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miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/__init__.pxd

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+from scipy.linalg cimport cython_blas, cython_lapack

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/__init__.py

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+"""
+====================================
+Linear algebra (:mod:`scipy.linalg`)
+====================================
+
+.. currentmodule:: scipy.linalg
+
+.. toctree::
+   :hidden:
+
+   linalg.blas
+   linalg.cython_blas
+   linalg.cython_lapack
+   linalg.interpolative
+   linalg.lapack
+
+Linear algebra functions.
+
+.. eventually, we should replace the numpy.linalg HTML link with just `numpy.linalg`
+
+.. seealso::
+
+   `numpy.linalg <https://www.numpy.org/devdocs/reference/routines.linalg.html>`__
+   for more linear algebra functions. Note that
+   although `scipy.linalg` imports most of them, identically named
+   functions from `scipy.linalg` may offer more or slightly differing
+   functionality.
+
+
+Basics
+======
+
+.. autosummary::
+   :toctree: generated/
+
+   inv - Find the inverse of a square matrix
+   solve - Solve a linear system of equations
+   solve_banded - Solve a banded linear system
+   solveh_banded - Solve a Hermitian or symmetric banded system
+   solve_circulant - Solve a circulant system
+   solve_triangular - Solve a triangular matrix
+   solve_toeplitz - Solve a toeplitz matrix
+   matmul_toeplitz - Multiply a Toeplitz matrix with an array.
+   det - Find the determinant of a square matrix
+   norm - Matrix and vector norm
+   lstsq - Solve a linear least-squares problem
+   pinv - Pseudo-inverse (Moore-Penrose) using lstsq
+   pinvh - Pseudo-inverse of hermitian matrix
+   kron - Kronecker product of two arrays
+   khatri_rao - Khatri-Rao product of two arrays
+   orthogonal_procrustes - Solve an orthogonal Procrustes problem
+   matrix_balance - Balance matrix entries with a similarity transformation
+   subspace_angles - Compute the subspace angles between two matrices
+   bandwidth - Return the lower and upper bandwidth of an array
+   issymmetric - Check if a square 2D array is symmetric
+   ishermitian - Check if a square 2D array is Hermitian
+   LinAlgError
+   LinAlgWarning
+
+Eigenvalue Problems
+===================
+
+.. autosummary::
+   :toctree: generated/
+
+   eig - Find the eigenvalues and eigenvectors of a square matrix
+   eigvals - Find just the eigenvalues of a square matrix
+   eigh - Find the e-vals and e-vectors of a Hermitian or symmetric matrix
+   eigvalsh - Find just the eigenvalues of a Hermitian or symmetric matrix
+   eig_banded - Find the eigenvalues and eigenvectors of a banded matrix
+   eigvals_banded - Find just the eigenvalues of a banded matrix
+   eigh_tridiagonal - Find the eigenvalues and eigenvectors of a tridiagonal matrix
+   eigvalsh_tridiagonal - Find just the eigenvalues of a tridiagonal matrix
+
+Decompositions
+==============
+
+.. autosummary::
+   :toctree: generated/
+
+   lu - LU decomposition of a matrix
+   lu_factor - LU decomposition returning unordered matrix and pivots
+   lu_solve - Solve Ax=b using back substitution with output of lu_factor
+   svd - Singular value decomposition of a matrix
+   svdvals - Singular values of a matrix
+   diagsvd - Construct matrix of singular values from output of svd
+   orth - Construct orthonormal basis for the range of A using svd
+   null_space - Construct orthonormal basis for the null space of A using svd
+   ldl - LDL.T decomposition of a Hermitian or a symmetric matrix.
+   cholesky - Cholesky decomposition of a matrix
+   cholesky_banded - Cholesky decomp. of a sym. or Hermitian banded matrix
+   cho_factor - Cholesky decomposition for use in solving a linear system
+   cho_solve - Solve previously factored linear system
+   cho_solve_banded - Solve previously factored banded linear system
+   polar - Compute the polar decomposition.
+   qr - QR decomposition of a matrix
+   qr_multiply - QR decomposition and multiplication by Q
+   qr_update - Rank k QR update
+   qr_delete - QR downdate on row or column deletion
+   qr_insert - QR update on row or column insertion
+   rq - RQ decomposition of a matrix
+   qz - QZ decomposition of a pair of matrices
+   ordqz - QZ decomposition of a pair of matrices with reordering
+   schur - Schur decomposition of a matrix
+   rsf2csf - Real to complex Schur form
+   hessenberg - Hessenberg form of a matrix
+   cdf2rdf - Complex diagonal form to real diagonal block form
+   cossin - Cosine sine decomposition of a unitary or orthogonal matrix
+
+.. seealso::
+
+   `scipy.linalg.interpolative` -- Interpolative matrix decompositions
+
+
+Matrix Functions
+================
+
+.. autosummary::
+   :toctree: generated/
+
+   expm - Matrix exponential
+   logm - Matrix logarithm
+   cosm - Matrix cosine
+   sinm - Matrix sine
+   tanm - Matrix tangent
+   coshm - Matrix hyperbolic cosine
+   sinhm - Matrix hyperbolic sine
+   tanhm - Matrix hyperbolic tangent
+   signm - Matrix sign
+   sqrtm - Matrix square root
+   funm - Evaluating an arbitrary matrix function
+   expm_frechet - Frechet derivative of the matrix exponential
+   expm_cond - Relative condition number of expm in the Frobenius norm
+   fractional_matrix_power - Fractional matrix power
+
+
+Matrix Equation Solvers
+=======================
+
+.. autosummary::
+   :toctree: generated/
+
+   solve_sylvester - Solve the Sylvester matrix equation
+   solve_continuous_are - Solve the continuous-time algebraic Riccati equation
+   solve_discrete_are - Solve the discrete-time algebraic Riccati equation
+   solve_continuous_lyapunov - Solve the continuous-time Lyapunov equation
+   solve_discrete_lyapunov - Solve the discrete-time Lyapunov equation
+
+
+Sketches and Random Projections
+===============================
+
+.. autosummary::
+   :toctree: generated/
+
+   clarkson_woodruff_transform - Applies the Clarkson Woodruff Sketch (a.k.a CountMin Sketch)
+
+Special Matrices
+================
+
+.. autosummary::
+   :toctree: generated/
+
+   block_diag - Construct a block diagonal matrix from submatrices
+   circulant - Circulant matrix
+   companion - Companion matrix
+   convolution_matrix - Convolution matrix
+   dft - Discrete Fourier transform matrix
+   fiedler - Fiedler matrix
+   fiedler_companion - Fiedler companion matrix
+   hadamard - Hadamard matrix of order 2**n
+   hankel - Hankel matrix
+   helmert - Helmert matrix
+   hilbert - Hilbert matrix
+   invhilbert - Inverse Hilbert matrix
+   leslie - Leslie matrix
+   pascal - Pascal matrix
+   invpascal - Inverse Pascal matrix
+   toeplitz - Toeplitz matrix
+
+Low-level routines
+==================
+
+.. autosummary::
+   :toctree: generated/
+
+   get_blas_funcs
+   get_lapack_funcs
+   find_best_blas_type
+
+.. seealso::
+
+   `scipy.linalg.blas` -- Low-level BLAS functions
+
+   `scipy.linalg.lapack` -- Low-level LAPACK functions
+
+   `scipy.linalg.cython_blas` -- Low-level BLAS functions for Cython
+
+   `scipy.linalg.cython_lapack` -- Low-level LAPACK functions for Cython
+
+"""  # noqa: E501
+
+from ._misc import *
+from ._cythonized_array_utils import *
+from ._basic import *
+from ._decomp import *
+from ._decomp_lu import *
+from ._decomp_ldl import *
+from ._decomp_cholesky import *
+from ._decomp_qr import *
+from ._decomp_qz import *
+from ._decomp_svd import *
+from ._decomp_schur import *
+from ._decomp_polar import *
+from ._matfuncs import *
+from .blas import *
+from .lapack import *
+from ._special_matrices import *
+from ._solvers import *
+from ._procrustes import *
+from ._decomp_update import *
+from ._sketches import *
+from ._decomp_cossin import *
+
+# Deprecated namespaces, to be removed in v2.0.0
+from . import (
+    decomp, decomp_cholesky, decomp_lu, decomp_qr, decomp_svd, decomp_schur,
+    basic, misc, special_matrices, matfuncs,
+)
+
+__all__ = [s for s in dir() if not s.startswith('_')]
+
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_basic.py

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+#
+# Author: Pearu Peterson, March 2002
+#
+# w/ additions by Travis Oliphant, March 2002
+#              and Jake Vanderplas, August 2012
+
+import warnings
+from warnings import warn
+from itertools import product
+import numpy as np
+from numpy import atleast_1d, atleast_2d
+from .lapack import get_lapack_funcs, _compute_lwork
+from ._misc import LinAlgError, _datacopied, LinAlgWarning
+from ._decomp import _asarray_validated
+from . import _decomp, _decomp_svd
+from ._solve_toeplitz import levinson
+from ._cythonized_array_utils import (find_det_from_lu, bandwidth, issymmetric,
+                                      ishermitian)
+
+__all__ = ['solve', 'solve_triangular', 'solveh_banded', 'solve_banded',
+           'solve_toeplitz', 'solve_circulant', 'inv', 'det', 'lstsq',
+           'pinv', 'pinvh', 'matrix_balance', 'matmul_toeplitz']
+
+
+# The numpy facilities for type-casting checks are too slow for small sized
+# arrays and eat away the time budget for the checkups. Here we set a
+# precomputed dict container of the numpy.can_cast() table.
+
+# It can be used to determine quickly what a dtype can be cast to LAPACK
+# compatible types, i.e., 'float32, float64, complex64, complex128'.
+# Then it can be checked via "casting_dict[arr.dtype.char]"
+lapack_cast_dict = {x: ''.join([y for y in 'fdFD' if np.can_cast(x, y)])
+                    for x in np.typecodes['All']}
+
+
+# Linear equations
+def _solve_check(n, info, lamch=None, rcond=None):
+    """ Check arguments during the different steps of the solution phase """
+    if info < 0:
+        raise ValueError(f'LAPACK reported an illegal value in {-info}-th argument.')
+    elif 0 < info:
+        raise LinAlgError('Matrix is singular.')
+
+    if lamch is None:
+        return
+    E = lamch('E')
+    if rcond < E:
+        warn(f'Ill-conditioned matrix (rcond={rcond:.6g}): '
+             'result may not be accurate.',
+             LinAlgWarning, stacklevel=3)
+
+
+def _find_matrix_structure(a):
+    n = a.shape[0]
+    n_below, n_above = bandwidth(a)
+
+    if n_below == n_above == 0:
+        kind = 'diagonal'
+    elif n_above == 0:
+        kind = 'lower triangular'
+    elif n_below == 0:
+        kind = 'upper triangular'
+    elif n_above <= 1 and n_below <= 1 and n > 3:
+        kind = 'tridiagonal'
+    elif np.issubdtype(a.dtype, np.complexfloating) and ishermitian(a):
+        kind = 'hermitian'
+    elif issymmetric(a):
+        kind = 'symmetric'
+    else:
+        kind = 'general'
+
+    return kind, n_below, n_above
+
+
+def solve(a, b, lower=False, overwrite_a=False,
+          overwrite_b=False, check_finite=True, assume_a=None,
+          transposed=False):
+    """
+    Solves the linear equation set ``a @ x == b`` for the unknown ``x``
+    for square `a` matrix.
+
+    If the data matrix is known to be a particular type then supplying the
+    corresponding string to ``assume_a`` key chooses the dedicated solver.
+    The available options are
+
+    ===================  ================================
+     diagonal             'diagonal'
+     tridiagonal          'tridiagonal'
+     banded               'banded'
+     upper triangular     'upper triangular'
+     lower triangular     'lower triangular'
+     symmetric            'symmetric' (or 'sym')
+     hermitian            'hermitian' (or 'her')
+     positive definite    'positive definite' (or 'pos')
+     general              'general' (or 'gen')
+    ===================  ================================
+
+    Parameters
+    ----------
+    a : (N, N) array_like
+        Square input data
+    b : (N, NRHS) array_like
+        Input data for the right hand side.
+    lower : bool, default: False
+        Ignored unless ``assume_a`` is one of ``'sym'``, ``'her'``, or ``'pos'``.
+        If True, the calculation uses only the data in the lower triangle of `a`;
+        entries above the diagonal are ignored. If False (default), the
+        calculation uses only the data in the upper triangle of `a`; entries
+        below the diagonal are ignored.
+    overwrite_a : bool, default: False
+        Allow overwriting data in `a` (may enhance performance).
+    overwrite_b : bool, default: False
+        Allow overwriting data in `b` (may enhance performance).
+    check_finite : bool, default: True
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+    assume_a : str, optional
+        Valid entries are described above.
+        If omitted or ``None``, checks are performed to identify structure so the
+        appropriate solver can be called.
+    transposed : bool, default: False
+        If True, solve ``a.T @ x == b``. Raises `NotImplementedError`
+        for complex `a`.
+
+    Returns
+    -------
+    x : (N, NRHS) ndarray
+        The solution array.
+
+    Raises
+    ------
+    ValueError
+        If size mismatches detected or input a is not square.
+    LinAlgError
+        If the matrix is singular.
+    LinAlgWarning
+        If an ill-conditioned input a is detected.
+    NotImplementedError
+        If transposed is True and input a is a complex matrix.
+
+    Notes
+    -----
+    If the input b matrix is a 1-D array with N elements, when supplied
+    together with an NxN input a, it is assumed as a valid column vector
+    despite the apparent size mismatch. This is compatible with the
+    numpy.dot() behavior and the returned result is still 1-D array.
+
+    The general, symmetric, Hermitian and positive definite solutions are
+    obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of
+    LAPACK respectively.
+
+    The datatype of the arrays define which solver is called regardless
+    of the values. In other words, even when the complex array entries have
+    precisely zero imaginary parts, the complex solver will be called based
+    on the data type of the array.
+
+    Examples
+    --------
+    Given `a` and `b`, solve for `x`:
+
+    >>> import numpy as np
+    >>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]])
+    >>> b = np.array([2, 4, -1])
+    >>> from scipy import linalg
+    >>> x = linalg.solve(a, b)
+    >>> x
+    array([ 2., -2.,  9.])
+    >>> np.dot(a, x) == b
+    array([ True,  True,  True], dtype=bool)
+
+    """
+    # Flags for 1-D or N-D right-hand side
+    b_is_1D = False
+
+    # check finite after determining structure
+    a1 = atleast_2d(_asarray_validated(a, check_finite=False))
+    b1 = atleast_1d(_asarray_validated(b, check_finite=False))
+    a1, b1 = _ensure_dtype_cdsz(a1, b1)
+    n = a1.shape[0]
+
+    overwrite_a = overwrite_a or _datacopied(a1, a)
+    overwrite_b = overwrite_b or _datacopied(b1, b)
+
+    if a1.shape[0] != a1.shape[1]:
+        raise ValueError('Input a needs to be a square matrix.')
+
+    if n != b1.shape[0]:
+        # Last chance to catch 1x1 scalar a and 1-D b arrays
+        if not (n == 1 and b1.size != 0):
+            raise ValueError('Input b has to have same number of rows as '
+                             'input a')
+
+    # accommodate empty arrays
+    if b1.size == 0:
+        dt = solve(np.eye(2, dtype=a1.dtype), np.ones(2, dtype=b1.dtype)).dtype
+        return np.empty_like(b1, dtype=dt)
+
+    # regularize 1-D b arrays to 2D
+    if b1.ndim == 1:
+        if n == 1:
+            b1 = b1[None, :]
+        else:
+            b1 = b1[:, None]
+        b_is_1D = True
+
+    if assume_a not in {None, 'diagonal', 'tridiagonal', 'banded', 'lower triangular',
+                        'upper triangular', 'symmetric', 'hermitian',
+                        'positive definite', 'general', 'sym', 'her', 'pos', 'gen'}:
+        raise ValueError(f'{assume_a} is not a recognized matrix structure')
+
+    # for a real matrix, describe it as "symmetric", not "hermitian"
+    # (lapack doesn't know what to do with real hermitian matrices)
+    if assume_a in {'hermitian', 'her'} and not np.iscomplexobj(a1):
+        assume_a = 'symmetric'
+
+    n_below, n_above = None, None
+    if assume_a is None:
+        assume_a, n_below, n_above = _find_matrix_structure(a1)
+
+    # Get the correct lamch function.
+    # The LAMCH functions only exists for S and D
+    # So for complex values we have to convert to real/double.
+    if a1.dtype.char in 'fF':  # single precision
+        lamch = get_lapack_funcs('lamch', dtype='f')
+    else:
+        lamch = get_lapack_funcs('lamch', dtype='d')
+
+    # Currently we do not have the other forms of the norm calculators
+    #   lansy, lanpo, lanhe.
+    # However, in any case they only reduce computations slightly...
+    if assume_a == 'diagonal':
+        _matrix_norm = _matrix_norm_diagonal
+    elif assume_a == 'tridiagonal':
+        _matrix_norm = _matrix_norm_tridiagonal
+    elif assume_a in {'lower triangular', 'upper triangular'}:
+        _matrix_norm = _matrix_norm_triangular(assume_a)
+    else:
+        _matrix_norm = _matrix_norm_general
+
+    # Since the I-norm and 1-norm are the same for symmetric matrices
+    # we can collect them all in this one call
+    # Note however, that when issuing 'gen' and form!='none', then
+    # the I-norm should be used
+    if transposed:
+        trans = 1
+        norm = 'I'
+        if np.iscomplexobj(a1):
+            raise NotImplementedError('scipy.linalg.solve can currently '
+                                      'not solve a^T x = b or a^H x = b '
+                                      'for complex matrices.')
+    else:
+        trans = 0
+        norm = '1'
+
+    anorm = _matrix_norm(norm, a1, check_finite)
+
+    info, rcond = 0, np.inf
+
+    # Generalized case 'gesv'
+    if assume_a in {'general', 'gen'}:
+        gecon, getrf, getrs = get_lapack_funcs(('gecon', 'getrf', 'getrs'),
+                                               (a1, b1))
+        lu, ipvt, info = getrf(a1, overwrite_a=overwrite_a)
+        _solve_check(n, info)
+        x, info = getrs(lu, ipvt, b1,
+                        trans=trans, overwrite_b=overwrite_b)
+        _solve_check(n, info)
+        rcond, info = gecon(lu, anorm, norm=norm)
+    # Hermitian case 'hesv'
+    elif assume_a in {'hermitian', 'her'}:
+        hecon, hesv, hesv_lw = get_lapack_funcs(('hecon', 'hesv',
+                                                 'hesv_lwork'), (a1, b1))
+        lwork = _compute_lwork(hesv_lw, n, lower)
+        lu, ipvt, x, info = hesv(a1, b1, lwork=lwork,
+                                 lower=lower,
+                                 overwrite_a=overwrite_a,
+                                 overwrite_b=overwrite_b)
+        _solve_check(n, info)
+        rcond, info = hecon(lu, ipvt, anorm)
+    # Symmetric case 'sysv'
+    elif assume_a in {'symmetric', 'sym'}:
+        sycon, sysv, sysv_lw = get_lapack_funcs(('sycon', 'sysv',
+                                                 'sysv_lwork'), (a1, b1))
+        lwork = _compute_lwork(sysv_lw, n, lower)
+        lu, ipvt, x, info = sysv(a1, b1, lwork=lwork,
+                                 lower=lower,
+                                 overwrite_a=overwrite_a,
+                                 overwrite_b=overwrite_b)
+        _solve_check(n, info)
+        rcond, info = sycon(lu, ipvt, anorm)
+    # Diagonal case
+    elif assume_a == 'diagonal':
+        diag_a = np.diag(a1)
+        x = (b1.T / diag_a).T
+        abs_diag_a = np.abs(diag_a)
+        rcond = abs_diag_a.min() / abs_diag_a.max()
+    # Tri-diagonal case
+    elif assume_a == 'tridiagonal':
+        a1 = a1.T if transposed else a1
+        dl, d, du = np.diag(a1, -1), np.diag(a1, 0), np.diag(a1, 1)
+        _gttrf, _gttrs, _gtcon = get_lapack_funcs(('gttrf', 'gttrs', 'gtcon'), (a1, b1))
+        dl, d, du, du2, ipiv, info = _gttrf(dl, d, du)
+        _solve_check(n, info)
+        x, info = _gttrs(dl, d, du, du2, ipiv, b1, overwrite_b=overwrite_b)
+        _solve_check(n, info)
+        rcond, info = _gtcon(dl, d, du, du2, ipiv, anorm)
+    # Banded case
+    elif assume_a == 'banded':
+        a1, n_below, n_above = ((a1.T, n_above, n_below) if transposed
+                                else (a1, n_below, n_above))
+        n_below, n_above = bandwidth(a1) if n_below is None else (n_below, n_above)
+        ab = _to_banded(n_below, n_above, a1)
+        gbsv, = get_lapack_funcs(('gbsv',), (a1, b1))
+        # Next two lines copied from `solve_banded`
+        a2 = np.zeros((2*n_below + n_above + 1, ab.shape[1]), dtype=gbsv.dtype)
+        a2[n_below:, :] = ab
+        _, _, x, info = gbsv(n_below, n_above, a2, b1,
+                             overwrite_ab=True, overwrite_b=overwrite_b)
+        _solve_check(n, info)
+        # TODO: wrap gbcon and use to get rcond
+    # Triangular case
+    elif assume_a in {'lower triangular', 'upper triangular'}:
+        lower = assume_a == 'lower triangular'
+        x, info = _solve_triangular(a1, b1, lower=lower, overwrite_b=overwrite_b,
+                                    trans=transposed)
+        _solve_check(n, info)
+        _trcon = get_lapack_funcs(('trcon'), (a1, b1))
+        rcond, info = _trcon(a1, uplo='L' if lower else 'U')
+    # Positive definite case 'posv'
+    else:
+        pocon, posv = get_lapack_funcs(('pocon', 'posv'),
+                                       (a1, b1))
+        lu, x, info = posv(a1, b1, lower=lower,
+                           overwrite_a=overwrite_a,
+                           overwrite_b=overwrite_b)
+        _solve_check(n, info)
+        rcond, info = pocon(lu, anorm)
+
+    _solve_check(n, info, lamch, rcond)
+
+    if b_is_1D:
+        x = x.ravel()
+
+    return x
+
+
+def _matrix_norm_diagonal(_, a, check_finite):
+    # Equivalent of dlange for diagonal matrix, assuming
+    # norm is either 'I' or '1' (really just not the Frobenius norm)
+    d = np.diag(a)
+    d = np.asarray_chkfinite(d) if check_finite else d
+    return np.abs(d).max()
+
+
+def _matrix_norm_tridiagonal(norm, a, check_finite):
+    # Equivalent of dlange for tridiagonal matrix, assuming
+    # norm is either 'I' or '1'
+    if norm == 'I':
+        a = a.T
+    # Context to avoid warning before error in cases like -inf + inf
+    with np.errstate(invalid='ignore'):
+        d = np.abs(np.diag(a))
+        d[1:] += np.abs(np.diag(a, 1))
+        d[:-1] += np.abs(np.diag(a, -1))
+    d = np.asarray_chkfinite(d) if check_finite else d
+    return d.max()
+
+
+def _matrix_norm_triangular(structure):
+    def fun(norm, a, check_finite):
+        a = np.asarray_chkfinite(a) if check_finite else a
+        lantr = get_lapack_funcs('lantr', (a,))
+        return lantr(norm, a, 'L' if structure == 'lower triangular' else 'U' )
+    return fun
+
+
+def _matrix_norm_general(norm, a, check_finite):
+    a = np.asarray_chkfinite(a) if check_finite else a
+    lange = get_lapack_funcs('lange', (a,))
+    return lange(norm, a)
+
+
+def _to_banded(n_below, n_above, a):
+    n = a.shape[0]
+    rows = n_above + n_below + 1
+    ab = np.zeros((rows, n), dtype=a.dtype)
+    ab[n_above] = np.diag(a)
+    for i in range(1, n_above + 1):
+        ab[n_above - i, i:] = np.diag(a, i)
+    for i in range(1, n_below + 1):
+        ab[n_above + i, :-i] = np.diag(a, -i)
+    return ab
+
+
+def _ensure_dtype_cdsz(*arrays):
+    # Ensure that the dtype of arrays is one of the standard types
+    # compatible with LAPACK functions (single or double precision
+    # real or complex).
+    dtype = np.result_type(*arrays)
+    if not np.issubdtype(dtype, np.inexact):
+        return (array.astype(np.float64) for array in arrays)
+    complex = np.issubdtype(dtype, np.complexfloating)
+    if np.finfo(dtype).bits <= 32:
+        dtype = np.complex64 if complex else np.float32
+    elif np.finfo(dtype).bits >= 64:
+        dtype = np.complex128 if complex else np.float64
+    return (array.astype(dtype, copy=False) for array in arrays)
+
+
+def solve_triangular(a, b, trans=0, lower=False, unit_diagonal=False,
+                     overwrite_b=False, check_finite=True):
+    """
+    Solve the equation ``a x = b`` for `x`, assuming a is a triangular matrix.
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        A triangular matrix
+    b : (M,) or (M, N) array_like
+        Right-hand side matrix in ``a x = b``
+    lower : bool, optional
+        Use only data contained in the lower triangle of `a`.
+        Default is to use upper triangle.
+    trans : {0, 1, 2, 'N', 'T', 'C'}, optional
+        Type of system to solve:
+
+        ========  =========
+        trans     system
+        ========  =========
+        0 or 'N'  a x  = b
+        1 or 'T'  a^T x = b
+        2 or 'C'  a^H x = b
+        ========  =========
+    unit_diagonal : bool, optional
+        If True, diagonal elements of `a` are assumed to be 1 and
+        will not be referenced.
+    overwrite_b : bool, optional
+        Allow overwriting data in `b` (may enhance performance)
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    x : (M,) or (M, N) ndarray
+        Solution to the system ``a x = b``.  Shape of return matches `b`.
+
+    Raises
+    ------
+    LinAlgError
+        If `a` is singular
+
+    Notes
+    -----
+    .. versionadded:: 0.9.0
+
+    Examples
+    --------
+    Solve the lower triangular system a x = b, where::
+
+             [3  0  0  0]       [4]
+        a =  [2  1  0  0]   b = [2]
+             [1  0  1  0]       [4]
+             [1  1  1  1]       [2]
+
+    >>> import numpy as np
+    >>> from scipy.linalg import solve_triangular
+    >>> a = np.array([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]])
+    >>> b = np.array([4, 2, 4, 2])
+    >>> x = solve_triangular(a, b, lower=True)
+    >>> x
+    array([ 1.33333333, -0.66666667,  2.66666667, -1.33333333])
+    >>> a.dot(x)  # Check the result
+    array([ 4.,  2.,  4.,  2.])
+
+    """
+
+    a1 = _asarray_validated(a, check_finite=check_finite)
+    b1 = _asarray_validated(b, check_finite=check_finite)
+
+    if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
+        raise ValueError('expected square matrix')
+
+    if a1.shape[0] != b1.shape[0]:
+        raise ValueError(f'shapes of a {a1.shape} and b {b1.shape} are incompatible')
+
+    # accommodate empty arrays
+    if b1.size == 0:
+        dt_nonempty = solve_triangular(
+            np.eye(2, dtype=a1.dtype), np.ones(2, dtype=b1.dtype)
+        ).dtype
+        return np.empty_like(b1, dtype=dt_nonempty)
+
+    overwrite_b = overwrite_b or _datacopied(b1, b)
+
+    x, _ = _solve_triangular(a1, b1, trans, lower, unit_diagonal, overwrite_b)
+    return x
+
+
+# solve_triangular without the input validation
+def _solve_triangular(a1, b1, trans=0, lower=False, unit_diagonal=False,
+                      overwrite_b=False):
+
+    trans = {'N': 0, 'T': 1, 'C': 2}.get(trans, trans)
+    trtrs, = get_lapack_funcs(('trtrs',), (a1, b1))
+    if a1.flags.f_contiguous or trans == 2:
+        x, info = trtrs(a1, b1, overwrite_b=overwrite_b, lower=lower,
+                        trans=trans, unitdiag=unit_diagonal)
+    else:
+        # transposed system is solved since trtrs expects Fortran ordering
+        x, info = trtrs(a1.T, b1, overwrite_b=overwrite_b, lower=not lower,
+                        trans=not trans, unitdiag=unit_diagonal)
+
+    if info == 0:
+        return x, info
+    if info > 0:
+        raise LinAlgError("singular matrix: resolution failed at diagonal %d" %
+                          (info-1))
+    raise ValueError('illegal value in %dth argument of internal trtrs' %
+                     (-info))
+
+
+def solve_banded(l_and_u, ab, b, overwrite_ab=False, overwrite_b=False,
+                 check_finite=True):
+    """
+    Solve the equation a x = b for x, assuming a is banded matrix.
+
+    The matrix a is stored in `ab` using the matrix diagonal ordered form::
+
+        ab[u + i - j, j] == a[i,j]
+
+    Example of `ab` (shape of a is (6,6), `u` =1, `l` =2)::
+
+        *    a01  a12  a23  a34  a45
+        a00  a11  a22  a33  a44  a55
+        a10  a21  a32  a43  a54   *
+        a20  a31  a42  a53   *    *
+
+    Parameters
+    ----------
+    (l, u) : (integer, integer)
+        Number of non-zero lower and upper diagonals
+    ab : (`l` + `u` + 1, M) array_like
+        Banded matrix
+    b : (M,) or (M, K) array_like
+        Right-hand side
+    overwrite_ab : bool, optional
+        Discard data in `ab` (may enhance performance)
+    overwrite_b : bool, optional
+        Discard data in `b` (may enhance performance)
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    x : (M,) or (M, K) ndarray
+        The solution to the system a x = b. Returned shape depends on the
+        shape of `b`.
+
+    Examples
+    --------
+    Solve the banded system a x = b, where::
+
+            [5  2 -1  0  0]       [0]
+            [1  4  2 -1  0]       [1]
+        a = [0  1  3  2 -1]   b = [2]
+            [0  0  1  2  2]       [2]
+            [0  0  0  1  1]       [3]
+
+    There is one nonzero diagonal below the main diagonal (l = 1), and
+    two above (u = 2). The diagonal banded form of the matrix is::
+
+             [*  * -1 -1 -1]
+        ab = [*  2  2  2  2]
+             [5  4  3  2  1]
+             [1  1  1  1  *]
+
+    >>> import numpy as np
+    >>> from scipy.linalg import solve_banded
+    >>> ab = np.array([[0,  0, -1, -1, -1],
+    ...                [0,  2,  2,  2,  2],
+    ...                [5,  4,  3,  2,  1],
+    ...                [1,  1,  1,  1,  0]])
+    >>> b = np.array([0, 1, 2, 2, 3])
+    >>> x = solve_banded((1, 2), ab, b)
+    >>> x
+    array([-2.37288136,  3.93220339, -4.        ,  4.3559322 , -1.3559322 ])
+
+    """
+
+    a1 = _asarray_validated(ab, check_finite=check_finite, as_inexact=True)
+    b1 = _asarray_validated(b, check_finite=check_finite, as_inexact=True)
+
+    # Validate shapes.
+    if a1.shape[-1] != b1.shape[0]:
+        raise ValueError("shapes of ab and b are not compatible.")
+
+    (nlower, nupper) = l_and_u
+    if nlower + nupper + 1 != a1.shape[0]:
+        raise ValueError("invalid values for the number of lower and upper "
+                         "diagonals: l+u+1 (%d) does not equal ab.shape[0] "
+                         "(%d)" % (nlower + nupper + 1, ab.shape[0]))
+
+    # accommodate empty arrays
+    if b1.size == 0:
+        dt = solve(np.eye(1, dtype=a1.dtype), np.ones(1, dtype=b1.dtype)).dtype
+        return np.empty_like(b1, dtype=dt)
+
+    overwrite_b = overwrite_b or _datacopied(b1, b)
+    if a1.shape[-1] == 1:
+        b2 = np.array(b1, copy=(not overwrite_b))
+        # a1.shape[-1] == 1 -> original matrix is 1x1. Typically, the user
+        # will pass u = l = 0 and `a1` will be 1x1. However, the rest of the
+        # function works with unnecessary rows in `a1` as long as
+        # `a1[u + i - j, j] == a[i,j]`. In the 1x1 case, we want i = j = 0,
+        # so the diagonal is in row `u` of `a1`. See gh-8906.
+        b2 /= a1[nupper, 0]
+        return b2
+    if nlower == nupper == 1:
+        overwrite_ab = overwrite_ab or _datacopied(a1, ab)
+        gtsv, = get_lapack_funcs(('gtsv',), (a1, b1))
+        du = a1[0, 1:]
+        d = a1[1, :]
+        dl = a1[2, :-1]
+        du2, d, du, x, info = gtsv(dl, d, du, b1, overwrite_ab, overwrite_ab,
+                                   overwrite_ab, overwrite_b)
+    else:
+        gbsv, = get_lapack_funcs(('gbsv',), (a1, b1))
+        a2 = np.zeros((2*nlower + nupper + 1, a1.shape[1]), dtype=gbsv.dtype)
+        a2[nlower:, :] = a1
+        lu, piv, x, info = gbsv(nlower, nupper, a2, b1, overwrite_ab=True,
+                                overwrite_b=overwrite_b)
+    if info == 0:
+        return x
+    if info > 0:
+        raise LinAlgError("singular matrix")
+    raise ValueError('illegal value in %d-th argument of internal '
+                     'gbsv/gtsv' % -info)
+
+
+def solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False,
+                  check_finite=True):
+    """
+    Solve equation a x = b. a is Hermitian positive-definite banded matrix.
+
+    Uses Thomas' Algorithm, which is more efficient than standard LU
+    factorization, but should only be used for Hermitian positive-definite
+    matrices.
+
+    The matrix ``a`` is stored in `ab` either in lower diagonal or upper
+    diagonal ordered form:
+
+        ab[u + i - j, j] == a[i,j]        (if upper form; i <= j)
+        ab[    i - j, j] == a[i,j]        (if lower form; i >= j)
+
+    Example of `ab` (shape of ``a`` is (6, 6), number of upper diagonals,
+    ``u`` =2)::
+
+        upper form:
+        *   *   a02 a13 a24 a35
+        *   a01 a12 a23 a34 a45
+        a00 a11 a22 a33 a44 a55
+
+        lower form:
+        a00 a11 a22 a33 a44 a55
+        a10 a21 a32 a43 a54 *
+        a20 a31 a42 a53 *   *
+
+    Cells marked with * are not used.
+
+    Parameters
+    ----------
+    ab : (``u`` + 1, M) array_like
+        Banded matrix
+    b : (M,) or (M, K) array_like
+        Right-hand side
+    overwrite_ab : bool, optional
+        Discard data in `ab` (may enhance performance)
+    overwrite_b : bool, optional
+        Discard data in `b` (may enhance performance)
+    lower : bool, optional
+        Is the matrix in the lower form. (Default is upper form)
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    x : (M,) or (M, K) ndarray
+        The solution to the system ``a x = b``. Shape of return matches shape
+        of `b`.
+
+    Notes
+    -----
+    In the case of a non-positive definite matrix ``a``, the solver
+    `solve_banded` may be used.
+
+    Examples
+    --------
+    Solve the banded system ``A x = b``, where::
+
+            [ 4  2 -1  0  0  0]       [1]
+            [ 2  5  2 -1  0  0]       [2]
+        A = [-1  2  6  2 -1  0]   b = [2]
+            [ 0 -1  2  7  2 -1]       [3]
+            [ 0  0 -1  2  8  2]       [3]
+            [ 0  0  0 -1  2  9]       [3]
+
+    >>> import numpy as np
+    >>> from scipy.linalg import solveh_banded
+
+    ``ab`` contains the main diagonal and the nonzero diagonals below the
+    main diagonal. That is, we use the lower form:
+
+    >>> ab = np.array([[ 4,  5,  6,  7, 8, 9],
+    ...                [ 2,  2,  2,  2, 2, 0],
+    ...                [-1, -1, -1, -1, 0, 0]])
+    >>> b = np.array([1, 2, 2, 3, 3, 3])
+    >>> x = solveh_banded(ab, b, lower=True)
+    >>> x
+    array([ 0.03431373,  0.45938375,  0.05602241,  0.47759104,  0.17577031,
+            0.34733894])
+
+
+    Solve the Hermitian banded system ``H x = b``, where::
+
+            [ 8   2-1j   0     0  ]        [ 1  ]
+        H = [2+1j  5     1j    0  ]    b = [1+1j]
+            [ 0   -1j    9   -2-1j]        [1-2j]
+            [ 0    0   -2+1j   6  ]        [ 0  ]
+
+    In this example, we put the upper diagonals in the array ``hb``:
+
+    >>> hb = np.array([[0, 2-1j, 1j, -2-1j],
+    ...                [8,  5,    9,   6  ]])
+    >>> b = np.array([1, 1+1j, 1-2j, 0])
+    >>> x = solveh_banded(hb, b)
+    >>> x
+    array([ 0.07318536-0.02939412j,  0.11877624+0.17696461j,
+            0.10077984-0.23035393j, -0.00479904-0.09358128j])
+
+    """
+    a1 = _asarray_validated(ab, check_finite=check_finite)
+    b1 = _asarray_validated(b, check_finite=check_finite)
+
+    # Validate shapes.
+    if a1.shape[-1] != b1.shape[0]:
+        raise ValueError("shapes of ab and b are not compatible.")
+
+    # accommodate empty arrays
+    if b1.size == 0:
+        dt = solve(np.eye(1, dtype=a1.dtype), np.ones(1, dtype=b1.dtype)).dtype
+        return np.empty_like(b1, dtype=dt)
+
+    overwrite_b = overwrite_b or _datacopied(b1, b)
+    overwrite_ab = overwrite_ab or _datacopied(a1, ab)
+
+    if a1.shape[0] == 2:
+        ptsv, = get_lapack_funcs(('ptsv',), (a1, b1))
+        if lower:
+            d = a1[0, :].real
+            e = a1[1, :-1]
+        else:
+            d = a1[1, :].real
+            e = a1[0, 1:].conj()
+        d, du, x, info = ptsv(d, e, b1, overwrite_ab, overwrite_ab,
+                              overwrite_b)
+    else:
+        pbsv, = get_lapack_funcs(('pbsv',), (a1, b1))
+        c, x, info = pbsv(a1, b1, lower=lower, overwrite_ab=overwrite_ab,
+                          overwrite_b=overwrite_b)
+    if info > 0:
+        raise LinAlgError("%dth leading minor not positive definite" % info)
+    if info < 0:
+        raise ValueError('illegal value in %dth argument of internal '
+                         'pbsv' % -info)
+    return x
+
+
+def solve_toeplitz(c_or_cr, b, check_finite=True):
+    r"""Solve a Toeplitz system using Levinson Recursion
+
+    The Toeplitz matrix has constant diagonals, with c as its first column
+    and r as its first row. If r is not given, ``r == conjugate(c)`` is
+    assumed.
+
+    .. warning::
+
+        Beginning in SciPy 1.17, multidimensional input will be treated as a batch,
+        not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments
+        before passing them to `solve_toeplitz`.
+
+    Parameters
+    ----------
+    c_or_cr : array_like or tuple of (array_like, array_like)
+        The vector ``c``, or a tuple of arrays (``c``, ``r``). If not
+        supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
+        real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
+        of the Toeplitz matrix is ``[c[0], r[1:]]``.
+    b : (M,) or (M, K) array_like
+        Right-hand side in ``T x = b``.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (result entirely NaNs) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    x : (M,) or (M, K) ndarray
+        The solution to the system ``T x = b``. Shape of return matches shape
+        of `b`.
+
+    See Also
+    --------
+    toeplitz : Toeplitz matrix
+
+    Notes
+    -----
+    The solution is computed using Levinson-Durbin recursion, which is faster
+    than generic least-squares methods, but can be less numerically stable.
+
+    Examples
+    --------
+    Solve the Toeplitz system T x = b, where::
+
+            [ 1 -1 -2 -3]       [1]
+        T = [ 3  1 -1 -2]   b = [2]
+            [ 6  3  1 -1]       [2]
+            [10  6  3  1]       [5]
+
+    To specify the Toeplitz matrix, only the first column and the first
+    row are needed.
+
+    >>> import numpy as np
+    >>> c = np.array([1, 3, 6, 10])    # First column of T
+    >>> r = np.array([1, -1, -2, -3])  # First row of T
+    >>> b = np.array([1, 2, 2, 5])
+
+    >>> from scipy.linalg import solve_toeplitz, toeplitz
+    >>> x = solve_toeplitz((c, r), b)
+    >>> x
+    array([ 1.66666667, -1.        , -2.66666667,  2.33333333])
+
+    Check the result by creating the full Toeplitz matrix and
+    multiplying it by `x`.  We should get `b`.
+
+    >>> T = toeplitz(c, r)
+    >>> T.dot(x)
+    array([ 1.,  2.,  2.,  5.])
+
+    """
+    # If numerical stability of this algorithm is a problem, a future
+    # developer might consider implementing other O(N^2) Toeplitz solvers,
+    # such as GKO (https://www.jstor.org/stable/2153371) or Bareiss.
+
+    r, c, b, dtype, b_shape = _validate_args_for_toeplitz_ops(
+        c_or_cr, b, check_finite, keep_b_shape=True)
+
+    # accommodate empty arrays
+    if b.size == 0:
+        return np.empty_like(b)
+
+    # Form a 1-D array of values to be used in the matrix, containing a
+    # reversed copy of r[1:], followed by c.
+    vals = np.concatenate((r[-1:0:-1], c))
+    if b is None:
+        raise ValueError('illegal value, `b` is a required argument')
+
+    if b.ndim == 1:
+        x, _ = levinson(vals, np.ascontiguousarray(b))
+    else:
+        x = np.column_stack([levinson(vals, np.ascontiguousarray(b[:, i]))[0]
+                             for i in range(b.shape[1])])
+        x = x.reshape(*b_shape)
+
+    return x
+
+
+def _get_axis_len(aname, a, axis):
+    ax = axis
+    if ax < 0:
+        ax += a.ndim
+    if 0 <= ax < a.ndim:
+        return a.shape[ax]
+    raise ValueError(f"'{aname}axis' entry is out of bounds")
+
+
+def solve_circulant(c, b, singular='raise', tol=None,
+                    caxis=-1, baxis=0, outaxis=0):
+    """Solve C x = b for x, where C is a circulant matrix.
+
+    `C` is the circulant matrix associated with the vector `c`.
+
+    The system is solved by doing division in Fourier space. The
+    calculation is::
+
+        x = ifft(fft(b) / fft(c))
+
+    where `fft` and `ifft` are the fast Fourier transform and its inverse,
+    respectively. For a large vector `c`, this is *much* faster than
+    solving the system with the full circulant matrix.
+
+    Parameters
+    ----------
+    c : array_like
+        The coefficients of the circulant matrix.
+    b : array_like
+        Right-hand side matrix in ``a x = b``.
+    singular : str, optional
+        This argument controls how a near singular circulant matrix is
+        handled.  If `singular` is "raise" and the circulant matrix is
+        near singular, a `LinAlgError` is raised. If `singular` is
+        "lstsq", the least squares solution is returned. Default is "raise".
+    tol : float, optional
+        If any eigenvalue of the circulant matrix has an absolute value
+        that is less than or equal to `tol`, the matrix is considered to be
+        near singular. If not given, `tol` is set to::
+
+            tol = abs_eigs.max() * abs_eigs.size * np.finfo(np.float64).eps
+
+        where `abs_eigs` is the array of absolute values of the eigenvalues
+        of the circulant matrix.
+    caxis : int
+        When `c` has dimension greater than 1, it is viewed as a collection
+        of circulant vectors. In this case, `caxis` is the axis of `c` that
+        holds the vectors of circulant coefficients.
+    baxis : int
+        When `b` has dimension greater than 1, it is viewed as a collection
+        of vectors. In this case, `baxis` is the axis of `b` that holds the
+        right-hand side vectors.
+    outaxis : int
+        When `c` or `b` are multidimensional, the value returned by
+        `solve_circulant` is multidimensional. In this case, `outaxis` is
+        the axis of the result that holds the solution vectors.
+
+    Returns
+    -------
+    x : ndarray
+        Solution to the system ``C x = b``.
+
+    Raises
+    ------
+    LinAlgError
+        If the circulant matrix associated with `c` is near singular.
+
+    See Also
+    --------
+    circulant : circulant matrix
+
+    Notes
+    -----
+    For a 1-D vector `c` with length `m`, and an array `b`
+    with shape ``(m, ...)``,
+
+        solve_circulant(c, b)
+
+    returns the same result as
+
+        solve(circulant(c), b)
+
+    where `solve` and `circulant` are from `scipy.linalg`.
+
+    .. versionadded:: 0.16.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import solve_circulant, solve, circulant, lstsq
+
+    >>> c = np.array([2, 2, 4])
+    >>> b = np.array([1, 2, 3])
+    >>> solve_circulant(c, b)
+    array([ 0.75, -0.25,  0.25])
+
+    Compare that result to solving the system with `scipy.linalg.solve`:
+
+    >>> solve(circulant(c), b)
+    array([ 0.75, -0.25,  0.25])
+
+    A singular example:
+
+    >>> c = np.array([1, 1, 0, 0])
+    >>> b = np.array([1, 2, 3, 4])
+
+    Calling ``solve_circulant(c, b)`` will raise a `LinAlgError`.  For the
+    least square solution, use the option ``singular='lstsq'``:
+
+    >>> solve_circulant(c, b, singular='lstsq')
+    array([ 0.25,  1.25,  2.25,  1.25])
+
+    Compare to `scipy.linalg.lstsq`:
+
+    >>> x, resid, rnk, s = lstsq(circulant(c), b)
+    >>> x
+    array([ 0.25,  1.25,  2.25,  1.25])
+
+    A broadcasting example:
+
+    Suppose we have the vectors of two circulant matrices stored in an array
+    with shape (2, 5), and three `b` vectors stored in an array with shape
+    (3, 5).  For example,
+
+    >>> c = np.array([[1.5, 2, 3, 0, 0], [1, 1, 4, 3, 2]])
+    >>> b = np.arange(15).reshape(-1, 5)
+
+    We want to solve all combinations of circulant matrices and `b` vectors,
+    with the result stored in an array with shape (2, 3, 5). When we
+    disregard the axes of `c` and `b` that hold the vectors of coefficients,
+    the shapes of the collections are (2,) and (3,), respectively, which are
+    not compatible for broadcasting. To have a broadcast result with shape
+    (2, 3), we add a trivial dimension to `c`: ``c[:, np.newaxis, :]`` has
+    shape (2, 1, 5). The last dimension holds the coefficients of the
+    circulant matrices, so when we call `solve_circulant`, we can use the
+    default ``caxis=-1``. The coefficients of the `b` vectors are in the last
+    dimension of the array `b`, so we use ``baxis=-1``. If we use the
+    default `outaxis`, the result will have shape (5, 2, 3), so we'll use
+    ``outaxis=-1`` to put the solution vectors in the last dimension.
+
+    >>> x = solve_circulant(c[:, np.newaxis, :], b, baxis=-1, outaxis=-1)
+    >>> x.shape
+    (2, 3, 5)
+    >>> np.set_printoptions(precision=3)  # For compact output of numbers.
+    >>> x
+    array([[[-0.118,  0.22 ,  1.277, -0.142,  0.302],
+            [ 0.651,  0.989,  2.046,  0.627,  1.072],
+            [ 1.42 ,  1.758,  2.816,  1.396,  1.841]],
+           [[ 0.401,  0.304,  0.694, -0.867,  0.377],
+            [ 0.856,  0.758,  1.149, -0.412,  0.831],
+            [ 1.31 ,  1.213,  1.603,  0.042,  1.286]]])
+
+    Check by solving one pair of `c` and `b` vectors (cf. ``x[1, 1, :]``):
+
+    >>> solve_circulant(c[1], b[1, :])
+    array([ 0.856,  0.758,  1.149, -0.412,  0.831])
+
+    """
+    c = np.atleast_1d(c)
+    nc = _get_axis_len("c", c, caxis)
+    b = np.atleast_1d(b)
+    nb = _get_axis_len("b", b, baxis)
+    if nc != nb:
+        raise ValueError(f'Shapes of c {c.shape} and b {b.shape} are incompatible')
+
+    # accommodate empty arrays
+    if b.size == 0:
+        dt = solve_circulant(np.arange(3, dtype=c.dtype),
+                             np.ones(3, dtype=b.dtype)).dtype
+        return np.empty_like(b, dtype=dt)
+
+    fc = np.fft.fft(np.moveaxis(c, caxis, -1), axis=-1)
+    abs_fc = np.abs(fc)
+    if tol is None:
+        # This is the same tolerance as used in np.linalg.matrix_rank.
+        tol = abs_fc.max(axis=-1) * nc * np.finfo(np.float64).eps
+        if tol.shape != ():
+            tol.shape = tol.shape + (1,)
+        else:
+            tol = np.atleast_1d(tol)
+
+    near_zeros = abs_fc <= tol
+    is_near_singular = np.any(near_zeros)
+    if is_near_singular:
+        if singular == 'raise':
+            raise LinAlgError("near singular circulant matrix.")
+        else:
+            # Replace the small values with 1 to avoid errors in the
+            # division fb/fc below.
+            fc[near_zeros] = 1
+
+    fb = np.fft.fft(np.moveaxis(b, baxis, -1), axis=-1)
+
+    q = fb / fc
+
+    if is_near_singular:
+        # `near_zeros` is a boolean array, same shape as `c`, that is
+        # True where `fc` is (near) zero. `q` is the broadcasted result
+        # of fb / fc, so to set the values of `q` to 0 where `fc` is near
+        # zero, we use a mask that is the broadcast result of an array
+        # of True values shaped like `b` with `near_zeros`.
+        mask = np.ones_like(b, dtype=bool) & near_zeros
+        q[mask] = 0
+
+    x = np.fft.ifft(q, axis=-1)
+    if not (np.iscomplexobj(c) or np.iscomplexobj(b)):
+        x = x.real
+    if outaxis != -1:
+        x = np.moveaxis(x, -1, outaxis)
+    return x
+
+
+# matrix inversion
+def inv(a, overwrite_a=False, check_finite=True):
+    """
+    Compute the inverse of a matrix.
+
+    Parameters
+    ----------
+    a : array_like
+        Square matrix to be inverted.
+    overwrite_a : bool, optional
+        Discard data in `a` (may improve performance). Default is False.
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    ainv : ndarray
+        Inverse of the matrix `a`.
+
+    Raises
+    ------
+    LinAlgError
+        If `a` is singular.
+    ValueError
+        If `a` is not square, or not 2D.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import linalg
+    >>> a = np.array([[1., 2.], [3., 4.]])
+    >>> linalg.inv(a)
+    array([[-2. ,  1. ],
+           [ 1.5, -0.5]])
+    >>> np.dot(a, linalg.inv(a))
+    array([[ 1.,  0.],
+           [ 0.,  1.]])
+
+    """
+    a1 = _asarray_validated(a, check_finite=check_finite)
+    if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
+        raise ValueError('expected square matrix')
+
+    # accommodate empty square matrices
+    if a1.size == 0:
+        dt = inv(np.eye(2, dtype=a1.dtype)).dtype
+        return np.empty_like(a1, dtype=dt)
+
+    overwrite_a = overwrite_a or _datacopied(a1, a)
+    getrf, getri, getri_lwork = get_lapack_funcs(('getrf', 'getri',
+                                                  'getri_lwork'),
+                                                 (a1,))
+    lu, piv, info = getrf(a1, overwrite_a=overwrite_a)
+    if info == 0:
+        lwork = _compute_lwork(getri_lwork, a1.shape[0])
+
+        # XXX: the following line fixes curious SEGFAULT when
+        # benchmarking 500x500 matrix inverse. This seems to
+        # be a bug in LAPACK ?getri routine because if lwork is
+        # minimal (when using lwork[0] instead of lwork[1]) then
+        # all tests pass. Further investigation is required if
+        # more such SEGFAULTs occur.
+        lwork = int(1.01 * lwork)
+        inv_a, info = getri(lu, piv, lwork=lwork, overwrite_lu=1)
+    if info > 0:
+        raise LinAlgError("singular matrix")
+    if info < 0:
+        raise ValueError('illegal value in %d-th argument of internal '
+                         'getrf|getri' % -info)
+    return inv_a
+
+
+# Determinant
+
+def det(a, overwrite_a=False, check_finite=True):
+    """
+    Compute the determinant of a matrix
+
+    The determinant is a scalar that is a function of the associated square
+    matrix coefficients. The determinant value is zero for singular matrices.
+
+    Parameters
+    ----------
+    a : (..., M, M) array_like
+        Input array to compute determinants for.
+    overwrite_a : bool, optional
+        Allow overwriting data in a (may enhance performance).
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    det : (...) float or complex
+        Determinant of `a`. For stacked arrays, a scalar is returned for each
+        (m, m) slice in the last two dimensions of the input. For example, an
+        input of shape (p, q, m, m) will produce a result of shape (p, q). If
+        all dimensions are 1 a scalar is returned regardless of ndim.
+
+    Notes
+    -----
+    The determinant is computed by performing an LU factorization of the
+    input with LAPACK routine 'getrf', and then calculating the product of
+    diagonal entries of the U factor.
+
+    Even if the input array is single precision (float32 or complex64), the
+    result will be returned in double precision (float64 or complex128) to
+    prevent overflows.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import linalg
+    >>> a = np.array([[1,2,3], [4,5,6], [7,8,9]])  # A singular matrix
+    >>> linalg.det(a)
+    0.0
+    >>> b = np.array([[0,2,3], [4,5,6], [7,8,9]])
+    >>> linalg.det(b)
+    3.0
+    >>> # An array with the shape (3, 2, 2, 2)
+    >>> c = np.array([[[[1., 2.], [3., 4.]],
+    ...                [[5., 6.], [7., 8.]]],
+    ...               [[[9., 10.], [11., 12.]],
+    ...                [[13., 14.], [15., 16.]]],
+    ...               [[[17., 18.], [19., 20.]],
+    ...                [[21., 22.], [23., 24.]]]])
+    >>> linalg.det(c)  # The resulting shape is (3, 2)
+    array([[-2., -2.],
+           [-2., -2.],
+           [-2., -2.]])
+    >>> linalg.det(c[0, 0])  # Confirm the (0, 0) slice, [[1, 2], [3, 4]]
+    -2.0
+    """
+    # The goal is to end up with a writable contiguous array to pass to Cython
+
+    # First we check and make arrays.
+    a1 = np.asarray_chkfinite(a) if check_finite else np.asarray(a)
+    if a1.ndim < 2:
+        raise ValueError('The input array must be at least two-dimensional.')
+    if a1.shape[-1] != a1.shape[-2]:
+        raise ValueError('Last 2 dimensions of the array must be square'
+                         f' but received shape {a1.shape}.')
+
+    # Also check if dtype is LAPACK compatible
+    if a1.dtype.char not in 'fdFD':
+        dtype_char = lapack_cast_dict[a1.dtype.char]
+        if not dtype_char:  # No casting possible
+            raise TypeError(f'The dtype "{a1.dtype.name}" cannot be cast '
+                            'to float(32, 64) or complex(64, 128).')
+
+        a1 = a1.astype(dtype_char[0])  # makes a copy, free to scratch
+        overwrite_a = True
+
+    # Empty array has determinant 1 because math.
+    if min(*a1.shape) == 0:
+        dtyp = np.float64 if a1.dtype.char not in 'FD' else np.complex128
+        if a1.ndim == 2:
+            return dtyp(1.0)
+        else:
+            return np.ones(shape=a1.shape[:-2], dtype=dtyp)
+
+    # Scalar case
+    if a1.shape[-2:] == (1, 1):
+        a1 = a1[..., 0, 0]
+        if a1.ndim == 0:
+            a1 = a1[()]
+        # Convert float32 to float64, and complex64 to complex128.
+        if a1.dtype.char in 'dD':
+            return a1
+        return a1.astype('d') if a1.dtype.char == 'f' else a1.astype('D')
+
+    # Then check overwrite permission
+    if not _datacopied(a1, a):  # "a"  still alive through "a1"
+        if not overwrite_a:
+            # Data belongs to "a" so make a copy
+            a1 = a1.copy(order='C')
+        #  else: Do nothing we'll use "a" if possible
+    # else:  a1 has its own data thus free to scratch
+
+    # Then layout checks, might happen that overwrite is allowed but original
+    # array was read-only or non-C-contiguous.
+    if not (a1.flags['C_CONTIGUOUS'] and a1.flags['WRITEABLE']):
+        a1 = a1.copy(order='C')
+
+    if a1.ndim == 2:
+        det = find_det_from_lu(a1)
+        # Convert float, complex to NumPy scalars
+        return (np.float64(det) if np.isrealobj(det) else np.complex128(det))
+
+    # loop over the stacked array, and avoid overflows for single precision
+    # Cf. np.linalg.det(np.diag([1e+38, 1e+38]).astype(np.float32))
+    dtype_char = a1.dtype.char
+    if dtype_char in 'fF':
+        dtype_char = 'd' if dtype_char.islower() else 'D'
+
+    det = np.empty(a1.shape[:-2], dtype=dtype_char)
+    for ind in product(*[range(x) for x in a1.shape[:-2]]):
+        det[ind] = find_det_from_lu(a1[ind])
+    return det
+
+
+# Linear Least Squares
+def lstsq(a, b, cond=None, overwrite_a=False, overwrite_b=False,
+          check_finite=True, lapack_driver=None):
+    """
+    Compute least-squares solution to equation Ax = b.
+
+    Compute a vector x such that the 2-norm ``|b - A x|`` is minimized.
+
+    Parameters
+    ----------
+    a : (M, N) array_like
+        Left-hand side array
+    b : (M,) or (M, K) array_like
+        Right hand side array
+    cond : float, optional
+        Cutoff for 'small' singular values; used to determine effective
+        rank of a. Singular values smaller than
+        ``cond * largest_singular_value`` are considered zero.
+    overwrite_a : bool, optional
+        Discard data in `a` (may enhance performance). Default is False.
+    overwrite_b : bool, optional
+        Discard data in `b` (may enhance performance). Default is False.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+    lapack_driver : str, optional
+        Which LAPACK driver is used to solve the least-squares problem.
+        Options are ``'gelsd'``, ``'gelsy'``, ``'gelss'``. Default
+        (``'gelsd'``) is a good choice.  However, ``'gelsy'`` can be slightly
+        faster on many problems.  ``'gelss'`` was used historically.  It is
+        generally slow but uses less memory.
+
+        .. versionadded:: 0.17.0
+
+    Returns
+    -------
+    x : (N,) or (N, K) ndarray
+        Least-squares solution.
+    residues : (K,) ndarray or float
+        Square of the 2-norm for each column in ``b - a x``, if ``M > N`` and
+        ``rank(A) == n`` (returns a scalar if ``b`` is 1-D). Otherwise a
+        (0,)-shaped array is returned.
+    rank : int
+        Effective rank of `a`.
+    s : (min(M, N),) ndarray or None
+        Singular values of `a`. The condition number of ``a`` is
+        ``s[0] / s[-1]``.
+
+    Raises
+    ------
+    LinAlgError
+        If computation does not converge.
+
+    ValueError
+        When parameters are not compatible.
+
+    See Also
+    --------
+    scipy.optimize.nnls : linear least squares with non-negativity constraint
+
+    Notes
+    -----
+    When ``'gelsy'`` is used as a driver, `residues` is set to a (0,)-shaped
+    array and `s` is always ``None``.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import lstsq
+    >>> import matplotlib.pyplot as plt
+
+    Suppose we have the following data:
+
+    >>> x = np.array([1, 2.5, 3.5, 4, 5, 7, 8.5])
+    >>> y = np.array([0.3, 1.1, 1.5, 2.0, 3.2, 6.6, 8.6])
+
+    We want to fit a quadratic polynomial of the form ``y = a + b*x**2``
+    to this data.  We first form the "design matrix" M, with a constant
+    column of 1s and a column containing ``x**2``:
+
+    >>> M = x[:, np.newaxis]**[0, 2]
+    >>> M
+    array([[  1.  ,   1.  ],
+           [  1.  ,   6.25],
+           [  1.  ,  12.25],
+           [  1.  ,  16.  ],
+           [  1.  ,  25.  ],
+           [  1.  ,  49.  ],
+           [  1.  ,  72.25]])
+
+    We want to find the least-squares solution to ``M.dot(p) = y``,
+    where ``p`` is a vector with length 2 that holds the parameters
+    ``a`` and ``b``.
+
+    >>> p, res, rnk, s = lstsq(M, y)
+    >>> p
+    array([ 0.20925829,  0.12013861])
+
+    Plot the data and the fitted curve.
+
+    >>> plt.plot(x, y, 'o', label='data')
+    >>> xx = np.linspace(0, 9, 101)
+    >>> yy = p[0] + p[1]*xx**2
+    >>> plt.plot(xx, yy, label='least squares fit, $y = a + bx^2$')
+    >>> plt.xlabel('x')
+    >>> plt.ylabel('y')
+    >>> plt.legend(framealpha=1, shadow=True)
+    >>> plt.grid(alpha=0.25)
+    >>> plt.show()
+
+    """
+    a1 = _asarray_validated(a, check_finite=check_finite)
+    b1 = _asarray_validated(b, check_finite=check_finite)
+    if len(a1.shape) != 2:
+        raise ValueError('Input array a should be 2D')
+    m, n = a1.shape
+    if len(b1.shape) == 2:
+        nrhs = b1.shape[1]
+    else:
+        nrhs = 1
+    if m != b1.shape[0]:
+        raise ValueError('Shape mismatch: a and b should have the same number'
+                         f' of rows ({m} != {b1.shape[0]}).')
+    if m == 0 or n == 0:  # Zero-sized problem, confuses LAPACK
+        x = np.zeros((n,) + b1.shape[1:], dtype=np.common_type(a1, b1))
+        if n == 0:
+            residues = np.linalg.norm(b1, axis=0)**2
+        else:
+            residues = np.empty((0,))
+        return x, residues, 0, np.empty((0,))
+
+    driver = lapack_driver
+    if driver is None:
+        driver = lstsq.default_lapack_driver
+    if driver not in ('gelsd', 'gelsy', 'gelss'):
+        raise ValueError(f'LAPACK driver "{driver}" is not found')
+
+    lapack_func, lapack_lwork = get_lapack_funcs((driver,
+                                                 f'{driver}_lwork'),
+                                                 (a1, b1))
+    real_data = True if (lapack_func.dtype.kind == 'f') else False
+
+    if m < n:
+        # need to extend b matrix as it will be filled with
+        # a larger solution matrix
+        if len(b1.shape) == 2:
+            b2 = np.zeros((n, nrhs), dtype=lapack_func.dtype)
+            b2[:m, :] = b1
+        else:
+            b2 = np.zeros(n, dtype=lapack_func.dtype)
+            b2[:m] = b1
+        b1 = b2
+
+    overwrite_a = overwrite_a or _datacopied(a1, a)
+    overwrite_b = overwrite_b or _datacopied(b1, b)
+
+    if cond is None:
+        cond = np.finfo(lapack_func.dtype).eps
+
+    if driver in ('gelss', 'gelsd'):
+        if driver == 'gelss':
+            lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
+            v, x, s, rank, work, info = lapack_func(a1, b1, cond, lwork,
+                                                    overwrite_a=overwrite_a,
+                                                    overwrite_b=overwrite_b)
+
+        elif driver == 'gelsd':
+            if real_data:
+                lwork, iwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
+                x, s, rank, info = lapack_func(a1, b1, lwork,
+                                               iwork, cond, False, False)
+            else:  # complex data
+                lwork, rwork, iwork = _compute_lwork(lapack_lwork, m, n,
+                                                     nrhs, cond)
+                x, s, rank, info = lapack_func(a1, b1, lwork, rwork, iwork,
+                                               cond, False, False)
+        if info > 0:
+            raise LinAlgError("SVD did not converge in Linear Least Squares")
+        if info < 0:
+            raise ValueError('illegal value in %d-th argument of internal %s'
+                             % (-info, lapack_driver))
+        resids = np.asarray([], dtype=x.dtype)
+        if m > n:
+            x1 = x[:n]
+            if rank == n:
+                resids = np.sum(np.abs(x[n:])**2, axis=0)
+            x = x1
+        return x, resids, rank, s
+
+    elif driver == 'gelsy':
+        lwork = _compute_lwork(lapack_lwork, m, n, nrhs, cond)
+        jptv = np.zeros((a1.shape[1], 1), dtype=np.int32)
+        v, x, j, rank, info = lapack_func(a1, b1, jptv, cond,
+                                          lwork, False, False)
+        if info < 0:
+            raise ValueError("illegal value in %d-th argument of internal "
+                             "gelsy" % -info)
+        if m > n:
+            x1 = x[:n]
+            x = x1
+        return x, np.array([], x.dtype), rank, None
+
+
+lstsq.default_lapack_driver = 'gelsd'
+
+
+def pinv(a, *, atol=None, rtol=None, return_rank=False, check_finite=True):
+    """
+    Compute the (Moore-Penrose) pseudo-inverse of a matrix.
+
+    Calculate a generalized inverse of a matrix using its
+    singular-value decomposition ``U @ S @ V`` in the economy mode and picking
+    up only the columns/rows that are associated with significant singular
+    values.
+
+    If ``s`` is the maximum singular value of ``a``, then the
+    significance cut-off value is determined by ``atol + rtol * s``. Any
+    singular value below this value is assumed insignificant.
+
+    Parameters
+    ----------
+    a : (M, N) array_like
+        Matrix to be pseudo-inverted.
+    atol : float, optional
+        Absolute threshold term, default value is 0.
+
+        .. versionadded:: 1.7.0
+
+    rtol : float, optional
+        Relative threshold term, default value is ``max(M, N) * eps`` where
+        ``eps`` is the machine precision value of the datatype of ``a``.
+
+        .. versionadded:: 1.7.0
+
+    return_rank : bool, optional
+        If True, return the effective rank of the matrix.
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    B : (N, M) ndarray
+        The pseudo-inverse of matrix `a`.
+    rank : int
+        The effective rank of the matrix. Returned if `return_rank` is True.
+
+    Raises
+    ------
+    LinAlgError
+        If SVD computation does not converge.
+
+    See Also
+    --------
+    pinvh : Moore-Penrose pseudoinverse of a hermitian matrix.
+
+    Notes
+    -----
+    If ``A`` is invertible then the Moore-Penrose pseudoinverse is exactly
+    the inverse of ``A`` [1]_. If ``A`` is not invertible then the
+    Moore-Penrose pseudoinverse computes the ``x`` solution to ``Ax = b`` such
+    that ``||Ax - b||`` is minimized [1]_.
+
+    References
+    ----------
+    .. [1] Penrose, R. (1956). On best approximate solutions of linear matrix
+           equations. Mathematical Proceedings of the Cambridge Philosophical
+           Society, 52(1), 17-19. doi:10.1017/S0305004100030929
+
+    Examples
+    --------
+
+    Given an ``m x n`` matrix ``A`` and an ``n x m`` matrix ``B`` the four
+    Moore-Penrose conditions are:
+
+    1. ``ABA = A`` (``B`` is a generalized inverse of ``A``),
+    2. ``BAB = B`` (``A`` is a generalized inverse of ``B``),
+    3. ``(AB)* = AB`` (``AB`` is hermitian),
+    4. ``(BA)* = BA`` (``BA`` is hermitian) [1]_.
+
+    Here, ``A*`` denotes the conjugate transpose. The Moore-Penrose
+    pseudoinverse is a unique ``B`` that satisfies all four of these
+    conditions and exists for any ``A``. Note that, unlike the standard
+    matrix inverse, ``A`` does not have to be a square matrix or have
+    linearly independent columns/rows.
+
+    As an example, we can calculate the Moore-Penrose pseudoinverse of a
+    random non-square matrix and verify it satisfies the four conditions.
+
+    >>> import numpy as np
+    >>> from scipy import linalg
+    >>> rng = np.random.default_rng()
+    >>> A = rng.standard_normal((9, 6))
+    >>> B = linalg.pinv(A)
+    >>> np.allclose(A @ B @ A, A)  # Condition 1
+    True
+    >>> np.allclose(B @ A @ B, B)  # Condition 2
+    True
+    >>> np.allclose((A @ B).conj().T, A @ B)  # Condition 3
+    True
+    >>> np.allclose((B @ A).conj().T, B @ A)  # Condition 4
+    True
+
+    """
+    a = _asarray_validated(a, check_finite=check_finite)
+    u, s, vh = _decomp_svd.svd(a, full_matrices=False, check_finite=False)
+    t = u.dtype.char.lower()
+    maxS = np.max(s, initial=0.)
+
+    atol = 0. if atol is None else atol
+    rtol = max(a.shape) * np.finfo(t).eps if (rtol is None) else rtol
+
+    if (atol < 0.) or (rtol < 0.):
+        raise ValueError("atol and rtol values must be positive.")
+
+    val = atol + maxS * rtol
+    rank = np.sum(s > val)
+
+    u = u[:, :rank]
+    u /= s[:rank]
+    B = (u @ vh[:rank]).conj().T
+
+    if return_rank:
+        return B, rank
+    else:
+        return B
+
+
+def pinvh(a, atol=None, rtol=None, lower=True, return_rank=False,
+          check_finite=True):
+    """
+    Compute the (Moore-Penrose) pseudo-inverse of a Hermitian matrix.
+
+    Calculate a generalized inverse of a complex Hermitian/real symmetric
+    matrix using its eigenvalue decomposition and including all eigenvalues
+    with 'large' absolute value.
+
+    Parameters
+    ----------
+    a : (N, N) array_like
+        Real symmetric or complex hermetian matrix to be pseudo-inverted
+
+    atol : float, optional
+        Absolute threshold term, default value is 0.
+
+        .. versionadded:: 1.7.0
+
+    rtol : float, optional
+        Relative threshold term, default value is ``N * eps`` where
+        ``eps`` is the machine precision value of the datatype of ``a``.
+
+        .. versionadded:: 1.7.0
+
+    lower : bool, optional
+        Whether the pertinent array data is taken from the lower or upper
+        triangle of `a`. (Default: lower)
+    return_rank : bool, optional
+        If True, return the effective rank of the matrix.
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    B : (N, N) ndarray
+        The pseudo-inverse of matrix `a`.
+    rank : int
+        The effective rank of the matrix.  Returned if `return_rank` is True.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue algorithm does not converge.
+
+    See Also
+    --------
+    pinv : Moore-Penrose pseudoinverse of a matrix.
+
+    Examples
+    --------
+
+    For a more detailed example see `pinv`.
+
+    >>> import numpy as np
+    >>> from scipy.linalg import pinvh
+    >>> rng = np.random.default_rng()
+    >>> a = rng.standard_normal((9, 6))
+    >>> a = np.dot(a, a.T)
+    >>> B = pinvh(a)
+    >>> np.allclose(a, a @ B @ a)
+    True
+    >>> np.allclose(B, B @ a @ B)
+    True
+
+    """
+    a = _asarray_validated(a, check_finite=check_finite)
+    s, u = _decomp.eigh(a, lower=lower, check_finite=False, driver='ev')
+    t = u.dtype.char.lower()
+    maxS = np.max(np.abs(s), initial=0.)
+
+    atol = 0. if atol is None else atol
+    rtol = max(a.shape) * np.finfo(t).eps if (rtol is None) else rtol
+
+    if (atol < 0.) or (rtol < 0.):
+        raise ValueError("atol and rtol values must be positive.")
+
+    val = atol + maxS * rtol
+    above_cutoff = (abs(s) > val)
+
+    psigma_diag = 1.0 / s[above_cutoff]
+    u = u[:, above_cutoff]
+
+    B = (u * psigma_diag) @ u.conj().T
+
+    if return_rank:
+        return B, len(psigma_diag)
+    else:
+        return B
+
+
+def matrix_balance(A, permute=True, scale=True, separate=False,
+                   overwrite_a=False):
+    """
+    Compute a diagonal similarity transformation for row/column balancing.
+
+    The balancing tries to equalize the row and column 1-norms by applying
+    a similarity transformation such that the magnitude variation of the
+    matrix entries is reflected to the scaling matrices.
+
+    Moreover, if enabled, the matrix is first permuted to isolate the upper
+    triangular parts of the matrix and, again if scaling is also enabled,
+    only the remaining subblocks are subjected to scaling.
+
+    The balanced matrix satisfies the following equality
+
+    .. math::
+
+                        B = T^{-1} A T
+
+    The scaling coefficients are approximated to the nearest power of 2
+    to avoid round-off errors.
+
+    Parameters
+    ----------
+    A : (n, n) array_like
+        Square data matrix for the balancing.
+    permute : bool, optional
+        The selector to define whether permutation of A is also performed
+        prior to scaling.
+    scale : bool, optional
+        The selector to turn on and off the scaling. If False, the matrix
+        will not be scaled.
+    separate : bool, optional
+        This switches from returning a full matrix of the transformation
+        to a tuple of two separate 1-D permutation and scaling arrays.
+    overwrite_a : bool, optional
+        This is passed to xGEBAL directly. Essentially, overwrites the result
+        to the data. It might increase the space efficiency. See LAPACK manual
+        for details. This is False by default.
+
+    Returns
+    -------
+    B : (n, n) ndarray
+        Balanced matrix
+    T : (n, n) ndarray
+        A possibly permuted diagonal matrix whose nonzero entries are
+        integer powers of 2 to avoid numerical truncation errors.
+    scale, perm : (n,) ndarray
+        If ``separate`` keyword is set to True then instead of the array
+        ``T`` above, the scaling and the permutation vectors are given
+        separately as a tuple without allocating the full array ``T``.
+
+    Notes
+    -----
+    This algorithm is particularly useful for eigenvalue and matrix
+    decompositions and in many cases it is already called by various
+    LAPACK routines.
+
+    The algorithm is based on the well-known technique of [1]_ and has
+    been modified to account for special cases. See [2]_ for details
+    which have been implemented since LAPACK v3.5.0. Before this version
+    there are corner cases where balancing can actually worsen the
+    conditioning. See [3]_ for such examples.
+
+    The code is a wrapper around LAPACK's xGEBAL routine family for matrix
+    balancing.
+
+    .. versionadded:: 0.19.0
+
+    References
+    ----------
+    .. [1] B.N. Parlett and C. Reinsch, "Balancing a Matrix for
+       Calculation of Eigenvalues and Eigenvectors", Numerische Mathematik,
+       Vol.13(4), 1969, :doi:`10.1007/BF02165404`
+    .. [2] R. James, J. Langou, B.R. Lowery, "On matrix balancing and
+       eigenvector computation", 2014, :arxiv:`1401.5766`
+    .. [3] D.S. Watkins. A case where balancing is harmful.
+       Electron. Trans. Numer. Anal, Vol.23, 2006.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import linalg
+    >>> x = np.array([[1,2,0], [9,1,0.01], [1,2,10*np.pi]])
+
+    >>> y, permscale = linalg.matrix_balance(x)
+    >>> np.abs(x).sum(axis=0) / np.abs(x).sum(axis=1)
+    array([ 3.66666667,  0.4995005 ,  0.91312162])
+
+    >>> np.abs(y).sum(axis=0) / np.abs(y).sum(axis=1)
+    array([ 1.2       ,  1.27041742,  0.92658316])  # may vary
+
+    >>> permscale  # only powers of 2 (0.5 == 2^(-1))
+    array([[  0.5,   0. ,  0. ],  # may vary
+           [  0. ,   1. ,  0. ],
+           [  0. ,   0. ,  1. ]])
+
+    """
+
+    A = np.atleast_2d(_asarray_validated(A, check_finite=True))
+
+    if not np.equal(*A.shape):
+        raise ValueError('The data matrix for balancing should be square.')
+
+    # accommodate empty arrays
+    if A.size == 0:
+        b_n, t_n = matrix_balance(np.eye(2, dtype=A.dtype))
+        B = np.empty_like(A, dtype=b_n.dtype)
+        if separate:
+            scaling = np.ones_like(A, shape=len(A))
+            perm = np.arange(len(A))
+            return B, (scaling, perm)
+        return B, np.empty_like(A, dtype=t_n.dtype)
+
+    gebal = get_lapack_funcs(('gebal'), (A,))
+    B, lo, hi, ps, info = gebal(A, scale=scale, permute=permute,
+                                overwrite_a=overwrite_a)
+
+    if info < 0:
+        raise ValueError('xGEBAL exited with the internal error '
+                         f'"illegal value in argument number {-info}.". See '
+                         'LAPACK documentation for the xGEBAL error codes.')
+
+    # Separate the permutations from the scalings and then convert to int
+    scaling = np.ones_like(ps, dtype=float)
+    scaling[lo:hi+1] = ps[lo:hi+1]
+
+    # gebal uses 1-indexing
+    ps = ps.astype(int, copy=False) - 1
+    n = A.shape[0]
+    perm = np.arange(n)
+
+    # LAPACK permutes with the ordering n --> hi, then 0--> lo
+    if hi < n:
+        for ind, x in enumerate(ps[hi+1:][::-1], 1):
+            if n-ind == x:
+                continue
+            perm[[x, n-ind]] = perm[[n-ind, x]]
+
+    if lo > 0:
+        for ind, x in enumerate(ps[:lo]):
+            if ind == x:
+                continue
+            perm[[x, ind]] = perm[[ind, x]]
+
+    if separate:
+        return B, (scaling, perm)
+
+    # get the inverse permutation
+    iperm = np.empty_like(perm)
+    iperm[perm] = np.arange(n)
+
+    return B, np.diag(scaling)[iperm, :]
+
+
+def _validate_args_for_toeplitz_ops(c_or_cr, b, check_finite, keep_b_shape,
+                                    enforce_square=True):
+    """Validate arguments and format inputs for toeplitz functions
+
+    Parameters
+    ----------
+    c_or_cr : array_like or tuple of (array_like, array_like)
+        The vector ``c``, or a tuple of arrays (``c``, ``r``). Whatever the
+        actual shape of ``c``, it will be converted to a 1-D array. If not
+        supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
+        real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
+        of the Toeplitz matrix is ``[c[0], r[1:]]``. Whatever the actual shape
+        of ``r``, it will be converted to a 1-D array.
+    b : (M,) or (M, K) array_like
+        Right-hand side in ``T x = b``.
+    check_finite : bool
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (result entirely NaNs) if the inputs do contain infinities or NaNs.
+    keep_b_shape : bool
+        Whether to convert a (M,) dimensional b into a (M, 1) dimensional
+        matrix.
+    enforce_square : bool, optional
+        If True (default), this verifies that the Toeplitz matrix is square.
+
+    Returns
+    -------
+    r : array
+        1d array corresponding to the first row of the Toeplitz matrix.
+    c: array
+        1d array corresponding to the first column of the Toeplitz matrix.
+    b: array
+        (M,), (M, 1) or (M, K) dimensional array, post validation,
+        corresponding to ``b``.
+    dtype: numpy datatype
+        ``dtype`` stores the datatype of ``r``, ``c`` and ``b``. If any of
+        ``r``, ``c`` or ``b`` are complex, ``dtype`` is ``np.complex128``,
+        otherwise, it is ``np.float``.
+    b_shape: tuple
+        Shape of ``b`` after passing it through ``_asarray_validated``.
+
+    """
+
+    if isinstance(c_or_cr, tuple):
+        c, r = c_or_cr
+        c = _asarray_validated(c, check_finite=check_finite)
+        r = _asarray_validated(r, check_finite=check_finite)
+    else:
+        c = _asarray_validated(c_or_cr, check_finite=check_finite)
+        r = c.conjugate()
+
+    if c.ndim > 1 or r.ndim > 1:
+        msg = ("Beginning in SciPy 1.17, multidimensional input will be treated as a "
+               "batch, not `ravel`ed. To preserve the existing behavior and silence "
+               "this warning, `ravel` arguments before passing them to "
+               "`toeplitz`, `matmul_toeplitz`, and `solve_toeplitz`.")
+        warnings.warn(msg, FutureWarning, stacklevel=2)
+        c = c.ravel()
+        r = r.ravel()
+
+    if b is None:
+        raise ValueError('`b` must be an array, not None.')
+
+    b = _asarray_validated(b, check_finite=check_finite)
+    b_shape = b.shape
+
+    is_not_square = r.shape[0] != c.shape[0]
+    if (enforce_square and is_not_square) or b.shape[0] != r.shape[0]:
+        raise ValueError('Incompatible dimensions.')
+
+    is_cmplx = np.iscomplexobj(r) or np.iscomplexobj(c) or np.iscomplexobj(b)
+    dtype = np.complex128 if is_cmplx else np.float64
+    r, c, b = (np.asarray(i, dtype=dtype) for i in (r, c, b))
+
+    if b.ndim == 1 and not keep_b_shape:
+        b = b.reshape(-1, 1)
+    elif b.ndim != 1:
+        b = b.reshape(b.shape[0], -1 if b.size > 0 else 0)
+
+    return r, c, b, dtype, b_shape
+
+
+def matmul_toeplitz(c_or_cr, x, check_finite=False, workers=None):
+    r"""Efficient Toeplitz Matrix-Matrix Multiplication using FFT
+
+    This function returns the matrix multiplication between a Toeplitz
+    matrix and a dense matrix.
+
+    The Toeplitz matrix has constant diagonals, with c as its first column
+    and r as its first row. If r is not given, ``r == conjugate(c)`` is
+    assumed.
+
+    .. warning::
+
+        Beginning in SciPy 1.17, multidimensional input will be treated as a batch,
+        not ``ravel``\ ed. To preserve the existing behavior, ``ravel`` arguments
+        before passing them to `matmul_toeplitz`.
+
+    Parameters
+    ----------
+    c_or_cr : array_like or tuple of (array_like, array_like)
+        The vector ``c``, or a tuple of arrays (``c``, ``r``). If not
+        supplied, ``r = conjugate(c)`` is assumed; in this case, if c[0] is
+        real, the Toeplitz matrix is Hermitian. r[0] is ignored; the first row
+        of the Toeplitz matrix is ``[c[0], r[1:]]``.
+    x : (M,) or (M, K) array_like
+        Matrix with which to multiply.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (result entirely NaNs) if the inputs do contain infinities or NaNs.
+    workers : int, optional
+        To pass to scipy.fft.fft and ifft. Maximum number of workers to use
+        for parallel computation. If negative, the value wraps around from
+        ``os.cpu_count()``. See scipy.fft.fft for more details.
+
+    Returns
+    -------
+    T @ x : (M,) or (M, K) ndarray
+        The result of the matrix multiplication ``T @ x``. Shape of return
+        matches shape of `x`.
+
+    See Also
+    --------
+    toeplitz : Toeplitz matrix
+    solve_toeplitz : Solve a Toeplitz system using Levinson Recursion
+
+    Notes
+    -----
+    The Toeplitz matrix is embedded in a circulant matrix and the FFT is used
+    to efficiently calculate the matrix-matrix product.
+
+    Because the computation is based on the FFT, integer inputs will
+    result in floating point outputs.  This is unlike NumPy's `matmul`,
+    which preserves the data type of the input.
+
+    This is partly based on the implementation that can be found in [1]_,
+    licensed under the MIT license. More information about the method can be
+    found in reference [2]_. References [3]_ and [4]_ have more reference
+    implementations in Python.
+
+    .. versionadded:: 1.6.0
+
+    References
+    ----------
+    .. [1] Jacob R Gardner, Geoff Pleiss, David Bindel, Kilian
+       Q Weinberger, Andrew Gordon Wilson, "GPyTorch: Blackbox Matrix-Matrix
+       Gaussian Process Inference with GPU Acceleration" with contributions
+       from Max Balandat and Ruihan Wu. Available online:
+       https://github.com/cornellius-gp/gpytorch
+
+    .. [2] J. Demmel, P. Koev, and X. Li, "A Brief Survey of Direct Linear
+       Solvers". In Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der
+       Vorst, editors. Templates for the Solution of Algebraic Eigenvalue
+       Problems: A Practical Guide. SIAM, Philadelphia, 2000. Available at:
+       http://www.netlib.org/utk/people/JackDongarra/etemplates/node384.html
+
+    .. [3] R. Scheibler, E. Bezzam, I. Dokmanic, Pyroomacoustics: A Python
+       package for audio room simulations and array processing algorithms,
+       Proc. IEEE ICASSP, Calgary, CA, 2018.
+       https://github.com/LCAV/pyroomacoustics/blob/pypi-release/
+       pyroomacoustics/adaptive/util.py
+
+    .. [4] Marano S, Edwards B, Ferrari G and Fah D (2017), "Fitting
+       Earthquake Spectra: Colored Noise and Incomplete Data", Bulletin of
+       the Seismological Society of America., January, 2017. Vol. 107(1),
+       pp. 276-291.
+
+    Examples
+    --------
+    Multiply the Toeplitz matrix T with matrix x::
+
+            [ 1 -1 -2 -3]       [1 10]
+        T = [ 3  1 -1 -2]   x = [2 11]
+            [ 6  3  1 -1]       [2 11]
+            [10  6  3  1]       [5 19]
+
+    To specify the Toeplitz matrix, only the first column and the first
+    row are needed.
+
+    >>> import numpy as np
+    >>> c = np.array([1, 3, 6, 10])    # First column of T
+    >>> r = np.array([1, -1, -2, -3])  # First row of T
+    >>> x = np.array([[1, 10], [2, 11], [2, 11], [5, 19]])
+
+    >>> from scipy.linalg import toeplitz, matmul_toeplitz
+    >>> matmul_toeplitz((c, r), x)
+    array([[-20., -80.],
+           [ -7.,  -8.],
+           [  9.,  85.],
+           [ 33., 218.]])
+
+    Check the result by creating the full Toeplitz matrix and
+    multiplying it by ``x``.
+
+    >>> toeplitz(c, r) @ x
+    array([[-20, -80],
+           [ -7,  -8],
+           [  9,  85],
+           [ 33, 218]])
+
+    The full matrix is never formed explicitly, so this routine
+    is suitable for very large Toeplitz matrices.
+
+    >>> n = 1000000
+    >>> matmul_toeplitz([1] + [0]*(n-1), np.ones(n))
+    array([1., 1., 1., ..., 1., 1., 1.], shape=(1000000,))
+
+    """
+
+    from ..fft import fft, ifft, rfft, irfft
+
+    r, c, x, dtype, x_shape = _validate_args_for_toeplitz_ops(
+        c_or_cr, x, check_finite, keep_b_shape=False, enforce_square=False)
+    n, m = x.shape
+
+    T_nrows = len(c)
+    T_ncols = len(r)
+    p = T_nrows + T_ncols - 1  # equivalent to len(embedded_col)
+    return_shape = (T_nrows,) if len(x_shape) == 1 else (T_nrows, m)
+
+    # accommodate empty arrays
+    if x.size == 0:
+        return np.empty_like(x, shape=return_shape)
+
+    embedded_col = np.concatenate((c, r[-1:0:-1]))
+
+    if np.iscomplexobj(embedded_col) or np.iscomplexobj(x):
+        fft_mat = fft(embedded_col, axis=0, workers=workers).reshape(-1, 1)
+        fft_x = fft(x, n=p, axis=0, workers=workers)
+
+        mat_times_x = ifft(fft_mat*fft_x, axis=0,
+                           workers=workers)[:T_nrows, :]
+    else:
+        # Real inputs; using rfft is faster
+        fft_mat = rfft(embedded_col, axis=0, workers=workers).reshape(-1, 1)
+        fft_x = rfft(x, n=p, axis=0, workers=workers)
+
+        mat_times_x = irfft(fft_mat*fft_x, axis=0,
+                            workers=workers, n=p)[:T_nrows, :]
+
+    return mat_times_x.reshape(*return_shape)

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_blas_subroutines.h

@@ -0,0 +1,164 @@
+/*
+This file was generated by _generate_pyx.py.
+Do not edit this file directly.
+*/
+
+#include "npy_cblas.h"
+#include "fortran_defs.h"
+
+#ifdef __cplusplus
+extern "C" {
+#endif
+
+void BLAS_FUNC(caxpy)(int *n, npy_complex64 *ca, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy);
+void BLAS_FUNC(ccopy)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy);
+void F_FUNC(cdotcwrp,CDOTCWRP)(npy_complex64 *out, int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy);
+void F_FUNC(cdotuwrp,CDOTUWRP)(npy_complex64 *out, int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy);
+void BLAS_FUNC(cgbmv)(char *trans, int *m, int *n, int *kl, int *ku, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy);
+void BLAS_FUNC(cgemm)(char *transa, char *transb, int *m, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc);
+void BLAS_FUNC(cgemv)(char *trans, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy);
+void BLAS_FUNC(cgerc)(int *m, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda);
+void BLAS_FUNC(cgeru)(int *m, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda);
+void BLAS_FUNC(chbmv)(char *uplo, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy);
+void BLAS_FUNC(chemm)(char *side, char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc);
+void BLAS_FUNC(chemv)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy);
+void BLAS_FUNC(cher)(char *uplo, int *n, float *alpha, npy_complex64 *x, int *incx, npy_complex64 *a, int *lda);
+void BLAS_FUNC(cher2)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *a, int *lda);
+void BLAS_FUNC(cher2k)(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, float *beta, npy_complex64 *c, int *ldc);
+void BLAS_FUNC(cherk)(char *uplo, char *trans, int *n, int *k, float *alpha, npy_complex64 *a, int *lda, float *beta, npy_complex64 *c, int *ldc);
+void BLAS_FUNC(chpmv)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *ap, npy_complex64 *x, int *incx, npy_complex64 *beta, npy_complex64 *y, int *incy);
+void BLAS_FUNC(chpr)(char *uplo, int *n, float *alpha, npy_complex64 *x, int *incx, npy_complex64 *ap);
+void BLAS_FUNC(chpr2)(char *uplo, int *n, npy_complex64 *alpha, npy_complex64 *x, int *incx, npy_complex64 *y, int *incy, npy_complex64 *ap);
+void BLAS_FUNC(crotg)(npy_complex64 *ca, npy_complex64 *cb, float *c, npy_complex64 *s);
+void BLAS_FUNC(cscal)(int *n, npy_complex64 *ca, npy_complex64 *cx, int *incx);
+void BLAS_FUNC(csrot)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy, float *c, float *s);
+void BLAS_FUNC(csscal)(int *n, float *sa, npy_complex64 *cx, int *incx);
+void BLAS_FUNC(cswap)(int *n, npy_complex64 *cx, int *incx, npy_complex64 *cy, int *incy);
+void BLAS_FUNC(csymm)(char *side, char *uplo, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc);
+void BLAS_FUNC(csyr2k)(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb, npy_complex64 *beta, npy_complex64 *c, int *ldc);
+void BLAS_FUNC(csyrk)(char *uplo, char *trans, int *n, int *k, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *beta, npy_complex64 *c, int *ldc);
+void BLAS_FUNC(ctbmv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx);
+void BLAS_FUNC(ctbsv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx);
+void BLAS_FUNC(ctpmv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *ap, npy_complex64 *x, int *incx);
+void BLAS_FUNC(ctpsv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *ap, npy_complex64 *x, int *incx);
+void BLAS_FUNC(ctrmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb);
+void BLAS_FUNC(ctrmv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx);
+void BLAS_FUNC(ctrsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex64 *alpha, npy_complex64 *a, int *lda, npy_complex64 *b, int *ldb);
+void BLAS_FUNC(ctrsv)(char *uplo, char *trans, char *diag, int *n, npy_complex64 *a, int *lda, npy_complex64 *x, int *incx);
+double BLAS_FUNC(dasum)(int *n, double *dx, int *incx);
+void BLAS_FUNC(daxpy)(int *n, double *da, double *dx, int *incx, double *dy, int *incy);
+double BLAS_FUNC(dcabs1)(npy_complex128 *z);
+void BLAS_FUNC(dcopy)(int *n, double *dx, int *incx, double *dy, int *incy);
+double BLAS_FUNC(ddot)(int *n, double *dx, int *incx, double *dy, int *incy);
+void BLAS_FUNC(dgbmv)(char *trans, int *m, int *n, int *kl, int *ku, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy);
+void BLAS_FUNC(dgemm)(char *transa, char *transb, int *m, int *n, int *k, double *alpha, double *a, int *lda, double *b, int *ldb, double *beta, double *c, int *ldc);
+void BLAS_FUNC(dgemv)(char *trans, int *m, int *n, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy);
+void BLAS_FUNC(dger)(int *m, int *n, double *alpha, double *x, int *incx, double *y, int *incy, double *a, int *lda);
+double BLAS_FUNC(dnrm2)(int *n, double *x, int *incx);
+void BLAS_FUNC(drot)(int *n, double *dx, int *incx, double *dy, int *incy, double *c, double *s);
+void BLAS_FUNC(drotg)(double *da, double *db, double *c, double *s);
+void BLAS_FUNC(drotm)(int *n, double *dx, int *incx, double *dy, int *incy, double *dparam);
+void BLAS_FUNC(drotmg)(double *dd1, double *dd2, double *dx1, double *dy1, double *dparam);
+void BLAS_FUNC(dsbmv)(char *uplo, int *n, int *k, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy);
+void BLAS_FUNC(dscal)(int *n, double *da, double *dx, int *incx);
+double BLAS_FUNC(dsdot)(int *n, float *sx, int *incx, float *sy, int *incy);
+void BLAS_FUNC(dspmv)(char *uplo, int *n, double *alpha, double *ap, double *x, int *incx, double *beta, double *y, int *incy);
+void BLAS_FUNC(dspr)(char *uplo, int *n, double *alpha, double *x, int *incx, double *ap);
+void BLAS_FUNC(dspr2)(char *uplo, int *n, double *alpha, double *x, int *incx, double *y, int *incy, double *ap);
+void BLAS_FUNC(dswap)(int *n, double *dx, int *incx, double *dy, int *incy);
+void BLAS_FUNC(dsymm)(char *side, char *uplo, int *m, int *n, double *alpha, double *a, int *lda, double *b, int *ldb, double *beta, double *c, int *ldc);
+void BLAS_FUNC(dsymv)(char *uplo, int *n, double *alpha, double *a, int *lda, double *x, int *incx, double *beta, double *y, int *incy);
+void BLAS_FUNC(dsyr)(char *uplo, int *n, double *alpha, double *x, int *incx, double *a, int *lda);
+void BLAS_FUNC(dsyr2)(char *uplo, int *n, double *alpha, double *x, int *incx, double *y, int *incy, double *a, int *lda);
+void BLAS_FUNC(dsyr2k)(char *uplo, char *trans, int *n, int *k, double *alpha, double *a, int *lda, double *b, int *ldb, double *beta, double *c, int *ldc);
+void BLAS_FUNC(dsyrk)(char *uplo, char *trans, int *n, int *k, double *alpha, double *a, int *lda, double *beta, double *c, int *ldc);
+void BLAS_FUNC(dtbmv)(char *uplo, char *trans, char *diag, int *n, int *k, double *a, int *lda, double *x, int *incx);
+void BLAS_FUNC(dtbsv)(char *uplo, char *trans, char *diag, int *n, int *k, double *a, int *lda, double *x, int *incx);
+void BLAS_FUNC(dtpmv)(char *uplo, char *trans, char *diag, int *n, double *ap, double *x, int *incx);
+void BLAS_FUNC(dtpsv)(char *uplo, char *trans, char *diag, int *n, double *ap, double *x, int *incx);
+void BLAS_FUNC(dtrmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, double *alpha, double *a, int *lda, double *b, int *ldb);
+void BLAS_FUNC(dtrmv)(char *uplo, char *trans, char *diag, int *n, double *a, int *lda, double *x, int *incx);
+void BLAS_FUNC(dtrsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, double *alpha, double *a, int *lda, double *b, int *ldb);
+void BLAS_FUNC(dtrsv)(char *uplo, char *trans, char *diag, int *n, double *a, int *lda, double *x, int *incx);
+double BLAS_FUNC(dzasum)(int *n, npy_complex128 *zx, int *incx);
+double BLAS_FUNC(dznrm2)(int *n, npy_complex128 *x, int *incx);
+int BLAS_FUNC(icamax)(int *n, npy_complex64 *cx, int *incx);
+int BLAS_FUNC(idamax)(int *n, double *dx, int *incx);
+int BLAS_FUNC(isamax)(int *n, float *sx, int *incx);
+int BLAS_FUNC(izamax)(int *n, npy_complex128 *zx, int *incx);
+int BLAS_FUNC(lsame)(char *ca, char *cb);
+float BLAS_FUNC(sasum)(int *n, float *sx, int *incx);
+void BLAS_FUNC(saxpy)(int *n, float *sa, float *sx, int *incx, float *sy, int *incy);
+float BLAS_FUNC(scasum)(int *n, npy_complex64 *cx, int *incx);
+float BLAS_FUNC(scnrm2)(int *n, npy_complex64 *x, int *incx);
+void BLAS_FUNC(scopy)(int *n, float *sx, int *incx, float *sy, int *incy);
+float BLAS_FUNC(sdot)(int *n, float *sx, int *incx, float *sy, int *incy);
+float BLAS_FUNC(sdsdot)(int *n, float *sb, float *sx, int *incx, float *sy, int *incy);
+void BLAS_FUNC(sgbmv)(char *trans, int *m, int *n, int *kl, int *ku, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy);
+void BLAS_FUNC(sgemm)(char *transa, char *transb, int *m, int *n, int *k, float *alpha, float *a, int *lda, float *b, int *ldb, float *beta, float *c, int *ldc);
+void BLAS_FUNC(sgemv)(char *trans, int *m, int *n, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy);
+void BLAS_FUNC(sger)(int *m, int *n, float *alpha, float *x, int *incx, float *y, int *incy, float *a, int *lda);
+float BLAS_FUNC(snrm2)(int *n, float *x, int *incx);
+void BLAS_FUNC(srot)(int *n, float *sx, int *incx, float *sy, int *incy, float *c, float *s);
+void BLAS_FUNC(srotg)(float *sa, float *sb, float *c, float *s);
+void BLAS_FUNC(srotm)(int *n, float *sx, int *incx, float *sy, int *incy, float *sparam);
+void BLAS_FUNC(srotmg)(float *sd1, float *sd2, float *sx1, float *sy1, float *sparam);
+void BLAS_FUNC(ssbmv)(char *uplo, int *n, int *k, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy);
+void BLAS_FUNC(sscal)(int *n, float *sa, float *sx, int *incx);
+void BLAS_FUNC(sspmv)(char *uplo, int *n, float *alpha, float *ap, float *x, int *incx, float *beta, float *y, int *incy);
+void BLAS_FUNC(sspr)(char *uplo, int *n, float *alpha, float *x, int *incx, float *ap);
+void BLAS_FUNC(sspr2)(char *uplo, int *n, float *alpha, float *x, int *incx, float *y, int *incy, float *ap);
+void BLAS_FUNC(sswap)(int *n, float *sx, int *incx, float *sy, int *incy);
+void BLAS_FUNC(ssymm)(char *side, char *uplo, int *m, int *n, float *alpha, float *a, int *lda, float *b, int *ldb, float *beta, float *c, int *ldc);
+void BLAS_FUNC(ssymv)(char *uplo, int *n, float *alpha, float *a, int *lda, float *x, int *incx, float *beta, float *y, int *incy);
+void BLAS_FUNC(ssyr)(char *uplo, int *n, float *alpha, float *x, int *incx, float *a, int *lda);
+void BLAS_FUNC(ssyr2)(char *uplo, int *n, float *alpha, float *x, int *incx, float *y, int *incy, float *a, int *lda);
+void BLAS_FUNC(ssyr2k)(char *uplo, char *trans, int *n, int *k, float *alpha, float *a, int *lda, float *b, int *ldb, float *beta, float *c, int *ldc);
+void BLAS_FUNC(ssyrk)(char *uplo, char *trans, int *n, int *k, float *alpha, float *a, int *lda, float *beta, float *c, int *ldc);
+void BLAS_FUNC(stbmv)(char *uplo, char *trans, char *diag, int *n, int *k, float *a, int *lda, float *x, int *incx);
+void BLAS_FUNC(stbsv)(char *uplo, char *trans, char *diag, int *n, int *k, float *a, int *lda, float *x, int *incx);
+void BLAS_FUNC(stpmv)(char *uplo, char *trans, char *diag, int *n, float *ap, float *x, int *incx);
+void BLAS_FUNC(stpsv)(char *uplo, char *trans, char *diag, int *n, float *ap, float *x, int *incx);
+void BLAS_FUNC(strmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, float *alpha, float *a, int *lda, float *b, int *ldb);
+void BLAS_FUNC(strmv)(char *uplo, char *trans, char *diag, int *n, float *a, int *lda, float *x, int *incx);
+void BLAS_FUNC(strsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, float *alpha, float *a, int *lda, float *b, int *ldb);
+void BLAS_FUNC(strsv)(char *uplo, char *trans, char *diag, int *n, float *a, int *lda, float *x, int *incx);
+void BLAS_FUNC(zaxpy)(int *n, npy_complex128 *za, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy);
+void BLAS_FUNC(zcopy)(int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy);
+void F_FUNC(zdotcwrp,ZDOTCWRP)(npy_complex128 *out, int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy);
+void F_FUNC(zdotuwrp,ZDOTUWRP)(npy_complex128 *out, int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy);
+void BLAS_FUNC(zdrot)(int *n, npy_complex128 *cx, int *incx, npy_complex128 *cy, int *incy, double *c, double *s);
+void BLAS_FUNC(zdscal)(int *n, double *da, npy_complex128 *zx, int *incx);
+void BLAS_FUNC(zgbmv)(char *trans, int *m, int *n, int *kl, int *ku, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy);
+void BLAS_FUNC(zgemm)(char *transa, char *transb, int *m, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc);
+void BLAS_FUNC(zgemv)(char *trans, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy);
+void BLAS_FUNC(zgerc)(int *m, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda);
+void BLAS_FUNC(zgeru)(int *m, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda);
+void BLAS_FUNC(zhbmv)(char *uplo, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy);
+void BLAS_FUNC(zhemm)(char *side, char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc);
+void BLAS_FUNC(zhemv)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy);
+void BLAS_FUNC(zher)(char *uplo, int *n, double *alpha, npy_complex128 *x, int *incx, npy_complex128 *a, int *lda);
+void BLAS_FUNC(zher2)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *a, int *lda);
+void BLAS_FUNC(zher2k)(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, double *beta, npy_complex128 *c, int *ldc);
+void BLAS_FUNC(zherk)(char *uplo, char *trans, int *n, int *k, double *alpha, npy_complex128 *a, int *lda, double *beta, npy_complex128 *c, int *ldc);
+void BLAS_FUNC(zhpmv)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *ap, npy_complex128 *x, int *incx, npy_complex128 *beta, npy_complex128 *y, int *incy);
+void BLAS_FUNC(zhpr)(char *uplo, int *n, double *alpha, npy_complex128 *x, int *incx, npy_complex128 *ap);
+void BLAS_FUNC(zhpr2)(char *uplo, int *n, npy_complex128 *alpha, npy_complex128 *x, int *incx, npy_complex128 *y, int *incy, npy_complex128 *ap);
+void BLAS_FUNC(zrotg)(npy_complex128 *ca, npy_complex128 *cb, double *c, npy_complex128 *s);
+void BLAS_FUNC(zscal)(int *n, npy_complex128 *za, npy_complex128 *zx, int *incx);
+void BLAS_FUNC(zswap)(int *n, npy_complex128 *zx, int *incx, npy_complex128 *zy, int *incy);
+void BLAS_FUNC(zsymm)(char *side, char *uplo, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc);
+void BLAS_FUNC(zsyr2k)(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb, npy_complex128 *beta, npy_complex128 *c, int *ldc);
+void BLAS_FUNC(zsyrk)(char *uplo, char *trans, int *n, int *k, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *beta, npy_complex128 *c, int *ldc);
+void BLAS_FUNC(ztbmv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx);
+void BLAS_FUNC(ztbsv)(char *uplo, char *trans, char *diag, int *n, int *k, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx);
+void BLAS_FUNC(ztpmv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *ap, npy_complex128 *x, int *incx);
+void BLAS_FUNC(ztpsv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *ap, npy_complex128 *x, int *incx);
+void BLAS_FUNC(ztrmm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb);
+void BLAS_FUNC(ztrmv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx);
+void BLAS_FUNC(ztrsm)(char *side, char *uplo, char *transa, char *diag, int *m, int *n, npy_complex128 *alpha, npy_complex128 *a, int *lda, npy_complex128 *b, int *ldb);
+void BLAS_FUNC(ztrsv)(char *uplo, char *trans, char *diag, int *n, npy_complex128 *a, int *lda, npy_complex128 *x, int *incx);
+
+#ifdef __cplusplus
+}
+#endif

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pxd

@@ -0,0 +1,40 @@
+cimport numpy as cnp
+
+ctypedef fused lapack_t:
+    float
+    double
+    (float complex)
+    (double complex)
+
+ctypedef fused lapack_cz_t:
+    (float complex)
+    (double complex)
+
+ctypedef fused lapack_sd_t:
+    float
+    double
+
+ctypedef fused np_numeric_t:
+    cnp.int8_t
+    cnp.int16_t
+    cnp.int32_t
+    cnp.int64_t
+    cnp.uint8_t
+    cnp.uint16_t
+    cnp.uint32_t
+    cnp.uint64_t
+    cnp.float32_t
+    cnp.float64_t
+    cnp.longdouble_t
+    cnp.complex64_t
+    cnp.complex128_t
+
+ctypedef fused np_complex_numeric_t:
+    cnp.complex64_t
+    cnp.complex128_t
+
+
+cdef void swap_c_and_f_layout(lapack_t *a, lapack_t *b, int r, int c) noexcept nogil
+cdef (int, int) band_check_internal_c(np_numeric_t[:, ::1]A) noexcept nogil
+cdef bint is_sym_her_real_c_internal(np_numeric_t[:, ::1]A) noexcept nogil
+cdef bint is_sym_her_complex_c_internal(np_complex_numeric_t[:, ::1]A) noexcept nogil

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_cythonized_array_utils.pyi

@@ -0,0 +1,16 @@
+from numpy.typing import NDArray
+from typing import Any
+
+def bandwidth(a: NDArray[Any]) -> tuple[int, int]: ...
+
+def issymmetric(
+    a: NDArray[Any],
+    atol: None | float = ...,
+    rtol: None | float = ...,
+) -> bool: ...
+
+def ishermitian(
+    a: NDArray[Any],
+    atol: None | float = ...,
+    rtol: None | float = ...,
+) -> bool: ...

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_decomp.py

@@ -0,0 +1,1632 @@
+#
+# Author: Pearu Peterson, March 2002
+#
+# additions by Travis Oliphant, March 2002
+# additions by Eric Jones,      June 2002
+# additions by Johannes Loehnert, June 2006
+# additions by Bart Vandereycken, June 2006
+# additions by Andrew D Straw, May 2007
+# additions by Tiziano Zito, November 2008
+#
+# April 2010: Functions for LU, QR, SVD, Schur, and Cholesky decompositions
+# were moved to their own files. Still in this file are functions for
+# eigenstuff and for the Hessenberg form.
+
+__all__ = ['eig', 'eigvals', 'eigh', 'eigvalsh',
+           'eig_banded', 'eigvals_banded',
+           'eigh_tridiagonal', 'eigvalsh_tridiagonal', 'hessenberg', 'cdf2rdf']
+
+import numpy as np
+from numpy import (array, isfinite, inexact, nonzero, iscomplexobj,
+                   flatnonzero, conj, asarray, argsort, empty,
+                   iscomplex, zeros, einsum, eye, inf)
+# Local imports
+from scipy._lib._util import _asarray_validated
+from ._misc import LinAlgError, _datacopied, norm
+from .lapack import get_lapack_funcs, _compute_lwork
+
+
+_I = np.array(1j, dtype='F')
+
+
+def _make_complex_eigvecs(w, vin, dtype):
+    """
+    Produce complex-valued eigenvectors from LAPACK DGGEV real-valued output
+    """
+    # - see LAPACK man page DGGEV at ALPHAI
+    v = np.array(vin, dtype=dtype)
+    m = (w.imag > 0)
+    m[:-1] |= (w.imag[1:] < 0)  # workaround for LAPACK bug, cf. ticket #709
+    for i in flatnonzero(m):
+        v.imag[:, i] = vin[:, i+1]
+        conj(v[:, i], v[:, i+1])
+    return v
+
+
+def _make_eigvals(alpha, beta, homogeneous_eigvals):
+    if homogeneous_eigvals:
+        if beta is None:
+            return np.vstack((alpha, np.ones_like(alpha)))
+        else:
+            return np.vstack((alpha, beta))
+    else:
+        if beta is None:
+            return alpha
+        else:
+            w = np.empty_like(alpha)
+            alpha_zero = (alpha == 0)
+            beta_zero = (beta == 0)
+            beta_nonzero = ~beta_zero
+            w[beta_nonzero] = alpha[beta_nonzero]/beta[beta_nonzero]
+            # Use np.inf for complex values too since
+            # 1/np.inf = 0, i.e., it correctly behaves as projective
+            # infinity.
+            w[~alpha_zero & beta_zero] = np.inf
+            if np.all(alpha.imag == 0):
+                w[alpha_zero & beta_zero] = np.nan
+            else:
+                w[alpha_zero & beta_zero] = complex(np.nan, np.nan)
+            return w
+
+
+def _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
+            homogeneous_eigvals):
+    ggev, = get_lapack_funcs(('ggev',), (a1, b1))
+    cvl, cvr = left, right
+    res = ggev(a1, b1, lwork=-1)
+    lwork = res[-2][0].real.astype(np.int_)
+    if ggev.typecode in 'cz':
+        alpha, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr, lwork,
+                                               overwrite_a, overwrite_b)
+        w = _make_eigvals(alpha, beta, homogeneous_eigvals)
+    else:
+        alphar, alphai, beta, vl, vr, work, info = ggev(a1, b1, cvl, cvr,
+                                                        lwork, overwrite_a,
+                                                        overwrite_b)
+        alpha = alphar + _I * alphai
+        w = _make_eigvals(alpha, beta, homogeneous_eigvals)
+    _check_info(info, 'generalized eig algorithm (ggev)')
+
+    only_real = np.all(w.imag == 0.0)
+    if not (ggev.typecode in 'cz' or only_real):
+        t = w.dtype.char
+        if left:
+            vl = _make_complex_eigvecs(w, vl, t)
+        if right:
+            vr = _make_complex_eigvecs(w, vr, t)
+
+    # the eigenvectors returned by the lapack function are NOT normalized
+    for i in range(vr.shape[0]):
+        if right:
+            vr[:, i] /= norm(vr[:, i])
+        if left:
+            vl[:, i] /= norm(vl[:, i])
+
+    if not (left or right):
+        return w
+    if left:
+        if right:
+            return w, vl, vr
+        return w, vl
+    return w, vr
+
+
+def eig(a, b=None, left=False, right=True, overwrite_a=False,
+        overwrite_b=False, check_finite=True, homogeneous_eigvals=False):
+    """
+    Solve an ordinary or generalized eigenvalue problem of a square matrix.
+
+    Find eigenvalues w and right or left eigenvectors of a general matrix::
+
+        a   vr[:,i] = w[i]        b   vr[:,i]
+        a.H vl[:,i] = w[i].conj() b.H vl[:,i]
+
+    where ``.H`` is the Hermitian conjugation.
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        A complex or real matrix whose eigenvalues and eigenvectors
+        will be computed.
+    b : (M, M) array_like, optional
+        Right-hand side matrix in a generalized eigenvalue problem.
+        Default is None, identity matrix is assumed.
+    left : bool, optional
+        Whether to calculate and return left eigenvectors.  Default is False.
+    right : bool, optional
+        Whether to calculate and return right eigenvectors.  Default is True.
+    overwrite_a : bool, optional
+        Whether to overwrite `a`; may improve performance.  Default is False.
+    overwrite_b : bool, optional
+        Whether to overwrite `b`; may improve performance.  Default is False.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+    homogeneous_eigvals : bool, optional
+        If True, return the eigenvalues in homogeneous coordinates.
+        In this case ``w`` is a (2, M) array so that::
+
+            w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
+
+        Default is False.
+
+    Returns
+    -------
+    w : (M,) or (2, M) double or complex ndarray
+        The eigenvalues, each repeated according to its
+        multiplicity. The shape is (M,) unless
+        ``homogeneous_eigvals=True``.
+    vl : (M, M) double or complex ndarray
+        The left eigenvector corresponding to the eigenvalue
+        ``w[i]`` is the column ``vl[:,i]``. Only returned if ``left=True``.
+        The left eigenvector is not normalized.
+    vr : (M, M) double or complex ndarray
+        The normalized right eigenvector corresponding to the eigenvalue
+        ``w[i]`` is the column ``vr[:,i]``.  Only returned if ``right=True``.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge.
+
+    See Also
+    --------
+    eigvals : eigenvalues of general arrays
+    eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays.
+    eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
+        band matrices
+    eigh_tridiagonal : eigenvalues and right eiegenvectors for
+        symmetric/Hermitian tridiagonal matrices
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import linalg
+    >>> a = np.array([[0., -1.], [1., 0.]])
+    >>> linalg.eigvals(a)
+    array([0.+1.j, 0.-1.j])
+
+    >>> b = np.array([[0., 1.], [1., 1.]])
+    >>> linalg.eigvals(a, b)
+    array([ 1.+0.j, -1.+0.j])
+
+    >>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
+    >>> linalg.eigvals(a, homogeneous_eigvals=True)
+    array([[3.+0.j, 8.+0.j, 7.+0.j],
+           [1.+0.j, 1.+0.j, 1.+0.j]])
+
+    >>> a = np.array([[0., -1.], [1., 0.]])
+    >>> linalg.eigvals(a) == linalg.eig(a)[0]
+    array([ True,  True])
+    >>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector
+    array([[-0.70710678+0.j        , -0.70710678-0.j        ],
+           [-0.        +0.70710678j, -0.        -0.70710678j]])
+    >>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector
+    array([[0.70710678+0.j        , 0.70710678-0.j        ],
+           [0.        -0.70710678j, 0.        +0.70710678j]])
+
+
+
+    """
+    a1 = _asarray_validated(a, check_finite=check_finite)
+    if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
+        raise ValueError('expected square matrix')
+
+    # accommodate square empty matrices
+    if a1.size == 0:
+        w_n, vr_n = eig(np.eye(2, dtype=a1.dtype))
+        w = np.empty_like(a1, shape=(0,), dtype=w_n.dtype)
+        w = _make_eigvals(w, None, homogeneous_eigvals)
+        vl = np.empty_like(a1, shape=(0, 0), dtype=vr_n.dtype)
+        vr = np.empty_like(a1, shape=(0, 0), dtype=vr_n.dtype)
+        if not (left or right):
+            return w
+        if left:
+            if right:
+                return w, vl, vr
+            return w, vl
+        return w, vr
+
+    overwrite_a = overwrite_a or (_datacopied(a1, a))
+    if b is not None:
+        b1 = _asarray_validated(b, check_finite=check_finite)
+        overwrite_b = overwrite_b or _datacopied(b1, b)
+        if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
+            raise ValueError('expected square matrix')
+        if b1.shape != a1.shape:
+            raise ValueError('a and b must have the same shape')
+        return _geneig(a1, b1, left, right, overwrite_a, overwrite_b,
+                       homogeneous_eigvals)
+
+    geev, geev_lwork = get_lapack_funcs(('geev', 'geev_lwork'), (a1,))
+    compute_vl, compute_vr = left, right
+
+    lwork = _compute_lwork(geev_lwork, a1.shape[0],
+                           compute_vl=compute_vl,
+                           compute_vr=compute_vr)
+
+    if geev.typecode in 'cz':
+        w, vl, vr, info = geev(a1, lwork=lwork,
+                               compute_vl=compute_vl,
+                               compute_vr=compute_vr,
+                               overwrite_a=overwrite_a)
+        w = _make_eigvals(w, None, homogeneous_eigvals)
+    else:
+        wr, wi, vl, vr, info = geev(a1, lwork=lwork,
+                                    compute_vl=compute_vl,
+                                    compute_vr=compute_vr,
+                                    overwrite_a=overwrite_a)
+        w = wr + _I * wi
+        w = _make_eigvals(w, None, homogeneous_eigvals)
+
+    _check_info(info, 'eig algorithm (geev)',
+                positive='did not converge (only eigenvalues '
+                         'with order >= %d have converged)')
+
+    only_real = np.all(w.imag == 0.0)
+    if not (geev.typecode in 'cz' or only_real):
+        t = w.dtype.char
+        if left:
+            vl = _make_complex_eigvecs(w, vl, t)
+        if right:
+            vr = _make_complex_eigvecs(w, vr, t)
+    if not (left or right):
+        return w
+    if left:
+        if right:
+            return w, vl, vr
+        return w, vl
+    return w, vr
+
+
+def eigh(a, b=None, *, lower=True, eigvals_only=False, overwrite_a=False,
+         overwrite_b=False, type=1, check_finite=True, subset_by_index=None,
+         subset_by_value=None, driver=None):
+    """
+    Solve a standard or generalized eigenvalue problem for a complex
+    Hermitian or real symmetric matrix.
+
+    Find eigenvalues array ``w`` and optionally eigenvectors array ``v`` of
+    array ``a``, where ``b`` is positive definite such that for every
+    eigenvalue λ (i-th entry of w) and its eigenvector ``vi`` (i-th column of
+    ``v``) satisfies::
+
+                      a @ vi = λ * b @ vi
+        vi.conj().T @ a @ vi = λ
+        vi.conj().T @ b @ vi = 1
+
+    In the standard problem, ``b`` is assumed to be the identity matrix.
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        A complex Hermitian or real symmetric matrix whose eigenvalues and
+        eigenvectors will be computed.
+    b : (M, M) array_like, optional
+        A complex Hermitian or real symmetric definite positive matrix in.
+        If omitted, identity matrix is assumed.
+    lower : bool, optional
+        Whether the pertinent array data is taken from the lower or upper
+        triangle of ``a`` and, if applicable, ``b``. (Default: lower)
+    eigvals_only : bool, optional
+        Whether to calculate only eigenvalues and no eigenvectors.
+        (Default: both are calculated)
+    subset_by_index : iterable, optional
+        If provided, this two-element iterable defines the start and the end
+        indices of the desired eigenvalues (ascending order and 0-indexed).
+        To return only the second smallest to fifth smallest eigenvalues,
+        ``[1, 4]`` is used. ``[n-3, n-1]`` returns the largest three. Only
+        available with "evr", "evx", and "gvx" drivers. The entries are
+        directly converted to integers via ``int()``.
+    subset_by_value : iterable, optional
+        If provided, this two-element iterable defines the half-open interval
+        ``(a, b]`` that, if any, only the eigenvalues between these values
+        are returned. Only available with "evr", "evx", and "gvx" drivers. Use
+        ``np.inf`` for the unconstrained ends.
+    driver : str, optional
+        Defines which LAPACK driver should be used. Valid options are "ev",
+        "evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for
+        generalized (where b is not None) problems. See the Notes section.
+        The default for standard problems is "evr". For generalized problems,
+        "gvd" is used for full set, and "gvx" for subset requested cases.
+    type : int, optional
+        For the generalized problems, this keyword specifies the problem type
+        to be solved for ``w`` and ``v`` (only takes 1, 2, 3 as possible
+        inputs)::
+
+            1 =>     a @ v = w @ b @ v
+            2 => a @ b @ v = w @ v
+            3 => b @ a @ v = w @ v
+
+        This keyword is ignored for standard problems.
+    overwrite_a : bool, optional
+        Whether to overwrite data in ``a`` (may improve performance). Default
+        is False.
+    overwrite_b : bool, optional
+        Whether to overwrite data in ``b`` (may improve performance). Default
+        is False.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    w : (N,) ndarray
+        The N (N<=M) selected eigenvalues, in ascending order, each
+        repeated according to its multiplicity.
+    v : (M, N) ndarray
+        The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is
+        the column ``v[:,i]``. Only returned if ``eigvals_only=False``.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge, an error occurred, or
+        b matrix is not definite positive. Note that if input matrices are
+        not symmetric or Hermitian, no error will be reported but results will
+        be wrong.
+
+    See Also
+    --------
+    eigvalsh : eigenvalues of symmetric or Hermitian arrays
+    eig : eigenvalues and right eigenvectors for non-symmetric arrays
+    eigh_tridiagonal : eigenvalues and right eiegenvectors for
+        symmetric/Hermitian tridiagonal matrices
+
+    Notes
+    -----
+    This function does not check the input array for being Hermitian/symmetric
+    in order to allow for representing arrays with only their upper/lower
+    triangular parts. Also, note that even though not taken into account,
+    finiteness check applies to the whole array and unaffected by "lower"
+    keyword.
+
+    This function uses LAPACK drivers for computations in all possible keyword
+    combinations, prefixed with ``sy`` if arrays are real and ``he`` if
+    complex, e.g., a float array with "evr" driver is solved via
+    "syevr", complex arrays with "gvx" driver problem is solved via "hegvx"
+    etc.
+
+    As a brief summary, the slowest and the most robust driver is the
+    classical ``<sy/he>ev`` which uses symmetric QR. ``<sy/he>evr`` is seen as
+    the optimal choice for the most general cases. However, there are certain
+    occasions that ``<sy/he>evd`` computes faster at the expense of more
+    memory usage. ``<sy/he>evx``, while still being faster than ``<sy/he>ev``,
+    often performs worse than the rest except when very few eigenvalues are
+    requested for large arrays though there is still no performance guarantee.
+
+    Note that the underlying LAPACK algorithms are different depending on whether
+    `eigvals_only` is True or False --- thus the eigenvalues may differ
+    depending on whether eigenvectors are requested or not. The difference is
+    generally of the order of machine epsilon times the largest eigenvalue,
+    so is likely only visible for zero or nearly zero eigenvalues.
+
+    For the generalized problem, normalization with respect to the given
+    type argument::
+
+            type 1 and 3 :      v.conj().T @ a @ v = w
+            type 2       : inv(v).conj().T @ a @ inv(v) = w
+
+            type 1 or 2  :      v.conj().T @ b @ v  = I
+            type 3       : v.conj().T @ inv(b) @ v  = I
+
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import eigh
+    >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
+    >>> w, v = eigh(A)
+    >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
+    True
+
+    Request only the eigenvalues
+
+    >>> w = eigh(A, eigvals_only=True)
+
+    Request eigenvalues that are less than 10.
+
+    >>> A = np.array([[34, -4, -10, -7, 2],
+    ...               [-4, 7, 2, 12, 0],
+    ...               [-10, 2, 44, 2, -19],
+    ...               [-7, 12, 2, 79, -34],
+    ...               [2, 0, -19, -34, 29]])
+    >>> eigh(A, eigvals_only=True, subset_by_value=[-np.inf, 10])
+    array([6.69199443e-07, 9.11938152e+00])
+
+    Request the second smallest eigenvalue and its eigenvector
+
+    >>> w, v = eigh(A, subset_by_index=[1, 1])
+    >>> w
+    array([9.11938152])
+    >>> v.shape  # only a single column is returned
+    (5, 1)
+
+    """
+    # set lower
+    uplo = 'L' if lower else 'U'
+    # Set job for Fortran routines
+    _job = 'N' if eigvals_only else 'V'
+
+    drv_str = [None, "ev", "evd", "evr", "evx", "gv", "gvd", "gvx"]
+    if driver not in drv_str:
+        raise ValueError('"{}" is unknown. Possible values are "None", "{}".'
+                         ''.format(driver, '", "'.join(drv_str[1:])))
+
+    a1 = _asarray_validated(a, check_finite=check_finite)
+    if len(a1.shape) != 2 or a1.shape[0] != a1.shape[1]:
+        raise ValueError('expected square "a" matrix')
+
+    # accommodate square empty matrices
+    if a1.size == 0:
+        w_n, v_n = eigh(np.eye(2, dtype=a1.dtype))
+
+        w = np.empty_like(a1, shape=(0,), dtype=w_n.dtype)
+        v = np.empty_like(a1, shape=(0, 0), dtype=v_n.dtype)
+        if eigvals_only:
+            return w
+        else:
+            return w, v
+
+    overwrite_a = overwrite_a or (_datacopied(a1, a))
+    cplx = True if iscomplexobj(a1) else False
+    n = a1.shape[0]
+    drv_args = {'overwrite_a': overwrite_a}
+
+    if b is not None:
+        b1 = _asarray_validated(b, check_finite=check_finite)
+        overwrite_b = overwrite_b or _datacopied(b1, b)
+        if len(b1.shape) != 2 or b1.shape[0] != b1.shape[1]:
+            raise ValueError('expected square "b" matrix')
+
+        if b1.shape != a1.shape:
+            raise ValueError(f"wrong b dimensions {b1.shape}, should be {a1.shape}")
+
+        if type not in [1, 2, 3]:
+            raise ValueError('"type" keyword only accepts 1, 2, and 3.')
+
+        cplx = True if iscomplexobj(b1) else (cplx or False)
+        drv_args.update({'overwrite_b': overwrite_b, 'itype': type})
+
+    subset = (subset_by_index is not None) or (subset_by_value is not None)
+
+    # Both subsets can't be given
+    if subset_by_index and subset_by_value:
+        raise ValueError('Either index or value subset can be requested.')
+
+    # Check indices if given
+    if subset_by_index:
+        lo, hi = (int(x) for x in subset_by_index)
+        if not (0 <= lo <= hi < n):
+            raise ValueError('Requested eigenvalue indices are not valid. '
+                             f'Valid range is [0, {n-1}] and start <= end, but '
+                             f'start={lo}, end={hi} is given')
+        # fortran is 1-indexed
+        drv_args.update({'range': 'I', 'il': lo + 1, 'iu': hi + 1})
+
+    if subset_by_value:
+        lo, hi = subset_by_value
+        if not (-inf <= lo < hi <= inf):
+            raise ValueError('Requested eigenvalue bounds are not valid. '
+                             'Valid range is (-inf, inf) and low < high, but '
+                             f'low={lo}, high={hi} is given')
+
+        drv_args.update({'range': 'V', 'vl': lo, 'vu': hi})
+
+    # fix prefix for lapack routines
+    pfx = 'he' if cplx else 'sy'
+
+    # decide on the driver if not given
+    # first early exit on incompatible choice
+    if driver:
+        if b is None and (driver in ["gv", "gvd", "gvx"]):
+            raise ValueError(f'{driver} requires input b array to be supplied '
+                             'for generalized eigenvalue problems.')
+        if (b is not None) and (driver in ['ev', 'evd', 'evr', 'evx']):
+            raise ValueError(f'"{driver}" does not accept input b array '
+                             'for standard eigenvalue problems.')
+        if subset and (driver in ["ev", "evd", "gv", "gvd"]):
+            raise ValueError(f'"{driver}" cannot compute subsets of eigenvalues')
+
+    # Default driver is evr and gvd
+    else:
+        driver = "evr" if b is None else ("gvx" if subset else "gvd")
+
+    lwork_spec = {
+                  'syevd': ['lwork', 'liwork'],
+                  'syevr': ['lwork', 'liwork'],
+                  'heevd': ['lwork', 'liwork', 'lrwork'],
+                  'heevr': ['lwork', 'lrwork', 'liwork'],
+                  }
+
+    if b is None:  # Standard problem
+        drv, drvlw = get_lapack_funcs((pfx + driver, pfx+driver+'_lwork'),
+                                      [a1])
+        clw_args = {'n': n, 'lower': lower}
+        if driver == 'evd':
+            clw_args.update({'compute_v': 0 if _job == "N" else 1})
+
+        lw = _compute_lwork(drvlw, **clw_args)
+        # Multiple lwork vars
+        if isinstance(lw, tuple):
+            lwork_args = dict(zip(lwork_spec[pfx+driver], lw))
+        else:
+            lwork_args = {'lwork': lw}
+
+        drv_args.update({'lower': lower, 'compute_v': 0 if _job == "N" else 1})
+        w, v, *other_args, info = drv(a=a1, **drv_args, **lwork_args)
+
+    else:  # Generalized problem
+        # 'gvd' doesn't have lwork query
+        if driver == "gvd":
+            drv = get_lapack_funcs(pfx + "gvd", [a1, b1])
+            lwork_args = {}
+        else:
+            drv, drvlw = get_lapack_funcs((pfx + driver, pfx+driver+'_lwork'),
+                                          [a1, b1])
+            # generalized drivers use uplo instead of lower
+            lw = _compute_lwork(drvlw, n, uplo=uplo)
+            lwork_args = {'lwork': lw}
+
+        drv_args.update({'uplo': uplo, 'jobz': _job})
+
+        w, v, *other_args, info = drv(a=a1, b=b1, **drv_args, **lwork_args)
+
+    # m is always the first extra argument
+    w = w[:other_args[0]] if subset else w
+    v = v[:, :other_args[0]] if (subset and not eigvals_only) else v
+
+    # Check if we had a  successful exit
+    if info == 0:
+        if eigvals_only:
+            return w
+        else:
+            return w, v
+    else:
+        if info < -1:
+            raise LinAlgError(f'Illegal value in argument {-info} of internal '
+                              f'{drv.typecode + pfx + driver}')
+        elif info > n:
+            raise LinAlgError(f'The leading minor of order {info-n} of B is not '
+                              'positive definite. The factorization of B '
+                              'could not be completed and no eigenvalues '
+                              'or eigenvectors were computed.')
+        else:
+            drv_err = {'ev': 'The algorithm failed to converge; {} '
+                             'off-diagonal elements of an intermediate '
+                             'tridiagonal form did not converge to zero.',
+                       'evx': '{} eigenvectors failed to converge.',
+                       'evd': 'The algorithm failed to compute an eigenvalue '
+                              'while working on the submatrix lying in rows '
+                              'and columns {0}/{1} through mod({0},{1}).',
+                       'evr': 'Internal Error.'
+                       }
+            if driver in ['ev', 'gv']:
+                msg = drv_err['ev'].format(info)
+            elif driver in ['evx', 'gvx']:
+                msg = drv_err['evx'].format(info)
+            elif driver in ['evd', 'gvd']:
+                if eigvals_only:
+                    msg = drv_err['ev'].format(info)
+                else:
+                    msg = drv_err['evd'].format(info, n+1)
+            else:
+                msg = drv_err['evr']
+
+            raise LinAlgError(msg)
+
+
+_conv_dict = {0: 0, 1: 1, 2: 2,
+              'all': 0, 'value': 1, 'index': 2,
+              'a': 0, 'v': 1, 'i': 2}
+
+
+def _check_select(select, select_range, max_ev, max_len):
+    """Check that select is valid, convert to Fortran style."""
+    if isinstance(select, str):
+        select = select.lower()
+    try:
+        select = _conv_dict[select]
+    except KeyError as e:
+        raise ValueError('invalid argument for select') from e
+    vl, vu = 0., 1.
+    il = iu = 1
+    if select != 0:  # (non-all)
+        sr = asarray(select_range)
+        if sr.ndim != 1 or sr.size != 2 or sr[1] < sr[0]:
+            raise ValueError('select_range must be a 2-element array-like '
+                             'in nondecreasing order')
+        if select == 1:  # (value)
+            vl, vu = sr
+            if max_ev == 0:
+                max_ev = max_len
+        else:  # 2 (index)
+            if sr.dtype.char.lower() not in 'hilqp':
+                raise ValueError(
+                    f'when using select="i", select_range must '
+                    f'contain integers, got dtype {sr.dtype} ({sr.dtype.char})'
+                )
+            # translate Python (0 ... N-1) into Fortran (1 ... N) with + 1
+            il, iu = sr + 1
+            if min(il, iu) < 1 or max(il, iu) > max_len:
+                raise ValueError('select_range out of bounds')
+            max_ev = iu - il + 1
+    return select, vl, vu, il, iu, max_ev
+
+
+def eig_banded(a_band, lower=False, eigvals_only=False, overwrite_a_band=False,
+               select='a', select_range=None, max_ev=0, check_finite=True):
+    """
+    Solve real symmetric or complex Hermitian band matrix eigenvalue problem.
+
+    Find eigenvalues w and optionally right eigenvectors v of a::
+
+        a v[:,i] = w[i] v[:,i]
+        v.H v    = identity
+
+    The matrix a is stored in a_band either in lower diagonal or upper
+    diagonal ordered form:
+
+        a_band[u + i - j, j] == a[i,j]        (if upper form; i <= j)
+        a_band[    i - j, j] == a[i,j]        (if lower form; i >= j)
+
+    where u is the number of bands above the diagonal.
+
+    Example of a_band (shape of a is (6,6), u=2)::
+
+        upper form:
+        *   *   a02 a13 a24 a35
+        *   a01 a12 a23 a34 a45
+        a00 a11 a22 a33 a44 a55
+
+        lower form:
+        a00 a11 a22 a33 a44 a55
+        a10 a21 a32 a43 a54 *
+        a20 a31 a42 a53 *   *
+
+    Cells marked with * are not used.
+
+    Parameters
+    ----------
+    a_band : (u+1, M) array_like
+        The bands of the M by M matrix a.
+    lower : bool, optional
+        Is the matrix in the lower form. (Default is upper form)
+    eigvals_only : bool, optional
+        Compute only the eigenvalues and no eigenvectors.
+        (Default: calculate also eigenvectors)
+    overwrite_a_band : bool, optional
+        Discard data in a_band (may enhance performance)
+    select : {'a', 'v', 'i'}, optional
+        Which eigenvalues to calculate
+
+        ======  ========================================
+        select  calculated
+        ======  ========================================
+        'a'     All eigenvalues
+        'v'     Eigenvalues in the interval (min, max]
+        'i'     Eigenvalues with indices min <= i <= max
+        ======  ========================================
+    select_range : (min, max), optional
+        Range of selected eigenvalues
+    max_ev : int, optional
+        For select=='v', maximum number of eigenvalues expected.
+        For other values of select, has no meaning.
+
+        In doubt, leave this parameter untouched.
+
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    w : (M,) ndarray
+        The eigenvalues, in ascending order, each repeated according to its
+        multiplicity.
+    v : (M, M) float or complex ndarray
+        The normalized eigenvector corresponding to the eigenvalue w[i] is
+        the column v[:,i]. Only returned if ``eigvals_only=False``.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge.
+
+    See Also
+    --------
+    eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
+    eig : eigenvalues and right eigenvectors of general arrays.
+    eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
+    eigh_tridiagonal : eigenvalues and right eigenvectors for
+        symmetric/Hermitian tridiagonal matrices
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import eig_banded
+    >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
+    >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
+    >>> w, v = eig_banded(Ab, lower=True)
+    >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
+    True
+    >>> w = eig_banded(Ab, lower=True, eigvals_only=True)
+    >>> w
+    array([-4.26200532, -2.22987175,  3.95222349, 12.53965359])
+
+    Request only the eigenvalues between ``[-3, 4]``
+
+    >>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4])
+    >>> w
+    array([-2.22987175,  3.95222349])
+
+    """
+    if eigvals_only or overwrite_a_band:
+        a1 = _asarray_validated(a_band, check_finite=check_finite)
+        overwrite_a_band = overwrite_a_band or (_datacopied(a1, a_band))
+    else:
+        a1 = array(a_band)
+        if issubclass(a1.dtype.type, inexact) and not isfinite(a1).all():
+            raise ValueError("array must not contain infs or NaNs")
+        overwrite_a_band = 1
+
+    if len(a1.shape) != 2:
+        raise ValueError('expected a 2-D array')
+
+    # accommodate square empty matrices
+    if a1.size == 0:
+        w_n, v_n = eig_banded(np.array([[0, 0], [1, 1]], dtype=a1.dtype))
+
+        w = np.empty_like(a1, shape=(0,), dtype=w_n.dtype)
+        v = np.empty_like(a1, shape=(0, 0), dtype=v_n.dtype)
+        if eigvals_only:
+            return w
+        else:
+            return w, v
+
+    select, vl, vu, il, iu, max_ev = _check_select(
+        select, select_range, max_ev, a1.shape[1])
+
+    del select_range
+    if select == 0:
+        if a1.dtype.char in 'GFD':
+            # FIXME: implement this somewhen, for now go with builtin values
+            # FIXME: calc optimal lwork by calling ?hbevd(lwork=-1)
+            #        or by using calc_lwork.f ???
+            # lwork = calc_lwork.hbevd(bevd.typecode, a1.shape[0], lower)
+            internal_name = 'hbevd'
+        else:  # a1.dtype.char in 'fd':
+            # FIXME: implement this somewhen, for now go with builtin values
+            #         see above
+            # lwork = calc_lwork.sbevd(bevd.typecode, a1.shape[0], lower)
+            internal_name = 'sbevd'
+        bevd, = get_lapack_funcs((internal_name,), (a1,))
+        w, v, info = bevd(a1, compute_v=not eigvals_only,
+                          lower=lower, overwrite_ab=overwrite_a_band)
+    else:  # select in [1, 2]
+        if eigvals_only:
+            max_ev = 1
+        # calculate optimal abstol for dsbevx (see manpage)
+        if a1.dtype.char in 'fF':  # single precision
+            lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='f'),))
+        else:
+            lamch, = get_lapack_funcs(('lamch',), (array(0, dtype='d'),))
+        abstol = 2 * lamch('s')
+        if a1.dtype.char in 'GFD':
+            internal_name = 'hbevx'
+        else:  # a1.dtype.char in 'gfd'
+            internal_name = 'sbevx'
+        bevx, = get_lapack_funcs((internal_name,), (a1,))
+        w, v, m, ifail, info = bevx(
+            a1, vl, vu, il, iu, compute_v=not eigvals_only, mmax=max_ev,
+            range=select, lower=lower, overwrite_ab=overwrite_a_band,
+            abstol=abstol)
+        # crop off w and v
+        w = w[:m]
+        if not eigvals_only:
+            v = v[:, :m]
+    _check_info(info, internal_name)
+
+    if eigvals_only:
+        return w
+    return w, v
+
+
+def eigvals(a, b=None, overwrite_a=False, check_finite=True,
+            homogeneous_eigvals=False):
+    """
+    Compute eigenvalues from an ordinary or generalized eigenvalue problem.
+
+    Find eigenvalues of a general matrix::
+
+        a   vr[:,i] = w[i]        b   vr[:,i]
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        A complex or real matrix whose eigenvalues and eigenvectors
+        will be computed.
+    b : (M, M) array_like, optional
+        Right-hand side matrix in a generalized eigenvalue problem.
+        If omitted, identity matrix is assumed.
+    overwrite_a : bool, optional
+        Whether to overwrite data in a (may improve performance)
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities
+        or NaNs.
+    homogeneous_eigvals : bool, optional
+        If True, return the eigenvalues in homogeneous coordinates.
+        In this case ``w`` is a (2, M) array so that::
+
+            w[1,i] a vr[:,i] = w[0,i] b vr[:,i]
+
+        Default is False.
+
+    Returns
+    -------
+    w : (M,) or (2, M) double or complex ndarray
+        The eigenvalues, each repeated according to its multiplicity
+        but not in any specific order. The shape is (M,) unless
+        ``homogeneous_eigvals=True``.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge
+
+    See Also
+    --------
+    eig : eigenvalues and right eigenvectors of general arrays.
+    eigvalsh : eigenvalues of symmetric or Hermitian arrays
+    eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
+    eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
+        matrices
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import linalg
+    >>> a = np.array([[0., -1.], [1., 0.]])
+    >>> linalg.eigvals(a)
+    array([0.+1.j, 0.-1.j])
+
+    >>> b = np.array([[0., 1.], [1., 1.]])
+    >>> linalg.eigvals(a, b)
+    array([ 1.+0.j, -1.+0.j])
+
+    >>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
+    >>> linalg.eigvals(a, homogeneous_eigvals=True)
+    array([[3.+0.j, 8.+0.j, 7.+0.j],
+           [1.+0.j, 1.+0.j, 1.+0.j]])
+
+    """
+    return eig(a, b=b, left=0, right=0, overwrite_a=overwrite_a,
+               check_finite=check_finite,
+               homogeneous_eigvals=homogeneous_eigvals)
+
+
+def eigvalsh(a, b=None, *, lower=True, overwrite_a=False,
+             overwrite_b=False, type=1, check_finite=True, subset_by_index=None,
+             subset_by_value=None, driver=None):
+    """
+    Solves a standard or generalized eigenvalue problem for a complex
+    Hermitian or real symmetric matrix.
+
+    Find eigenvalues array ``w`` of array ``a``, where ``b`` is positive
+    definite such that for every eigenvalue λ (i-th entry of w) and its
+    eigenvector vi (i-th column of v) satisfies::
+
+                      a @ vi = λ * b @ vi
+        vi.conj().T @ a @ vi = λ
+        vi.conj().T @ b @ vi = 1
+
+    In the standard problem, b is assumed to be the identity matrix.
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        A complex Hermitian or real symmetric matrix whose eigenvalues will
+        be computed.
+    b : (M, M) array_like, optional
+        A complex Hermitian or real symmetric definite positive matrix in.
+        If omitted, identity matrix is assumed.
+    lower : bool, optional
+        Whether the pertinent array data is taken from the lower or upper
+        triangle of ``a`` and, if applicable, ``b``. (Default: lower)
+    overwrite_a : bool, optional
+        Whether to overwrite data in ``a`` (may improve performance). Default
+        is False.
+    overwrite_b : bool, optional
+        Whether to overwrite data in ``b`` (may improve performance). Default
+        is False.
+    type : int, optional
+        For the generalized problems, this keyword specifies the problem type
+        to be solved for ``w`` and ``v`` (only takes 1, 2, 3 as possible
+        inputs)::
+
+            1 =>     a @ v = w @ b @ v
+            2 => a @ b @ v = w @ v
+            3 => b @ a @ v = w @ v
+
+        This keyword is ignored for standard problems.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+    subset_by_index : iterable, optional
+        If provided, this two-element iterable defines the start and the end
+        indices of the desired eigenvalues (ascending order and 0-indexed).
+        To return only the second smallest to fifth smallest eigenvalues,
+        ``[1, 4]`` is used. ``[n-3, n-1]`` returns the largest three. Only
+        available with "evr", "evx", and "gvx" drivers. The entries are
+        directly converted to integers via ``int()``.
+    subset_by_value : iterable, optional
+        If provided, this two-element iterable defines the half-open interval
+        ``(a, b]`` that, if any, only the eigenvalues between these values
+        are returned. Only available with "evr", "evx", and "gvx" drivers. Use
+        ``np.inf`` for the unconstrained ends.
+    driver : str, optional
+        Defines which LAPACK driver should be used. Valid options are "ev",
+        "evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for
+        generalized (where b is not None) problems. See the Notes section of
+        `scipy.linalg.eigh`.
+
+    Returns
+    -------
+    w : (N,) ndarray
+        The N (N<=M) selected eigenvalues, in ascending order, each
+        repeated according to its multiplicity.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge, an error occurred, or
+        b matrix is not definite positive. Note that if input matrices are
+        not symmetric or Hermitian, no error will be reported but results will
+        be wrong.
+
+    See Also
+    --------
+    eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
+    eigvals : eigenvalues of general arrays
+    eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
+    eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
+        matrices
+
+    Notes
+    -----
+    This function does not check the input array for being Hermitian/symmetric
+    in order to allow for representing arrays with only their upper/lower
+    triangular parts.
+
+    This function serves as a one-liner shorthand for `scipy.linalg.eigh` with
+    the option ``eigvals_only=True`` to get the eigenvalues and not the
+    eigenvectors. Here it is kept as a legacy convenience. It might be
+    beneficial to use the main function to have full control and to be a bit
+    more pythonic.
+
+    Examples
+    --------
+    For more examples see `scipy.linalg.eigh`.
+
+    >>> import numpy as np
+    >>> from scipy.linalg import eigvalsh
+    >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
+    >>> w = eigvalsh(A)
+    >>> w
+    array([-3.74637491, -0.76263923,  6.08502336, 12.42399079])
+
+    """
+    return eigh(a, b=b, lower=lower, eigvals_only=True, overwrite_a=overwrite_a,
+                overwrite_b=overwrite_b, type=type, check_finite=check_finite,
+                subset_by_index=subset_by_index, subset_by_value=subset_by_value,
+                driver=driver)
+
+
+def eigvals_banded(a_band, lower=False, overwrite_a_band=False,
+                   select='a', select_range=None, check_finite=True):
+    """
+    Solve real symmetric or complex Hermitian band matrix eigenvalue problem.
+
+    Find eigenvalues w of a::
+
+        a v[:,i] = w[i] v[:,i]
+        v.H v    = identity
+
+    The matrix a is stored in a_band either in lower diagonal or upper
+    diagonal ordered form:
+
+        a_band[u + i - j, j] == a[i,j]        (if upper form; i <= j)
+        a_band[    i - j, j] == a[i,j]        (if lower form; i >= j)
+
+    where u is the number of bands above the diagonal.
+
+    Example of a_band (shape of a is (6,6), u=2)::
+
+        upper form:
+        *   *   a02 a13 a24 a35
+        *   a01 a12 a23 a34 a45
+        a00 a11 a22 a33 a44 a55
+
+        lower form:
+        a00 a11 a22 a33 a44 a55
+        a10 a21 a32 a43 a54 *
+        a20 a31 a42 a53 *   *
+
+    Cells marked with * are not used.
+
+    Parameters
+    ----------
+    a_band : (u+1, M) array_like
+        The bands of the M by M matrix a.
+    lower : bool, optional
+        Is the matrix in the lower form. (Default is upper form)
+    overwrite_a_band : bool, optional
+        Discard data in a_band (may enhance performance)
+    select : {'a', 'v', 'i'}, optional
+        Which eigenvalues to calculate
+
+        ======  ========================================
+        select  calculated
+        ======  ========================================
+        'a'     All eigenvalues
+        'v'     Eigenvalues in the interval (min, max]
+        'i'     Eigenvalues with indices min <= i <= max
+        ======  ========================================
+    select_range : (min, max), optional
+        Range of selected eigenvalues
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    w : (M,) ndarray
+        The eigenvalues, in ascending order, each repeated according to its
+        multiplicity.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge.
+
+    See Also
+    --------
+    eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
+        band matrices
+    eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
+        matrices
+    eigvals : eigenvalues of general arrays
+    eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
+    eig : eigenvalues and right eigenvectors for non-symmetric arrays
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import eigvals_banded
+    >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]])
+    >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]])
+    >>> w = eigvals_banded(Ab, lower=True)
+    >>> w
+    array([-4.26200532, -2.22987175,  3.95222349, 12.53965359])
+    """
+    return eig_banded(a_band, lower=lower, eigvals_only=1,
+                      overwrite_a_band=overwrite_a_band, select=select,
+                      select_range=select_range, check_finite=check_finite)
+
+
+def eigvalsh_tridiagonal(d, e, select='a', select_range=None,
+                         check_finite=True, tol=0., lapack_driver='auto'):
+    """
+    Solve eigenvalue problem for a real symmetric tridiagonal matrix.
+
+    Find eigenvalues `w` of ``a``::
+
+        a v[:,i] = w[i] v[:,i]
+        v.H v    = identity
+
+    For a real symmetric matrix ``a`` with diagonal elements `d` and
+    off-diagonal elements `e`.
+
+    Parameters
+    ----------
+    d : ndarray, shape (ndim,)
+        The diagonal elements of the array.
+    e : ndarray, shape (ndim-1,)
+        The off-diagonal elements of the array.
+    select : {'a', 'v', 'i'}, optional
+        Which eigenvalues to calculate
+
+        ======  ========================================
+        select  calculated
+        ======  ========================================
+        'a'     All eigenvalues
+        'v'     Eigenvalues in the interval (min, max]
+        'i'     Eigenvalues with indices min <= i <= max
+        ======  ========================================
+    select_range : (min, max), optional
+        Range of selected eigenvalues
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+    tol : float
+        The absolute tolerance to which each eigenvalue is required
+        (only used when ``lapack_driver='stebz'``).
+        An eigenvalue (or cluster) is considered to have converged if it
+        lies in an interval of this width. If <= 0. (default),
+        the value ``eps*|a|`` is used where eps is the machine precision,
+        and ``|a|`` is the 1-norm of the matrix ``a``.
+    lapack_driver : str
+        LAPACK function to use, can be 'auto', 'stemr', 'stebz',  'sterf',
+        or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'``
+        and 'stebz' otherwise. 'sterf' and 'stev' can only be used when
+        ``select='a'``.
+
+    Returns
+    -------
+    w : (M,) ndarray
+        The eigenvalues, in ascending order, each repeated according to its
+        multiplicity.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge.
+
+    See Also
+    --------
+    eigh_tridiagonal : eigenvalues and right eiegenvectors for
+        symmetric/Hermitian tridiagonal matrices
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import eigvalsh_tridiagonal, eigvalsh
+    >>> d = 3*np.ones(4)
+    >>> e = -1*np.ones(3)
+    >>> w = eigvalsh_tridiagonal(d, e)
+    >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
+    >>> w2 = eigvalsh(A)  # Verify with other eigenvalue routines
+    >>> np.allclose(w - w2, np.zeros(4))
+    True
+    """
+    return eigh_tridiagonal(
+        d, e, eigvals_only=True, select=select, select_range=select_range,
+        check_finite=check_finite, tol=tol, lapack_driver=lapack_driver)
+
+
+def eigh_tridiagonal(d, e, eigvals_only=False, select='a', select_range=None,
+                     check_finite=True, tol=0., lapack_driver='auto'):
+    """
+    Solve eigenvalue problem for a real symmetric tridiagonal matrix.
+
+    Find eigenvalues `w` and optionally right eigenvectors `v` of ``a``::
+
+        a v[:,i] = w[i] v[:,i]
+        v.H v    = identity
+
+    For a real symmetric matrix ``a`` with diagonal elements `d` and
+    off-diagonal elements `e`.
+
+    Parameters
+    ----------
+    d : ndarray, shape (ndim,)
+        The diagonal elements of the array.
+    e : ndarray, shape (ndim-1,)
+        The off-diagonal elements of the array.
+    eigvals_only : bool, optional
+        Compute only the eigenvalues and no eigenvectors.
+        (Default: calculate also eigenvectors)
+    select : {'a', 'v', 'i'}, optional
+        Which eigenvalues to calculate
+
+        ======  ========================================
+        select  calculated
+        ======  ========================================
+        'a'     All eigenvalues
+        'v'     Eigenvalues in the interval (min, max]
+        'i'     Eigenvalues with indices min <= i <= max
+        ======  ========================================
+    select_range : (min, max), optional
+        Range of selected eigenvalues
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+    tol : float
+        The absolute tolerance to which each eigenvalue is required
+        (only used when 'stebz' is the `lapack_driver`).
+        An eigenvalue (or cluster) is considered to have converged if it
+        lies in an interval of this width. If <= 0. (default),
+        the value ``eps*|a|`` is used where eps is the machine precision,
+        and ``|a|`` is the 1-norm of the matrix ``a``.
+    lapack_driver : str
+        LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf',
+        or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'``
+        and 'stebz' otherwise. When 'stebz' is used to find the eigenvalues and
+        ``eigvals_only=False``, then a second LAPACK call (to ``?STEIN``) is
+        used to find the corresponding eigenvectors. 'sterf' can only be
+        used when ``eigvals_only=True`` and ``select='a'``. 'stev' can only
+        be used when ``select='a'``.
+
+    Returns
+    -------
+    w : (M,) ndarray
+        The eigenvalues, in ascending order, each repeated according to its
+        multiplicity.
+    v : (M, M) ndarray
+        The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is
+        the column ``v[:,i]``. Only returned if ``eigvals_only=False``.
+
+    Raises
+    ------
+    LinAlgError
+        If eigenvalue computation does not converge.
+
+    See Also
+    --------
+    eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
+        matrices
+    eig : eigenvalues and right eigenvectors for non-symmetric arrays
+    eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
+    eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
+        band matrices
+
+    Notes
+    -----
+    This function makes use of LAPACK ``S/DSTEMR`` routines.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import eigh_tridiagonal
+    >>> d = 3*np.ones(4)
+    >>> e = -1*np.ones(3)
+    >>> w, v = eigh_tridiagonal(d, e)
+    >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
+    >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
+    True
+    """
+    d = _asarray_validated(d, check_finite=check_finite)
+    e = _asarray_validated(e, check_finite=check_finite)
+    for check in (d, e):
+        if check.ndim != 1:
+            raise ValueError('expected a 1-D array')
+        if check.dtype.char in 'GFD':  # complex
+            raise TypeError('Only real arrays currently supported')
+    if d.size != e.size + 1:
+        raise ValueError(f'd ({d.size}) must have one more element than e ({e.size})')
+    select, vl, vu, il, iu, _ = _check_select(
+        select, select_range, 0, d.size)
+    if not isinstance(lapack_driver, str):
+        raise TypeError('lapack_driver must be str')
+    drivers = ('auto', 'stemr', 'sterf', 'stebz', 'stev')
+    if lapack_driver not in drivers:
+        raise ValueError(f'lapack_driver must be one of {drivers}, '
+                         f'got {lapack_driver}')
+    if lapack_driver == 'auto':
+        lapack_driver = 'stemr' if select == 0 else 'stebz'
+
+    # Quick exit for 1x1 case
+    if len(d) == 1:
+        if select == 1 and (not (vl < d[0] <= vu)):  # request by value
+            w = array([])
+            v = empty([1, 0], dtype=d.dtype)
+        else:  # all and request by index
+            w = array([d[0]], dtype=d.dtype)
+            v = array([[1.]], dtype=d.dtype)
+
+        if eigvals_only:
+            return w
+        else:
+            return w, v
+
+    func, = get_lapack_funcs((lapack_driver,), (d, e))
+    compute_v = not eigvals_only
+    if lapack_driver == 'sterf':
+        if select != 0:
+            raise ValueError('sterf can only be used when select == "a"')
+        if not eigvals_only:
+            raise ValueError('sterf can only be used when eigvals_only is '
+                             'True')
+        w, info = func(d, e)
+        m = len(w)
+    elif lapack_driver == 'stev':
+        if select != 0:
+            raise ValueError('stev can only be used when select == "a"')
+        w, v, info = func(d, e, compute_v=compute_v)
+        m = len(w)
+    elif lapack_driver == 'stebz':
+        tol = float(tol)
+        internal_name = 'stebz'
+        stebz, = get_lapack_funcs((internal_name,), (d, e))
+        # If getting eigenvectors, needs to be block-ordered (B) instead of
+        # matrix-ordered (E), and we will reorder later
+        order = 'E' if eigvals_only else 'B'
+        m, w, iblock, isplit, info = stebz(d, e, select, vl, vu, il, iu, tol,
+                                           order)
+    else:   # 'stemr'
+        # ?STEMR annoyingly requires size N instead of N-1
+        e_ = empty(e.size+1, e.dtype)
+        e_[:-1] = e
+        stemr_lwork, = get_lapack_funcs(('stemr_lwork',), (d, e))
+        lwork, liwork, info = stemr_lwork(d, e_, select, vl, vu, il, iu,
+                                          compute_v=compute_v)
+        _check_info(info, 'stemr_lwork')
+        m, w, v, info = func(d, e_, select, vl, vu, il, iu,
+                             compute_v=compute_v, lwork=lwork, liwork=liwork)
+    _check_info(info, lapack_driver + ' (eigh_tridiagonal)')
+    w = w[:m]
+    if eigvals_only:
+        return w
+    else:
+        # Do we still need to compute the eigenvalues?
+        if lapack_driver == 'stebz':
+            func, = get_lapack_funcs(('stein',), (d, e))
+            v, info = func(d, e, w, iblock, isplit)
+            _check_info(info, 'stein (eigh_tridiagonal)',
+                        positive='%d eigenvectors failed to converge')
+            # Convert block-order to matrix-order
+            order = argsort(w)
+            w, v = w[order], v[:, order]
+        else:
+            v = v[:, :m]
+        return w, v
+
+
+def _check_info(info, driver, positive='did not converge (LAPACK info=%d)'):
+    """Check info return value."""
+    if info < 0:
+        raise ValueError('illegal value in argument %d of internal %s'
+                         % (-info, driver))
+    if info > 0 and positive:
+        raise LinAlgError(("%s " + positive) % (driver, info,))
+
+
+def hessenberg(a, calc_q=False, overwrite_a=False, check_finite=True):
+    """
+    Compute Hessenberg form of a matrix.
+
+    The Hessenberg decomposition is::
+
+        A = Q H Q^H
+
+    where `Q` is unitary/orthogonal and `H` has only zero elements below
+    the first sub-diagonal.
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        Matrix to bring into Hessenberg form.
+    calc_q : bool, optional
+        Whether to compute the transformation matrix.  Default is False.
+    overwrite_a : bool, optional
+        Whether to overwrite `a`; may improve performance.
+        Default is False.
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    H : (M, M) ndarray
+        Hessenberg form of `a`.
+    Q : (M, M) ndarray
+        Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``.
+        Only returned if ``calc_q=True``.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import hessenberg
+    >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
+    >>> H, Q = hessenberg(A, calc_q=True)
+    >>> H
+    array([[  2.        , -11.65843866,   1.42005301,   0.25349066],
+           [ -9.94987437,  14.53535354,  -5.31022304,   2.43081618],
+           [  0.        ,  -1.83299243,   0.38969961,  -0.51527034],
+           [  0.        ,   0.        ,  -3.83189513,   1.07494686]])
+    >>> np.allclose(Q @ H @ Q.conj().T - A, np.zeros((4, 4)))
+    True
+    """
+    a1 = _asarray_validated(a, check_finite=check_finite)
+    if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
+        raise ValueError('expected square matrix')
+    overwrite_a = overwrite_a or (_datacopied(a1, a))
+
+    if a1.size == 0:
+        h3 = hessenberg(np.eye(3, dtype=a1.dtype))
+        h = np.empty(a1.shape, dtype=h3.dtype)
+        if not calc_q:
+            return h
+        else:
+            h3, q3 = hessenberg(np.eye(3, dtype=a1.dtype), calc_q=True)
+            q = np.empty(a1.shape, dtype=q3.dtype)
+            h = np.empty(a1.shape, dtype=h3.dtype)
+            return h, q
+
+    # if 2x2 or smaller: already in Hessenberg
+    if a1.shape[0] <= 2:
+        if calc_q:
+            return a1, eye(a1.shape[0])
+        return a1
+
+    gehrd, gebal, gehrd_lwork = get_lapack_funcs(('gehrd', 'gebal',
+                                                  'gehrd_lwork'), (a1,))
+    ba, lo, hi, pivscale, info = gebal(a1, permute=0, overwrite_a=overwrite_a)
+    _check_info(info, 'gebal (hessenberg)', positive=False)
+    n = len(a1)
+
+    lwork = _compute_lwork(gehrd_lwork, ba.shape[0], lo=lo, hi=hi)
+
+    hq, tau, info = gehrd(ba, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
+    _check_info(info, 'gehrd (hessenberg)', positive=False)
+    h = np.triu(hq, -1)
+    if not calc_q:
+        return h
+
+    # use orghr/unghr to compute q
+    orghr, orghr_lwork = get_lapack_funcs(('orghr', 'orghr_lwork'), (a1,))
+    lwork = _compute_lwork(orghr_lwork, n, lo=lo, hi=hi)
+
+    q, info = orghr(a=hq, tau=tau, lo=lo, hi=hi, lwork=lwork, overwrite_a=1)
+    _check_info(info, 'orghr (hessenberg)', positive=False)
+    return h, q
+
+
+def cdf2rdf(w, v):
+    """
+    Converts complex eigenvalues ``w`` and eigenvectors ``v`` to real
+    eigenvalues in a block diagonal form ``wr`` and the associated real
+    eigenvectors ``vr``, such that::
+
+        vr @ wr = X @ vr
+
+    continues to hold, where ``X`` is the original array for which ``w`` and
+    ``v`` are the eigenvalues and eigenvectors.
+
+    .. versionadded:: 1.1.0
+
+    Parameters
+    ----------
+    w : (..., M) array_like
+        Complex or real eigenvalues, an array or stack of arrays
+
+        Conjugate pairs must not be interleaved, else the wrong result
+        will be produced. So ``[1+1j, 1, 1-1j]`` will give a correct result,
+        but ``[1+1j, 2+1j, 1-1j, 2-1j]`` will not.
+
+    v : (..., M, M) array_like
+        Complex or real eigenvectors, a square array or stack of square arrays.
+
+    Returns
+    -------
+    wr : (..., M, M) ndarray
+        Real diagonal block form of eigenvalues
+    vr : (..., M, M) ndarray
+        Real eigenvectors associated with ``wr``
+
+    See Also
+    --------
+    eig : Eigenvalues and right eigenvectors for non-symmetric arrays
+    rsf2csf : Convert real Schur form to complex Schur form
+
+    Notes
+    -----
+    ``w``, ``v`` must be the eigenstructure for some *real* matrix ``X``.
+    For example, obtained by ``w, v = scipy.linalg.eig(X)`` or
+    ``w, v = numpy.linalg.eig(X)`` in which case ``X`` can also represent
+    stacked arrays.
+
+    .. versionadded:: 1.1.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]])
+    >>> X
+    array([[ 1,  2,  3],
+           [ 0,  4,  5],
+           [ 0, -5,  4]])
+
+    >>> from scipy import linalg
+    >>> w, v = linalg.eig(X)
+    >>> w
+    array([ 1.+0.j,  4.+5.j,  4.-5.j])
+    >>> v
+    array([[ 1.00000+0.j     , -0.01906-0.40016j, -0.01906+0.40016j],
+           [ 0.00000+0.j     ,  0.00000-0.64788j,  0.00000+0.64788j],
+           [ 0.00000+0.j     ,  0.64788+0.j     ,  0.64788-0.j     ]])
+
+    >>> wr, vr = linalg.cdf2rdf(w, v)
+    >>> wr
+    array([[ 1.,  0.,  0.],
+           [ 0.,  4.,  5.],
+           [ 0., -5.,  4.]])
+    >>> vr
+    array([[ 1.     ,  0.40016, -0.01906],
+           [ 0.     ,  0.64788,  0.     ],
+           [ 0.     ,  0.     ,  0.64788]])
+
+    >>> vr @ wr
+    array([[ 1.     ,  1.69593,  1.9246 ],
+           [ 0.     ,  2.59153,  3.23942],
+           [ 0.     , -3.23942,  2.59153]])
+    >>> X @ vr
+    array([[ 1.     ,  1.69593,  1.9246 ],
+           [ 0.     ,  2.59153,  3.23942],
+           [ 0.     , -3.23942,  2.59153]])
+    """
+    w, v = _asarray_validated(w), _asarray_validated(v)
+
+    # check dimensions
+    if w.ndim < 1:
+        raise ValueError('expected w to be at least 1D')
+    if v.ndim < 2:
+        raise ValueError('expected v to be at least 2D')
+    if v.ndim != w.ndim + 1:
+        raise ValueError('expected eigenvectors array to have exactly one '
+                         'dimension more than eigenvalues array')
+
+    # check shapes
+    n = w.shape[-1]
+    M = w.shape[:-1]
+    if v.shape[-2] != v.shape[-1]:
+        raise ValueError('expected v to be a square matrix or stacked square '
+                         'matrices: v.shape[-2] = v.shape[-1]')
+    if v.shape[-1] != n:
+        raise ValueError('expected the same number of eigenvalues as '
+                         'eigenvectors')
+
+    # get indices for each first pair of complex eigenvalues
+    complex_mask = iscomplex(w)
+    n_complex = complex_mask.sum(axis=-1)
+
+    # check if all complex eigenvalues have conjugate pairs
+    if not (n_complex % 2 == 0).all():
+        raise ValueError('expected complex-conjugate pairs of eigenvalues')
+
+    # find complex indices
+    idx = nonzero(complex_mask)
+    idx_stack = idx[:-1]
+    idx_elem = idx[-1]
+
+    # filter them to conjugate indices, assuming pairs are not interleaved
+    j = idx_elem[0::2]
+    k = idx_elem[1::2]
+    stack_ind = ()
+    for i in idx_stack:
+        # should never happen, assuming nonzero orders by the last axis
+        assert (i[0::2] == i[1::2]).all(), \
+                "Conjugate pair spanned different arrays!"
+        stack_ind += (i[0::2],)
+
+    # all eigenvalues to diagonal form
+    wr = zeros(M + (n, n), dtype=w.real.dtype)
+    di = range(n)
+    wr[..., di, di] = w.real
+
+    # complex eigenvalues to real block diagonal form
+    wr[stack_ind + (j, k)] = w[stack_ind + (j,)].imag
+    wr[stack_ind + (k, j)] = w[stack_ind + (k,)].imag
+
+    # compute real eigenvectors associated with real block diagonal eigenvalues
+    u = zeros(M + (n, n), dtype=np.cdouble)
+    u[..., di, di] = 1.0
+    u[stack_ind + (j, j)] = 0.5j
+    u[stack_ind + (j, k)] = 0.5
+    u[stack_ind + (k, j)] = -0.5j
+    u[stack_ind + (k, k)] = 0.5
+
+    # multiply matrices v and u (equivalent to v @ u)
+    vr = einsum('...ij,...jk->...ik', v, u).real
+
+    return wr, vr

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_decomp_cholesky.py

@@ -0,0 +1,398 @@
+"""Cholesky decomposition functions."""
+
+import numpy as np
+from numpy import asarray_chkfinite, asarray, atleast_2d, empty_like
+
+# Local imports
+from ._misc import LinAlgError, _datacopied
+from .lapack import get_lapack_funcs
+
+__all__ = ['cholesky', 'cho_factor', 'cho_solve', 'cholesky_banded',
+           'cho_solve_banded']
+
+
+def _cholesky(a, lower=False, overwrite_a=False, clean=True,
+              check_finite=True):
+    """Common code for cholesky() and cho_factor()."""
+
+    a1 = asarray_chkfinite(a) if check_finite else asarray(a)
+    a1 = atleast_2d(a1)
+
+    # Dimension check
+    if a1.ndim != 2:
+        raise ValueError(f'Input array needs to be 2D but received a {a1.ndim}d-array.')
+    # Squareness check
+    if a1.shape[0] != a1.shape[1]:
+        raise ValueError('Input array is expected to be square but has '
+                         f'the shape: {a1.shape}.')
+
+    # Quick return for square empty array
+    if a1.size == 0:
+        dt = cholesky(np.eye(1, dtype=a1.dtype)).dtype
+        return empty_like(a1, dtype=dt), lower
+
+    overwrite_a = overwrite_a or _datacopied(a1, a)
+    potrf, = get_lapack_funcs(('potrf',), (a1,))
+    c, info = potrf(a1, lower=lower, overwrite_a=overwrite_a, clean=clean)
+    if info > 0:
+        raise LinAlgError("%d-th leading minor of the array is not positive "
+                          "definite" % info)
+    if info < 0:
+        raise ValueError(f'LAPACK reported an illegal value in {-info}-th argument'
+                         'on entry to "POTRF".')
+    return c, lower
+
+
+def cholesky(a, lower=False, overwrite_a=False, check_finite=True):
+    """
+    Compute the Cholesky decomposition of a matrix.
+
+    Returns the Cholesky decomposition, :math:`A = L L^*` or
+    :math:`A = U^* U` of a Hermitian positive-definite matrix A.
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        Matrix to be decomposed
+    lower : bool, optional
+        Whether to compute the upper- or lower-triangular Cholesky
+        factorization. During decomposition, only the selected half of the
+        matrix is referenced. Default is upper-triangular.
+    overwrite_a : bool, optional
+        Whether to overwrite data in `a` (may improve performance).
+    check_finite : bool, optional
+        Whether to check that the entire input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    c : (M, M) ndarray
+        Upper- or lower-triangular Cholesky factor of `a`.
+
+    Raises
+    ------
+    LinAlgError : if decomposition fails.
+
+    Notes
+    -----
+    During the finiteness check (if selected), the entire matrix `a` is
+    checked. During decomposition, `a` is assumed to be symmetric or Hermitian
+    (as applicable), and only the half selected by option `lower` is referenced.
+    Consequently, if `a` is asymmetric/non-Hermitian, `cholesky` may still
+    succeed if the symmetric/Hermitian matrix represented by the selected half
+    is positive definite, yet it may fail if an element in the other half is
+    non-finite.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import cholesky
+    >>> a = np.array([[1,-2j],[2j,5]])
+    >>> L = cholesky(a, lower=True)
+    >>> L
+    array([[ 1.+0.j,  0.+0.j],
+           [ 0.+2.j,  1.+0.j]])
+    >>> L @ L.T.conj()
+    array([[ 1.+0.j,  0.-2.j],
+           [ 0.+2.j,  5.+0.j]])
+
+    """
+    c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=True,
+                         check_finite=check_finite)
+    return c
+
+
+def cho_factor(a, lower=False, overwrite_a=False, check_finite=True):
+    """
+    Compute the Cholesky decomposition of a matrix, to use in cho_solve
+
+    Returns a matrix containing the Cholesky decomposition,
+    ``A = L L*`` or ``A = U* U`` of a Hermitian positive-definite matrix `a`.
+    The return value can be directly used as the first parameter to cho_solve.
+
+    .. warning::
+        The returned matrix also contains random data in the entries not
+        used by the Cholesky decomposition. If you need to zero these
+        entries, use the function `cholesky` instead.
+
+    Parameters
+    ----------
+    a : (M, M) array_like
+        Matrix to be decomposed
+    lower : bool, optional
+        Whether to compute the upper or lower triangular Cholesky factorization.
+        During decomposition, only the selected half of the matrix is referenced.
+        (Default: upper-triangular)
+    overwrite_a : bool, optional
+        Whether to overwrite data in a (may improve performance)
+    check_finite : bool, optional
+        Whether to check that the entire input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    c : (M, M) ndarray
+        Matrix whose upper or lower triangle contains the Cholesky factor
+        of `a`. Other parts of the matrix contain random data.
+    lower : bool
+        Flag indicating whether the factor is in the lower or upper triangle
+
+    Raises
+    ------
+    LinAlgError
+        Raised if decomposition fails.
+
+    See Also
+    --------
+    cho_solve : Solve a linear set equations using the Cholesky factorization
+                of a matrix.
+
+    Notes
+    -----
+    During the finiteness check (if selected), the entire matrix `a` is
+    checked. During decomposition, `a` is assumed to be symmetric or Hermitian
+    (as applicable), and only the half selected by option `lower` is referenced.
+    Consequently, if `a` is asymmetric/non-Hermitian, `cholesky` may still
+    succeed if the symmetric/Hermitian matrix represented by the selected half
+    is positive definite, yet it may fail if an element in the other half is
+    non-finite.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import cho_factor
+    >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]])
+    >>> c, low = cho_factor(A)
+    >>> c
+    array([[3.        , 1.        , 0.33333333, 1.66666667],
+           [3.        , 2.44948974, 1.90515869, -0.27216553],
+           [1.        , 5.        , 2.29330749, 0.8559528 ],
+           [5.        , 1.        , 2.        , 1.55418563]])
+    >>> np.allclose(np.triu(c).T @ np. triu(c) - A, np.zeros((4, 4)))
+    True
+
+    """
+    c, lower = _cholesky(a, lower=lower, overwrite_a=overwrite_a, clean=False,
+                         check_finite=check_finite)
+    return c, lower
+
+
+def cho_solve(c_and_lower, b, overwrite_b=False, check_finite=True):
+    """Solve the linear equations A x = b, given the Cholesky factorization of A.
+
+    Parameters
+    ----------
+    (c, lower) : tuple, (array, bool)
+        Cholesky factorization of a, as given by cho_factor
+    b : array
+        Right-hand side
+    overwrite_b : bool, optional
+        Whether to overwrite data in b (may improve performance)
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    x : array
+        The solution to the system A x = b
+
+    See Also
+    --------
+    cho_factor : Cholesky factorization of a matrix
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import cho_factor, cho_solve
+    >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]])
+    >>> c, low = cho_factor(A)
+    >>> x = cho_solve((c, low), [1, 1, 1, 1])
+    >>> np.allclose(A @ x - [1, 1, 1, 1], np.zeros(4))
+    True
+
+    """
+    (c, lower) = c_and_lower
+    if check_finite:
+        b1 = asarray_chkfinite(b)
+        c = asarray_chkfinite(c)
+    else:
+        b1 = asarray(b)
+        c = asarray(c)
+
+    if c.ndim != 2 or c.shape[0] != c.shape[1]:
+        raise ValueError("The factored matrix c is not square.")
+    if c.shape[1] != b1.shape[0]:
+        raise ValueError(f"incompatible dimensions ({c.shape} and {b1.shape})")
+
+    # accommodate empty arrays
+    if b1.size == 0:
+        dt = cho_solve((np.eye(2, dtype=b1.dtype), True),
+                        np.ones(2, dtype=c.dtype)).dtype
+        return empty_like(b1, dtype=dt)
+
+    overwrite_b = overwrite_b or _datacopied(b1, b)
+
+    potrs, = get_lapack_funcs(('potrs',), (c, b1))
+    x, info = potrs(c, b1, lower=lower, overwrite_b=overwrite_b)
+    if info != 0:
+        raise ValueError('illegal value in %dth argument of internal potrs'
+                         % -info)
+    return x
+
+
+def cholesky_banded(ab, overwrite_ab=False, lower=False, check_finite=True):
+    """
+    Cholesky decompose a banded Hermitian positive-definite matrix
+
+    The matrix a is stored in ab either in lower-diagonal or upper-
+    diagonal ordered form::
+
+        ab[u + i - j, j] == a[i,j]        (if upper form; i <= j)
+        ab[    i - j, j] == a[i,j]        (if lower form; i >= j)
+
+    Example of ab (shape of a is (6,6), u=2)::
+
+        upper form:
+        *   *   a02 a13 a24 a35
+        *   a01 a12 a23 a34 a45
+        a00 a11 a22 a33 a44 a55
+
+        lower form:
+        a00 a11 a22 a33 a44 a55
+        a10 a21 a32 a43 a54 *
+        a20 a31 a42 a53 *   *
+
+    Parameters
+    ----------
+    ab : (u + 1, M) array_like
+        Banded matrix
+    overwrite_ab : bool, optional
+        Discard data in ab (may enhance performance)
+    lower : bool, optional
+        Is the matrix in the lower form. (Default is upper form)
+    check_finite : bool, optional
+        Whether to check that the input matrix contains only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    c : (u + 1, M) ndarray
+        Cholesky factorization of a, in the same banded format as ab
+
+    See Also
+    --------
+    cho_solve_banded :
+        Solve a linear set equations, given the Cholesky factorization
+        of a banded Hermitian.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import cholesky_banded
+    >>> from numpy import allclose, zeros, diag
+    >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
+    >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
+    >>> A = A + A.conj().T + np.diag(Ab[2, :])
+    >>> c = cholesky_banded(Ab)
+    >>> C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :])
+    >>> np.allclose(C.conj().T @ C - A, np.zeros((5, 5)))
+    True
+
+    """
+    if check_finite:
+        ab = asarray_chkfinite(ab)
+    else:
+        ab = asarray(ab)
+
+    # accommodate square empty matrices
+    if ab.size == 0:
+        dt = cholesky_banded(np.array([[0, 0], [1, 1]], dtype=ab.dtype)).dtype
+        return empty_like(ab, dtype=dt)
+
+    pbtrf, = get_lapack_funcs(('pbtrf',), (ab,))
+    c, info = pbtrf(ab, lower=lower, overwrite_ab=overwrite_ab)
+    if info > 0:
+        raise LinAlgError("%d-th leading minor not positive definite" % info)
+    if info < 0:
+        raise ValueError('illegal value in %d-th argument of internal pbtrf'
+                         % -info)
+    return c
+
+
+def cho_solve_banded(cb_and_lower, b, overwrite_b=False, check_finite=True):
+    """
+    Solve the linear equations ``A x = b``, given the Cholesky factorization of
+    the banded Hermitian ``A``.
+
+    Parameters
+    ----------
+    (cb, lower) : tuple, (ndarray, bool)
+        `cb` is the Cholesky factorization of A, as given by cholesky_banded.
+        `lower` must be the same value that was given to cholesky_banded.
+    b : array_like
+        Right-hand side
+    overwrite_b : bool, optional
+        If True, the function will overwrite the values in `b`.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    x : array
+        The solution to the system A x = b
+
+    See Also
+    --------
+    cholesky_banded : Cholesky factorization of a banded matrix
+
+    Notes
+    -----
+
+    .. versionadded:: 0.8.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import cholesky_banded, cho_solve_banded
+    >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]])
+    >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1)
+    >>> A = A + A.conj().T + np.diag(Ab[2, :])
+    >>> c = cholesky_banded(Ab)
+    >>> x = cho_solve_banded((c, False), np.ones(5))
+    >>> np.allclose(A @ x - np.ones(5), np.zeros(5))
+    True
+
+    """
+    (cb, lower) = cb_and_lower
+    if check_finite:
+        cb = asarray_chkfinite(cb)
+        b = asarray_chkfinite(b)
+    else:
+        cb = asarray(cb)
+        b = asarray(b)
+
+    # Validate shapes.
+    if cb.shape[-1] != b.shape[0]:
+        raise ValueError("shapes of cb and b are not compatible.")
+
+    # accommodate empty arrays
+    if b.size == 0:
+        m = cholesky_banded(np.array([[0, 0], [1, 1]], dtype=cb.dtype))
+        dt = cho_solve_banded((m, True), np.ones(2, dtype=b.dtype)).dtype
+        return empty_like(b, dtype=dt)
+
+    pbtrs, = get_lapack_funcs(('pbtrs',), (cb, b))
+    x, info = pbtrs(cb, b, lower=lower, overwrite_b=overwrite_b)
+    if info > 0:
+        raise LinAlgError("%dth leading minor not positive definite" % info)
+    if info < 0:
+        raise ValueError('illegal value in %dth argument of internal pbtrs'
+                         % -info)
+    return x

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_decomp_cossin.py

@@ -0,0 +1,221 @@
+from collections.abc import Iterable
+import numpy as np
+
+from scipy._lib._util import _asarray_validated
+from scipy.linalg import block_diag, LinAlgError
+from .lapack import _compute_lwork, get_lapack_funcs
+
+__all__ = ['cossin']
+
+
+def cossin(X, p=None, q=None, separate=False,
+           swap_sign=False, compute_u=True, compute_vh=True):
+    """
+    Compute the cosine-sine (CS) decomposition of an orthogonal/unitary matrix.
+
+    X is an ``(m, m)`` orthogonal/unitary matrix, partitioned as the following
+    where upper left block has the shape of ``(p, q)``::
+
+                                   ┌                   ┐
+                                   │ I  0  0 │ 0  0  0 │
+        ┌           ┐   ┌         ┐│ 0  C  0 │ 0 -S  0 │┌         ┐*
+        │ X11 │ X12 │   │ U1 │    ││ 0  0  0 │ 0  0 -I ││ V1 │    │
+        │ ────┼──── │ = │────┼────││─────────┼─────────││────┼────│
+        │ X21 │ X22 │   │    │ U2 ││ 0  0  0 │ I  0  0 ││    │ V2 │
+        └           ┘   └         ┘│ 0  S  0 │ 0  C  0 │└         ┘
+                                   │ 0  0  I │ 0  0  0 │
+                                   └                   ┘
+
+    ``U1``, ``U2``, ``V1``, ``V2`` are square orthogonal/unitary matrices of
+    dimensions ``(p,p)``, ``(m-p,m-p)``, ``(q,q)``, and ``(m-q,m-q)``
+    respectively, and ``C`` and ``S`` are ``(r, r)`` nonnegative diagonal
+    matrices satisfying ``C^2 + S^2 = I`` where ``r = min(p, m-p, q, m-q)``.
+
+    Moreover, the rank of the identity matrices are ``min(p, q) - r``,
+    ``min(p, m - q) - r``, ``min(m - p, q) - r``, and ``min(m - p, m - q) - r``
+    respectively.
+
+    X can be supplied either by itself and block specifications p, q or its
+    subblocks in an iterable from which the shapes would be derived. See the
+    examples below.
+
+    Parameters
+    ----------
+    X : array_like, iterable
+        complex unitary or real orthogonal matrix to be decomposed, or iterable
+        of subblocks ``X11``, ``X12``, ``X21``, ``X22``, when ``p``, ``q`` are
+        omitted.
+    p : int, optional
+        Number of rows of the upper left block ``X11``, used only when X is
+        given as an array.
+    q : int, optional
+        Number of columns of the upper left block ``X11``, used only when X is
+        given as an array.
+    separate : bool, optional
+        if ``True``, the low level components are returned instead of the
+        matrix factors, i.e. ``(u1,u2)``, ``theta``, ``(v1h,v2h)`` instead of
+        ``u``, ``cs``, ``vh``.
+    swap_sign : bool, optional
+        if ``True``, the ``-S``, ``-I`` block will be the bottom left,
+        otherwise (by default) they will be in the upper right block.
+    compute_u : bool, optional
+        if ``False``, ``u`` won't be computed and an empty array is returned.
+    compute_vh : bool, optional
+        if ``False``, ``vh`` won't be computed and an empty array is returned.
+
+    Returns
+    -------
+    u : ndarray
+        When ``compute_u=True``, contains the block diagonal orthogonal/unitary
+        matrix consisting of the blocks ``U1`` (``p`` x ``p``) and ``U2``
+        (``m-p`` x ``m-p``) orthogonal/unitary matrices. If ``separate=True``,
+        this contains the tuple of ``(U1, U2)``.
+    cs : ndarray
+        The cosine-sine factor with the structure described above.
+         If ``separate=True``, this contains the ``theta`` array containing the
+         angles in radians.
+    vh : ndarray
+        When ``compute_vh=True`, contains the block diagonal orthogonal/unitary
+        matrix consisting of the blocks ``V1H`` (``q`` x ``q``) and ``V2H``
+        (``m-q`` x ``m-q``) orthogonal/unitary matrices. If ``separate=True``,
+        this contains the tuple of ``(V1H, V2H)``.
+
+    References
+    ----------
+    .. [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
+           Algorithms, 50(1):33-65, 2009.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.linalg import cossin
+    >>> from scipy.stats import unitary_group
+    >>> x = unitary_group.rvs(4)
+    >>> u, cs, vdh = cossin(x, p=2, q=2)
+    >>> np.allclose(x, u @ cs @ vdh)
+    True
+
+    Same can be entered via subblocks without the need of ``p`` and ``q``. Also
+    let's skip the computation of ``u``
+
+    >>> ue, cs, vdh = cossin((x[:2, :2], x[:2, 2:], x[2:, :2], x[2:, 2:]),
+    ...                      compute_u=False)
+    >>> print(ue)
+    []
+    >>> np.allclose(x, u @ cs @ vdh)
+    True
+
+    """
+
+    if p or q:
+        p = 1 if p is None else int(p)
+        q = 1 if q is None else int(q)
+        X = _asarray_validated(X, check_finite=True)
+        if not np.equal(*X.shape):
+            raise ValueError("Cosine Sine decomposition only supports square"
+                             f" matrices, got {X.shape}")
+        m = X.shape[0]
+        if p >= m or p <= 0:
+            raise ValueError(f"invalid p={p}, 0<p<{X.shape[0]} must hold")
+        if q >= m or q <= 0:
+            raise ValueError(f"invalid q={q}, 0<q<{X.shape[0]} must hold")
+
+        x11, x12, x21, x22 = X[:p, :q], X[:p, q:], X[p:, :q], X[p:, q:]
+    elif not isinstance(X, Iterable):
+        raise ValueError("When p and q are None, X must be an Iterable"
+                         " containing the subblocks of X")
+    else:
+        if len(X) != 4:
+            raise ValueError("When p and q are None, exactly four arrays"
+                             f" should be in X, got {len(X)}")
+
+        x11, x12, x21, x22 = (np.atleast_2d(x) for x in X)
+        for name, block in zip(["x11", "x12", "x21", "x22"],
+                               [x11, x12, x21, x22]):
+            if block.shape[1] == 0:
+                raise ValueError(f"{name} can't be empty")
+        p, q = x11.shape
+        mmp, mmq = x22.shape
+
+        if x12.shape != (p, mmq):
+            raise ValueError(f"Invalid x12 dimensions: desired {(p, mmq)}, "
+                             f"got {x12.shape}")
+
+        if x21.shape != (mmp, q):
+            raise ValueError(f"Invalid x21 dimensions: desired {(mmp, q)}, "
+                             f"got {x21.shape}")
+
+        if p + mmp != q + mmq:
+            raise ValueError("The subblocks have compatible sizes but "
+                             "don't form a square array (instead they form a"
+                              f" {p + mmp}x{q + mmq} array). This might be "
+                              "due to missing p, q arguments.")
+
+        m = p + mmp
+
+    cplx = any([np.iscomplexobj(x) for x in [x11, x12, x21, x22]])
+    driver = "uncsd" if cplx else "orcsd"
+    csd, csd_lwork = get_lapack_funcs([driver, driver + "_lwork"],
+                                      [x11, x12, x21, x22])
+    lwork = _compute_lwork(csd_lwork, m=m, p=p, q=q)
+    lwork_args = ({'lwork': lwork[0], 'lrwork': lwork[1]} if cplx else
+                  {'lwork': lwork})
+    *_, theta, u1, u2, v1h, v2h, info = csd(x11=x11, x12=x12, x21=x21, x22=x22,
+                                            compute_u1=compute_u,
+                                            compute_u2=compute_u,
+                                            compute_v1t=compute_vh,
+                                            compute_v2t=compute_vh,
+                                            trans=False, signs=swap_sign,
+                                            **lwork_args)
+
+    method_name = csd.typecode + driver
+    if info < 0:
+        raise ValueError(f'illegal value in argument {-info} '
+                         f'of internal {method_name}')
+    if info > 0:
+        raise LinAlgError(f"{method_name} did not converge: {info}")
+
+    if separate:
+        return (u1, u2), theta, (v1h, v2h)
+
+    U = block_diag(u1, u2)
+    VDH = block_diag(v1h, v2h)
+
+    # Construct the middle factor CS
+    c = np.diag(np.cos(theta))
+    s = np.diag(np.sin(theta))
+    r = min(p, q, m - p, m - q)
+    n11 = min(p, q) - r
+    n12 = min(p, m - q) - r
+    n21 = min(m - p, q) - r
+    n22 = min(m - p, m - q) - r
+    Id = np.eye(np.max([n11, n12, n21, n22, r]), dtype=theta.dtype)
+    CS = np.zeros((m, m), dtype=theta.dtype)
+
+    CS[:n11, :n11] = Id[:n11, :n11]
+
+    xs = n11 + r
+    xe = n11 + r + n12
+    ys = n11 + n21 + n22 + 2 * r
+    ye = n11 + n21 + n22 + 2 * r + n12
+    CS[xs: xe, ys:ye] = Id[:n12, :n12] if swap_sign else -Id[:n12, :n12]
+
+    xs = p + n22 + r
+    xe = p + n22 + r + + n21
+    ys = n11 + r
+    ye = n11 + r + n21
+    CS[xs:xe, ys:ye] = -Id[:n21, :n21] if swap_sign else Id[:n21, :n21]
+
+    CS[p:p + n22, q:q + n22] = Id[:n22, :n22]
+    CS[n11:n11 + r, n11:n11 + r] = c
+    CS[p + n22:p + n22 + r, n11 + r + n21 + n22:2 * r + n11 + n21 + n22] = c
+
+    xs = n11
+    xe = n11 + r
+    ys = n11 + n21 + n22 + r
+    ye = n11 + n21 + n22 + 2 * r
+    CS[xs:xe, ys:ye] = s if swap_sign else -s
+
+    CS[p + n22:p + n22 + r, n11:n11 + r] = -s if swap_sign else s
+
+    return U, CS, VDH

miniconda3/envs/ladir/lib/python3.10/site-packages/scipy/linalg/_decomp_ldl.py

@@ -0,0 +1,353 @@
+from warnings import warn
+
+import numpy as np
+from numpy import (atleast_2d, arange, zeros_like, imag, diag,
+                   iscomplexobj, tril, triu, argsort, empty_like)
+from scipy._lib._util import ComplexWarning
+from ._decomp import _asarray_validated
+from .lapack import get_lapack_funcs, _compute_lwork
+
+__all__ = ['ldl']
+
+
+def ldl(A, lower=True, hermitian=True, overwrite_a=False, check_finite=True):
+    """ Computes the LDLt or Bunch-Kaufman factorization of a symmetric/
+    hermitian matrix.
+
+    This function returns a block diagonal matrix D consisting blocks of size
+    at most 2x2 and also a possibly permuted unit lower triangular matrix
+    ``L`` such that the factorization ``A = L D L^H`` or ``A = L D L^T``
+    holds. If `lower` is False then (again possibly permuted) upper
+    triangular matrices are returned as outer factors.
+
+    The permutation array can be used to triangularize the outer factors
+    simply by a row shuffle, i.e., ``lu[perm, :]`` is an upper/lower
+    triangular matrix. This is also equivalent to multiplication with a
+    permutation matrix ``P.dot(lu)``, where ``P`` is a column-permuted
+    identity matrix ``I[:, perm]``.
+
+    Depending on the value of the boolean `lower`, only upper or lower
+    triangular part of the input array is referenced. Hence, a triangular
+    matrix on entry would give the same result as if the full matrix is
+    supplied.
+
+    Parameters
+    ----------
+    A : array_like
+        Square input array
+    lower : bool, optional
+        This switches between the lower and upper triangular outer factors of
+        the factorization. Lower triangular (``lower=True``) is the default.
+    hermitian : bool, optional
+        For complex-valued arrays, this defines whether ``A = A.conj().T`` or
+        ``A = A.T`` is assumed. For real-valued arrays, this switch has no
+        effect.
+    overwrite_a : bool, optional
+        Allow overwriting data in `A` (may enhance performance). The default
+        is False.
+    check_finite : bool, optional
+        Whether to check that the input matrices contain only finite numbers.
+        Disabling may give a performance gain, but may result in problems
+        (crashes, non-termination) if the inputs do contain infinities or NaNs.
+
+    Returns
+    -------
+    lu : ndarray
+        The (possibly) permuted upper/lower triangular outer factor of the
+        factorization.
+    d : ndarray
+        The block diagonal multiplier of the factorization.
+    perm : ndarray
+        The row-permutation index array that brings lu into triangular form.
+
+    Raises
+    ------
+    ValueError
+        If input array is not square.
+    ComplexWarning
+        If a complex-valued array with nonzero imaginary parts on the
+        diagonal is given and hermitian is set to True.
+
+    See Also
+    --------
+    cholesky, lu
+
+    Notes
+    -----
+    This function uses ``?SYTRF`` routines for symmetric matrices and
+    ``?HETRF`` routines for Hermitian matrices from LAPACK. See [1]_ for
+    the algorithm details.
+
+    Depending on the `lower` keyword value, only lower or upper triangular
+    part of the input array is referenced. Moreover, this keyword also defines
+    the structure of the outer factors of the factorization.
+
+    .. versionadded:: 1.1.0
+
+    References
+    ----------
+    .. [1] J.R. Bunch, L. Kaufman, Some stable methods for calculating
+       inertia and solving symmetric linear systems, Math. Comput. Vol.31,
+       1977. :doi:`10.2307/2005787`
+
+    Examples
+    --------
+    Given an upper triangular array ``a`` that represents the full symmetric
+    array with its entries, obtain ``l``, 'd' and the permutation vector `perm`:
+
+    >>> import numpy as np
+    >>> from scipy.linalg import ldl
+    >>> a = np.array([[2, -1, 3], [0, 2, 0], [0, 0, 1]])
+    >>> lu, d, perm = ldl(a, lower=0) # Use the upper part
+    >>> lu
+    array([[ 0. ,  0. ,  1. ],
+           [ 0. ,  1. , -0.5],
+           [ 1. ,  1. ,  1.5]])
+    >>> d
+    array([[-5. ,  0. ,  0. ],
+           [ 0. ,  1.5,  0. ],
+           [ 0. ,  0. ,  2. ]])
+    >>> perm
+    array([2, 1, 0])
+    >>> lu[perm, :]
+    array([[ 1. ,  1. ,  1.5],
+           [ 0. ,  1. , -0.5],
+           [ 0. ,  0. ,  1. ]])
+    >>> lu.dot(d).dot(lu.T)
+    array([[ 2., -1.,  3.],
+           [-1.,  2.,  0.],
+           [ 3.,  0.,  1.]])
+
+    """
+    a = atleast_2d(_asarray_validated(A, check_finite=check_finite))
+    if a.shape[0] != a.shape[1]:
+        raise ValueError('The input array "a" should be square.')
+    # Return empty arrays for empty square input
+    if a.size == 0:
+        return empty_like(a), empty_like(a), np.array([], dtype=int)
+
+    n = a.shape[0]
+    r_or_c = complex if iscomplexobj(a) else float
+
+    # Get the LAPACK routine
+    if r_or_c is complex and hermitian:
+        s, sl = 'hetrf', 'hetrf_lwork'
+        if np.any(imag(diag(a))):
+            warn('scipy.linalg.ldl():\nThe imaginary parts of the diagonal'
+                 'are ignored. Use "hermitian=False" for factorization of'
+                 'complex symmetric arrays.', ComplexWarning, stacklevel=2)
+    else:
+        s, sl = 'sytrf', 'sytrf_lwork'
+
+    solver, solver_lwork = get_lapack_funcs((s, sl), (a,))
+    lwork = _compute_lwork(solver_lwork, n, lower=lower)
+    ldu, piv, info = solver(a, lwork=lwork, lower=lower,
+                            overwrite_a=overwrite_a)
+    if info < 0:
+        raise ValueError(f'{s.upper()} exited with the internal error "illegal value '
+                         f'in argument number {-info}". See LAPACK documentation '
+                         'for the error codes.')
+
+    swap_arr, pivot_arr = _ldl_sanitize_ipiv(piv, lower=lower)
+    d, lu = _ldl_get_d_and_l(ldu, pivot_arr, lower=lower, hermitian=hermitian)
+    lu, perm = _ldl_construct_tri_factor(lu, swap_arr, pivot_arr, lower=lower)
+
+    return lu, d, perm
+
+
+def _ldl_sanitize_ipiv(a, lower=True):
+    """
+    This helper function takes the rather strangely encoded permutation array
+    returned by the LAPACK routines ?(HE/SY)TRF and converts it into
+    regularized permutation and diagonal pivot size format.
+
+    Since FORTRAN uses 1-indexing and LAPACK uses different start points for
+    upper and lower formats there are certain offsets in the indices used
+    below.
+
+    Let's assume a result where the matrix is 6x6 and there are two 2x2
+    and two 1x1 blocks reported by the routine. To ease the coding efforts,
+    we still populate a 6-sized array and fill zeros as the following ::
+
+        pivots = [2, 0, 2, 0, 1, 1]
+
+    This denotes a diagonal matrix of the form ::
+
+        [x x        ]
+        [x x        ]
+        [    x x    ]
+        [    x x    ]
+        [        x  ]
+        [          x]
+
+    In other words, we write 2 when the 2x2 block is first encountered and
+    automatically write 0 to the next entry and skip the next spin of the
+    loop. Thus, a separate counter or array appends to keep track of block
+    sizes are avoided. If needed, zeros can be filtered out later without
+    losing the block structure.
+
+    Parameters
+    ----------
+    a : ndarray
+        The permutation array ipiv returned by LAPACK
+    lower : bool, optional
+        The switch to select whether upper or lower triangle is chosen in
+        the LAPACK call.
+
+    Returns
+    -------
+    swap_ : ndarray
+        The array that defines the row/column swap operations. For example,
+        if row two is swapped with row four, the result is [0, 3, 2, 3].
+    pivots : ndarray
+        The array that defines the block diagonal structure as given above.
+
+    """
+    n = a.size
+    swap_ = arange(n)
+    pivots = zeros_like(swap_, dtype=int)
+    skip_2x2 = False
+
+    # Some upper/lower dependent offset values
+    # range (s)tart, r(e)nd, r(i)ncrement
+    x, y, rs, re, ri = (1, 0, 0, n, 1) if lower else (-1, -1, n-1, -1, -1)
+
+    for ind in range(rs, re, ri):
+        # If previous spin belonged already to a 2x2 block
+        if skip_2x2:
+            skip_2x2 = False
+            continue
+
+        cur_val = a[ind]
+        # do we have a 1x1 block or not?
+        if cur_val > 0:
+            if cur_val != ind+1:
+                # Index value != array value --> permutation required
+                swap_[ind] = swap_[cur_val-1]
+            pivots[ind] = 1
+        # Not.
+        elif cur_val < 0 and cur_val == a[ind+x]:
+            # first neg entry of 2x2 block identifier
+            if -cur_val != ind+2:
+                # Index value != array value --> permutation required
+                swap_[ind+x] = swap_[-cur_val-1]
+            pivots[ind+y] = 2
+            skip_2x2 = True
+        else:  # Doesn't make sense, give up
+            raise ValueError('While parsing the permutation array '
+                             'in "scipy.linalg.ldl", invalid entries '
+                             'found. The array syntax is invalid.')
+    return swap_, pivots
+
+
+def _ldl_get_d_and_l(ldu, pivs, lower=True, hermitian=True):
+    """
+    Helper function to extract the diagonal and triangular matrices for
+    LDL.T factorization.
+
+    Parameters
+    ----------
+    ldu : ndarray
+        The compact output returned by the LAPACK routing
+    pivs : ndarray
+        The sanitized array of {0, 1, 2} denoting the sizes of the pivots. For
+        every 2 there is a succeeding 0.
+    lower : bool, optional
+        If set to False, upper triangular part is considered.
+    hermitian : bool, optional
+        If set to False a symmetric complex array is assumed.
+
+    Returns
+    -------
+    d : ndarray
+        The block diagonal matrix.
+    lu : ndarray
+        The upper/lower triangular matrix
+    """
+    is_c = iscomplexobj(ldu)
+    d = diag(diag(ldu))
+    n = d.shape[0]
+    blk_i = 0  # block index
+
+    # row/column offsets for selecting sub-, super-diagonal
+    x, y = (1, 0) if lower else (0, 1)
+
+    lu = tril(ldu, -1) if lower else triu(ldu, 1)
+    diag_inds = arange(n)
+    lu[diag_inds, diag_inds] = 1
+
+    for blk in pivs[pivs != 0]:
+        # increment the block index and check for 2s
+        # if 2 then copy the off diagonals depending on uplo
+        inc = blk_i + blk
+
+        if blk == 2:
+            d[blk_i+x, blk_i+y] = ldu[blk_i+x, blk_i+y]
+            # If Hermitian matrix is factorized, the cross-offdiagonal element
+            # should be conjugated.
+            if is_c and hermitian:
+                d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y].conj()
+            else:
+                d[blk_i+y, blk_i+x] = ldu[blk_i+x, blk_i+y]
+
+            lu[blk_i+x, blk_i+y] = 0.
+        blk_i = inc
+
+    return d, lu
+
+
+def _ldl_construct_tri_factor(lu, swap_vec, pivs, lower=True):
+    """
+    Helper function to construct explicit outer factors of LDL factorization.
+
+    If lower is True the permuted factors are multiplied as L(1)*L(2)*...*L(k).
+    Otherwise, the permuted factors are multiplied as L(k)*...*L(2)*L(1). See
+    LAPACK documentation for more details.
+
+    Parameters
+    ----------
+    lu : ndarray
+        The triangular array that is extracted from LAPACK routine call with
+        ones on the diagonals.
+    swap_vec : ndarray
+        The array that defines the row swapping indices. If the kth entry is m
+        then rows k,m are swapped. Notice that the mth entry is not necessarily
+        k to avoid undoing the swapping.
+    pivs : ndarray
+        The array that defines the block diagonal structure returned by
+        _ldl_sanitize_ipiv().
+    lower : bool, optional
+        The boolean to switch between lower and upper triangular structure.
+
+    Returns
+    -------
+    lu : ndarray
+        The square outer factor which satisfies the L * D * L.T = A
+    perm : ndarray
+        The permutation vector that brings the lu to the triangular form
+
+    Notes
+    -----
+    Note that the original argument "lu" is overwritten.
+
+    """
+    n = lu.shape[0]
+    perm = arange(n)
+    # Setup the reading order of the permutation matrix for upper/lower
+    rs, re, ri = (n-1, -1, -1) if lower else (0, n, 1)
+
+    for ind in range(rs, re, ri):
+        s_ind = swap_vec[ind]
+        if s_ind != ind:
+            # Column start and end positions
+            col_s = ind if lower else 0
+            col_e = n if lower else ind+1
+
+            # If we stumble upon a 2x2 block include both cols in the perm.
+            if pivs[ind] == (0 if lower else 2):
+                col_s += -1 if lower else 0
+                col_e += 0 if lower else 1
+            lu[[s_ind, ind], col_s:col_e] = lu[[ind, s_ind], col_s:col_e]
+            perm[[s_ind, ind]] = perm[[ind, s_ind]]
+
+    return lu, argsort(perm)