+    [c1, c2] = as_series([c1, c2])
+    if len(c1) > len(c2):
+        c1[:c2.size] -= c2
+        ret = c1
+    else:
+        c2 = -c2
+        c2[:c1.size] += c1
+        ret = c2
+    return trimseq(ret)
+
+
+def _fit(vander_f, x, y, deg, rcond=None, full=False, w=None):
+    """
+    Helper function used to implement the ``<type>fit`` functions.
+
+    Parameters
+    ----------
+    vander_f : function(array_like, int) -> ndarray
+        The 1d vander function, such as ``polyvander``
+    c1, c2
+        See the ``<type>fit`` functions for more detail
+    """
+    x = np.asarray(x) + 0.0
+    y = np.asarray(y) + 0.0
+    deg = np.asarray(deg)
+
+    # check arguments.
+    if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0:
+        raise TypeError("deg must be an int or non-empty 1-D array of int")
+    if deg.min() < 0:
+        raise ValueError("expected deg >= 0")
+    if x.ndim != 1:
+        raise TypeError("expected 1D vector for x")
+    if x.size == 0:
+        raise TypeError("expected non-empty vector for x")
+    if y.ndim < 1 or y.ndim > 2:
+        raise TypeError("expected 1D or 2D array for y")
+    if len(x) != len(y):
+        raise TypeError("expected x and y to have same length")
+
+    if deg.ndim == 0:
+        lmax = deg
+        order = lmax + 1
+        van = vander_f(x, lmax)
+    else:
+        deg = np.sort(deg)
+        lmax = deg[-1]
+        order = len(deg)
+        van = vander_f(x, lmax)[:, deg]
+
+    # set up the least squares matrices in transposed form
+    lhs = van.T
+    rhs = y.T
+    if w is not None:
+        w = np.asarray(w) + 0.0
+        if w.ndim != 1:
+            raise TypeError("expected 1D vector for w")
+        if len(x) != len(w):
+            raise TypeError("expected x and w to have same length")
+        # apply weights. Don't use inplace operations as they
+        # can cause problems with NA.
+        lhs = lhs * w
+        rhs = rhs * w
+
+    # set rcond
+    if rcond is None:
+        rcond = len(x)*np.finfo(x.dtype).eps
+
+    # Determine the norms of the design matrix columns.
+    if issubclass(lhs.dtype.type, np.complexfloating):
+        scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1))
+    else:
+        scl = np.sqrt(np.square(lhs).sum(1))
+    scl[scl == 0] = 1
+
+    # Solve the least squares problem.
+    c, resids, rank, s = np.linalg.lstsq(lhs.T/scl, rhs.T, rcond)
+    c = (c.T/scl).T
+
+    # Expand c to include non-fitted coefficients which are set to zero
+    if deg.ndim > 0:
+        if c.ndim == 2:
+            cc = np.zeros((lmax+1, c.shape[1]), dtype=c.dtype)
+        else:
+            cc = np.zeros(lmax+1, dtype=c.dtype)
+        cc[deg] = c
+        c = cc
+
+    # warn on rank reduction
+    if rank != order and not full:
+        msg = "The fit may be poorly conditioned"
+        warnings.warn(msg, RankWarning, stacklevel=2)
+
+    if full:
+        return c, [resids, rank, s, rcond]
+    else:
+        return c
+
+
+def _pow(mul_f, c, pow, maxpower):
+    """
+    Helper function used to implement the ``<type>pow`` functions.
+
+    Parameters
+    ----------
+    mul_f : function(array_like, array_like) -> ndarray
+        The ``<type>mul`` function, such as ``polymul``
+    c : array_like
+        1-D array of array of series coefficients
+    pow, maxpower
+        See the ``<type>pow`` functions for more detail
+    """
+    # c is a trimmed copy
+    [c] = as_series([c])
+    power = int(pow)
+    if power != pow or power < 0:
+        raise ValueError("Power must be a non-negative integer.")
+    elif maxpower is not None and power > maxpower:
+        raise ValueError("Power is too large")
+    elif power == 0:
+        return np.array([1], dtype=c.dtype)
+    elif power == 1:
+        return c
+    else:
+        # This can be made more efficient by using powers of two
+        # in the usual way.
+        prd = c
+        for i in range(2, power + 1):
+            prd = mul_f(prd, c)
+        return prd
+
+
+def _deprecate_as_int(x, desc):
+    """
+    Like `operator.index`, but emits a deprecation warning when passed a float
+
+    Parameters
+    ----------
+    x : int-like, or float with integral value
+        Value to interpret as an integer
+    desc : str
+        description to include in any error message
+
+    Raises
+    ------
+    TypeError : if x is a non-integral float or non-numeric
+    DeprecationWarning : if x is an integral float
+    """
+    try:
+        return operator.index(x)
+    except TypeError as e:
+        # Numpy 1.17.0, 2019-03-11
+        try:
+            ix = int(x)
+        except TypeError:
+            pass
+        else:
+            if ix == x:
+                warnings.warn(
+                    f"In future, this will raise TypeError, as {desc} will "
+                    "need to be an integer not just an integral float.",
+                    DeprecationWarning,
+                    stacklevel=3
+                )
+                return ix
+
+        raise TypeError(f"{desc} must be an integer") from e
+
+
+def format_float(x, parens=False):
+    if not np.issubdtype(type(x), np.floating):
+        return str(x)
+
+    opts = np.get_printoptions()
+
+    if np.isnan(x):
+        return opts['nanstr']
+    elif np.isinf(x):
+        return opts['infstr']
+
+    exp_format = False
+    if x != 0:
+        a = absolute(x)
+        if a >= 1.e8 or a < 10**min(0, -(opts['precision']-1)//2):
+            exp_format = True
+
+    trim, unique = '0', True
+    if opts['floatmode'] == 'fixed':
+        trim, unique = 'k', False
+
+    if exp_format:
+        s = dragon4_scientific(x, precision=opts['precision'],
+                               unique=unique, trim=trim, 
+                               sign=opts['sign'] == '+')
+        if parens:
+            s = '(' + s + ')'
+    else:
+        s = dragon4_positional(x, precision=opts['precision'],
+                               fractional=True,
+                               unique=unique, trim=trim,
+                               sign=opts['sign'] == '+')
+    return s

lib/python3.12/site-packages/numpy/polynomial/polyutils.pyi

@@ -0,0 +1,11 @@
+__all__: list[str]
+
+class RankWarning(UserWarning): ...
+
+def trimseq(seq): ...
+def as_series(alist, trim=...): ...
+def trimcoef(c, tol=...): ...
+def getdomain(x): ...
+def mapparms(old, new): ...
+def mapdomain(x, old, new): ...
+def format_float(x, parens=...): ...

lib/python3.12/site-packages/numpy/polynomial/setup.py

@@ -0,0 +1,10 @@
+def configuration(parent_package='',top_path=None):
+    from numpy.distutils.misc_util import Configuration
+    config = Configuration('polynomial', parent_package, top_path)
+    config.add_subpackage('tests')
+    config.add_data_files('*.pyi')
+    return config
+
+if __name__ == '__main__':
+    from numpy.distutils.core import setup
+    setup(configuration=configuration)