Upload folder using huggingface_hub (part 15)

.gitattributes

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.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_batch.py

@@ -0,0 +1,742 @@
+import inspect
+import pytest
+import numpy as np
+from numpy.testing import assert_allclose
+from scipy import linalg, sparse
+
+
+real_floating = [np.float32, np.float64]
+complex_floating = [np.complex64, np.complex128]
+floating = real_floating + complex_floating
+
+
+def get_random(shape, *, dtype, rng):
+    A = rng.random(shape)
+    if np.issubdtype(dtype, np.complexfloating):
+        A = A + rng.random(shape) * 1j
+    return A.astype(dtype)
+
+def get_nearly_hermitian(shape, dtype, atol, rng):
+    # Generate a batch of nearly Hermitian matrices with specified
+    # `shape` and `dtype`. `atol` controls the level of noise in
+    # Hermitian-ness to by generated by `rng`.
+    A = rng.random(shape).astype(dtype)
+    At = np.conj(A.swapaxes(-1, -2))
+    noise = rng.standard_normal(size=A.shape).astype(dtype) * atol
+    return A + At + noise
+
+
+class TestBatch:
+    # Test batch support for most linalg functions
+
+    def batch_test(self, fun, arrays, *, core_dim=2, n_out=1, kwargs=None, dtype=None,
+                   broadcast=True, check_kwargs=True):
+        # Check that all outputs of batched call `fun(A, **kwargs)` are the same
+        # as if we loop over the separate vectors/matrices in `A`. Also check
+        # that `fun` accepts `A` by position or keyword and that results are
+        # identical. This is important because the name of the array argument
+        # is manually specified to the decorator, and it's easy to mess up.
+        # However, this makes it hard to test positional arguments passed
+        # after the array, so we test that separately for a few functions to
+        # make sure the decorator is working as it should.
+
+        kwargs = {} if kwargs is None else kwargs
+        parameters = list(inspect.signature(fun).parameters.keys())
+        arrays = (arrays,) if not isinstance(arrays, tuple) else arrays
+
+        # Identical results when passing argument by keyword or position
+        res2 = fun(*arrays, **kwargs)
+        if check_kwargs:
+            res1 = fun(**dict(zip(parameters, arrays)), **kwargs)
+            for out1, out2 in zip(res1, res2):  # even a single array is iterable...
+                np.testing.assert_equal(out1, out2)
+
+        # Check results vs looping over
+        res = (res2,) if n_out == 1 else res2
+        # This is not the general behavior (only batch dimensions get
+        # broadcasted by the decorator) but it's easier for testing.
+        if broadcast:
+            arrays = np.broadcast_arrays(*arrays)
+        batch_shape = arrays[0].shape[:-core_dim]
+        for i in range(batch_shape[0]):
+            for j in range(batch_shape[1]):
+                arrays_ij = (array[i, j] for array in arrays)
+                ref = fun(*arrays_ij, **kwargs)
+                # see gh-25508:
+                if (np.dtype(np.int_).itemsize == 4 and
+                     n_out > 1 and isinstance(ref[0], int)):
+                     ref = tuple(np.asarray(refk, dtype=np.int64) for refk in ref)
+                else:
+                     ref = ((np.asarray(ref),) if n_out == 1 else
+                            tuple(np.asarray(refk) for refk in ref))
+                for k in range(n_out):
+                    assert_allclose(res[k][i, j], ref[k])
+                    assert np.shape(res[k][i, j]) == ref[k].shape
+
+        for k in range(len(ref)):
+            out_dtype = ref[k].dtype if dtype is None else dtype
+            assert res[k].dtype == out_dtype
+
+        return res2  # return original, non-tuplized result
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_expm_cond(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = rng.random((5, 3, 4, 4)).astype(dtype)
+        self.batch_test(linalg.expm_cond, A)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_issymmetric(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_nearly_hermitian((5, 3, 4, 4), dtype, 3e-4, rng)
+        res = self.batch_test(linalg.issymmetric, A, kwargs=dict(atol=1e-3))
+        assert not np.all(res)  # ensure test is not trivial: not all True or False;
+        assert np.any(res)      # also confirms that `atol` is passed to issymmetric
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_ishermitian(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_nearly_hermitian((5, 3, 4, 4), dtype, 3e-4, rng)
+        res = self.batch_test(linalg.ishermitian, A, kwargs=dict(atol=1e-3))
+        assert not np.all(res)  # ensure test is not trivial: not all True or False;
+        assert np.any(res)      # also confirms that `atol` is passed to ishermitian
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_diagsvd(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = rng.random((5, 3, 4)).astype(dtype)
+        res1 = self.batch_test(linalg.diagsvd, A, kwargs=dict(M=6, N=4), core_dim=1)
+        # test that `M, N` can be passed by position
+        res2 = linalg.diagsvd(A, 6, 4)
+        np.testing.assert_equal(res1, res2)
+
+    @pytest.mark.parametrize('fun', [linalg.inv, linalg.sqrtm, linalg.signm,
+                                     linalg.sinm, linalg.cosm, linalg.tanhm,
+                                     linalg.sinhm, linalg.coshm, linalg.tanhm,
+                                     linalg.pinv, linalg.pinvh, linalg.orth])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_matmat(self, fun, dtype):  # matrix in, matrix out
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+
+        # sqrtm can return complex output for real input resulting in i/o type
+        # mismatch. Nudge the eigenvalues to positive side to avoid this.
+        if fun == linalg.sqrtm:
+            A = A + 3*np.eye(4, dtype=dtype)
+
+        self.batch_test(fun, A)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_null_space(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 6), dtype=dtype, rng=rng)
+        self.batch_test(linalg.null_space, A)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_funm(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 4, 3, 3), dtype=dtype, rng=rng)
+        self.batch_test(linalg.funm, A, kwargs=dict(func=np.sin))
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_fractional_matrix_power(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 4, 3, 3), dtype=dtype, rng=rng)
+        res1 = self.batch_test(linalg.fractional_matrix_power, A, kwargs={'t':1.5})
+        # test that `t` can be passed by position
+        res2 = linalg.fractional_matrix_power(A, 1.5)
+        np.testing.assert_equal(res1, res2)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_logm(self, dtype):
+        # One test failed absolute tolerance with default random seed
+        rng = np.random.default_rng(89940026998903887141749720079406074936)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+        A = A + 3*np.eye(4)  # avoid complex output for real input
+        res1 = self.batch_test(linalg.logm, A)
+        # test that `disp` can be passed by position
+        res2 = linalg.logm(A)
+        for res1i, res2i in zip(res1, res2):
+            np.testing.assert_equal(res1i, res2i)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_pinv(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+        self.batch_test(linalg.pinv, A, n_out=2, kwargs=dict(return_rank=True))
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_matrix_balance(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+        self.batch_test(linalg.matrix_balance, A, n_out=2)
+        self.batch_test(linalg.matrix_balance, A, n_out=2, kwargs={'separate':True})
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_bandwidth(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((4, 4), dtype=dtype, rng=rng)
+        A = np.asarray([np.triu(A, k) for k in range(-3, 3)]).reshape((2, 3, 4, 4))
+        self.batch_test(linalg.bandwidth, A, n_out=2)
+
+    @pytest.mark.parametrize('fun_n_out', [(linalg.cholesky, 1), (linalg.ldl, 3),
+                                           (linalg.cho_factor, 2)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_ldl_cholesky(self, fun_n_out, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        fun, n_out = fun_n_out
+        A = get_nearly_hermitian((5, 3, 4, 4), dtype, 0, rng)  # exactly Hermitian
+        A = A + 4*np.eye(4, dtype=dtype)  # ensure positive definite for Cholesky
+        self.batch_test(fun, A, n_out=n_out)
+
+    @pytest.mark.parametrize('compute_uv', [False, True])
+    @pytest.mark.parametrize('full_matrices', [False, True])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_svd(self, compute_uv, full_matrices, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 2, 4), dtype=dtype, rng=rng)
+        n_out = 3 if compute_uv else 1
+        self.batch_test(
+            linalg.svd, A, n_out=n_out,
+            kwargs=dict(compute_uv=compute_uv, full_matrices=full_matrices)
+        )
+
+        A = get_random((5, 3, 2, 0), dtype=dtype, rng=rng)
+        self.batch_test(
+            linalg.svd, A, n_out=n_out,
+            kwargs=dict(compute_uv=compute_uv, full_matrices=full_matrices)
+        )
+
+    @pytest.mark.parametrize('fun', [linalg.polar, linalg.rq])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_polar_qr_rq(self, fun, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 2, 4), dtype=dtype, rng=rng)
+        self.batch_test(fun, A, n_out=2)
+
+    @pytest.mark.parametrize('pivoting', [True, False])
+    @pytest.mark.parametrize('dtype', floating)
+    @pytest.mark.parametrize('mode', ['full', 'economic', 'r'])
+    def test_qr(self, mode, dtype, pivoting):
+        rng = np.random.default_rng(12345)
+        a = get_random((5, 3, 2, 4), dtype=dtype, rng=rng)
+        self.batch_test(lambda x: linalg.qr(x, mode=mode, pivoting=pivoting), a,
+                        n_out=(1 if mode == 'r' else 2) + 1 if pivoting else 0)
+
+    @pytest.mark.parametrize('pivoting', [True, False])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_qr_raw(self, dtype, pivoting):
+        rng = np.random.default_rng(12345)
+        a = get_random((5, 3, 2, 4), dtype=dtype, rng=rng)
+        (q_raw, tau), r, *other = linalg.qr(a, mode="raw", pivoting=pivoting)
+
+        # return argument contains a tuple, hence `batch_test` is not applicable.
+        for i in range(a.shape[0]):
+            for j in range(a.shape[1]):
+                (q_raw_ij, tau_ij), r_ij, *other_ij = linalg.qr(
+                    a[i, j, :, :], mode="raw", pivoting=pivoting
+                )
+
+                assert_allclose(q_raw[i, j, :, :], q_raw_ij)
+                assert_allclose(tau[i, j, :], tau_ij)
+                assert_allclose(r[i, j, :, :], r_ij)
+
+                if pivoting:
+                    assert_allclose(other[0][i, j, :], other_ij[0])
+
+
+    @pytest.mark.parametrize('cdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_qr_multiply(self, cdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        c = get_random(cdim, dtype=dtype, rng=rng)
+        res = linalg.qr_multiply(A, c, mode='left')
+        q, r = linalg.qr(A)
+        ref = q @ c
+        atol = 1e-6 if dtype in {np.float32, np.complex64} else 1e-12
+        assert_allclose(res[0], ref, atol=atol)
+        assert_allclose(res[1], r, atol=atol)
+
+    @pytest.mark.parametrize('uvdim', [[(5,), (3,)], [(4, 5, 2), (4, 3, 2)]])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_qr_update(self, uvdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        udim, vdim = uvdim
+        A = get_random((4, 5, 3), dtype=dtype, rng=rng)
+        u = get_random(udim, dtype=dtype, rng=rng)
+        v = get_random(vdim, dtype=dtype, rng=rng)
+        q, r = linalg.qr(A)
+        res = linalg.qr_update(q, r, u, v)
+        for i in range(4):
+            qi, ri = q[i], r[i]
+            ui, vi = (u, v) if u.ndim == 1 else (u[i], v[i])
+            ref_i = linalg.qr_update(qi, ri, ui, vi)
+            assert_allclose(res[0][i], ref_i[0])
+            assert_allclose(res[1][i], ref_i[1])
+
+    @pytest.mark.parametrize('udim', [(5,), (4, 3, 5)])
+    @pytest.mark.parametrize('kdim', [(), (4,)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_qr_insert(self, udim, kdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((4, 5, 5), dtype=dtype, rng=rng)
+        u = get_random(udim, dtype=dtype, rng=rng)
+        k = rng.integers(0, 5, size=kdim)
+        q, r = linalg.qr(A)
+        res = linalg.qr_insert(q, r, u, k)
+        for i in range(4):
+            qi, ri = q[i], r[i]
+            ki = k if k.ndim == 0 else k[i]
+            ui = u if u.ndim == 1 else u[i]
+            ref_i = linalg.qr_insert(qi, ri, ui, ki)
+            assert_allclose(res[0][i], ref_i[0])
+            assert_allclose(res[1][i], ref_i[1])
+
+    @pytest.mark.parametrize('kdim', [(), (4,)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_qr_delete(self, kdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((4, 5, 5), dtype=dtype, rng=rng)
+        k = rng.integers(0, 4, size=kdim)
+        q, r = linalg.qr(A)
+        res = linalg.qr_delete(q, r, k)
+        for i in range(4):
+            qi, ri = q[i], r[i]
+            ki = k if k.ndim == 0 else k[i]
+            ref_i = linalg.qr_delete(qi, ri, ki)
+            assert_allclose(res[0][i], ref_i[0])
+            assert_allclose(res[1][i], ref_i[1])
+
+    @pytest.mark.parametrize('fun', [linalg.schur, linalg.lu_factor])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_schur_lu(self, fun, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+        self.batch_test(fun, A, n_out=2)
+
+    @pytest.mark.parametrize('calc_q', [False, True])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_hessenberg(self, calc_q, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+        n_out = 2 if calc_q else 1
+        self.batch_test(linalg.hessenberg, A, n_out=n_out, kwargs=dict(calc_q=calc_q))
+
+    @pytest.mark.parametrize('eigvals_only', [False, True])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_eig_banded(self, eigvals_only, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+        n_out = 1 if eigvals_only else 2
+        self.batch_test(linalg.eig_banded, A, n_out=n_out,
+                        kwargs=dict(eigvals_only=eigvals_only))
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_eigvals_banded(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 4), dtype=dtype, rng=rng)
+        self.batch_test(linalg.eigvals_banded, A)
+
+    @pytest.mark.parametrize('two_in', [False, True])
+    @pytest.mark.parametrize('fun_n_nout', [(linalg.eigh, 1), (linalg.eigh, 2),
+                                            (linalg.eigvalsh, 1), (linalg.eigvals, 1)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_eigh(self, two_in, fun_n_nout, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        fun, n_out = fun_n_nout
+        A = get_nearly_hermitian((1, 3, 4, 4), dtype, 0, rng)  # exactly Hermitian
+        B = get_nearly_hermitian((2, 1, 4, 4), dtype, 0, rng)  # exactly Hermitian
+        B = B + 4*np.eye(4).astype(dtype)  # needs to be positive definite
+        args = (A, B) if two_in else (A,)
+        kwargs = dict(eigvals_only=True) if (n_out == 1 and fun==linalg.eigh) else {}
+        self.batch_test(fun, args, n_out=n_out, kwargs=kwargs)
+
+    @pytest.mark.parametrize('compute_expm', [False, True])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_expm_frechet(self, compute_expm, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((1, 3, 4, 4), dtype=dtype, rng=rng)
+        E = get_random((2, 1, 4, 4), dtype=dtype, rng=rng)
+        n_out = 2 if compute_expm else 1
+        self.batch_test(linalg.expm_frechet, (A, E), n_out=n_out,
+                        kwargs=dict(compute_expm=compute_expm))
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_subspace_angles(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((1, 3, 4, 3), dtype=dtype, rng=rng)
+        B = get_random((2, 1, 4, 3), dtype=dtype, rng=rng)
+        self.batch_test(linalg.subspace_angles, (A, B))
+        # just to show that A and B don't need to be broadcastable
+        M, N, K = 4, 5, 3
+        A = get_random((1, 3, M, N), dtype=dtype, rng=rng)
+        B = get_random((2, 1, M, K), dtype=dtype, rng=rng)
+        assert linalg.subspace_angles(A, B).shape == (2, 3, min(N, K))
+
+    @pytest.mark.parametrize('fun', [linalg.svdvals])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_svdvals(self, fun, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 4, 5), dtype=dtype, rng=rng)
+        self.batch_test(fun, A)
+
+    @pytest.mark.parametrize('fun_n_out', [(linalg.orthogonal_procrustes, 2),
+                                           (linalg.khatri_rao, 1),
+                                           (linalg.solve_continuous_lyapunov, 1),
+                                           (linalg.solve_discrete_lyapunov, 1),
+                                           (linalg.qz, 4),
+                                           (linalg.ordqz, 6)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_two_generic_matrix_inputs(self, fun_n_out, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        fun, n_out = fun_n_out
+        A = get_random((2, 3, 4, 4), dtype=dtype, rng=rng)
+        B = get_random((2, 3, 4, 4), dtype=dtype, rng=rng)
+        self.batch_test(fun, (A, B), n_out=n_out)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_cossin(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        p, q = 3, 4
+        X = get_random((2, 3, 10, 10), dtype=dtype, rng=rng)
+        x11, x12, x21, x22 = (X[..., :p, :q], X[..., :p, q:],
+                              X[..., p:, :q], X[..., p:, q:])
+        res = linalg.cossin(X, p, q)
+        ref = linalg.cossin((x11, x12, x21, x22))
+        for res_i, ref_i in zip(res, ref):
+            np.testing.assert_equal(res_i, ref_i)
+
+        for j in range(2):
+            for k in range(3):
+                ref_jk = linalg.cossin(X[j, k], p, q)
+                for res_i, ref_ijk in zip(res, ref_jk):
+                    np.testing.assert_equal(res_i[j, k], ref_ijk)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_sylvester(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        B = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        C = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        self.batch_test(linalg.solve_sylvester, (A, B, C))
+
+    @pytest.mark.parametrize('fun', [linalg.solve_continuous_are,
+                                     linalg.solve_discrete_are])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_are(self, fun, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        a = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        b = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        q = get_nearly_hermitian((2, 3, 5, 5), dtype=dtype, atol=0, rng=rng)
+        r = get_nearly_hermitian((2, 3, 5, 5), dtype=dtype, atol=0, rng=rng)
+        a = a + 5*np.eye(5)  # making these positive definite seems to help
+        b = b + 5*np.eye(5)
+        q = q + 5*np.eye(5)
+        r = r + 5*np.eye(5)
+        e = np.eye(5)
+        s = np.zeros((5, 5))
+        self.batch_test(fun, (a, b, q, r))
+        self.batch_test(fun, (a, b, q, r, e))
+        self.batch_test(fun, (a, b, q, r, e, s))
+
+        res = fun(a, b, q, r)
+        ref = fun(a, b, q, r, s=s)
+        np.testing.assert_allclose(res, ref)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_rsf2cs(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 4, 4), dtype=dtype, rng=rng)
+        T, Z = linalg.schur(A)
+        self.batch_test(linalg.rsf2csf, (T, Z), n_out=2)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_cholesky_banded(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        ab = get_random((5, 4, 3, 6), dtype=dtype, rng=rng)
+        ab[..., -1, :] = 10  # make diagonal dominant
+        self.batch_test(linalg.cholesky_banded, ab)
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_block_diag(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        a = get_random((1, 3, 1, 3), dtype=dtype, rng=rng)
+        b = get_random((2, 1, 3, 6), dtype=dtype, rng=rng)
+        c = get_random((1, 1, 3, 2), dtype=dtype, rng=rng)
+
+        # batch_test doesn't have the logic to broadcast just the batch shapes,
+        # so do it manually.
+        a2 = np.broadcast_to(a, (2, 3, 1, 3))
+        b2 = np.broadcast_to(b, (2, 3, 3, 6))
+        c2 = np.broadcast_to(c, (2, 3, 3, 2))
+        ref = self.batch_test(linalg.block_diag, (a2, b2, c2),
+                              check_kwargs=False, broadcast=False)
+
+        # Check that `block_diag` broadcasts the batch shapes as expected.
+        res = linalg.block_diag(a, b, c)
+        assert_allclose(res, ref)
+
+    @pytest.mark.parametrize('fun_n_out', [(linalg.eigh_tridiagonal, 2),
+                                           (linalg.eigvalsh_tridiagonal, 1)])
+    @pytest.mark.parametrize('dtype', real_floating)
+    # "Only real arrays currently supported"
+    def test_eigh_tridiagonal(self, fun_n_out, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        fun, n_out = fun_n_out
+        d = get_random((3, 4, 5), dtype=dtype, rng=rng)
+        e = get_random((3, 4, 4), dtype=dtype, rng=rng)
+        self.batch_test(fun, (d, e), core_dim=1, n_out=n_out, broadcast=False)
+
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_solve(self, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        x = linalg.solve(A, b)
+        if len(bdim) == 1:
+            x = x[..., np.newaxis]
+            b = b[..., np.newaxis]
+        assert_allclose(A @ x - b, 0, atol=2e-6)
+        atol = 3e-6
+        if dtype is np.float32:
+            atol = 4e-6
+        assert_allclose(x, np.linalg.solve(A, b), atol=atol)
+
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_lu_solve(self, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        lu_and_piv = linalg.lu_factor(A)
+        x = linalg.lu_solve(lu_and_piv, b)
+        if len(bdim) == 1:
+            x = x[..., np.newaxis]
+            b = b[..., np.newaxis]
+        assert_allclose(A @ x - b, 0, atol=2e-6)
+        atol = 3e-6
+        if dtype is np.float32:
+            atol = 4e-6
+        assert_allclose(x, np.linalg.solve(A, b), atol=atol)
+
+    @pytest.mark.parametrize('l_and_u', [(1, 1), ([2, 1, 0], [0, 1 , 2])])
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_solve_banded(self, l_and_u, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        l, u = l_and_u
+        ab = get_random((2, 3, 3, 5), dtype=dtype, rng=rng)
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        x = linalg.solve_banded((l, u), ab, b)
+        for i in range(2):
+            for j in range(3):
+                bij = b if len(bdim) <= 2 else b[i, j]
+                lj = l if np.ndim(l) == 0 else l[j]
+                uj = u if np.ndim(u) == 0 else u[j]
+                xij = linalg.solve_banded((lj, uj), ab[i, j], bij)
+                assert_allclose(x[i, j], xij)
+
+    @pytest.mark.parametrize('separate_r', [False, True])
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_solve_toeplitz(self, separate_r, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        c = get_random((2, 3, 5), dtype=dtype, rng=rng)
+        r = get_random((2, 3, 5), dtype=dtype, rng=rng)
+        c_or_cr = (c, r) if separate_r else c
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        x = linalg.solve_toeplitz(c_or_cr, b)
+        for i in range(2):
+            for j in range(3):
+                bij = b if len(bdim) <= 2 else b[i, j]
+                c_or_cr_ij = (c[i, j], r[i, j]) if separate_r else c[i, j]
+                xij = linalg.solve_toeplitz(c_or_cr_ij, bij)
+                assert_allclose(x[i, j], xij)
+
+    @pytest.mark.parametrize('separate_r', [False, True])
+    @pytest.mark.parametrize('xdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_matmul_toeplitz(self, separate_r, xdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        c = get_random((2, 3, 5), dtype=dtype, rng=rng)
+        r = get_random((2, 3, 5), dtype=dtype, rng=rng)
+        c_or_cr = (c, r) if separate_r else c
+        x = get_random(xdim, dtype=dtype, rng=rng)
+        res = linalg.matmul_toeplitz(c_or_cr, x)
+        if separate_r:
+            ref = linalg.toeplitz(c, r) @ x
+        else:
+            ref = linalg.toeplitz(c) @ x
+        atol = 1e-6 if dtype in {np.float32, np.complex64} else 1e-12
+        assert_allclose(res, ref, atol=atol)
+
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_cho_solve(self, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_nearly_hermitian((2, 3, 5, 5), dtype=dtype, atol=0, rng=rng)
+        A = A + 5*np.eye(5)
+        c_and_lower = linalg.cho_factor(A)
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        x = linalg.cho_solve(c_and_lower, b)
+        if len(bdim) == 1:
+            x = x[..., np.newaxis]
+            b = b[..., np.newaxis]
+        assert_allclose(A @ x - b, 0, atol=1e-6)
+        assert_allclose(x, np.linalg.solve(A, b), atol=2e-6)
+
+    @pytest.mark.parametrize('lower', [False, True])
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_cho_solve_banded(self, lower, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 3, 5), dtype=dtype, rng=rng)
+        row_diag = 0 if lower else -1
+        A[:, :, row_diag] = 10
+        cb = linalg.cholesky_banded(A, lower=lower)
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        x = linalg.cho_solve_banded((cb, lower), b)
+        for i in range(2):
+            for j in range(3):
+                bij = b if len(bdim) <= 2 else b[i, j]
+                xij = linalg.cho_solve_banded((cb[i, j], lower), bij)
+                assert_allclose(x[i, j], xij)
+
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_solveh_banded(self, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 3, 5), dtype=dtype, rng=rng)
+        A[:, :, -1] = 10
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        x = linalg.solveh_banded(A, b)
+        for i in range(2):
+            for j in range(3):
+                bij = b if len(bdim) <= 2 else b[i, j]
+                xij = linalg.solveh_banded(A[i, j], bij)
+                assert_allclose(x[i, j], xij)
+
+    @pytest.mark.parametrize('bdim', [(5,), (5, 4), (2, 3, 5, 4)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_solve_triangular(self, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 5, 5), dtype=dtype, rng=rng)
+        A = np.tril(A)
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        x = linalg.solve_triangular(A, b, lower=True)
+        if len(bdim) == 1:
+            x = x[..., np.newaxis]
+            b = b[..., np.newaxis]
+        atol = 1e-10 if dtype in (np.complex128, np.float64) else 2e-4
+        assert_allclose(A @ x - b, 0, atol=atol)
+        assert_allclose(x, np.linalg.solve(A, b), atol=5*atol)
+
+    @pytest.mark.parametrize('bdim', [(4,), (4, 3), (2, 3, 4, 3)])
+    @pytest.mark.parametrize('dtype', floating)
+    def test_lstsq(self, bdim, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((2, 3, 4, 5), dtype=dtype, rng=rng)
+        b = get_random(bdim, dtype=dtype, rng=rng)
+        res = linalg.lstsq(A, b)
+        x = res[0]
+        if len(bdim) == 1:
+            x = x[..., np.newaxis]
+            b = b[..., np.newaxis]
+        assert_allclose(A @ x - b, 0, atol=2e-6)
+        assert len(res) == 4
+
+    @pytest.mark.parametrize('dtype', floating)
+    def test_clarkson_woodruff_transform(self, dtype):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 6), dtype=dtype, rng=rng)
+        self.batch_test(linalg.clarkson_woodruff_transform, A,
+                        kwargs=dict(sketch_size=3, rng=311224))
+
+    def test_clarkson_woodruff_transform_sparse(self):
+        rng = np.random.default_rng(8342310302941288912051)
+        A = get_random((5, 3, 4, 6), dtype=np.float64, rng=rng)
+        A = sparse.coo_array(A)
+        message = "Batch support for sparse arrays is not available."
+        with pytest.raises(NotImplementedError, match=message):
+            linalg.clarkson_woodruff_transform(A, sketch_size=3, rng=rng)
+
+    @pytest.mark.parametrize('f, args', [
+        (linalg.toeplitz, (np.ones((0, 4)),)),
+    ])
+    def test_zero_size_batch(self, f, args):
+        message = "does not support zero-size batches."
+        with pytest.raises(ValueError, match=message):
+            f(*args)
+
+
+@pytest.mark.parametrize(
+    "func, core_shape",
+    [
+     (linalg.solve, (3, 3)),
+     (linalg.solve_triangular, (3, 3)),
+     (lambda a, b: linalg.lstsq(a, b)[0], (3, 2)),
+    ],
+    ids=["solve", "solve_triangular", "lstsq",]
+)
+def test_shapes_solve_like(func, core_shape):
+    # test shapes for solve-like functions, where the 2nd argument may have trailing
+    # dimensions
+    m, n = core_shape
+
+    # ### 1. a.ndim == 2 ###
+
+    # 1.1 : b.ndim == 1, the "scalar" case
+    a = np.eye(m, n)
+    b = np.ones(m)
+
+    x = func(a, b)
+    assert x.shape == (n,)
+
+    # 1.2 : b.ndim == 2 : b has trailing dims
+    b = np.ones((m, 8))
+    x = func(a, b)
+    assert x.shape == (n, 8)
+
+    # 1.3 : b.ndim == 3 : b has trailing dims *and* batch dims
+    b = np.ones((4, m, 8))
+    x = func(a, b)
+    assert x.shape == (4, n, 8)
+
+    # 1.4 : b.ndim == 2 : b has batch dims *but* no trailing dims
+    b = np.ones((4, m))
+    pattern = "Shape mismatch|incompatible shapes|shapes of a|shape mismatch"
+    with pytest.raises(ValueError, match=pattern):
+        # fails to broadcast `b` vs `a` (to fix: append a length-1 trailing dim)
+        func(a, b)
+
+    # ### 2. a.ndim > 2 ###
+    a = np.broadcast_to(a, (5, 4, *a.shape))  # batch_shape = (5, 4)
+
+    # 2.1 : b is 1D
+    b = np.ones(m)
+    x = func(a, b)
+    assert x.shape == (5, 4, n)
+
+    # 2.2 : b.ndim == 2, has trailing dims
+    b = np.ones((m, 8))
+    x = func(a, b)
+    assert x.shape == (5, 4, n, 8)
+
+    # 2.3 : b has both trailing and batch dims
+    b = np.ones((5, 4, m, 8))
+    x = func(a, b)
+    assert x.shape == (5, 4, n, 8)
+
+    # 2.3.1 : b has trailing dims and (broadcastable) batch dims
+    b = np.ones((4, m, 8))
+    x = func(a, b)
+    assert x.shape == (5, 4, n, 8)
+
+    # 2.4 : b has batch dims but no trailing dims
+    b = np.ones((5, 4, m))
+    pattern = "Shape mismatch|incompatible shapes|shapes of a|shape mismatch"
+    with pytest.raises(ValueError, match=pattern):
+        # fails to broadcast `b` vs `a` (to fix: append a length-1 trailing dim)
+        func(a, b)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_blas.py

@@ -0,0 +1,1133 @@
+#
+# Created by: Pearu Peterson, April 2002
+#
+
+import math
+import pytest
+import numpy as np
+from numpy.testing import (assert_equal, assert_almost_equal,
+                           assert_array_almost_equal, assert_allclose)
+from pytest import raises as assert_raises
+
+from numpy import (arange, triu, tril, zeros, tril_indices, ones,
+                   diag, append, eye, nonzero)
+
+import scipy
+from scipy.linalg import get_blas_funcs, toeplitz, solve
+from scipy.linalg.blas import HAS_ILP64
+
+try:
+    from scipy.linalg import _fblas as fblas
+    FBLAS_ERROR = fblas.__fblas_error
+    HAS_LP64_FBLAS = True
+except ImportError:
+    fblas = None
+    HAS_LP64_FBLAS = False
+
+try:
+    from scipy.linalg import _fblas_64 as fblas_64
+    FBLAS_ERROR = fblas_64.__fblas_64_error
+except ImportError:
+    fblas_64 = None
+
+
+REAL_DTYPES = [np.float32, np.float64]
+COMPLEX_DTYPES = [np.complex64, np.complex128]
+DTYPES = REAL_DTYPES + COMPLEX_DTYPES
+
+
+def test_get_blas_funcs():
+    # check that it returns Fortran code for arrays that are
+    # fortran-ordered
+    f1, f2, f3 = get_blas_funcs(
+        ('axpy', 'axpy', 'axpy'),
+        (np.empty((2, 2), dtype=np.complex64, order='F'),
+         np.empty((2, 2), dtype=np.complex128, order='C'))
+        )
+
+    # get_blas_funcs will choose libraries depending on most generic
+    # array
+    assert_equal(f1.typecode, 'z')
+    assert_equal(f2.typecode, 'z')
+
+    # check defaults.
+    f1 = get_blas_funcs('rotg')
+    assert_equal(f1.typecode, 'd')
+
+    # check also dtype interface
+    f1 = get_blas_funcs('gemm', dtype=np.complex64)
+    assert_equal(f1.typecode, 'c')
+    f1 = get_blas_funcs('gemm', dtype='F')
+    assert_equal(f1.typecode, 'c')
+
+    # extended precision complex
+    f1 = get_blas_funcs('gemm', dtype=np.clongdouble)
+    assert_equal(f1.typecode, 'z')
+
+    # check safe complex upcasting
+    f1 = get_blas_funcs('axpy',
+                        (np.empty((2, 2), dtype=np.float64),
+                         np.empty((2, 2), dtype=np.complex64))
+                        )
+    assert_equal(f1.typecode, 'z')
+
+
+def test_get_blas_funcs_ilp_preferred():
+    # "preferred" mean ILP64 if available LP64 otherwise
+    gemm = get_blas_funcs('gemm', (np.eye(3),), ilp64="preferred")
+    assert gemm.int_dtype == np.int64 if HAS_ILP64 else np.int32
+    assert gemm.module_name == 'fblas_64' if HAS_ILP64 else 'fblas'
+
+    # default is "preferred"
+    gemm = get_blas_funcs('gemm', (np.eye(3),))
+    assert gemm.int_dtype == np.int64 if HAS_ILP64 else np.int32
+    assert gemm.module_name == 'fblas_64' if HAS_ILP64 else 'fblas'
+
+
+def test_get_blas_funcs_ilp_true():
+    # True is ILP64 or fail if not available
+    if HAS_ILP64:
+        gemm = get_blas_funcs('gemm', (np.eye(3),), ilp64=True)
+        assert gemm.int_dtype == np.int64
+        assert gemm.module_name == 'fblas_64'
+    else:
+        with pytest.raises(RuntimeError):
+            get_blas_funcs('gemm', (np.eye(3),), ilp64=True)
+
+
+def test_get_blas_funcs_ilp_false():
+    # False is LP64 or fail if not available
+    if HAS_LP64_FBLAS:
+        gemm = get_blas_funcs('gemm', (np.eye(3),), ilp64=False)
+        assert gemm.int_dtype == np.int32
+        assert gemm.module_name == 'fblas'
+    else:
+        with pytest.raises(ValueError):
+            get_blas_funcs('gemm', (np.eye(3),), ilp64=False)
+
+
+def test_get_blas_funcs_alias():
+    # check alias for get_blas_funcs
+    f, g = get_blas_funcs(('nrm2', 'dot'), dtype=np.complex64)
+    assert f.typecode == 'c'
+    assert g.typecode == 'c'
+
+    f, g, h = get_blas_funcs(('dot', 'dotc', 'dotu'), dtype=np.float64)
+    assert f is g
+    assert f is h
+
+
+def _dt_from_prefix(prefix):
+    """Array dtype from a blas-style prefix."""
+    if prefix.startswith('i'):
+        prefix = prefix[1:]   # isamax, idamax etc
+    elif prefix in ['sc', 'dz']:
+        prefix = prefix[0]   # scasum, dzasum
+    elif prefix in ['cs', 'zd']:
+        prefix = prefix[0]  # zdscal, csscal
+
+    dt_map = {'z': np.complex128, 'c': np.complex64, 'd': np.float64, 's': np.float32}
+    return dt_map[prefix]
+
+
+def parametrize_blas(func_name, prefixes, modules=None):
+    """Parametrize a test over BLAS prefixes, "sdcz", and over the BLAS modules,
+    `fblas,fblas_64.
+
+    Given a func_name "gemm", this generates a pytest parametrization over up to 8
+    variants: sgemm, dgemm, cgemm, zgemm, each loaded from `fblas` (i.e. 32-bit LP64
+    variants) and `fblas_64` (i.e. 64-bit ILP64 variants)..
+
+    If a module is not available, the pytest parameter has a skip mark, so that the
+    test is skipped with a descriptive message.
+
+    Parameters
+    ----------
+    func_name : str
+        The base name of a BLAS function, e.g. "gemm"
+    prefixes : str or sequence
+        BLAS prefixes to prepend to `func_name`.
+        E.g. ``func_name='gemm', prefixes='cz'`` generates `cgemm` and `zgemm`.
+    modules : sequence of ``(module, str)`` pairs
+        BLAS modules to fetch functions from. By default, use `fblas` (LP64 variant)
+        and fblas_64 (ILP64 variant).
+    """
+    if modules is None:
+        modules = [(fblas, "fblas"), (fblas_64, "fblas_64")]
+
+    params = []
+    for mod, mod_name in modules:
+        for prefix in prefixes:
+            dtype = _dt_from_prefix(prefix)
+            if mod is None:
+                param_ = pytest.param(
+                    None, dtype,
+                    id=f"{mod_name}.{prefix}{func_name}",
+                    marks=pytest.mark.skip(reason=f"{mod_name} is not available")
+                )
+            else:
+                # Fetch the BLAS function from the BLAS module. NB: if the name is not
+                # found in the module, it's a hard failure (all names must be present).
+                f = getattr(mod, prefix + func_name)
+                param_ = pytest.param(f, dtype, id=f"{mod_name}.{prefix}{func_name}")
+
+            params.append(param_)
+
+    return pytest.mark.parametrize("f, dtype", params)
+
+
+class TestFBLAS1Simple:
+
+    @parametrize_blas("axpy", "sdcz")
+    def test_axpy(self, f, dtype):
+        assert_array_almost_equal(f([1, 2, 3], [2, -1, 3], a=5),
+                                  [7, 9, 18])
+        if dtype in COMPLEX_DTYPES:
+            assert_array_almost_equal(f([1, 2j, 3], [2, -1, 3], a=5),
+                                      [7, 10j-1, 18])
+
+    @parametrize_blas("copy", "sdcz")
+    def test_copy(self, f, dtype):
+        assert_array_almost_equal(f([3, 4, 5], [8]*3), [3, 4, 5])
+        if dtype in COMPLEX_DTYPES:
+            assert_array_almost_equal(f([3, 4j, 5+3j], [8]*3), [3, 4j, 5+3j])
+
+    @parametrize_blas("asum", ["s", "d", "sc", "dz"])
+    def test_asum(self, f, dtype):
+        assert_almost_equal(f([3, -4, 5]), 12)
+        if dtype in COMPLEX_DTYPES:
+            assert_almost_equal(f([3j, -4, 3-4j]), 14)
+
+    @parametrize_blas("dot", "sd")
+    def test_dot(self, f, dtype):
+        assert_almost_equal(f([3, -4, 5], [2, 5, 1]), -9)
+
+    @parametrize_blas("dotu", "cz")
+    def test_dotu(self, f, dtype):
+        assert_almost_equal(f([3j, -4, 3-4j], [2, 3, 1]), -9+2j)
+
+    @parametrize_blas("dotc", "cz")
+    def test_dotc(self, f, dtype):
+        assert_almost_equal(f([3j, -4, 3-4j], [2, 3j, 1]), 3-14j)
+
+    @parametrize_blas("nrm2", ["s", "d", "sc", "dz"])
+    def test_nrm2(self, f, dtype):
+        assert_almost_equal(f([3, -4, 5]), math.sqrt(50))
+        if dtype in COMPLEX_DTYPES:
+            assert_almost_equal(f([3j, -4, 3-4j]), math.sqrt(50))
+
+    @parametrize_blas("scal", ["s", "d", "cs", "zd"])
+    def test_scal(self, f, dtype):
+        assert_array_almost_equal(f(2, [3, -4, 5]), [6, -8, 10])
+        if dtype in COMPLEX_DTYPES:
+            assert_array_almost_equal(f(3, [3j, -4, 3-4j]), [9j, -12, 9-12j])
+
+    @parametrize_blas("swap", "sdcz")
+    def test_swap(self, f, dtype):
+        x, y = [2, 3, 1], [-2, 3, 7]
+        x1, y1 = f(x, y)
+        assert_array_almost_equal(x1, y)
+        assert_array_almost_equal(y1, x)
+
+        if dtype in COMPLEX_DTYPES:
+            x, y = [2, 3j, 1], [-2, 3, 7-3j]
+            x1, y1 = f(x, y)
+            assert_array_almost_equal(x1, y)
+            assert_array_almost_equal(y1, x)
+
+    @parametrize_blas("amax", ["is", "id", "ic", "iz"])
+    def test_amax(self, f, dtype):
+        assert_equal(f([-2, 4, 3]), 1)
+        if dtype in COMPLEX_DTYPES:
+            assert_equal(f([-5, 4+3j, 6]), 1)
+
+    # XXX: need tests for rot,rotm,rotg,rotmg
+
+
+class TestFBLAS2Simple:
+    @parametrize_blas("gemv", "sdcz")
+    def test_gemv(self, f, dtype):
+        assert_array_almost_equal(f(3, [[3]], [-4]), [-36])
+        assert_array_almost_equal(f(3, [[3]], [-4], 3, [5]), [-21])
+        if dtype in COMPLEX_DTYPES:
+            assert_array_almost_equal(f(3j, [[3-4j]], [-4]), [-48-36j])
+            assert_array_almost_equal(f(3j, [[3-4j]], [-4], 3, [5j]),
+                                      [-48-21j])
+
+    @parametrize_blas("ger", "sd")
+    def test_ger(self, f, dtype):
+        assert_array_almost_equal(f(1, [1, 2], [3, 4]), [[3, 4], [6, 8]])
+        assert_array_almost_equal(f(2, [1, 2, 3], [3, 4]),
+                                  [[6, 8], [12, 16], [18, 24]])
+        assert_array_almost_equal(f(1, [1, 2], [3, 4],
+                                  a=[[1, 2], [3, 4]]), [[4, 6], [9, 12]])
+        if dtype in COMPLEX_DTYPES:
+            assert_array_almost_equal(f(1, [1j, 2], [3, 4]),
+                                      [[3j, 4j], [6, 8]])
+            assert_array_almost_equal(f(2, [1j, 2j, 3j], [3j, 4j]),
+                                      [[6, 8], [12, 16], [18, 24]])
+
+    @parametrize_blas("geru", "cz")
+    def test_geru(self, f, dtype):
+        assert_array_almost_equal(f(1, [1j, 2], [3, 4]),
+                                  [[3j, 4j], [6, 8]])
+        assert_array_almost_equal(f(-2, [1j, 2j, 3j], [3j, 4j]),
+                                  [[6, 8], [12, 16], [18, 24]])
+
+    @parametrize_blas("gerc", "cz")
+    def test_gerc(self, f, dtype):
+        assert_array_almost_equal(f(1, [1j, 2], [3, 4]),
+                                  [[3j, 4j], [6, 8]])
+        assert_array_almost_equal(f(2, [1j, 2j, 3j], [3j, 4j]),
+                                  [[6, 8], [12, 16], [18, 24]])
+
+    @parametrize_blas("syr", "sdcz")
+    def test_syr(self, f, dtype):
+        x = np.arange(1, 5, dtype='d')
+        resx = np.triu(x[:, np.newaxis] * x)
+        resx_reverse = np.triu(x[::-1, np.newaxis] * x[::-1])
+        y = np.linspace(0, 8.5, 17, endpoint=False)
+        z = np.arange(1, 9, dtype='d').view('D')
+        resz = np.triu(z[:, np.newaxis] * z)
+        resz_reverse = np.triu(z[::-1, np.newaxis] * z[::-1])
+        w = np.c_[np.zeros(4), z, np.zeros(4)].ravel()
+
+        rtol = np.finfo(dtype).eps
+
+        assert_allclose(f(1.0, x), resx, rtol=rtol)
+        assert_allclose(f(1.0, x, lower=True), resx.T, rtol=rtol)
+        assert_allclose(f(1.0, y, incx=2, offx=2, n=4), resx, rtol=rtol)
+        # negative increments imply reversed vectors in blas
+        assert_allclose(f(1.0, y, incx=-2, offx=2, n=4),
+                        resx_reverse, rtol=rtol)
+
+        if dtype in COMPLEX_DTYPES:
+            assert_allclose(f(1.0, z), resz, rtol=rtol)
+            assert_allclose(f(1.0, z, lower=True), resz.T, rtol=rtol)
+            assert_allclose(f(1.0, w, incx=3, offx=1, n=4), resz, rtol=rtol)
+            # negative increments imply reversed vectors in blas
+            assert_allclose(f(1.0, w, incx=-3, offx=1, n=4),
+                            resz_reverse, rtol=rtol)
+
+            a = np.zeros((4, 4), dtype, 'F')
+            b = f(1.0, z, a=a, overwrite_a=True)
+            assert_allclose(a, resz, rtol=rtol)
+            b = f(2.0, z, a=a)
+            assert a is not b
+            assert_allclose(b, 3*resz, rtol=rtol)
+
+        else:
+            a = np.zeros((4, 4), dtype, 'F')
+            b = f(1.0, x, a=a, overwrite_a=True)
+            assert_allclose(a, resx, rtol=rtol)
+            b = f(2.0, x, a=a)
+            assert a is not b
+            assert_allclose(b, 3*resx, rtol=rtol)
+
+        assert_raises(Exception, f, 1.0, x, incx=0)
+        assert_raises(Exception, f, 1.0, x, offx=5)
+        assert_raises(Exception, f, 1.0, x, offx=-2)
+        assert_raises(Exception, f, 1.0, x, n=-2)
+        assert_raises(Exception, f, 1.0, x, n=5)
+        assert_raises(Exception, f, 1.0, x, lower=2)
+        assert_raises(Exception, f, 1.0, x, a=np.zeros((2, 2), 'd', 'F'))
+
+    @parametrize_blas("her", "cz")
+    def test_her(self, f, dtype):
+        x = np.arange(1, 5, dtype='d')
+        z = np.arange(1, 9, dtype='d').view('D')
+        rehz = np.triu(z[:, np.newaxis] * z.conj())
+        rehz_reverse = np.triu(z[::-1, np.newaxis] * z[::-1].conj())
+        w = np.c_[np.zeros(4), z, np.zeros(4)].ravel()
+
+        rtol = np.finfo(dtype).eps
+
+        assert_allclose(f(1.0, z), rehz, rtol=rtol)
+        assert_allclose(f(1.0, z, lower=True), rehz.T.conj(), rtol=rtol)
+        assert_allclose(f(1.0, w, incx=3, offx=1, n=4), rehz, rtol=rtol)
+        # negative increments imply reversed vectors in blas
+        assert_allclose(f(1.0, w, incx=-3, offx=1, n=4),
+                        rehz_reverse, rtol=rtol)
+
+        a = np.zeros((4, 4), dtype, 'F')
+        b = f(1.0, z, a=a, overwrite_a=True)
+        assert_allclose(a, rehz, rtol=rtol)
+
+        b = f(2.0, z, a=a)
+        assert a is not b
+        assert_allclose(b, 3*rehz, rtol=rtol)
+
+        assert_raises(Exception, f, 1.0, x, incx=0)
+        assert_raises(Exception, f, 1.0, x, offx=5)
+        assert_raises(Exception, f, 1.0, x, offx=-2)
+        assert_raises(Exception, f, 1.0, x, n=-2)
+        assert_raises(Exception, f, 1.0, x, n=5)
+        assert_raises(Exception, f, 1.0, x, lower=2)
+        assert_raises(Exception, f, 1.0, x, a=np.zeros((2, 2), 'd', 'F'))
+
+    @parametrize_blas("syr2", "sd")
+    def test_syr2(self, f, dtype):
+        x = np.arange(1, 5, dtype='d')
+        y = np.arange(5, 9, dtype='d')
+        resxy = np.triu(x[:, np.newaxis] * y + y[:, np.newaxis] * x)
+        resxy_reverse = np.triu(x[::-1, np.newaxis] * y[::-1]
+                                + y[::-1, np.newaxis] * x[::-1])
+
+        q = np.linspace(0, 8.5, 17, endpoint=False)
+        rtol = np.finfo(dtype).eps
+
+        assert_allclose(f(1.0, x, y), resxy, rtol=rtol)
+        assert_allclose(f(1.0, x, y, n=3), resxy[:3, :3], rtol=rtol)
+        assert_allclose(f(1.0, x, y, lower=True), resxy.T, rtol=rtol)
+
+        assert_allclose(f(1.0, q, q, incx=2, offx=2, incy=2, offy=10),
+                        resxy, rtol=rtol)
+        assert_allclose(f(1.0, q, q, incx=2, offx=2, incy=2, offy=10, n=3),
+                        resxy[:3, :3], rtol=rtol)
+        # negative increments imply reversed vectors in blas
+        assert_allclose(f(1.0, q, q, incx=-2, offx=2, incy=-2, offy=10),
+                        resxy_reverse, rtol=rtol)
+
+        a = np.zeros((4, 4), dtype, 'F')
+        b = f(1.0, x, y, a=a, overwrite_a=True)
+        assert_allclose(a, resxy, rtol=rtol)
+
+        b = f(2.0, x, y, a=a)
+        assert a is not b
+        assert_allclose(b, 3*resxy, rtol=rtol)
+
+        assert_raises(Exception, f, 1.0, x, y, incx=0)
+        assert_raises(Exception, f, 1.0, x, y, offx=5)
+        assert_raises(Exception, f, 1.0, x, y, offx=-2)
+        assert_raises(Exception, f, 1.0, x, y, incy=0)
+        assert_raises(Exception, f, 1.0, x, y, offy=5)
+        assert_raises(Exception, f, 1.0, x, y, offy=-2)
+        assert_raises(Exception, f, 1.0, x, y, n=-2)
+        assert_raises(Exception, f, 1.0, x, y, n=5)
+        assert_raises(Exception, f, 1.0, x, y, lower=2)
+        assert_raises(Exception, f, 1.0, x, y, a=np.zeros((2, 2), 'd', 'F'))
+
+    @parametrize_blas("her2", "cz")
+    def test_her2(self, f, dtype):
+        x = np.arange(1, 9, dtype='d').view('D')
+        y = np.arange(9, 17, dtype='d').view('D')
+        resxy = x[:, np.newaxis] * y.conj() + y[:, np.newaxis] * x.conj()
+        resxy = np.triu(resxy)
+
+        resxy_reverse = x[::-1, np.newaxis] * y[::-1].conj()
+        resxy_reverse += y[::-1, np.newaxis] * x[::-1].conj()
+        resxy_reverse = np.triu(resxy_reverse)
+
+        u = np.c_[np.zeros(4), x, np.zeros(4)].ravel()
+        v = np.c_[np.zeros(4), y, np.zeros(4)].ravel()
+
+        rtol = np.finfo(dtype).eps
+
+        assert_allclose(f(1.0, x, y), resxy, rtol=rtol)
+        assert_allclose(f(1.0, x, y, n=3), resxy[:3, :3], rtol=rtol)
+        assert_allclose(f(1.0, x, y, lower=True), resxy.T.conj(),
+                        rtol=rtol)
+
+        assert_allclose(f(1.0, u, v, incx=3, offx=1, incy=3, offy=1),
+                        resxy, rtol=rtol)
+        assert_allclose(f(1.0, u, v, incx=3, offx=1, incy=3, offy=1, n=3),
+                        resxy[:3, :3], rtol=rtol)
+        # negative increments imply reversed vectors in blas
+        assert_allclose(f(1.0, u, v, incx=-3, offx=1, incy=-3, offy=1),
+                        resxy_reverse, rtol=rtol)
+
+        a = np.zeros((4, 4), dtype, 'F')
+        b = f(1.0, x, y, a=a, overwrite_a=True)
+        assert_allclose(a, resxy, rtol=rtol)
+
+        b = f(2.0, x, y, a=a)
+        assert a is not b
+        assert_allclose(b, 3*resxy, rtol=rtol)
+
+        assert_raises(Exception, f, 1.0, x, y, incx=0)
+        assert_raises(Exception, f, 1.0, x, y, offx=5)
+        assert_raises(Exception, f, 1.0, x, y, offx=-2)
+        assert_raises(Exception, f, 1.0, x, y, incy=0)
+        assert_raises(Exception, f, 1.0, x, y, offy=5)
+        assert_raises(Exception, f, 1.0, x, y, offy=-2)
+        assert_raises(Exception, f, 1.0, x, y, n=-2)
+        assert_raises(Exception, f, 1.0, x, y, n=5)
+        assert_raises(Exception, f, 1.0, x, y, lower=2)
+        assert_raises(Exception, f, 1.0, x, y,
+                      a=np.zeros((2, 2), 'd', 'F'))
+
+    @parametrize_blas("gbmv", "sdcz")
+    def test_gbmv(self, f, dtype):
+        rng = np.random.default_rng(1234)
+        n = 7
+        m = 5
+        kl = 1
+        ku = 2
+        # fake a banded matrix via toeplitz
+        A = toeplitz(append(rng.random(kl+1), zeros(m-kl-1)),
+                     append(rng.random(ku+1), zeros(n-ku-1)))
+        A = A.astype(dtype)
+        Ab = zeros((kl+ku+1, n), dtype=dtype)
+
+        # Form the banded storage
+        Ab[2, :5] = A[0, 0]  # diag
+        Ab[1, 1:6] = A[0, 1]  # sup1
+        Ab[0, 2:7] = A[0, 2]  # sup2
+        Ab[3, :4] = A[1, 0]  # sub1
+
+        x = rng.random(n).astype(dtype)
+        y = rng.random(m).astype(dtype)
+        alpha, beta = dtype(3), dtype(-5)
+
+        func = f
+        y1 = func(m=m, n=n, ku=ku, kl=kl, alpha=alpha, a=Ab,
+                  x=x, y=y, beta=beta)
+        y2 = alpha * A.dot(x) + beta * y
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(m=m, n=n, ku=ku, kl=kl, alpha=alpha, a=Ab,
+                  x=y, y=x, beta=beta, trans=1)
+        y2 = alpha * A.T.dot(y) + beta * x
+        assert_array_almost_equal(y1, y2)
+
+    @pytest.mark.parametrize("ilp64", [True, False])
+    @pytest.mark.parametrize("dtype", DTYPES)
+    def test_sbmv_hbmv(self, dtype, ilp64):
+        mod = fblas_64 if ilp64 else fblas
+        if mod is None:
+            mod_name = "fblas_64" if ilp64 else "fblas"
+            pytest.skip(f"{mod_name} is not available")
+
+        rng = np.random.default_rng(1234)
+        n = 6
+        k = 2
+        A = zeros((n, n), dtype=dtype)
+        Ab = zeros((k+1, n), dtype=dtype)
+
+        # Form the array and its packed banded storage
+        A[arange(n), arange(n)] = rng.random(n)
+        for ind2 in range(1, k+1):
+            temp = rng.random(n-ind2)
+            A[arange(n-ind2), arange(ind2, n)] = temp
+            Ab[-1-ind2, ind2:] = temp
+        A = A.astype(dtype)
+        if dtype in COMPLEX_DTYPES:
+            A += A.conj().T
+            func, = get_blas_funcs(('hbmv',), dtype=dtype, ilp64="preferred")
+        else:
+            A += A.T
+            func, = get_blas_funcs(('sbmv',), dtype=dtype, ilp64="preferred")
+
+        Ab[-1, :] = diag(A)
+        x = rng.random(n).astype(dtype)
+        y = rng.random(n).astype(dtype)
+        alpha, beta = dtype(1.25), dtype(3)
+
+        y1 = func(k=k, alpha=alpha, a=Ab, x=x, y=y, beta=beta)
+        y2 = alpha * A.dot(x) + beta * y
+        assert_array_almost_equal(y1, y2)
+
+    @pytest.mark.parametrize("ilp64", [True, False])
+    @pytest.mark.parametrize("fname,dtype", [
+        *[('spmv', dtype) for dtype in REAL_DTYPES + COMPLEX_DTYPES],
+        *[('hpmv', dtype) for dtype in COMPLEX_DTYPES],
+    ])
+    def test_spmv_hpmv(self, fname, dtype, ilp64):
+        mod = fblas_64 if ilp64 else fblas
+        if mod is None:
+            mod_name = "fblas_64" if ilp64 else "fblas"
+            pytest.skip(f"{mod_name} is not available")
+
+        rng = np.random.default_rng(1234)
+        n = 3
+        A = rng.random((n, n)).astype(dtype)
+        if dtype in COMPLEX_DTYPES:
+            A += rng.random((n, n))*1j
+        A += A.T if fname == 'spmv' else A.conj().T
+        c, r = tril_indices(n)
+        Ap = A[r, c]
+        x = rng.random(n).astype(dtype)
+        y = rng.random(n).astype(dtype)
+        xlong = arange(2*n).astype(dtype)
+        ylong = ones(2*n).astype(dtype)
+        alpha, beta = dtype(1.25), dtype(2)
+
+        func, = get_blas_funcs((fname,), dtype=dtype, ilp64="preferred")
+        y1 = func(n=n, alpha=alpha, ap=Ap, x=x, y=y, beta=beta)
+        y2 = alpha * A.dot(x) + beta * y
+        assert_array_almost_equal(y1, y2)
+
+        # Test inc and offsets
+        y1 = func(n=n-1, alpha=alpha, beta=beta, x=xlong, y=ylong, ap=Ap,
+                  incx=2, incy=2, offx=n, offy=n)
+        y2 = (alpha * A[:-1, :-1]).dot(xlong[3::2]) + beta * ylong[3::2]
+        assert_array_almost_equal(y1[3::2], y2)
+        assert_almost_equal(y1[4], ylong[4])
+
+    @pytest.mark.parametrize("ilp64", [True, False])
+    @pytest.mark.parametrize("fname,dtype", [
+        *[('spr', dtype) for dtype in REAL_DTYPES + COMPLEX_DTYPES],
+        *[('hpr', dtype) for dtype in COMPLEX_DTYPES],
+    ])
+    def test_spr_hpr(self, fname, dtype, ilp64):
+        mod = fblas_64 if ilp64 else fblas
+        if mod is None:
+            mod_name = "fblas_64" if ilp64 else "fblas"
+            pytest.skip(f"{mod_name} is not available")
+
+        rng = np.random.default_rng(1234)
+        n = 3
+        A = rng.random((n, n)).astype(dtype)
+        if dtype in COMPLEX_DTYPES:
+            A += rng.random((n, n))*1j
+        A += A.T if fname == 'spr' else A.conj().T
+        c, r = tril_indices(n)
+        Ap = A[r, c]
+        x = rng.random(n).astype(dtype)
+
+        alpha = np.finfo(dtype).dtype.type(2.5)
+        if fname == 'hpr':
+            func, = get_blas_funcs(('hpr',), dtype=dtype, ilp64="preferred")
+            y2 = alpha * x[:, None].dot(x[None, :].conj()) + A
+        else:
+            func, = get_blas_funcs(('spr',), dtype=dtype, ilp64="preferred")
+            y2 = alpha * x[:, None].dot(x[None, :]) + A
+
+        y1 = func(n=n, alpha=alpha, ap=Ap, x=x)
+        y1f = zeros((3, 3), dtype=dtype)
+        y1f[r, c] = y1
+        y1f[c, r] = y1.conj() if fname == 'hpr' else y1
+        assert_array_almost_equal(y1f, y2)
+
+    @pytest.mark.parametrize("ilp64", [True, False])
+    @pytest.mark.parametrize("dtype", DTYPES)
+    def test_spr2_hpr2(self, dtype, ilp64):
+        mod = fblas_64 if ilp64 else fblas
+        if mod is None:
+            mod_name = "fblas_64" if ilp64 else "fblas"
+            pytest.skip(f"{mod_name} is not available")
+
+        rng = np.random.default_rng(1234)
+        n = 3
+        A = rng.random((n, n)).astype(dtype)
+        if dtype in COMPLEX_DTYPES:
+            A += rng.random((n, n))*1j
+            A += A.conj().T
+            func, = get_blas_funcs(('hpr2',), dtype=dtype, ilp64="preferred")
+        else:
+            A += A.T
+            func, = get_blas_funcs(('spr2',), dtype=dtype, ilp64="preferred")
+
+        c, r = tril_indices(n)
+        Ap = A[r, c]
+        x = rng.random(n).astype(dtype)
+        y = rng.random(n).astype(dtype)
+        alpha = dtype(2)
+
+        u = alpha.conj() * x[:, None].dot(y[None, :].conj())
+        y2 = A + u + u.conj().T
+        y1 = func(n=n, alpha=alpha, x=x, y=y, ap=Ap)
+        y1f = zeros((3, 3), dtype=dtype)
+        y1f[r, c] = y1
+        y1f[[1, 2, 2], [0, 0, 1]] = y1[[1, 3, 4]].conj()
+        assert_array_almost_equal(y1f, y2)
+
+    @parametrize_blas('tbmv', 'sdcz')
+    def test_tbmv(self, f, dtype):
+        rng = np.random.default_rng(1234)
+        n = 10
+        k = 3
+        x = rng.random(n).astype(dtype)
+        A = zeros((n, n), dtype=dtype)
+        # Banded upper triangular array
+        for sup in range(k+1):
+            A[arange(n-sup), arange(sup, n)] = rng.random(n-sup)
+
+        # Add complex parts for c,z
+        if dtype in COMPLEX_DTYPES:
+            A[nonzero(A)] += 1j * rng.random((k+1)*n-(k*(k+1)//2)).astype(dtype)
+
+        # Form the banded storage
+        Ab = zeros((k+1, n), dtype=dtype)
+        for row in range(k+1):
+            Ab[-row-1, row:] = diag(A, k=row)
+        func = f
+
+        y1 = func(k=k, a=Ab, x=x)
+        y2 = A.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(k=k, a=Ab, x=x, diag=1)
+        A[arange(n), arange(n)] = dtype(1)
+        y2 = A.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(k=k, a=Ab, x=x, diag=1, trans=1)
+        y2 = A.T.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(k=k, a=Ab, x=x, diag=1, trans=2)
+        y2 = A.conj().T.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+    @parametrize_blas('tbsv', 'sdcz')
+    def test_tbsv(self, f, dtype):
+        rng = np.random.default_rng(12345)
+        n = 6
+        k = 3
+        x = rng.random(n).astype(dtype)
+        A = zeros((n, n), dtype=dtype)
+        # Banded upper triangular array
+        for sup in range(k+1):
+            A[arange(n-sup), arange(sup, n)] = rng.random(n-sup)
+
+        # Add complex parts for c,z
+        if dtype in COMPLEX_DTYPES:
+            A[nonzero(A)] += 1j * rng.random((k+1)*n-(k*(k+1)//2)).astype(dtype)
+
+        # Form the banded storage
+        Ab = zeros((k+1, n), dtype=dtype)
+        for row in range(k+1):
+            Ab[-row-1, row:] = diag(A, k=row)
+        func = f
+
+        y1 = func(k=k, a=Ab, x=x)
+        y2 = solve(A, x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(k=k, a=Ab, x=x, diag=1)
+        A[arange(n), arange(n)] = dtype(1)
+        y2 = solve(A, x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(k=k, a=Ab, x=x, diag=1, trans=1)
+        y2 = solve(A.T, x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(k=k, a=Ab, x=x, diag=1, trans=2)
+        y2 = solve(A.conj().T, x)
+        assert_array_almost_equal(y1, y2)
+
+    @parametrize_blas('tpmv', 'sdcz')
+    def test_tpmv(self, f, dtype):
+        rng = np.random.default_rng(1234)
+        n = 10
+        x = rng.random(n).astype(dtype)
+        # Upper triangular array
+        if dtype in COMPLEX_DTYPES:
+            A = triu(rng.random((n, n)) + rng.random((n, n))*1j)
+        else:
+            A = triu(rng.random((n, n)))
+
+        # Form the packed storage
+        c, r = tril_indices(n)
+        Ap = A[r, c]
+        func = f
+
+        y1 = func(n=n, ap=Ap, x=x)
+        y2 = A.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(n=n, ap=Ap, x=x, diag=1)
+        A[arange(n), arange(n)] = dtype(1)
+        y2 = A.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(n=n, ap=Ap, x=x, diag=1, trans=1)
+        y2 = A.T.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(n=n, ap=Ap, x=x, diag=1, trans=2)
+        y2 = A.conj().T.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+    @parametrize_blas('tpsv', 'sdcz')
+    def test_tpsv(self, f, dtype):
+        rng = np.random.default_rng(1234)
+        n = 10
+        x = rng.random(n).astype(dtype)
+        # Upper triangular array
+        if dtype in COMPLEX_DTYPES:
+            A = triu(rng.random((n, n)) + rng.random((n, n))*1j)
+        else:
+            A = triu(rng.random((n, n)))
+        A += eye(n)
+        # Form the packed storage
+        c, r = tril_indices(n)
+        Ap = A[r, c]
+        func = f
+
+        y1 = func(n=n, ap=Ap, x=x)
+        y2 = solve(A, x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(n=n, ap=Ap, x=x, diag=1)
+        A[arange(n), arange(n)] = dtype(1)
+        y2 = solve(A, x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(n=n, ap=Ap, x=x, diag=1, trans=1)
+        y2 = solve(A.T, x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(n=n, ap=Ap, x=x, diag=1, trans=2)
+        y2 = solve(A.conj().T, x)
+        assert_array_almost_equal(y1, y2)
+
+    @parametrize_blas('trmv', 'sdcz')
+    def test_trmv(self, f, dtype):
+        rng = np.random.default_rng(1234)
+        n = 3
+        A = (rng.random((n, n))+eye(n)).astype(dtype)
+        x = rng.random(3).astype(dtype)
+        func = f
+
+        y1 = func(a=A, x=x)
+        y2 = triu(A).dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(a=A, x=x, diag=1)
+        A[arange(n), arange(n)] = dtype(1)
+        y2 = triu(A).dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(a=A, x=x, diag=1, trans=1)
+        y2 = triu(A).T.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(a=A, x=x, diag=1, trans=2)
+        y2 = triu(A).conj().T.dot(x)
+        assert_array_almost_equal(y1, y2)
+
+    @parametrize_blas('trsv', 'sdcz')
+    def test_trsv(self, f, dtype):
+        rng = np.random.default_rng(1234)
+        n = 15
+        A = (rng.random((n, n))+eye(n)).astype(dtype)
+        x = rng.random(n).astype(dtype)
+        func = f
+
+        y1 = func(a=A, x=x)
+        y2 = solve(triu(A), x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(a=A, x=x, lower=1)
+        y2 = solve(tril(A), x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(a=A, x=x, diag=1)
+        A[arange(n), arange(n)] = dtype(1)
+        y2 = solve(triu(A), x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(a=A, x=x, diag=1, trans=1)
+        y2 = solve(triu(A).T, x)
+        assert_array_almost_equal(y1, y2)
+
+        y1 = func(a=A, x=x, diag=1, trans=2)
+        y2 = solve(triu(A).conj().T, x)
+        assert_array_almost_equal(y1, y2)
+
+
+class TestFBLAS3Simple:
+    @parametrize_blas("gemm", "sdcz")
+    def test_gemm(self, f, dtype):
+        assert_array_almost_equal(f(3, [3], [-4]), [[-36]])
+        assert_array_almost_equal(f(3, [3], [-4], 3, [5]), [-21])
+        if dtype in COMPLEX_DTYPES:
+            assert_array_almost_equal(f(3j, [3-4j], [-4]), [[-48-36j]])
+            assert_array_almost_equal(f(3j, [3-4j], [-4], 3, [5j]), [-48-21j])
+
+
+class TestBLAS3Symm:
+
+    def setup_method(self):
+        self.a = np.array([[1., 2.],
+                           [0., 1.]])
+        self.b = np.array([[1., 0., 3.],
+                           [0., -1., 2.]])
+        self.c = np.ones((2, 3))
+        self.t = np.array([[2., -1., 8.],
+                           [3., 0., 9.]])
+
+    @parametrize_blas("symm", "sdcz")
+    def test_symm(self, f, dtype):
+        res = f(a=self.a, b=self.b, c=self.c, alpha=1., beta=1.)
+        assert_array_almost_equal(res, self.t)
+
+        res = f(a=self.a.T, b=self.b, lower=1, c=self.c, alpha=1., beta=1.)
+        assert_array_almost_equal(res, self.t)
+
+        res = f(a=self.a, b=self.b.T, side=1, c=self.c.T,
+                alpha=1., beta=1.)
+        assert_array_almost_equal(res, self.t.T)
+
+    @parametrize_blas("symm", "sdcz")
+    def test_symm_wrong_side(self, f, dtype):
+        """`side=1` means C <- B*A, hence shapes of A and B are to be
+        compatible. Otherwise, f2py exception is raised.
+        """
+        # FIXME narrow down to _fblas.error
+        with pytest.raises(Exception):
+            f(a=self.a, b=self.b, alpha=1, side=1)
+
+    @parametrize_blas("symm", "sdcz")
+    def test_symm_wrong_uplo(self, f, dtype):
+        """SYMM only considers the upper/lower part of A. Hence setting
+        wrong value for `lower` (default is lower=0, meaning upper triangle)
+        gives a wrong result.
+        """
+        res = f(a=self.a, b=self.b, c=self.c, alpha=1., beta=1.)
+        assert np.allclose(res, self.t)
+        res = f(a=self.a, b=self.b, lower=1, c=self.c, alpha=1., beta=1.)
+        assert not np.allclose(res, self.t)
+
+
+class TestBLAS3Syrk:
+    def setup_method(self):
+        self.a = np.array([[1., 0.],
+                           [0., -2.],
+                           [2., 3.]])
+        self.t = np.array([[1., 0., 2.],
+                           [0., 4., -6.],
+                           [2., -6., 13.]])
+        self.tt = np.array([[5., 6.],
+                            [6., 13.]])
+
+    @parametrize_blas("syrk", "sdcz")
+    def test_syrk(self, f, dtype):
+        c = f(a=self.a, alpha=1.)
+        assert_array_almost_equal(np.triu(c), np.triu(self.t))
+
+        c = f(a=self.a, alpha=1., lower=1)
+        assert_array_almost_equal(np.tril(c), np.tril(self.t))
+
+        c0 = np.ones(self.t.shape)
+        c = f(a=self.a, alpha=1., beta=1., c=c0)
+        assert_array_almost_equal(np.triu(c), np.triu(self.t+c0))
+
+        c = f(a=self.a, alpha=1., trans=1)
+        assert_array_almost_equal(np.triu(c), np.triu(self.tt))
+
+    # prints '0-th dimension must be fixed to 3 but got 5',
+    # FIXME: suppress?
+    @parametrize_blas("syrk", "sdcz")
+    def test_syrk_wrong_c(self, f, dtype):
+        # FIXME narrow down to _fblas.error
+        with pytest.raises(Exception):
+            f(a=self.a, alpha=1., c=np.ones((5, 8)))
+        # if C is supplied, it must have compatible dimensions
+
+
+class TestBLAS3Syr2k:
+    def setup_method(self):
+        self.a = np.array([[1., 0.],
+                           [0., -2.],
+                           [2., 3.]])
+        self.b = np.array([[0., 1.],
+                           [1., 0.],
+                           [0, 1.]])
+        self.t = np.array([[0., -1., 3.],
+                           [-1., 0., 0.],
+                           [3., 0., 6.]])
+        self.tt = np.array([[0., 1.],
+                            [1., 6]])
+
+    @parametrize_blas("syr2k", "sdcz")
+    def test_syr2k(self, f, dtype):
+        c = f(a=self.a, b=self.b, alpha=1.)
+        assert_array_almost_equal(np.triu(c), np.triu(self.t))
+
+        c = f(a=self.a, b=self.b, alpha=1., lower=1)
+        assert_array_almost_equal(np.tril(c), np.tril(self.t))
+
+        c0 = np.ones(self.t.shape)
+        c = f(a=self.a, b=self.b, alpha=1., beta=1., c=c0)
+        assert_array_almost_equal(np.triu(c), np.triu(self.t+c0))
+
+        c = f(a=self.a, b=self.b, alpha=1., trans=1)
+        assert_array_almost_equal(np.triu(c), np.triu(self.tt))
+
+    # prints '0-th dimension must be fixed to 3 but got 5', FIXME: suppress?
+    @parametrize_blas("syr2k", "sdcz")
+    def test_syr2k_wrong_c(self, f, dtype):
+        with pytest.raises(Exception):
+            f(a=self.a, b=self.b, alpha=1., c=np.zeros((15, 8)))
+        # if C is supplied, it must have compatible dimensions
+
+
+class TestSyHe:
+    """Quick and simple tests for (zc)-symm, syrk, syr2k."""
+
+    def setup_method(self):
+        self.sigma_y = np.array([[0., -1.j],
+                                 [1.j, 0.]])
+
+    @parametrize_blas("symm", "zc")
+    def test_symm(self, f, dtype):
+        # NB: a is symmetric w/upper diag of ONLY
+        res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.)
+        assert_array_almost_equal(np.triu(res), np.diag([1, -1]))
+
+    @parametrize_blas("hemm", "zc")
+    def test_hemm(self, f, dtype):
+        # NB: a is hermitian w/upper diag of ONLY
+        res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.)
+        assert_array_almost_equal(np.triu(res), np.diag([1, 1]))
+
+    @parametrize_blas("syrk", "zc")
+    def test_syrk(self, f, dtype):
+        res = f(a=self.sigma_y, alpha=1.)
+        assert_array_almost_equal(np.triu(res), np.diag([-1, -1]))
+
+    @parametrize_blas("herk", "zc")
+    def test_herk(self, f, dtype):
+        res = f(a=self.sigma_y, alpha=1.)
+        assert_array_almost_equal(np.triu(res), np.diag([1, 1]))
+
+    @parametrize_blas("syr2k", "zc")
+    def test_syr2k_zr(self, f, dtype):
+        res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.)
+        assert_array_almost_equal(np.triu(res), 2.*np.diag([-1, -1]))
+
+    @parametrize_blas("her2k", "zc")
+    def test_her2k_zr(self, f, dtype):
+        res = f(a=self.sigma_y, b=self.sigma_y, alpha=1.)
+        assert_array_almost_equal(np.triu(res), 2.*np.diag([1, 1]))
+
+
+class TestTRMM:
+    """Quick and simple tests for *trmm."""
+
+    def setup_method(self):
+        self.a = np.array([[1., 2., ],
+                           [-2., 1.]])
+        self.b = np.array([[3., 4., -1.],
+                           [5., 6., -2.]])
+
+        self.a2 = np.array([[1, 1, 2, 3],
+                            [0, 1, 4, 5],
+                            [0, 0, 1, 6],
+                            [0, 0, 0, 1]], order="f")
+        self.b2 = np.array([[1, 4], [2, 5], [3, 6], [7, 8], [9, 10]],
+                           order="f")
+
+    @parametrize_blas("trmm", "sdzc")
+    def test_side(self, f, dtype):
+        trmm = f
+        # Provide large A array that works for side=1 but not 0 (see gh-10841)
+        assert_raises(Exception, trmm, 1.0, self.a2, self.b2)
+        res = trmm(1.0, self.a2.astype(dtype), self.b2.astype(dtype),
+                   side=1)
+        k = self.b2.shape[1]
+        assert_allclose(res, self.b2 @ self.a2[:k, :k], rtol=0.,
+                        atol=100*np.finfo(dtype).eps)
+
+    @parametrize_blas("trmm", "sdcz")
+    def test_ab(self, f, dtype):
+        result = f(1., self.a, self.b)
+        # default a is upper triangular
+        expected = np.array([[13., 16., -5.],
+                             [ 5.,  6., -2.]])
+        assert_array_almost_equal(result, expected)
+
+    @parametrize_blas("trmm", "sdcz")
+    def test_ab_lower(self, f, dtype):
+        result = f(1., self.a, self.b, lower=True)
+        expected = np.array([[ 3.,  4., -1.],
+                             [-1., -2.,  0.]])  # now a is lower triangular
+        assert_array_almost_equal(result, expected)
+
+    @parametrize_blas("trmm", "sdcz")
+    def test_b_overwrites(self, f, dtype):
+        # BLAS *trmm modifies B argument in-place.
+        # Here the default is to copy, but this can be overridden
+        b = self.b.astype(dtype)
+        for overwr in [True, False]:
+            bcopy = b.copy()
+            result = f(1., self.a, bcopy, overwrite_b=overwr)
+            # C-contiguous arrays are copied
+            assert not bcopy.flags.f_contiguous
+            assert not np.may_share_memory(bcopy, result)
+            assert_equal(bcopy, b)
+
+        bcopy = np.asfortranarray(b.copy())  # or just transpose it
+        result = f(1., self.a, bcopy, overwrite_b=True)
+        assert bcopy.flags.f_contiguous
+        assert np.may_share_memory(bcopy, result)
+        assert_array_almost_equal(bcopy, result)
+
+
+@parametrize_blas("trsm", "sdcz")
+def test_trsm(f, dtype):
+    rng = np.random.default_rng(1234)
+    tol = np.finfo(dtype).eps*1000
+    func = f
+
+    # Test protection against size mismatches
+    A = rng.random((4, 5)).astype(dtype)
+    B = rng.random((4, 4)).astype(dtype)
+    alpha = dtype(1)
+    assert_raises(Exception, func, alpha, A, B)
+    assert_raises(Exception, func, alpha, A.T, B)
+
+    n = 8
+    m = 7
+    alpha = dtype(-2.5)
+    if dtype in COMPLEX_DTYPES:
+        A = (rng.random((m, m)) + rng.random((m, m))*1j) + eye(m)
+    else:
+        A = rng.random((m, m)) + eye(m)
+    A = A.astype(dtype)
+    Au = triu(A)
+    Al = tril(A)
+    B1 = rng.random((m, n)).astype(dtype)
+    B2 = rng.random((n, m)).astype(dtype)
+
+    x1 = func(alpha=alpha, a=A, b=B1)
+    assert_equal(B1.shape, x1.shape)
+    x2 = solve(Au, alpha*B1)
+    assert_allclose(x1, x2, atol=tol)
+
+    x1 = func(alpha=alpha, a=A, b=B1, trans_a=1)
+    x2 = solve(Au.T, alpha*B1)
+    assert_allclose(x1, x2, atol=tol)
+
+    x1 = func(alpha=alpha, a=A, b=B1, trans_a=2)
+    x2 = solve(Au.conj().T, alpha*B1)
+    assert_allclose(x1, x2, atol=tol)
+
+    x1 = func(alpha=alpha, a=A, b=B1, diag=1)
+    Au[arange(m), arange(m)] = dtype(1)
+    x2 = solve(Au, alpha*B1)
+    assert_allclose(x1, x2, atol=tol)
+
+    x1 = func(alpha=alpha, a=A, b=B2, diag=1, side=1)
+    x2 = solve(Au.conj().T, alpha*B2.conj().T)
+    assert_allclose(x1, x2.conj().T, atol=tol)
+
+    x1 = func(alpha=alpha, a=A, b=B2, diag=1, side=1, lower=1)
+    Al[arange(m), arange(m)] = dtype(1)
+    x2 = solve(Al.conj().T, alpha*B2.conj().T)
+    assert_allclose(x1, x2.conj().T, atol=tol)
+
+
+@pytest.mark.xfail(run=False,
+                   reason="gh-16930")
+def test_gh_169309():
+    x = np.repeat(10, 9)
+    actual = scipy.linalg.blas.dnrm2(x, 5, 3, -1)
+    expected = math.sqrt(500)
+    assert_allclose(actual, expected)
+
+
+@pytest.mark.xfail(HAS_ILP64, reason='does not fail with ILP64 MKL')
+def test_dnrm2_neg_incx():
+    # check that dnrm2(..., incx < 0) raises
+    # XXX: remove the test after the lowest supported BLAS implements
+    # negative incx (new in LAPACK 3.10)
+    x = np.repeat(10, 9)
+    incx = -1
+    dnrm2 = scipy.linalg.blas.get_blas_funcs('nrm2', (x,), ilp64='preferred')
+    with assert_raises(FBLAS_ERROR):
+        dnrm2(x, 5, 3, incx)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_cython_abi.py

@@ -0,0 +1,120 @@
+# ---------------------------------------------------------------------------
+# AUTO-GENERATED - do not edit by hand.
+# Re-generate with:  python tools/generate_cython_abi_tests.py linalg
+#
+# Generated against SciPy 1.17.1 on 2026-03-29
+# Python 3.12.13
+# ---------------------------------------------------------------------------
+
+"""
+Regression test for the cython_blas / cython_lapack binary interface.
+
+The __pyx_capi__ dict of each module contains PyCapsule objects whose
+*name* string encodes the C-level function signature that Cython checks
+at import time when another module does ``cimport``.  Any change to
+those strings is a binary incompatibility that breaks every downstream
+package (e.g., scikit-learn, statsmodels) compiled against the previous release,
+without any compile-time warning.
+
+The expected signatures are stored in cython_abi_signatures.json.
+The test fails if:
+  * an existing name disappears, or
+  * an existing name's signature string changes.
+Adding new names is explicitly allowed (it is not a break).
+"""
+
+import ctypes
+import importlib
+import importlib.resources
+import json
+
+import pytest
+import scipy
+
+CYTHON_BLAS_ILP64 = (
+    scipy.show_config(mode='dicts')['Build Dependencies']['blas']['cython blas ilp64']
+)
+
+
+# ---------------------------------------------------------------------------
+# Helper
+# ---------------------------------------------------------------------------
+
+_get_capsule_name = ctypes.pythonapi.PyCapsule_GetName
+_get_capsule_name.restype = ctypes.c_char_p
+_get_capsule_name.argtypes = [ctypes.py_object]
+
+
+def _extract_capi(module):
+    """Return {name: signature} for every entry in module.__pyx_capi__."""
+    result = {}
+    for name, capsule in module.__pyx_capi__.items():
+        raw = _get_capsule_name(capsule)
+        if raw is not None:
+            result[name] = raw.decode('utf-8')
+    return result
+
+
+_sig_file = importlib.resources.files('scipy.linalg.tests').joinpath(
+    'cython_abi_signatures.json'
+)
+with importlib.resources.as_file(_sig_file) as _f:
+    _EXPECTED = json.loads(_f.read_text())
+
+
+@pytest.mark.skipif(
+    CYTHON_BLAS_ILP64,
+    reason='cython_blas/cython_lapack ABI depends on ILP64 BLAS',
+)
+def test_cython_blas_abi_stability():
+    """No existing cython_blas signature may change or disappear."""
+    expected = _EXPECTED['scipy.linalg.cython_blas']
+    mod = importlib.import_module('scipy.linalg.cython_blas')
+    actual = _extract_capi(mod)
+    errors = []
+    for name, expected_sig in expected.items():
+        if name not in actual:
+            errors.append(
+                f"REMOVED  {name!r} (was {expected_sig!r})"
+            )
+        elif actual[name] != expected_sig:
+            errors.append(
+                f"CHANGED  {name!r}\n"
+                f"  expected: {expected_sig!r}\n"
+                f"  actual:   {actual[name]!r}"
+            )
+    if errors:
+        joined = '\n'.join(errors)
+        pytest.fail(
+            f"scipy.linalg.cython_blas.__pyx_capi__ has {len(errors)} "
+            f"ABI breakage(s):\n{joined}"
+        )
+
+
+@pytest.mark.skipif(
+    CYTHON_BLAS_ILP64,
+    reason='cython_blas/cython_lapack ABI depends on ILP64 BLAS',
+)
+def test_cython_lapack_abi_stability():
+    """No existing cython_lapack signature may change or disappear."""
+    expected = _EXPECTED['scipy.linalg.cython_lapack']
+    mod = importlib.import_module('scipy.linalg.cython_lapack')
+    actual = _extract_capi(mod)
+    errors = []
+    for name, expected_sig in expected.items():
+        if name not in actual:
+            errors.append(
+                f"REMOVED  {name!r} (was {expected_sig!r})"
+            )
+        elif actual[name] != expected_sig:
+            errors.append(
+                f"CHANGED  {name!r}\n"
+                f"  expected: {expected_sig!r}\n"
+                f"  actual:   {actual[name]!r}"
+            )
+    if errors:
+        joined = '\n'.join(errors)
+        pytest.fail(
+            f"scipy.linalg.cython_lapack.__pyx_capi__ has {len(errors)} "
+            f"ABI breakage(s):\n{joined}"
+        )

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_cython_blas.py

@@ -0,0 +1,147 @@
+import numpy as np
+from numpy.testing import (assert_allclose,
+                           assert_equal)
+import scipy.linalg.cython_blas as blas
+
+
+class TestBlasInt:
+    """Tests for blas_int type used in cython_blas/cython_lapack."""
+
+    def test_blas_int_size(self):
+        """Verify blas_int size matches the build configuration."""
+        from scipy.__config__ import CONFIG
+        size = blas._blas_int_size()
+        cython_blas_ilp64 = CONFIG['Build Dependencies']['blas']['cython blas ilp64']
+        if cython_blas_ilp64:
+            assert size == 8, f"ILP64 build but blas_int is {size} bytes"
+        else:
+            assert size == 4, f"LP64 build but blas_int is {size} bytes"
+
+        # also verify blas_bint size
+        bint_sizes = blas._blas_bint_bint_sizes()  # (sizeof(blas_bint), sizeof(bint))
+        if cython_blas_ilp64:
+            assert bint_sizes == (8, 4)
+        else:
+            assert bint_sizes == (4, 4)
+
+    def test_dgemm_with_blas_int_dimensions(self):
+        """Test dgemm works correctly - exercises blas_int for dimensions."""
+        a = np.eye(3, dtype=np.float64)
+        b = np.arange(9, dtype=np.float64).reshape((3, 3))
+        c = np.empty((3, 3), dtype=np.float64)
+        blas._test_dgemm(1., a, b, 0., c)
+        assert_allclose(c, b)
+
+
+class TestDGEMM:
+
+    def test_transposes(self):
+
+        a = np.arange(12, dtype=np.float64).reshape((3, 4))[:2,:2]
+        b = np.arange(1, 13, dtype=np.float64).reshape((4, 3))[:2,:2]
+        c = np.empty((2, 4), dtype=np.float64)[:2,:2]
+
+        blas._test_dgemm(1., a, b, 0., c)
+        assert_allclose(c, a.dot(b))
+
+        blas._test_dgemm(1., a.T, b, 0., c)
+        assert_allclose(c, a.T.dot(b))
+
+        blas._test_dgemm(1., a, b.T, 0., c)
+        assert_allclose(c, a.dot(b.T))
+
+        blas._test_dgemm(1., a.T, b.T, 0., c)
+        assert_allclose(c, a.T.dot(b.T))
+
+        blas._test_dgemm(1., a, b, 0., c.T)
+        assert_allclose(c, a.dot(b).T)
+
+        blas._test_dgemm(1., a.T, b, 0., c.T)
+        assert_allclose(c, a.T.dot(b).T)
+
+        blas._test_dgemm(1., a, b.T, 0., c.T)
+        assert_allclose(c, a.dot(b.T).T)
+
+        blas._test_dgemm(1., a.T, b.T, 0., c.T)
+        assert_allclose(c, a.T.dot(b.T).T)
+
+    def test_shapes(self):
+        a = np.arange(6, dtype=np.float64).reshape((3, 2))
+        b = np.arange(-6, 2, dtype=np.float64).reshape((2, 4))
+        c = np.empty((3, 4), dtype=np.float64)
+
+        blas._test_dgemm(1., a, b, 0., c)
+        assert_allclose(c, a.dot(b))
+
+        blas._test_dgemm(1., b.T, a.T, 0., c.T)
+        assert_allclose(c, b.T.dot(a.T).T)
+
+class TestWfuncPointers:
+    """ Test the function pointers that are expected to fail on
+    Mac OS X without the additional entry statement in their definitions
+    in fblas_l1.pyf.src. """
+
+    def test_complex_args(self):
+
+        cx = np.array([.5 + 1.j, .25 - .375j, 12.5 - 4.j], np.complex64)
+        cy = np.array([.8 + 2.j, .875 - .625j, -1. + 2.j], np.complex64)
+
+        assert_allclose(blas._test_cdotc(cx, cy),
+                        -17.6468753815+21.3718757629j)
+        assert_allclose(blas._test_cdotu(cx, cy),
+                        -6.11562538147+30.3156242371j)
+
+        assert_equal(blas._test_icamax(cx), 3)
+
+        assert_allclose(blas._test_scasum(cx), 18.625)
+        assert_allclose(blas._test_scnrm2(cx), 13.1796483994)
+
+        assert_allclose(blas._test_cdotc(cx[::2], cy[::2]),
+                        -18.1000003815+21.2000007629j)
+        assert_allclose(blas._test_cdotu(cx[::2], cy[::2]),
+                        -6.10000038147+30.7999992371j)
+        assert_allclose(blas._test_scasum(cx[::2]), 18.)
+        assert_allclose(blas._test_scnrm2(cx[::2]), 13.1719398499)
+
+    def test_double_args(self):
+
+        x = np.array([5., -3, -.5], np.float64)
+        y = np.array([2, 1, .5], np.float64)
+
+        assert_allclose(blas._test_dasum(x), 8.5)
+        assert_allclose(blas._test_ddot(x, y), 6.75)
+        assert_allclose(blas._test_dnrm2(x), 5.85234975815)
+
+        assert_allclose(blas._test_dasum(x[::2]), 5.5)
+        assert_allclose(blas._test_ddot(x[::2], y[::2]), 9.75)
+        assert_allclose(blas._test_dnrm2(x[::2]), 5.0249376297)
+
+        assert_equal(blas._test_idamax(x), 1)
+
+    def test_float_args(self):
+
+        x = np.array([5., -3, -.5], np.float32)
+        y = np.array([2, 1, .5], np.float32)
+
+        assert_equal(blas._test_isamax(x), 1)
+
+        assert_allclose(blas._test_sasum(x), 8.5)
+        assert_allclose(blas._test_sdot(x, y), 6.75)
+        assert_allclose(blas._test_snrm2(x), 5.85234975815)
+
+        assert_allclose(blas._test_sasum(x[::2]), 5.5)
+        assert_allclose(blas._test_sdot(x[::2], y[::2]), 9.75)
+        assert_allclose(blas._test_snrm2(x[::2]), 5.0249376297)
+
+    def test_double_complex_args(self):
+
+        cx = np.array([.5 + 1.j, .25 - .375j, 13. - 4.j], np.complex128)
+        cy = np.array([.875 + 2.j, .875 - .625j, -1. + 2.j], np.complex128)
+
+        assert_equal(blas._test_izamax(cx), 3)
+
+        assert_allclose(blas._test_zdotc(cx, cy), -18.109375+22.296875j)
+        assert_allclose(blas._test_zdotu(cx, cy), -6.578125+31.390625j)
+
+        assert_allclose(blas._test_zdotc(cx[::2], cy[::2]), -18.5625+22.125j)
+        assert_allclose(blas._test_zdotu(cx[::2], cy[::2]), -6.5625+31.875j)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_cython_lapack.py

@@ -0,0 +1,56 @@
+from numpy.testing import assert_allclose
+from scipy.linalg import cython_lapack as cython_lapack
+from scipy.linalg import lapack
+
+
+class TestLamch:
+
+    def test_slamch(self):
+        lapack_slamch = lapack.get_lapack_funcs(
+            'lamch', dtype='float32', ilp64='preferred'
+        )
+        for c in [b'e', b's', b'b', b'p', b'n', b'r', b'm', b'u', b'l', b'o']:
+            assert_allclose(cython_lapack._test_slamch(c),
+                            lapack_slamch(c))
+
+    def test_dlamch(self):
+        lapack_dlamch = lapack.get_lapack_funcs(
+            'lamch', dtype='float64', ilp64='preferred'
+        )
+
+        for c in [b'e', b's', b'b', b'p', b'n', b'r', b'm', b'u', b'l', b'o']:
+            assert_allclose(cython_lapack._test_dlamch(c),
+                            lapack_dlamch(c))
+
+    def test_complex_ladiv(self):
+        cx = .5 + 1.j
+        cy = .875 + 2.j
+        assert_allclose(cython_lapack._test_zladiv(cy, cx), 1.95+0.1j)
+        assert_allclose(cython_lapack._test_cladiv(cy, cx), 1.95+0.1j)
+
+
+class TestBlasInt:
+    """Test for blas_int/blas_bint types used in cython_blas/cython_lapack.
+
+    Note similar test is in test_blas.py: both cython_blas.pyx and cython_lapack.pyx
+    define the types, so we smoke test that `blas_int` and `blas_bint` types
+    can be cimported from both `cython_blas` and `cython_lapack` extensions.
+    """
+
+    def test_blas_int_size(self):
+        """Verify blas_int and blas_bint sizes matches the build configuration."""
+        from scipy.__config__ import CONFIG
+        size = cython_lapack._blas_int_size()
+        cython_blas_ilp64 = CONFIG['Build Dependencies']['blas']['cython blas ilp64']
+        if cython_blas_ilp64:
+            assert size == 8, f"ILP64 build but blas_int is {size} bytes"
+        else:
+            assert size == 4, f"LP64 build but blas_int is {size} bytes"
+
+        # also verify blas_bint size
+        # bint_sizes is a 2-tuple, (sizeof(blas_bint), sizeof(bint))
+        bint_sizes = cython_lapack._blas_bint_bint_sizes()
+        if cython_blas_ilp64:
+            assert bint_sizes == (8, 4)
+        else:
+            assert bint_sizes == (4, 4)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_cythonized_array_utils.py

@@ -0,0 +1,161 @@
+import numpy as np
+from scipy.linalg import bandwidth, issymmetric, ishermitian
+from scipy.conftest import skip_xp_invalid_arg
+import pytest
+from pytest import raises
+from numpy.testing import assert_equal
+
+BANDWIDTH_DTYPES = (
+    np.bool,
+    np.int8, np.int16, np.int32, np.int64,
+    np.uint8, np.uint16, np.uint32, np.uint64,
+    np.float32, np.float64,
+    np.complex64, np.complex128,
+)
+
+
+@skip_xp_invalid_arg
+@pytest.mark.filterwarnings("ignore::DeprecationWarning")
+def test_bandwidth_dtypes():
+    n = 5
+    for dt in BANDWIDTH_DTYPES:
+        _ = bandwidth(np.zeros([n, n], dtype=dt))
+
+    unsupported_dtypes = [np.float16, np.longdouble, np.clongdouble,
+                          np.bytes_, np.str_, np.void, np.object_,
+                          np.datetime64, np.timedelta64]
+    for dt in unsupported_dtypes:
+        raises(TypeError, bandwidth, np.zeros([n, n], dtype=dt))
+
+
+@pytest.mark.parametrize('T', BANDWIDTH_DTYPES)
+def test_bandwidth_square_inputs(T):
+    n = 20
+    k = 4
+    R = np.zeros([n, n], dtype=T, order='F')
+    # form a banded matrix inplace
+    R[[x for x in range(n)], [x for x in range(n)]] = 1
+    R[[x for x in range(n-k)], [x for x in range(k, n)]] = 1
+    R[[x for x in range(1, n)], [x for x in range(n-1)]] = 1
+    R[[x for x in range(k, n)], [x for x in range(n-k)]] = 1
+    assert bandwidth(R) == (k, k)
+    A = np.array([
+        [1, 1, 0, 0, 0, 0, 0, 0],
+        [1, 0, 0, 0, 0, 0, 0, 0],
+        [0, 0, 0, 0, 0, 0, 0, 0],
+        [0, 0, 0, 0, 0, 0, 0, 0],
+        [0, 0, 0, 0, 0, 0, 0, 0],
+        [0, 0, 0, 0, 0, 1, 1, 1],
+        [0, 0, 0, 0, 0, 1, 0, 0],
+        [0, 0, 0, 0, 0, 1, 0, 0],
+    ])
+    assert bandwidth(A) == (2, 2)
+
+    A = np.array(
+        [
+            [[1, 0, 0], [0, 1, 0], [0, 0, 1]],  # diagonal
+            [[0, 0, 1], [0, 0, 0], [0, 0, 0]],  # upper triangular
+            [[0, 0, 0], [0, 0, 0], [1, 0, 0]],  # lower triangular
+            [[0, 0, 1], [0, 0, 0], [1, 0, 0]],  # full
+            [[0, 0, 1], [0, 0, 0], [0, 1, 0.]],  # upper hessenberg
+        ]
+    )
+    lo, hi = bandwidth(A)
+    assert_equal(lo, [0, 0, 2, 2, 1])
+    assert_equal(hi, [0, 2, 0, 2, 2])
+
+
+@skip_xp_invalid_arg
+@pytest.mark.parametrize('T', BANDWIDTH_DTYPES)
+def test_bandwidth_rect_inputs(T):
+    n, m = 10, 20
+    k = 5
+    R = np.zeros([n, m], dtype=T, order='F')
+    # form a banded matrix inplace
+    R[[x for x in range(n)], [x for x in range(n)]] = 1
+    R[[x for x in range(n-k)], [x for x in range(k, n)]] = 1
+    R[[x for x in range(1, n)], [x for x in range(n-1)]] = 1
+    R[[x for x in range(k, n)], [x for x in range(n-k)]] = 1
+    assert bandwidth(R) == (k, k)
+
+    R2 = np.tril(np.ones((2, 10, 2), dtype=T))
+    lo, hi = bandwidth(R2)
+    assert_equal(lo, [9, 9])
+    assert_equal(hi, [0, 0])
+
+    R3 = np.triu(np.ones((2, 10, 2), dtype=T))
+    lo, hi = bandwidth(R3)
+    assert_equal(lo, [0, 0])
+    assert_equal(hi, [1, 1])
+
+
+@skip_xp_invalid_arg
+@pytest.mark.filterwarnings("ignore::DeprecationWarning")
+def test_issymetric_ishermitian_dtypes():
+    n = 5
+    for t in np.typecodes['All']:
+        A = np.zeros([n, n], dtype=t)
+        if t in 'eUVOMm':
+            raises(TypeError, issymmetric, A)
+            raises(TypeError, ishermitian, A)
+        elif t == 'G':  # No-op test. On win these pass on others fail.
+            pass
+        else:
+            assert issymmetric(A)
+            assert ishermitian(A)
+
+
+def test_issymmetric_ishermitian_invalid_input():
+    A = np.array([1, 2, 3])
+    raises(ValueError, issymmetric, A)
+    raises(ValueError, ishermitian, A)
+    A = np.array([[[1, 2, 3], [4, 5, 6]]])
+    raises(ValueError, issymmetric, A)
+    raises(ValueError, ishermitian, A)
+    A = np.array([[1, 2, 3], [4, 5, 6]])
+    raises(ValueError, issymmetric, A)
+    raises(ValueError, ishermitian, A)
+
+
+def test_issymetric_complex_decimals():
+    A = np.arange(1, 10).astype(complex).reshape(3, 3)
+    A += np.arange(-4, 5).astype(complex).reshape(3, 3)*1j
+    # make entries decimal
+    A /= np.pi
+    A = A + A.T
+    assert issymmetric(A)
+
+
+def test_ishermitian_complex_decimals():
+    A = np.arange(1, 10).astype(complex).reshape(3, 3)
+    A += np.arange(-4, 5).astype(complex).reshape(3, 3)*1j
+    # make entries decimal
+    A /= np.pi
+    A = A + A.T.conj()
+    assert ishermitian(A)
+
+
+def test_issymmetric_approximate_results():
+    n = 20
+    rng = np.random.RandomState(123456789)
+    x = rng.uniform(high=5., size=[n, n])
+    y = x @ x.T  # symmetric
+    p = rng.standard_normal([n, n])
+    z = p @ y @ p.T
+    assert issymmetric(z, atol=1e-10)
+    assert issymmetric(z, atol=1e-10, rtol=0.)
+    assert issymmetric(z, atol=0., rtol=1e-12)
+    assert issymmetric(z, atol=1e-13, rtol=1e-12)
+
+
+def test_ishermitian_approximate_results():
+    n = 20
+    rng = np.random.RandomState(987654321)
+    x = rng.uniform(high=5., size=[n, n])
+    y = x @ x.T  # symmetric
+    p = rng.standard_normal([n, n]) + rng.standard_normal([n, n])*1j
+    z = p @ y @ p.conj().T
+    assert ishermitian(z, atol=1e-10)
+    assert ishermitian(z, atol=1e-10, rtol=0.)
+    assert ishermitian(z, atol=0., rtol=1e-12)
+    assert ishermitian(z, atol=1e-13, rtol=1e-12)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_decomp_cholesky.py

@@ -0,0 +1,452 @@
+import pytest
+import numpy as np
+from numpy.testing import assert_allclose
+from pytest import raises as assert_raises
+
+from numpy import array, transpose, zeros_like, empty
+from scipy.linalg import (cholesky, cholesky_banded, cho_solve_banded,
+     cho_factor, cho_solve, LinAlgError)
+
+from scipy.linalg._testutils import assert_no_overwrite
+from .test_basic import parametrize_overwrite_arg
+
+
+class TestCholesky:
+
+    def test_simple(self):
+        a = [[8, 2, 3], [2, 9, 3], [3, 3, 6]]
+        c = cholesky(a)
+        assert_allclose(c.T @ c, a, atol=1e-14)
+        c = transpose(c)
+        a = c @ c.T
+        assert_allclose(cholesky(a, lower=1), c, atol=1e-14)
+
+    def test_check_finite(self):
+        a = [[8, 2, 3], [2, 9, 3], [3, 3, 6]]
+        c = cholesky(a, check_finite=False)
+        assert_allclose(c.T @ c, a, atol=1e-14)
+        c = transpose(c)
+        a = c @ c.T
+        assert_allclose(cholesky(a, lower=1, check_finite=False), c, atol=1e-14)
+
+    def test_simple_complex(self):
+        m = array([[3+1j, 3+4j, 5], [0, 2+2j, 2+7j], [0, 0, 7+4j]])
+        a = np.conj(m.T) @ m
+        c = cholesky(a)
+        a1 = np.conj(c.T) @ c
+        assert_allclose(a, a1)
+        c = transpose(c)
+        a = c @ np.conj(c.T)
+        assert_allclose(cholesky(a, lower=1), c, atol=1e-14)
+
+    @pytest.mark.parametrize("n", [5, 10, 20])
+    @pytest.mark.parametrize("dtype", [np.float32, np.float64,
+                                        np.complex64, np.complex128])
+    @pytest.mark.parametrize("order", ["F", "C"])
+    def test_random(self, n, dtype, order):
+        atol = 5e-4 if dtype in [np.float32, np.complex64] else 1e-12
+        rng = np.random.default_rng(seed=12345)
+
+        for k in range(2):
+            m = rng.normal(size=(n, n)).astype(dtype)
+            if np.issubdtype(dtype, np.complexfloating):
+                m += 1j * rng.normal(size=(n, n)).astype(dtype)
+            for i in range(n):
+                m[i, i] = 20*(.1+m[i, i])
+
+            a = np.asarray(np.conj(m.T) @ m, order=order)
+
+            c = cholesky(a, lower=0)
+            a1 = np.conj(c.T) @ c
+            assert_allclose(a, a1, atol=atol)
+
+            c = transpose(c)
+            a = c @ np.conj(c.T)
+            assert_allclose(cholesky(a, lower=1), c, atol=atol)
+
+    @pytest.mark.xslow
+    def test_int_overflow(self):
+       # regression test for
+       # https://github.com/scipy/scipy/issues/17436
+       # the problem was an int overflow in zeroing out
+       # the unused triangular part
+       n = 47_000
+       x = np.eye(n, dtype=np.float64, order='F')
+       x[:4, :4] = np.array([[4, -2, 3, -1],
+                             [-2, 4, -3, 1],
+                             [3, -3, 5, 0],
+                             [-1, 1, 0, 5]])
+
+       cholesky(x, check_finite=False, overwrite_a=True)  # should not segfault
+
+    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
+    @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64])
+    def test_empty(self, dt, dt_b):
+        a = empty((0, 0), dtype=dt)
+
+        c = cholesky(a)
+        assert c.shape == (0, 0)
+        assert c.dtype == cholesky(np.eye(2, dtype=dt)).dtype
+
+        c_and_lower = (c, True)
+        b = np.asarray([], dtype=dt_b)
+        x = cho_solve(c_and_lower, b)
+        assert x.shape == (0,)
+        assert x.dtype == cho_solve((np.eye(2, dtype=dt), True),
+                                     np.ones(2, dtype=dt_b)).dtype
+
+        b = empty((0, 0), dtype=dt_b)
+        x = cho_solve(c_and_lower, b)
+        assert x.shape == (0, 0)
+        assert x.dtype == cho_solve((np.eye(2, dtype=dt), True),
+                                     np.ones(2, dtype=dt_b)).dtype
+
+        a1 = array([])
+        a2 = array([[]])
+        a3 = []
+        a4 = [[]]
+        for x in ([a1, a2, a3, a4]):
+            assert_raises(ValueError, cholesky, x)
+
+    @pytest.mark.parametrize("shape", [(3,), (2, 3), (3, 2)])
+    def test_invalid_shape(self, shape):
+        rng = np.random.default_rng(seed=12345)
+        a = rng.normal(size=shape)
+
+        with pytest.raises(ValueError, match="Expected a square matrix"):
+            cholesky(a)
+
+    @pytest.mark.parametrize("dtype", [np.float32, np.float64,
+                             np.complex64, np.complex128])
+    @pytest.mark.parametrize("order", ["C", "F"])
+    def test_non_posdef(self, dtype, order):
+        n = 10
+        rng = np.random.default_rng(seed=12345)
+
+        lmbd = np.diag(-np.abs(rng.normal(size=(n,))).astype(dtype))
+        x = rng.normal(size=(n, n)).astype(dtype)
+        if np.issubdtype(dtype, np.complexfloating):
+            x += 1j * rng.normal(size=(n, n)).astype(dtype)
+
+        a = np.asarray(x @ lmbd @ np.conj(x.T), order=order)
+
+        with pytest.raises(LinAlgError, match="Internal potrf return info"):
+            cholesky(a)
+
+    @parametrize_overwrite_arg
+    @pytest.mark.parametrize("dtype", [int, float])
+    @pytest.mark.parametrize("order", ["C", "F"])
+    def test_overwrite_args(self, overwrite_kw, dtype, order):
+        rng = np.random.default_rng(seed=12345)
+        n = 5
+
+        x = rng.normal(size=(n, n))
+        x = x.astype(dtype)
+        a = x @ x.T
+        a = a.astype(dtype, order=order)
+        a_ref = np.copy(a)
+
+        c = cholesky(a, **overwrite_kw)
+        overwrite_a = overwrite_kw.get("overwrite_a", False)
+        a_inplace = (overwrite_a and (dtype is not int)
+                     and (a.flags["F_CONTIGUOUS"] or a.flags["C_CONTIGUOUS"]))
+
+        assert np.shares_memory(a, c) == a_inplace
+        assert np.all(a == a_ref) != a_inplace
+
+    @pytest.mark.parametrize("lower", [True, False])
+    def test_overwrite_cleaning(self, lower):
+        # Since only half the array is copied over, this checks if the other
+        # half is correctly put to 0 when the input array is passed in instead.
+        rng = np.random.default_rng(seed=12345)
+        n = 5
+
+        x = rng.normal(size=(n, n))
+        a = np.asfortranarray(x @ x.T)
+        a_ref = np.copy(a)
+
+        c = cholesky(a, lower=lower, overwrite_a=True)
+
+        if lower:
+            assert_allclose(np.triu(a, k=1), np.zeros_like(a), atol=1e-14)
+            assert_allclose(c @ c.T, a_ref, atol=1e-14)
+        else:
+            assert_allclose(np.tril(a, k=-1), np.zeros_like(a), atol=1e-14)
+            assert_allclose(c.T @ c, a_ref, atol=1e-14)
+
+    @pytest.mark.parametrize("lower", [True, False])
+    @pytest.mark.parametrize("order", ['C', 'F', 'noncontig_C', 'noncontig_F'])
+    @pytest.mark.parametrize("overwrite_a", [True, False])
+    @pytest.mark.parametrize("dtype", [np.float64, np.complex128])
+    def test_nan_in_unused_triangle(self, lower, order, overwrite_a, dtype):
+        # The unused triangle should never be read, so NaN there must not
+        # affect the result.
+        rng = np.random.default_rng(seed=12345)
+        n = 5
+        x = rng.normal(size=(n, n))
+        if np.issubdtype(dtype, np.complexfloating):
+            x = x + 1j * rng.normal(size=(n, n))
+        a = (x @ x.conj().T + n * np.eye(n)).astype(dtype)
+        c_ref = cholesky(a, lower=lower)
+
+        # Prepare the array in the desired layout
+        if order.startswith('noncontig'):
+            # Unequal strides to probe strided path thoroughly
+            big_order = 'F' if order.endswith('_F') else 'C'
+            a_big = np.zeros((2*n, 3*n), dtype=dtype, order=big_order)
+            a_big[::2, ::3] = a
+            a_nan = a_big[::2, ::3]  # non-contiguous view
+            assert not a_nan.flags['C_CONTIGUOUS']
+            assert not a_nan.flags['F_CONTIGUOUS']
+        else:
+            a_nan = np.array(a, order=order)
+
+        # Put NaN in the unused triangle
+        if lower:
+            a_nan[np.triu_indices(n, 1)] = np.nan
+        else:
+            a_nan[np.tril_indices(n, -1)] = np.nan
+
+        c = cholesky(a_nan, lower=lower, overwrite_a=overwrite_a,
+                     check_finite=False)
+        assert_allclose(c, c_ref, atol=1e-14)
+        # Verify reconstruction
+        if lower:
+            assert_allclose(c @ c.conj().T, a, atol=1e-13)
+        else:
+            assert_allclose(c.conj().T @ c, a, atol=1e-13)
+
+
+class TestCholeskyBanded:
+    """Tests for cholesky_banded() and cho_solve_banded."""
+
+    def test_check_finite(self):
+        # Symmetric positive definite banded matrix `a`
+        a = array([[4.0, 1.0, 0.0, 0.0],
+                   [1.0, 4.0, 0.5, 0.0],
+                   [0.0, 0.5, 4.0, 0.2],
+                   [0.0, 0.0, 0.2, 4.0]])
+        # Banded storage form of `a`.
+        ab = array([[-1.0, 1.0, 0.5, 0.2],
+                    [4.0, 4.0, 4.0, 4.0]])
+        c = cholesky_banded(ab, lower=False, check_finite=False)
+        ufac = zeros_like(a)
+        ufac[list(range(4)), list(range(4))] = c[-1]
+        ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
+        assert_allclose(a, ufac.T @ ufac, atol=1e-14)
+
+        b = array([0.0, 0.5, 4.2, 4.2])
+        x = cho_solve_banded((c, False), b, check_finite=False)
+        assert_allclose(x, [0.0, 0.0, 1.0, 1.0], atol=1e-14)
+
+    def test_upper_real(self):
+        # Symmetric positive definite banded matrix `a`
+        a = array([[4.0, 1.0, 0.0, 0.0],
+                   [1.0, 4.0, 0.5, 0.0],
+                   [0.0, 0.5, 4.0, 0.2],
+                   [0.0, 0.0, 0.2, 4.0]])
+        # Banded storage form of `a`.
+        ab = array([[-1.0, 1.0, 0.5, 0.2],
+                    [4.0, 4.0, 4.0, 4.0]])
+        c = cholesky_banded(ab, lower=False)
+        ufac = zeros_like(a)
+        ufac[list(range(4)), list(range(4))] = c[-1]
+        ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
+        assert_allclose(a, ufac.T @ ufac, atol=1e-14)
+
+        b = array([0.0, 0.5, 4.2, 4.2])
+        x = cho_solve_banded((c, False), b)
+        assert_allclose(x, [0.0, 0.0, 1.0, 1.0], atol=1e-14)
+
+    def test_upper_complex(self):
+        # Hermitian positive definite banded matrix `a`
+        a = array([[4.0, 1.0, 0.0, 0.0],
+                   [1.0, 4.0, 0.5, 0.0],
+                   [0.0, 0.5, 4.0, -0.2j],
+                   [0.0, 0.0, 0.2j, 4.0]])
+        # Banded storage form of `a`.
+        ab = array([[-1.0, 1.0, 0.5, -0.2j],
+                    [4.0, 4.0, 4.0, 4.0]])
+        c = cholesky_banded(ab, lower=False)
+        ufac = zeros_like(a)
+        ufac[list(range(4)), list(range(4))] = c[-1]
+        ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
+        assert_allclose(a, np.conj(ufac.T) @ ufac, atol=1e-14)
+
+        b = array([0.0, 0.5, 4.0-0.2j, 0.2j + 4.0])
+        x = cho_solve_banded((c, False), b)
+        assert_allclose(x, [0.0, 0.0, 1.0, 1.0], atol=1e-14)
+
+    def test_lower_real(self):
+        # Symmetric positive definite banded matrix `a`
+        a = array([[4.0, 1.0, 0.0, 0.0],
+                   [1.0, 4.0, 0.5, 0.0],
+                   [0.0, 0.5, 4.0, 0.2],
+                   [0.0, 0.0, 0.2, 4.0]])
+        # Banded storage form of `a`.
+        ab = array([[4.0, 4.0, 4.0, 4.0],
+                    [1.0, 0.5, 0.2, -1.0]])
+        c = cholesky_banded(ab, lower=True)
+        lfac = zeros_like(a)
+        lfac[list(range(4)), list(range(4))] = c[0]
+        lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
+        assert_allclose(a, lfac @ lfac.T, atol=1e-14)
+
+        b = array([0.0, 0.5, 4.2, 4.2])
+        x = cho_solve_banded((c, True), b)
+        assert_allclose(x, [0.0, 0.0, 1.0, 1.0], atol=1e-14)
+
+    def test_lower_complex(self):
+        # Hermitian positive definite banded matrix `a`
+        a = array([[4.0, 1.0, 0.0, 0.0],
+                   [1.0, 4.0, 0.5, 0.0],
+                   [0.0, 0.5, 4.0, -0.2j],
+                   [0.0, 0.0, 0.2j, 4.0]])
+        # Banded storage form of `a`.
+        ab = array([[4.0, 4.0, 4.0, 4.0],
+                    [1.0, 0.5, 0.2j, -1.0]])
+        c = cholesky_banded(ab, lower=True)
+        lfac = zeros_like(a)
+        lfac[list(range(4)), list(range(4))] = c[0]
+        lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
+        assert_allclose(a, lfac @ np.conj(lfac.T), atol=1e-14)
+
+        b = array([0.0, 0.5j, 3.8j, 3.8])
+        x = cho_solve_banded((c, True), b)
+        assert_allclose(x, [0.0, 0.0, 1.0j, 1.0], atol=1e-14)
+
+    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
+    @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64])
+    def test_empty(self, dt, dt_b):
+        ab = empty((0, 0), dtype=dt)
+
+        cb = cholesky_banded(ab)
+        assert cb.shape == (0, 0)
+
+        m = cholesky_banded(np.array([[0, 0], [1, 1]], dtype=dt))
+        assert cb.dtype == m.dtype
+
+        cb_and_lower = (cb, True)
+        b = np.asarray([], dtype=dt_b)
+        x = cho_solve_banded(cb_and_lower, b)
+        assert x.shape == (0,)
+
+        dtype_nonempty = cho_solve_banded((m, True), np.ones(2, dtype=dt_b)).dtype
+        assert x.dtype == dtype_nonempty
+
+        b = empty((0, 0), dtype=dt_b)
+        x = cho_solve_banded(cb_and_lower, b)
+        assert x.shape == (0, 0)
+        assert x.dtype == dtype_nonempty
+
+
+class TestOverwrite:
+    def test_cholesky(self):
+        assert_no_overwrite(cholesky, [(3, 3)])
+
+    def test_cho_factor(self):
+        assert_no_overwrite(cho_factor, [(3, 3)])
+
+    def test_cho_solve(self):
+        x = array([[2, -1, 0], [-1, 2, -1], [0, -1, 2]])
+        xcho = cho_factor(x)
+        assert_no_overwrite(lambda b: cho_solve(xcho, b), [(3,)])
+
+    def test_cholesky_banded(self):
+        assert_no_overwrite(cholesky_banded, [(2, 3)])
+
+    def test_cho_solve_banded(self):
+        x = array([[0, -1, -1], [2, 2, 2]])
+        xcho = cholesky_banded(x)
+        assert_no_overwrite(lambda b: cho_solve_banded((xcho, False), b),
+                            [(3,)])
+
+
+class TestChoFactor:
+    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
+    def test_empty(self, dt):
+        a = np.empty((0, 0), dtype=dt)
+        x, lower = cho_factor(a)
+
+        assert x.shape == (0, 0)
+
+        xx, lower = cho_factor(np.eye(2, dtype=dt))
+        assert x.dtype == xx.dtype
+
+    @pytest.mark.parametrize("n", [5, 10, 20])
+    @pytest.mark.parametrize("dtype", [np.float32, np.float64,
+                                np.complex64, np.complex128])
+    @pytest.mark.parametrize("order", ["C", "F"])
+    @pytest.mark.parametrize("lower", [True, False])
+    def test_random(self, n, dtype, order, lower):
+        atol = 1e-5 if dtype in [np.float32, np.complex64] else 1e-12
+        rng = np.random.default_rng(seed=12345)
+
+        lmbd = np.diag(np.abs(rng.normal(size=(n,)).astype(dtype)))
+        x = rng.normal(size=(n, n)).astype(dtype)
+        if np.issubdtype(dtype, np.complexfloating):
+            x += 1j * rng.normal(size=(n, n)).astype(dtype)
+
+        a = np.asarray(x @ lmbd @ np.conj(x.T), order=order)
+        c = cholesky(a, lower=lower)
+        c_f, lower_out = cho_factor(a, lower=lower)
+
+        assert lower_out == lower
+
+        # Theoretically exactly equal
+        # For `cho_factor` the other half of the matrix is irrelevant.
+        if lower:
+            l = np.tril(c_f)
+            assert_allclose(c, l, atol=atol)
+            assert_allclose(a, l @ np.conj(l.T), atol=atol)
+        else:
+            u = np.triu(c_f)
+            assert_allclose(c, u, atol=atol)
+            assert_allclose(a, np.conj(u.T) @ u, atol=atol)
+
+    @pytest.mark.parametrize("shape", [(3,), (3, 2), (2, 3)])
+    def test_invalid_shape(self, shape):
+        rng = np.random.default_rng(seed=12345)
+        a = rng.normal(size=shape)
+
+        with pytest.raises(ValueError, match="Expected a square matrix or batch"):
+            cho_factor(a)
+
+    @pytest.mark.parametrize("dtype", [np.float32, np.float64,
+                             np.complex64, np.complex128])
+    @pytest.mark.parametrize("order", ["C", "F"])
+    def test_non_posdef(self, dtype, order):
+        n = 10
+        rng = np.random.default_rng(seed=12345)
+
+        lmbd = np.diag(-np.abs(rng.normal(size=(n,))).astype(dtype))
+        x = rng.normal(size=(n, n)).astype(dtype)
+        if np.issubdtype(dtype, np.complexfloating):
+            x += 1j * rng.normal(size=(n, n)).astype(dtype)
+
+        a = np.asarray(x @ lmbd @ np.conj(x.T), order=order)
+
+        with pytest.raises(LinAlgError, match="Internal potrf return info"):
+            cho_factor(a)
+
+    @parametrize_overwrite_arg
+    @pytest.mark.parametrize("dtype", [int, float])
+    @pytest.mark.parametrize("order", ["C", "F"])
+    @pytest.mark.parametrize("lower", [True, False])
+    def test_overwrite_args(self, overwrite_kw, dtype, order, lower):
+        rng = np.random.default_rng(seed=12345)
+        n = 5
+
+        x = rng.normal(size=(n, n))
+        x = x.astype(dtype)
+        a = x @ x.T
+        a = a.astype(dtype, order=order)
+        a_ref = np.copy(a)
+
+        c, lower = cho_factor(a, **overwrite_kw, lower=lower)
+        overwrite_a = overwrite_kw.get("overwrite_a", False)
+        a_inplace = (overwrite_a and (dtype is not int)
+                     and (a.flags["F_CONTIGUOUS"] or a.flags["C_CONTIGUOUS"]))
+
+        assert np.shares_memory(a, c) == a_inplace
+        assert np.all(a == a_ref) != a_inplace

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_decomp_cossin.py

@@ -0,0 +1,314 @@
+import pytest
+import numpy as np
+from numpy.random import default_rng
+from numpy.testing import assert_allclose
+
+from scipy import linalg
+from scipy.linalg.lapack import _compute_lwork
+from scipy.stats import ortho_group, unitary_group
+from scipy.linalg import cossin, get_lapack_funcs
+
+REAL_DTYPES = (np.float32, np.float64)
+COMPLEX_DTYPES = (np.complex64, np.complex128)
+DTYPES = REAL_DTYPES + COMPLEX_DTYPES
+
+
+@pytest.mark.parametrize('dtype_', DTYPES)
+@pytest.mark.parametrize('m, p, q',
+                         [
+                             (2, 1, 1),
+                             (3, 2, 1),
+                             (3, 1, 2),
+                             (4, 2, 2),
+                             (4, 1, 2),
+                             (40, 12, 20),
+                             (40, 30, 1),
+                             (40, 1, 30),
+                             (100, 50, 1),
+                             (100, 50, 50),
+                         ])
+@pytest.mark.parametrize('swap_sign', [True, False])
+def test_cossin(dtype_, m, p, q, swap_sign):
+    rng = default_rng(1708093570726217)
+    if dtype_ in COMPLEX_DTYPES:
+        x = np.array(unitary_group.rvs(m, random_state=rng), dtype=dtype_)
+    else:
+        x = np.array(ortho_group.rvs(m, random_state=rng), dtype=dtype_)
+
+    u, cs, vh = cossin(x, p, q,
+                       swap_sign=swap_sign)
+    assert_allclose(x, u @ cs @ vh, rtol=0., atol=m*1e3*np.finfo(dtype_).eps)
+    assert u.dtype == dtype_
+    # Test for float32 or float 64
+    assert cs.dtype == np.real(u).dtype
+    assert vh.dtype == dtype_
+
+    u, cs, vh = cossin([x[:p, :q], x[:p, q:], x[p:, :q], x[p:, q:]],
+                       swap_sign=swap_sign)
+    assert_allclose(x, u @ cs @ vh, rtol=0., atol=m*1e3*np.finfo(dtype_).eps)
+    assert u.dtype == dtype_
+    assert cs.dtype == np.real(u).dtype
+    assert vh.dtype == dtype_
+
+    _, cs2, vh2 = cossin(x, p, q,
+                         compute_u=False,
+                         swap_sign=swap_sign)
+    assert_allclose(cs, cs2, rtol=0., atol=10*np.finfo(dtype_).eps)
+    assert_allclose(vh, vh2, rtol=0., atol=10*np.finfo(dtype_).eps)
+
+    u2, cs2, _ = cossin(x, p, q,
+                        compute_vh=False,
+                        swap_sign=swap_sign)
+    assert_allclose(u, u2, rtol=0., atol=10*np.finfo(dtype_).eps)
+    assert_allclose(cs, cs2, rtol=0., atol=10*np.finfo(dtype_).eps)
+
+    _, cs2, _ = cossin(x, p, q,
+                       compute_u=False,
+                       compute_vh=False,
+                       swap_sign=swap_sign)
+    assert_allclose(cs, cs2, rtol=0., atol=10*np.finfo(dtype_).eps)
+
+
+def test_cossin_mixed_types():
+    rng = default_rng(1708093736390459)
+    x = np.array(ortho_group.rvs(4, random_state=rng), dtype=np.float64)
+    u, cs, vh = cossin([x[:2, :2],
+                        np.array(x[:2, 2:], dtype=np.complex128),
+                        x[2:, :2],
+                        x[2:, 2:]])
+
+    assert u.dtype == np.complex128
+    assert cs.dtype == np.float64
+    assert vh.dtype == np.complex128
+    assert_allclose(x, u @ cs @ vh, rtol=0.,
+                    atol=1e4 * np.finfo(np.complex128).eps)
+
+
+def test_cossin_error_incorrect_subblocks():
+    with pytest.raises(ValueError, match="be due to missing p, q arguments."):
+        cossin(([1, 2], [3, 4, 5], [6, 7], [8, 9, 10]))
+
+
+def test_cossin_error_empty_subblocks():
+    with pytest.raises(ValueError, match="x11.*empty"):
+        cossin(([], [], [], []))
+    with pytest.raises(ValueError, match="x12.*empty"):
+        cossin(([1, 2], [], [6, 7], [8, 9, 10]))
+    with pytest.raises(ValueError, match="x21.*empty"):
+        cossin(([1, 2], [3, 4, 5], [], [8, 9, 10]))
+    with pytest.raises(ValueError, match="x22.*empty"):
+        cossin(([1, 2], [3, 4, 5], [2], []))
+
+
+def test_cossin_error_missing_partitioning():
+    with pytest.raises(ValueError, match=".*exactly four arrays.* got 2"):
+        cossin(unitary_group.rvs(2))
+
+    with pytest.raises(ValueError, match=".*might be due to missing p, q"):
+        cossin(unitary_group.rvs(4))
+
+
+def test_cossin_error_non_iterable():
+    with pytest.raises(ValueError, match="containing the subblocks of X"):
+        cossin(12j)
+
+def test_cossin_error_invalid_shape():
+    # Invalid x12 dimensions
+    p, q = 3, 4
+    invalid_x12 = np.ones((p, q + 2))
+    valid_ones = np.ones((p, q))
+    with pytest.raises(ValueError, 
+            match=r"Invalid x12 dimensions: desired \(3, 4\), got \(3, 6\)"):
+        cossin((valid_ones, invalid_x12, valid_ones, valid_ones))
+
+    # Invalid x21 dimensions
+    invalid_x21 = np.ones(p + 2)
+    with pytest.raises(ValueError, 
+            match=r"Invalid x21 dimensions: desired \(3, 4\), got \(1, 5\)"):
+        cossin((valid_ones, valid_ones, invalid_x21, valid_ones))
+
+def test_cossin_error_non_square():
+    with pytest.raises(ValueError, match="only supports square"):
+        cossin(np.array([[1, 2]]), 1, 1)
+
+
+def test_cossin_error_partitioning():
+    x = np.array(ortho_group.rvs(4), dtype=np.float64)
+    with pytest.raises(ValueError, match="invalid p=0.*0<p<4.*"):
+        cossin(x, 0, 1)
+    with pytest.raises(ValueError, match="invalid p=4.*0<p<4.*"):
+        cossin(x, 4, 1)
+    with pytest.raises(ValueError, match="invalid q=-2.*0<q<4.*"):
+        cossin(x, 1, -2)
+    with pytest.raises(ValueError, match="invalid q=5.*0<q<4.*"):
+        cossin(x, 1, 5)
+
+
+@pytest.mark.parametrize("dtype_", DTYPES)
+def test_cossin_separate(dtype_):
+    rng = default_rng(1708093590167096)
+    m, p, q = 98, 37, 61
+
+    pfx = 'or' if dtype_ in REAL_DTYPES else 'un'
+    X = (ortho_group.rvs(m, random_state=rng) if pfx == 'or'
+         else unitary_group.rvs(m, random_state=rng))
+    X = np.array(X, dtype=dtype_)
+
+    drv, dlw = get_lapack_funcs((pfx + 'csd', pfx + 'csd_lwork'), [X])
+    lwval = _compute_lwork(dlw, m, p, q)
+    lwvals = {'lwork': lwval} if pfx == 'or' else dict(zip(['lwork',
+                                                            'lrwork'],
+                                                           lwval))
+
+    *_, theta, u1, u2, v1t, v2t, _ = \
+        drv(X[:p, :q], X[:p, q:], X[p:, :q], X[p:, q:], **lwvals)
+
+    (u1_2, u2_2), theta2, (v1t_2, v2t_2) = cossin(X, p, q, separate=True)
+
+    assert_allclose(u1_2, u1, rtol=0., atol=10*np.finfo(dtype_).eps)
+    assert_allclose(u2_2, u2, rtol=0., atol=10*np.finfo(dtype_).eps)
+    assert_allclose(v1t_2, v1t, rtol=0., atol=10*np.finfo(dtype_).eps)
+    assert_allclose(v2t_2, v2t, rtol=0., atol=10*np.finfo(dtype_).eps)
+    assert_allclose(theta2, theta, rtol=0., atol=10*np.finfo(dtype_).eps)
+
+
+@pytest.mark.parametrize("m", [2, 5, 10, 15, 20])
+@pytest.mark.parametrize("p", [1, 4, 9, 14, 19])
+@pytest.mark.parametrize("q", [1, 4, 9, 14, 19])
+@pytest.mark.parametrize("swap_sign", [True, False])
+def test_properties(m, p, q, swap_sign):
+    # Test all the properties advertised in `linalg.cossin` documentation.
+    # There may be some overlap with tests above, but this is sensitive to
+    # the bug reported in gh-19365 and more.
+    if (p >= m) or (q >= m):
+        pytest.skip("`0 < p < m` and `0 < q < m` must hold")
+
+    # Generate unitary input
+    rng = np.random.default_rng(329548272348596421)
+    X = unitary_group.rvs(m, random_state=rng)
+    np.testing.assert_allclose(X @ X.conj().T, np.eye(m), atol=1e-15)
+
+    # Perform the decomposition
+    u0, cs0, vh0 = linalg.cossin(X, p=p, q=q, separate=True, swap_sign=swap_sign)
+    u1, u2 = u0
+    v1, v2 = vh0
+    v1, v2 = v1.conj().T, v2.conj().T
+
+    # "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"
+    np.testing.assert_allclose(u1 @ u1.conj().T, np.eye(p), atol=1e-13)
+    np.testing.assert_allclose(u2 @ u2.conj().T, np.eye(m-p), atol=1e-13)
+    np.testing.assert_allclose(v1 @ v1.conj().T, np.eye(q), atol=1e-13)
+    np.testing.assert_allclose(v2 @ v2.conj().T, np.eye(m-q), atol=1e-13)
+
+    # "and C and S are (r, r) nonnegative diagonal matrices..."
+    C = np.diag(np.cos(cs0))
+    S = np.diag(np.sin(cs0))
+    # "...satisfying C^2 + S^2 = I where r = min(p, m-p, q, m-q)."
+    r = min(p, m-p, q, m-q)
+    np.testing.assert_allclose(C**2 + S**2, np.eye(r))
+
+    # "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."
+    I11 = np.eye(min(p, q) - r)
+    I12 = np.eye(min(p, m - q) - r)
+    I21 = np.eye(min(m - p, q) - r)
+    I22 = np.eye(min(m - p, m - q) - r)
+
+    # From:
+    #                            ┌                   ┐
+    #                            │ 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 │
+    #                            └                   ┘
+
+    # We can see that U and V are block diagonal matrices like so:
+    U = linalg.block_diag(u1, u2)
+    V = linalg.block_diag(v1, v2)
+
+    # And the center matrix, which we'll call Q here, must be:
+    Q11 = np.zeros((u1.shape[1], v1.shape[0]))
+    IC11 = linalg.block_diag(I11, C)
+    Q11[:IC11.shape[0], :IC11.shape[1]] = IC11
+
+    Q12 = np.zeros((u1.shape[1], v2.shape[0]))
+    SI12 = linalg.block_diag(S, I12) if swap_sign else linalg.block_diag(-S, -I12)
+    Q12[-SI12.shape[0]:, -SI12.shape[1]:] = SI12
+
+    Q21 = np.zeros((u2.shape[1], v1.shape[0]))
+    SI21 = linalg.block_diag(-S, -I21) if swap_sign else linalg.block_diag(S, I21)
+    Q21[-SI21.shape[0]:, -SI21.shape[1]:] = SI21
+
+    Q22 = np.zeros((u2.shape[1], v2.shape[0]))
+    IC22 = linalg.block_diag(I22, C)
+    Q22[:IC22.shape[0], :IC22.shape[1]] = IC22
+
+    Q = np.block([[Q11, Q12], [Q21, Q22]])
+
+    # Confirm that `cossin` decomposes `X` as shown
+    np.testing.assert_allclose(X, U @ Q @ V.conj().T)
+
+    # And check that `separate=False` agrees
+    U0, CS0, Vh0 = linalg.cossin(X, p=p, q=q, swap_sign=swap_sign)
+    np.testing.assert_allclose(U, U0)
+    np.testing.assert_allclose(Q, CS0)
+    np.testing.assert_allclose(V, Vh0.conj().T)
+
+    # Confirm that `compute_u`/`compute_vh` don't affect the results
+    kwargs = dict(p=p, q=q, swap_sign=swap_sign)
+
+    # `compute_u=False`
+    u, cs, vh = linalg.cossin(X, separate=True, compute_u=False, **kwargs)
+    assert u[0].shape == (0, 0)  # probably not ideal, but this is what it does
+    assert u[1].shape == (0, 0)
+    assert_allclose(cs, cs0, rtol=1e-15)
+    assert_allclose(vh[0], vh0[0], rtol=1e-15)
+    assert_allclose(vh[1], vh0[1], rtol=1e-15)
+
+    U, CS, Vh = linalg.cossin(X, compute_u=False, **kwargs)
+    assert U.shape == (0, 0)
+    assert_allclose(CS, CS0, rtol=1e-15)
+    assert_allclose(Vh, Vh0, rtol=1e-15)
+
+    # `compute_vh=False`
+    u, cs, vh = linalg.cossin(X, separate=True, compute_vh=False, **kwargs)
+    assert_allclose(u[0], u[0], rtol=1e-15)
+    assert_allclose(u[1], u[1], rtol=1e-15)
+    assert_allclose(cs, cs0, rtol=1e-15)
+    assert vh[0].shape == (0, 0)
+    assert vh[1].shape == (0, 0)
+
+    U, CS, Vh = linalg.cossin(X, compute_vh=False, **kwargs)
+    assert_allclose(U, U0, rtol=1e-15)
+    assert_allclose(CS, CS0, rtol=1e-15)
+    assert Vh.shape == (0, 0)
+
+    # `compute_u=False, compute_vh=False`
+    u, cs, vh = linalg.cossin(X, separate=True, compute_u=False,
+                              compute_vh=False, **kwargs)
+    assert u[0].shape == (0, 0)
+    assert u[1].shape == (0, 0)
+    assert_allclose(cs, cs0, rtol=1e-15)
+    assert vh[0].shape == (0, 0)
+    assert vh[1].shape == (0, 0)
+
+    U, CS, Vh = linalg.cossin(X, compute_u=False, compute_vh=False, **kwargs)
+    assert U.shape == (0, 0)
+    assert_allclose(CS, CS0, rtol=1e-15)
+    assert Vh.shape == (0, 0)
+
+
+def test_indexing_bug_gh19365():
+    # Regression test for gh-19365, which reported a bug with `separate=False`
+    rng = np.random.default_rng(32954827234421)
+    m = rng.integers(50, high=100)
+    p = rng.integers(10, 40)  # always p < m
+    q = rng.integers(m - p + 1, m - 1)  # always m-p < q < m
+    X = unitary_group.rvs(m, random_state=rng)  # random unitary matrix
+    U, D, Vt = linalg.cossin(X, p=p, q=q, separate=False)
+    assert np.allclose(U @ D @ Vt, X)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_decomp_ldl.py

@@ -0,0 +1,136 @@
+import numpy as np
+from numpy.testing import assert_array_almost_equal, assert_allclose, assert_
+from numpy import (array, eye, zeros, empty_like, empty, tril_indices_from,
+                   tril, triu_indices_from, spacing, float32, float64,
+                   complex64, complex128)
+from numpy.exceptions import ComplexWarning
+from scipy.linalg import ldl
+import pytest
+
+
+def test_args():
+    A = eye(3)
+    # Nonsquare array
+    with pytest.raises(ValueError):
+        ldl(A[:, :2])
+    # Complex matrix with imaginary diagonal entries with "hermitian=True"
+    with pytest.warns(ComplexWarning):
+        ldl(A*1j)
+
+
+def test_empty_array():
+    a = empty((0, 0), dtype=complex)
+    l, d, p = ldl(empty((0, 0)))
+    assert_array_almost_equal(l, empty_like(a))
+    assert_array_almost_equal(d, empty_like(a))
+    assert_array_almost_equal(p, array([], dtype=int))
+
+
+def test_simple():
+    a = array([[-0.39-0.71j, 5.14-0.64j, -7.86-2.96j, 3.80+0.92j],
+               [5.14-0.64j, 8.86+1.81j, -3.52+0.58j, 5.32-1.59j],
+               [-7.86-2.96j, -3.52+0.58j, -2.83-0.03j, -1.54-2.86j],
+               [3.80+0.92j, 5.32-1.59j, -1.54-2.86j, -0.56+0.12j]])
+    b = array([[5., 10, 1, 18],
+               [10., 2, 11, 1],
+               [1., 11, 19, 9],
+               [18., 1, 9, 0]])
+    c = array([[52., 97, 112, 107, 50],
+               [97., 114, 89, 98, 13],
+               [112., 89, 64, 33, 6],
+               [107., 98, 33, 60, 73],
+               [50., 13, 6, 73, 77]])
+
+    d = array([[2., 2, -4, 0, 4],
+               [2., -2, -2, 10, -8],
+               [-4., -2, 6, -8, -4],
+               [0., 10, -8, 6, -6],
+               [4., -8, -4, -6, 10]])
+    e = array([[-1.36+0.00j, 0+0j, 0+0j, 0+0j],
+               [1.58-0.90j, -8.87+0j, 0+0j, 0+0j],
+               [2.21+0.21j, -1.84+0.03j, -4.63+0j, 0+0j],
+               [3.91-1.50j, -1.78-1.18j, 0.11-0.11j, -1.84+0.00j]])
+    for x in (b, c, d):
+        l, d, p = ldl(x)
+        assert_allclose(l.dot(d).dot(l.T), x, atol=spacing(1000.), rtol=0)
+
+        u, d, p = ldl(x, lower=False)
+        assert_allclose(u.dot(d).dot(u.T), x, atol=spacing(1000.), rtol=0)
+
+    l, d, p = ldl(a, hermitian=False)
+    assert_allclose(l.dot(d).dot(l.T), a, atol=spacing(1000.), rtol=0)
+
+    u, d, p = ldl(a, lower=False, hermitian=False)
+    assert_allclose(u.dot(d).dot(u.T), a, atol=spacing(1000.), rtol=0)
+
+    # Use upper part for the computation and use the lower part for comparison
+    l, d, p = ldl(e.conj().T, lower=0)
+    assert_allclose(tril(l.dot(d).dot(l.conj().T)-e), zeros((4, 4)),
+                    atol=spacing(1000.), rtol=0)
+
+
+def test_permutations():
+    rng = np.random.default_rng(1234)
+    for _ in range(10):
+        n = rng.integers(1, 100)
+        # Random real/complex array
+        x = rng.random((n, n)) + 0 if rng.integers(2) else rng.random((n, n))*1j
+        x = x + x.conj().T
+        x += eye(n)*rng.integers(5, 1e6)
+        l_ind = tril_indices_from(x, k=-1)
+        u_ind = triu_indices_from(x, k=1)
+
+        # Test whether permutations lead to a triangular array
+        u, d, p = ldl(x, lower=0)
+        # lower part should be zero
+        assert_(not any(u[p, :][l_ind]), f'Spin {_} failed')
+
+        l, d, p = ldl(x, lower=1)
+        # upper part should be zero
+        assert_(not any(l[p, :][u_ind]), f'Spin {_} failed')
+
+
+@pytest.mark.parametrize("dtype", [float32, float64])
+@pytest.mark.parametrize("n", [30, 150])
+def test_ldl_type_size_combinations_real(n, dtype):
+    rng = np.random.default_rng(1234)
+    msg = (f"Failed for size: {n}, dtype: {dtype}")
+
+    x = rng.random((n, n)).astype(dtype)
+    x = x + x.T
+    x += eye(n, dtype=dtype)*dtype(rng.integers(5, 1e6))
+
+    l, d1, p = ldl(x)
+    u, d2, p = ldl(x, lower=0)
+    rtol = 1e-4 if dtype is float32 else 1e-10
+    assert_allclose(l.dot(d1).dot(l.T), x, rtol=rtol, err_msg=msg)
+    assert_allclose(u.dot(d2).dot(u.T), x, rtol=rtol, err_msg=msg)
+
+
+@pytest.mark.parametrize("dtype", [complex64, complex128])
+@pytest.mark.parametrize("n", [30, 150])
+def test_ldl_type_size_combinations_complex(n, dtype):
+    rng = np.random.default_rng(1234)
+    msg1 = (f"Her failed for size: {n}, dtype: {dtype}")
+    msg2 = (f"Sym failed for size: {n}, dtype: {dtype}")
+
+    # Complex hermitian upper/lower
+    x = (rng.random((n, n))+1j*rng.random((n, n))).astype(dtype)
+    x = x+x.conj().T
+    x += eye(n, dtype=dtype)*dtype(rng.integers(5, 1e6))
+
+    l, d1, p = ldl(x)
+    u, d2, p = ldl(x, lower=0)
+    rtol = 2e-4 if dtype is complex64 else 1e-10
+    assert_allclose(l.dot(d1).dot(l.conj().T), x, rtol=rtol, err_msg=msg1)
+    assert_allclose(u.dot(d2).dot(u.conj().T), x, rtol=rtol, err_msg=msg1)
+
+    # Complex symmetric upper/lower
+    x = (rng.random((n, n))+1j*rng.random((n, n))).astype(dtype)
+    x = x+x.T
+    x += eye(n, dtype=dtype)*dtype(rng.integers(5, 1e6))
+
+    l, d1, p = ldl(x, hermitian=0)
+    u, d2, p = ldl(x, lower=0, hermitian=0)
+    assert_allclose(l.dot(d1).dot(l.T), x, rtol=rtol, err_msg=msg2)
+    assert_allclose(u.dot(d2).dot(u.T), x, rtol=rtol, err_msg=msg2)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_decomp_lu.py

@@ -0,0 +1,310 @@
+import pytest
+from pytest import raises as assert_raises
+
+import numpy as np
+from scipy.linalg import lu, lu_factor, lu_solve, get_lapack_funcs, solve
+from numpy.testing import assert_allclose, assert_array_equal, assert_equal
+from scipy.linalg.lapack import HAS_ILP64
+
+
+REAL_DTYPES = [np.float32, np.float64]
+COMPLEX_DTYPES = [np.complex64, np.complex128]
+DTYPES = REAL_DTYPES + COMPLEX_DTYPES
+
+
+class TestLU:
+    def setup_method(self):
+        self.rng = np.random.default_rng(1682281250228846)
+
+    def test_old_lu_smoke_tests(self):
+        "Tests from old fortran based lu test suite"
+        a = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6]])
+        p, l, u = lu(a)
+        result_lu = np.array([[2., 5., 6.], [0.5, -0.5, 0.], [0.5, 1., 0.]])
+        assert_allclose(p, np.rot90(np.eye(3)))
+        assert_allclose(l, np.tril(result_lu, k=-1)+np.eye(3))
+        assert_allclose(u, np.triu(result_lu))
+
+        a = np.array([[1, 2, 3], [1, 2, 3], [2, 5j, 6]])
+        p, l, u = lu(a)
+        result_lu = np.array([[2., 5.j, 6.], [0.5, 2-2.5j, 0.], [0.5, 1., 0.]])
+        assert_allclose(p, np.rot90(np.eye(3)))
+        assert_allclose(l, np.tril(result_lu, k=-1)+np.eye(3))
+        assert_allclose(u, np.triu(result_lu))
+
+        b = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+        p, l, u = lu(b)
+        assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]]))
+        assert_allclose(l, np.array([[1, 0, 0], [1/7, 1, 0], [4/7, 0.5, 1]]))
+        assert_allclose(u, np.array([[7, 8, 9], [0, 6/7, 12/7], [0, 0, 0]]),
+                        rtol=0., atol=1e-14)
+
+        cb = np.array([[1.j, 2.j, 3.j], [4j, 5j, 6j], [7j, 8j, 9j]])
+        p, l, u = lu(cb)
+        assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]]))
+        assert_allclose(l, np.array([[1, 0, 0], [1/7, 1, 0], [4/7, 0.5, 1]]))
+        assert_allclose(u, np.array([[7, 8, 9], [0, 6/7, 12/7], [0, 0, 0]])*1j,
+                        rtol=0., atol=1e-14)
+
+        # Rectangular matrices
+        hrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]])
+        p, l, u = lu(hrect)
+        assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]]))
+        assert_allclose(l, np.array([[1, 0, 0], [1/9, 1, 0], [5/9, 0.5, 1]]))
+        assert_allclose(u, np.array([[9, 10, 12, 12], [0, 8/9,  15/9,  24/9],
+                                     [0, 0, -0.5, 0]]), rtol=0., atol=1e-14)
+
+        chrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]])*1.j
+        p, l, u = lu(chrect)
+        assert_allclose(p, np.array([[0, 1, 0], [0, 0, 1], [1, 0, 0]]))
+        assert_allclose(l, np.array([[1, 0, 0], [1/9, 1, 0], [5/9, 0.5, 1]]))
+        assert_allclose(u, np.array([[9, 10, 12, 12], [0, 8/9,  15/9,  24/9],
+                                     [0, 0, -0.5, 0]])*1j, rtol=0., atol=1e-14)
+
+        vrect = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]])
+        p, l, u = lu(vrect)
+        assert_allclose(p, np.eye(4)[[1, 3, 2, 0], :])
+        assert_allclose(l, np.array([[1., 0, 0], [0.1, 1, 0], [0.7, -0.5, 1],
+                                     [0.4, 0.25, 0.5]]))
+        assert_allclose(u, np.array([[10, 12, 12],
+                                     [0, 0.8, 1.8],
+                                     [0, 0,  1.5]]))
+
+        cvrect = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]])*1j
+        p, l, u = lu(cvrect)
+        assert_allclose(p, np.eye(4)[[1, 3, 2, 0], :])
+        assert_allclose(l, np.array([[1., 0, 0],
+                                     [0.1, 1, 0],
+                                     [0.7, -0.5, 1],
+                                     [0.4, 0.25, 0.5]]))
+        assert_allclose(u, np.array([[10, 12, 12],
+                                     [0, 0.8, 1.8],
+                                     [0, 0,  1.5]])*1j)
+
+    @pytest.mark.parametrize('shape', [[2, 2], [2, 4], [4, 2], [20, 20],
+                                       [20, 4], [4, 20], [3, 2, 9, 9],
+                                       [2, 2, 17, 5], [2, 2, 11, 7]])
+    def test_simple_lu_shapes_real_complex(self, shape):
+        a = self.rng.uniform(-10., 10., size=shape)
+        p, l, u = lu(a)
+        assert_allclose(a, p @ l @ u)
+        pl, u = lu(a, permute_l=True)
+        assert_allclose(a, pl @ u)
+
+        b = self.rng.uniform(-10., 10., size=shape)*1j
+        b += self.rng.uniform(-10, 10, size=shape)
+        pl, u = lu(b, permute_l=True)
+        assert_allclose(b, pl @ u)
+
+    @pytest.mark.parametrize('shape', [[2, 2], [2, 4], [4, 2], [20, 20],
+                                       [20, 4], [4, 20]])
+    def test_simple_lu_shapes_real_complex_2d_indices(self, shape):
+        a = self.rng.uniform(-10., 10., size=shape)
+        p, l, u = lu(a, p_indices=True)
+        assert_allclose(a, l[p, :] @ u)
+
+    def test_1by1_input_output(self):
+        a = self.rng.random([4, 5, 1, 1], dtype=np.float32)
+        p, l, u = lu(a, p_indices=True)
+        assert_allclose(p, np.zeros(shape=(4, 5, 1), dtype=int))
+        assert_allclose(l, np.ones(shape=(4, 5, 1, 1), dtype=np.float32))
+        assert_allclose(u, a)
+
+        a = self.rng.random([4, 5, 1, 1], dtype=np.float32)
+        p, l, u = lu(a)
+        assert_allclose(p, np.ones(shape=(4, 5, 1, 1), dtype=np.float32))
+        assert_allclose(l, np.ones(shape=(4, 5, 1, 1), dtype=np.float32))
+        assert_allclose(u, a)
+
+        pl, u = lu(a, permute_l=True)
+        assert_allclose(pl, np.ones(shape=(4, 5, 1, 1), dtype=np.float32))
+        assert_allclose(u, a)
+
+        a = self.rng.random([4, 5, 1, 1], dtype=np.float32)*np.complex64(1.j)
+        p, l, u = lu(a)
+        assert_allclose(p, np.ones(shape=(4, 5, 1, 1), dtype=np.complex64))
+        assert_allclose(l, np.ones(shape=(4, 5, 1, 1), dtype=np.complex64))
+        assert_allclose(u, a)
+
+    @pytest.mark.filterwarnings("ignore::DeprecationWarning")
+    def test_empty_edge_cases(self):
+        a = np.empty([0, 0])
+        p, l, u = lu(a)
+        assert_allclose(p, np.empty(shape=(0, 0), dtype=np.float64))
+        assert_allclose(l, np.empty(shape=(0, 0), dtype=np.float64))
+        assert_allclose(u, np.empty(shape=(0, 0), dtype=np.float64))
+
+        a = np.empty([0, 3], dtype=np.float16)
+        p, l, u = lu(a)
+        assert_allclose(p, np.empty(shape=(0, 0), dtype=np.float32))
+        assert_allclose(l, np.empty(shape=(0, 0), dtype=np.float32))
+        assert_allclose(u, np.empty(shape=(0, 3), dtype=np.float32))
+
+        a = np.empty([3, 0], dtype=np.complex64)
+        p, l, u = lu(a)
+        assert_allclose(p, np.empty(shape=(0, 0), dtype=np.float32))
+        assert_allclose(l, np.empty(shape=(3, 0), dtype=np.complex64))
+        assert_allclose(u, np.empty(shape=(0, 0), dtype=np.complex64))
+        p, l, u = lu(a, p_indices=True)
+        assert_allclose(p, np.empty(shape=(0,), dtype=int))
+        assert_allclose(l, np.empty(shape=(3, 0), dtype=np.complex64))
+        assert_allclose(u, np.empty(shape=(0, 0), dtype=np.complex64))
+        pl, u = lu(a, permute_l=True)
+        assert_allclose(pl, np.empty(shape=(3, 0), dtype=np.complex64))
+        assert_allclose(u, np.empty(shape=(0, 0), dtype=np.complex64))
+
+        a = np.empty([3, 0, 0], dtype=np.complex64)
+        p, l, u = lu(a)
+        assert_allclose(p, np.empty(shape=(3, 0, 0), dtype=np.float32))
+        assert_allclose(l, np.empty(shape=(3, 0, 0), dtype=np.complex64))
+        assert_allclose(u, np.empty(shape=(3, 0, 0), dtype=np.complex64))
+
+        a = np.empty([0, 0, 3])
+        p, l, u = lu(a)
+        assert_allclose(p, np.empty(shape=(0, 0, 0)))
+        assert_allclose(l, np.empty(shape=(0, 0, 0)))
+        assert_allclose(u, np.empty(shape=(0, 0, 3)))
+
+        with assert_raises(ValueError, match='at least two-dimensional'):
+            lu(np.array([]))
+
+        a = np.array([[]])
+        p, l, u = lu(a)
+        assert_allclose(p, np.empty(shape=(0, 0)))
+        assert_allclose(l, np.empty(shape=(1, 0)))
+        assert_allclose(u, np.empty(shape=(0, 0)))
+
+        a = np.array([[[]]])
+        p, l, u = lu(a)
+        assert_allclose(p, np.empty(shape=(1, 0, 0)))
+        assert_allclose(l, np.empty(shape=(1, 1, 0)))
+        assert_allclose(u, np.empty(shape=(1, 0, 0)))
+
+
+class TestLUFactor:
+    def setup_method(self):
+        self.rng = np.random.default_rng(1682281250228846)
+
+        self.a = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6]])
+        self.ca = np.array([[1, 2, 3], [1, 2, 3], [2, 5j, 6]])
+        # Those matrices are more robust to detect problems in permutation
+        # matrices than the ones above
+        self.b = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+        self.cb = np.array([[1j, 2j, 3j], [4j, 5j, 6j], [7j, 8j, 9j]])
+
+        # Rectangular matrices
+        self.hrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]])
+        self.chrect = np.array([[1, 2, 3, 4], [5, 6, 7, 8],
+                                [9, 10, 12, 12]]) * 1.j
+
+        self.vrect = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]])
+        self.cvrect = 1.j * np.array([[1, 2, 3],
+                                      [4, 5, 6],
+                                      [7, 8, 9],
+                                      [10, 12, 12]])
+
+        # Medium sizes matrices
+        self.med = self.rng.random((30, 40))
+        self.cmed = self.rng.random((30, 40)) + 1.j*self.rng.random((30, 40))
+
+    def _test_common_lu_factor(self, data):
+        l_and_u1, piv1 = lu_factor(data)
+        (getrf,) = get_lapack_funcs(("getrf",), (data,))
+        l_and_u2, piv2, _ = getrf(data, overwrite_a=False)
+        assert_allclose(l_and_u1, l_and_u2)
+        assert_allclose(piv1, piv2)
+
+    # Simple tests.
+    # For lu_factor gives a LinAlgWarning because these matrices are singular
+    def test_hrectangular(self):
+        self._test_common_lu_factor(self.hrect)
+
+    def test_vrectangular(self):
+        self._test_common_lu_factor(self.vrect)
+
+    def test_hrectangular_complex(self):
+        self._test_common_lu_factor(self.chrect)
+
+    def test_vrectangular_complex(self):
+        self._test_common_lu_factor(self.cvrect)
+
+    # Bigger matrices
+    def test_medium1(self):
+        """Check lu decomposition on medium size, rectangular matrix."""
+        self._test_common_lu_factor(self.med)
+
+    def test_medium1_complex(self):
+        """Check lu decomposition on medium size, rectangular matrix."""
+        self._test_common_lu_factor(self.cmed)
+
+    def test_check_finite(self):
+        p, l, u = lu(self.a, check_finite=False)
+        assert_allclose(p @ l @ u, self.a)
+
+    def test_simple_known(self):
+        # Ticket #1458
+        for order in ['C', 'F']:
+            A = np.array([[2, 1], [0, 1.]], order=order)
+            LU, P = lu_factor(A)
+            assert_allclose(LU, np.array([[2, 1], [0, 1]]))
+            assert_array_equal(P, np.array([0, 1]))
+
+    @pytest.mark.parametrize("m", [0, 1, 2])
+    @pytest.mark.parametrize("n", [0, 1, 2])
+    @pytest.mark.parametrize('dtype', DTYPES)
+    def test_shape_dtype(self, m, n,  dtype):
+        k = min(m, n)
+
+        a = np.eye(m, n, dtype=dtype)
+        lu, p = lu_factor(a)
+        assert_equal(lu.shape, (m, n))
+        assert_equal(lu.dtype, dtype)
+        assert_equal(p.shape, (k,))
+        assert_equal(p.dtype, np.int64 if HAS_ILP64 else np.int32)
+
+    @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)])
+    def test_empty(self, m, n):
+        a = np.zeros((m, n))
+        lu, p = lu_factor(a)
+        assert_allclose(lu, np.empty((m, n)))
+        assert_allclose(p, np.arange(0))
+
+
+class TestLUSolve:
+    def setup_method(self):
+        self.rng = np.random.default_rng(1682281250228846)
+
+    def test_lu(self):
+        a0 = self.rng.random((10, 10))
+        b = self.rng.random((10,))
+
+        for order in ['C', 'F']:
+            a = np.array(a0, order=order)
+            x1 = solve(a, b)
+            lu_a = lu_factor(a)
+            x2 = lu_solve(lu_a, b)
+            assert_allclose(x1, x2)
+
+    def test_check_finite(self):
+        a = self.rng.random((10, 10))
+        b = self.rng.random((10,))
+        x1 = solve(a, b)
+        lu_a = lu_factor(a, check_finite=False)
+        x2 = lu_solve(lu_a, b, check_finite=False)
+        assert_allclose(x1, x2)
+
+    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
+    @pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64])
+    def test_empty(self, dt, dt_b):
+        lu_and_piv = (np.empty((0, 0), dtype=dt), np.array([]))
+        b = np.asarray([], dtype=dt_b)
+        x = lu_solve(lu_and_piv, b)
+        assert x.shape == (0,)
+
+        m = lu_solve((np.eye(2, dtype=dt), [0, 1]), np.ones(2, dtype=dt_b))
+        assert x.dtype == m.dtype
+
+        b = np.empty((0, 0), dtype=dt_b)
+        x = lu_solve(lu_and_piv, b)
+        assert x.shape == (0, 0)
+        assert x.dtype == m.dtype

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_decomp_polar.py

@@ -0,0 +1,110 @@
+import pytest
+import numpy as np
+from numpy.linalg import norm
+from numpy.testing import (assert_, assert_allclose, assert_equal)
+from scipy.linalg import polar, eigh
+
+
+diag2 = np.array([[2, 0], [0, 3]])
+a13 = np.array([[1, 2, 2]])
+
+precomputed_cases = [
+    [[[0]], 'right', [[1]], [[0]]],
+    [[[0]], 'left', [[1]], [[0]]],
+    [[[9]], 'right', [[1]], [[9]]],
+    [[[9]], 'left', [[1]], [[9]]],
+    [diag2, 'right', np.eye(2), diag2],
+    [diag2, 'left', np.eye(2), diag2],
+    [a13, 'right', a13/norm(a13[0]), a13.T.dot(a13)/norm(a13[0])],
+]
+
+verify_cases = [
+    [[1, 2], [3, 4]],
+    [[1, 2, 3]],
+    [[1], [2], [3]],
+    [[1, 2, 3], [3, 4, 0]],
+    [[1, 2], [3, 4], [5, 5]],
+    [[1, 2], [3, 4+5j]],
+    [[1, 2, 3j]],
+    [[1], [2], [3j]],
+    [[1, 2, 3+2j], [3, 4-1j, -4j]],
+    [[1, 2], [3-2j, 4+0.5j], [5, 5]],
+    [[10000, 10, 1], [-1, 2, 3j], [0, 1, 2]],
+    np.empty((0, 0)),
+    np.empty((0, 2)),
+    np.empty((2, 0)),
+]
+
+
+def check_precomputed_polar(a, side, expected_u, expected_p):
+    # Compare the result of the polar decomposition to a
+    # precomputed result.
+    u, p = polar(a, side=side)
+    assert_allclose(u, expected_u, atol=1e-15)
+    assert_allclose(p, expected_p, atol=1e-15)
+
+
+def verify_polar(a):
+    # Compute the polar decomposition, and then verify that
+    # the result has all the expected properties.
+    product_atol = np.sqrt(np.finfo(float).eps)
+
+    aa = np.asarray(a)
+    m, n = aa.shape
+
+    u, p = polar(a, side='right')
+    assert_equal(u.shape, (m, n))
+    assert_equal(p.shape, (n, n))
+    # a = up
+    assert_allclose(u.dot(p), a, atol=product_atol)
+    if m >= n:
+        assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15)
+    else:
+        assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15)
+    # p is Hermitian positive semidefinite.
+    assert_allclose(p.conj().T, p)
+    evals = eigh(p, eigvals_only=True)
+    nonzero_evals = evals[abs(evals) > 1e-14]
+    assert_((nonzero_evals >= 0).all())
+
+    u, p = polar(a, side='left')
+    assert_equal(u.shape, (m, n))
+    assert_equal(p.shape, (m, m))
+    # a = pu
+    assert_allclose(p.dot(u), a, atol=product_atol)
+    if m >= n:
+        assert_allclose(u.conj().T.dot(u), np.eye(n), atol=1e-15)
+    else:
+        assert_allclose(u.dot(u.conj().T), np.eye(m), atol=1e-15)
+    # p is Hermitian positive semidefinite.
+    assert_allclose(p.conj().T, p)
+    evals = eigh(p, eigvals_only=True)
+    nonzero_evals = evals[abs(evals) > 1e-14]
+    assert_((nonzero_evals >= 0).all())
+
+
+def test_precomputed_cases():
+    for a, side, expected_u, expected_p in precomputed_cases:
+        check_precomputed_polar(a, side, expected_u, expected_p)
+
+
+def test_verify_cases():
+    for a in verify_cases:
+        verify_polar(a)
+
+@pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
+@pytest.mark.parametrize('shape',  [(0, 0), (0, 2), (2, 0)])
+@pytest.mark.parametrize('side', ['left', 'right'])
+def test_empty(dt, shape, side):
+    a = np.empty(shape, dtype=dt)
+    m, n = shape
+    p_shape = (m, m) if side == 'left' else (n, n)
+
+    u, p = polar(a, side=side)
+    u_n, p_n = polar(np.eye(5, dtype=dt))
+
+    assert_equal(u.dtype, u_n.dtype)
+    assert_equal(p.dtype, p_n.dtype)
+    assert u.shape == shape
+    assert p.shape == p_shape
+    assert np.all(p == 0)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_decomp_update.py

@@ -0,0 +1,1704 @@
+import itertools
+import pytest
+
+import numpy as np
+from numpy.testing import assert_, assert_allclose, assert_equal
+from pytest import raises as assert_raises
+from scipy import linalg
+import scipy.linalg._decomp_update as _decomp_update
+from scipy.linalg._decomp_update import qr_delete, qr_update, qr_insert
+
+def assert_unitary(a, rtol=None, atol=None, assert_sqr=True):
+    if rtol is None:
+        rtol = 10.0 ** -(np.finfo(a.dtype).precision-2)
+    if atol is None:
+        atol = 10*np.finfo(a.dtype).eps
+
+    if assert_sqr:
+        assert_(a.shape[0] == a.shape[1], 'unitary matrices must be square')
+    aTa = np.dot(a.T.conj(), a)
+    assert_allclose(aTa, np.eye(a.shape[1]), rtol=rtol, atol=atol)
+
+def assert_upper_tri(a, rtol=None, atol=None):
+    if rtol is None:
+        rtol = 10.0 ** -(np.finfo(a.dtype).precision-2)
+    if atol is None:
+        atol = 2*np.finfo(a.dtype).eps
+    mask = np.tri(a.shape[0], a.shape[1], -1, np.bool_)
+    assert_allclose(a[mask], 0.0, rtol=rtol, atol=atol)
+
+def check_qr(q, r, a, rtol, atol, assert_sqr=True):
+    assert_unitary(q, rtol, atol, assert_sqr)
+    assert_upper_tri(r, rtol, atol)
+    assert_allclose(q.dot(r), a, rtol=rtol, atol=atol)
+
+def make_strided(arrs):
+    strides = [(3, 7), (2, 2), (3, 4), (4, 2), (5, 4), (2, 3), (2, 1), (4, 5)]
+    kmax = len(strides)
+    k = 0
+    ret = []
+    for a in arrs:
+        if a.ndim == 1:
+            s = strides[k % kmax]
+            k += 1
+            base = np.zeros(s[0]*a.shape[0]+s[1], a.dtype)
+            view = base[s[1]::s[0]]
+            view[...] = a
+        elif a.ndim == 2:
+            s = strides[k % kmax]
+            t = strides[(k+1) % kmax]
+            k += 2
+            base = np.zeros((s[0]*a.shape[0]+s[1], t[0]*a.shape[1]+t[1]),
+                            a.dtype)
+            view = base[s[1]::s[0], t[1]::t[0]]
+            view[...] = a
+        else:
+            raise ValueError('make_strided only works for ndim = 1 or'
+                             ' 2 arrays')
+        ret.append(view)
+    return ret
+
+def negate_strides(arrs):
+    ret = []
+    for a in arrs:
+        b = np.zeros_like(a)
+        if b.ndim == 2:
+            b = b[::-1, ::-1]
+        elif b.ndim == 1:
+            b = b[::-1]
+        else:
+            raise ValueError('negate_strides only works for ndim = 1 or'
+                             ' 2 arrays')
+        b[...] = a
+        ret.append(b)
+    return ret
+
+def nonitemsize_strides(arrs):
+    out = []
+    for a in arrs:
+        a_dtype = a.dtype
+        b = np.zeros(a.shape, [('a', a_dtype), ('junk', 'S1')])
+        c = b.getfield(a_dtype)
+        c[...] = a
+        out.append(c)
+    return out
+
+
+def make_nonnative(arrs):
+    return [a.astype(a.dtype.newbyteorder()) for a in arrs]
+
+
+class BaseQRdeltas:
+    def setup_method(self):
+        self.rtol = 10.0 ** -(np.finfo(self.dtype).precision-2)
+        self.atol = 10 * np.finfo(self.dtype).eps
+
+    def generate(self, type, mode='full'):
+        rng = np.random.default_rng(29382)
+        shape = {'sqr': (8, 8), 'tall': (12, 7), 'fat': (7, 12),
+                 'Mx1': (8, 1), '1xN': (1, 8), '1x1': (1, 1)}[type]
+        a = rng.random(shape)
+        if np.iscomplexobj(self.dtype.type(1)):
+            b = rng.random(shape)
+            a = a + 1j * b
+        a = a.astype(self.dtype)
+        q, r = linalg.qr(a, mode=mode)
+        return a, q, r
+
+class BaseQRdelete(BaseQRdeltas):
+    def test_sqr_1_row(self):
+        a, q, r = self.generate('sqr')
+        for row in range(r.shape[0]):
+            q1, r1 = qr_delete(q, r, row, overwrite_qr=False)
+            a1 = np.delete(a, row, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_sqr_p_row(self):
+        a, q, r = self.generate('sqr')
+        for ndel in range(2, 6):
+            for row in range(a.shape[0]-ndel):
+                q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False)
+                a1 = np.delete(a, slice(row, row+ndel), 0)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_sqr_1_col(self):
+        a, q, r = self.generate('sqr')
+        for col in range(r.shape[1]):
+            q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False)
+            a1 = np.delete(a, col, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_sqr_p_col(self):
+        a, q, r = self.generate('sqr')
+        for ndel in range(2, 6):
+            for col in range(r.shape[1]-ndel):
+                q1, r1 = qr_delete(q, r, col, ndel, which='col',
+                                   overwrite_qr=False)
+                a1 = np.delete(a, slice(col, col+ndel), 1)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_1_row(self):
+        a, q, r = self.generate('tall')
+        for row in range(r.shape[0]):
+            q1, r1 = qr_delete(q, r, row, overwrite_qr=False)
+            a1 = np.delete(a, row, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_p_row(self):
+        a, q, r = self.generate('tall')
+        for ndel in range(2, 6):
+            for row in range(a.shape[0]-ndel):
+                q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False)
+                a1 = np.delete(a, slice(row, row+ndel), 0)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_1_col(self):
+        a, q, r = self.generate('tall')
+        for col in range(r.shape[1]):
+            q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False)
+            a1 = np.delete(a, col, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_p_col(self):
+        a, q, r = self.generate('tall')
+        for ndel in range(2, 6):
+            for col in range(r.shape[1]-ndel):
+                q1, r1 = qr_delete(q, r, col, ndel, which='col',
+                                   overwrite_qr=False)
+                a1 = np.delete(a, slice(col, col+ndel), 1)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_1_row(self):
+        a, q, r = self.generate('fat')
+        for row in range(r.shape[0]):
+            q1, r1 = qr_delete(q, r, row, overwrite_qr=False)
+            a1 = np.delete(a, row, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_p_row(self):
+        a, q, r = self.generate('fat')
+        for ndel in range(2, 6):
+            for row in range(a.shape[0]-ndel):
+                q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False)
+                a1 = np.delete(a, slice(row, row+ndel), 0)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_1_col(self):
+        a, q, r = self.generate('fat')
+        for col in range(r.shape[1]):
+            q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False)
+            a1 = np.delete(a, col, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_p_col(self):
+        a, q, r = self.generate('fat')
+        for ndel in range(2, 6):
+            for col in range(r.shape[1]-ndel):
+                q1, r1 = qr_delete(q, r, col, ndel, which='col',
+                                   overwrite_qr=False)
+                a1 = np.delete(a, slice(col, col+ndel), 1)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_economic_1_row(self):
+        # this test always starts and ends with an economic decomp.
+        a, q, r = self.generate('tall', 'economic')
+        for row in range(r.shape[0]):
+            q1, r1 = qr_delete(q, r, row, overwrite_qr=False)
+            a1 = np.delete(a, row, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    # for economic row deletes
+    # eco - prow = eco
+    # eco - prow = sqr
+    # eco - prow = fat
+    def base_economic_p_row_xxx(self, ndel):
+        a, q, r = self.generate('tall', 'economic')
+        for row in range(a.shape[0]-ndel):
+            q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False)
+            a1 = np.delete(a, slice(row, row+ndel), 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_economic_p_row_economic(self):
+        # (12, 7) - (3, 7) = (9,7) --> stays economic
+        self.base_economic_p_row_xxx(3)
+
+    def test_economic_p_row_sqr(self):
+        # (12, 7) - (5, 7) = (7, 7) --> becomes square
+        self.base_economic_p_row_xxx(5)
+
+    def test_economic_p_row_fat(self):
+        # (12, 7) - (7,7) = (5, 7) --> becomes fat
+        self.base_economic_p_row_xxx(7)
+
+    def test_economic_1_col(self):
+        a, q, r = self.generate('tall', 'economic')
+        for col in range(r.shape[1]):
+            q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False)
+            a1 = np.delete(a, col, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_economic_p_col(self):
+        a, q, r = self.generate('tall', 'economic')
+        for ndel in range(2, 6):
+            for col in range(r.shape[1]-ndel):
+                q1, r1 = qr_delete(q, r, col, ndel, which='col',
+                                   overwrite_qr=False)
+                a1 = np.delete(a, slice(col, col+ndel), 1)
+                check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_Mx1_1_row(self):
+        a, q, r = self.generate('Mx1')
+        for row in range(r.shape[0]):
+            q1, r1 = qr_delete(q, r, row, overwrite_qr=False)
+            a1 = np.delete(a, row, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_p_row(self):
+        a, q, r = self.generate('Mx1')
+        for ndel in range(2, 6):
+            for row in range(a.shape[0]-ndel):
+                q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False)
+                a1 = np.delete(a, slice(row, row+ndel), 0)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1xN_1_col(self):
+        a, q, r = self.generate('1xN')
+        for col in range(r.shape[1]):
+            q1, r1 = qr_delete(q, r, col, which='col', overwrite_qr=False)
+            a1 = np.delete(a, col, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1xN_p_col(self):
+        a, q, r = self.generate('1xN')
+        for ndel in range(2, 6):
+            for col in range(r.shape[1]-ndel):
+                q1, r1 = qr_delete(q, r, col, ndel, which='col',
+                                   overwrite_qr=False)
+                a1 = np.delete(a, slice(col, col+ndel), 1)
+                check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_economic_1_row(self):
+        a, q, r = self.generate('Mx1', 'economic')
+        for row in range(r.shape[0]):
+            q1, r1 = qr_delete(q, r, row, overwrite_qr=False)
+            a1 = np.delete(a, row, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_Mx1_economic_p_row(self):
+        a, q, r = self.generate('Mx1', 'economic')
+        for ndel in range(2, 6):
+            for row in range(a.shape[0]-ndel):
+                q1, r1 = qr_delete(q, r, row, ndel, overwrite_qr=False)
+                a1 = np.delete(a, slice(row, row+ndel), 0)
+                check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_delete_last_1_row(self):
+        # full and eco are the same for 1xN
+        a, q, r = self.generate('1xN')
+        q1, r1 = qr_delete(q, r, 0, 1, 'row')
+        assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype))
+        assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype))
+
+    def test_delete_last_p_row(self):
+        a, q, r = self.generate('tall', 'full')
+        q1, r1 = qr_delete(q, r, 0, a.shape[0], 'row')
+        assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype))
+        assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype))
+
+        a, q, r = self.generate('tall', 'economic')
+        q1, r1 = qr_delete(q, r, 0, a.shape[0], 'row')
+        assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype))
+        assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype))
+
+    def test_delete_last_1_col(self):
+        a, q, r = self.generate('Mx1', 'economic')
+        q1, r1 = qr_delete(q, r, 0, 1, 'col')
+        assert_equal(q1, np.ndarray(shape=(q.shape[0], 0), dtype=q.dtype))
+        assert_equal(r1, np.ndarray(shape=(0, 0), dtype=r.dtype))
+
+        a, q, r = self.generate('Mx1', 'full')
+        q1, r1 = qr_delete(q, r, 0, 1, 'col')
+        assert_unitary(q1)
+        assert_(q1.dtype == q.dtype)
+        assert_(q1.shape == q.shape)
+        assert_equal(r1, np.ndarray(shape=(r.shape[0], 0), dtype=r.dtype))
+
+    def test_delete_last_p_col(self):
+        a, q, r = self.generate('tall', 'full')
+        q1, r1 = qr_delete(q, r, 0, a.shape[1], 'col')
+        assert_unitary(q1)
+        assert_(q1.dtype == q.dtype)
+        assert_(q1.shape == q.shape)
+        assert_equal(r1, np.ndarray(shape=(r.shape[0], 0), dtype=r.dtype))
+
+        a, q, r = self.generate('tall', 'economic')
+        q1, r1 = qr_delete(q, r, 0, a.shape[1], 'col')
+        assert_equal(q1, np.ndarray(shape=(q.shape[0], 0), dtype=q.dtype))
+        assert_equal(r1, np.ndarray(shape=(0, 0), dtype=r.dtype))
+
+    def test_delete_1x1_row_col(self):
+        a, q, r = self.generate('1x1')
+        q1, r1 = qr_delete(q, r, 0, 1, 'row')
+        assert_equal(q1, np.ndarray(shape=(0, 0), dtype=q.dtype))
+        assert_equal(r1, np.ndarray(shape=(0, r.shape[1]), dtype=r.dtype))
+
+        a, q, r = self.generate('1x1')
+        q1, r1 = qr_delete(q, r, 0, 1, 'col')
+        assert_unitary(q1)
+        assert_(q1.dtype == q.dtype)
+        assert_(q1.shape == q.shape)
+        assert_equal(r1, np.ndarray(shape=(r.shape[0], 0), dtype=r.dtype))
+
+    # all full qr, row deletes and single column deletes should be able to
+    # handle any non negative strides. (only row and column vector
+    # operations are used.) p column delete require fortran ordered
+    # Q and R and will make a copy as necessary.  Economic qr row deletes
+    # require a contiguous q.
+
+    def base_non_simple_strides(self, adjust_strides, ks, p, which,
+                                overwriteable):
+        if which == 'row':
+            qind = (slice(p,None), slice(p,None))
+            rind = (slice(p,None), slice(None))
+        else:
+            qind = (slice(None), slice(None))
+            rind = (slice(None), slice(None,-p))
+
+        for type, k in itertools.product(['sqr', 'tall', 'fat'], ks):
+            a, q0, r0, = self.generate(type)
+            qs, rs = adjust_strides((q0, r0))
+            if p == 1:
+                a1 = np.delete(a, k, 0 if which == 'row' else 1)
+            else:
+                s = slice(k,k+p)
+                if k < 0:
+                    s = slice(k, k + p +
+                              (a.shape[0] if which == 'row' else a.shape[1]))
+                a1 = np.delete(a, s, 0 if which == 'row' else 1)
+
+            # for each variable, q, r we try with it strided and
+            # overwrite=False. Then we try with overwrite=True, and make
+            # sure that q and r are still overwritten.
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            q1, r1 = qr_delete(qs, r, k, p, which, False)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+            q1o, r1o = qr_delete(qs, r, k, p, which, True)
+            check_qr(q1o, r1o, a1, self.rtol, self.atol)
+            if overwriteable:
+                assert_allclose(q1o, qs[qind], rtol=self.rtol, atol=self.atol)
+                assert_allclose(r1o, r[rind], rtol=self.rtol, atol=self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            q2, r2 = qr_delete(q, rs, k, p, which, False)
+            check_qr(q2, r2, a1, self.rtol, self.atol)
+            q2o, r2o = qr_delete(q, rs, k, p, which, True)
+            check_qr(q2o, r2o, a1, self.rtol, self.atol)
+            if overwriteable:
+                assert_allclose(q2o, q[qind], rtol=self.rtol, atol=self.atol)
+                assert_allclose(r2o, rs[rind], rtol=self.rtol, atol=self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            # since some of these were consumed above
+            qs, rs = adjust_strides((q, r))
+            q3, r3 = qr_delete(qs, rs, k, p, which, False)
+            check_qr(q3, r3, a1, self.rtol, self.atol)
+            q3o, r3o = qr_delete(qs, rs, k, p, which, True)
+            check_qr(q3o, r3o, a1, self.rtol, self.atol)
+            if overwriteable:
+                assert_allclose(q2o, qs[qind], rtol=self.rtol, atol=self.atol)
+                assert_allclose(r3o, rs[rind], rtol=self.rtol, atol=self.atol)
+
+    def test_non_unit_strides_1_row(self):
+        self.base_non_simple_strides(make_strided, [0], 1, 'row', True)
+
+    def test_non_unit_strides_p_row(self):
+        self.base_non_simple_strides(make_strided, [0], 3, 'row', True)
+
+    def test_non_unit_strides_1_col(self):
+        self.base_non_simple_strides(make_strided, [0], 1, 'col', True)
+
+    def test_non_unit_strides_p_col(self):
+        self.base_non_simple_strides(make_strided, [0], 3, 'col', False)
+
+    def test_neg_strides_1_row(self):
+        self.base_non_simple_strides(negate_strides, [0], 1, 'row', False)
+
+    def test_neg_strides_p_row(self):
+        self.base_non_simple_strides(negate_strides, [0], 3, 'row', False)
+
+    def test_neg_strides_1_col(self):
+        self.base_non_simple_strides(negate_strides, [0], 1, 'col', False)
+
+    def test_neg_strides_p_col(self):
+        self.base_non_simple_strides(negate_strides, [0], 3, 'col', False)
+
+    def test_non_itemize_strides_1_row(self):
+        self.base_non_simple_strides(nonitemsize_strides, [0], 1, 'row', False)
+
+    def test_non_itemize_strides_p_row(self):
+        self.base_non_simple_strides(nonitemsize_strides, [0], 3, 'row', False)
+
+    def test_non_itemize_strides_1_col(self):
+        self.base_non_simple_strides(nonitemsize_strides, [0], 1, 'col', False)
+
+    def test_non_itemize_strides_p_col(self):
+        self.base_non_simple_strides(nonitemsize_strides, [0], 3, 'col', False)
+
+    def test_non_native_byte_order_1_row(self):
+        self.base_non_simple_strides(make_nonnative, [0], 1, 'row', False)
+
+    def test_non_native_byte_order_p_row(self):
+        self.base_non_simple_strides(make_nonnative, [0], 3, 'row', False)
+
+    def test_non_native_byte_order_1_col(self):
+        self.base_non_simple_strides(make_nonnative, [0], 1, 'col', False)
+
+    def test_non_native_byte_order_p_col(self):
+        self.base_non_simple_strides(make_nonnative, [0], 3, 'col', False)
+
+    def test_neg_k(self):
+        a, q, r = self.generate('sqr')
+        for k, p, w in itertools.product([-3, -7], [1, 3], ['row', 'col']):
+            q1, r1 = qr_delete(q, r, k, p, w, overwrite_qr=False)
+            if w == 'row':
+                a1 = np.delete(a, slice(k+a.shape[0], k+p+a.shape[0]), 0)
+            else:
+                a1 = np.delete(a, slice(k+a.shape[0], k+p+a.shape[1]), 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def base_overwrite_qr(self, which, p, test_C, test_F, mode='full'):
+        assert_sqr = True if mode == 'full' else False
+        if which == 'row':
+            qind = (slice(p,None), slice(p,None))
+            rind = (slice(p,None), slice(None))
+        else:
+            qind = (slice(None), slice(None))
+            rind = (slice(None), slice(None,-p))
+        a, q0, r0 = self.generate('sqr', mode)
+        if p == 1:
+            a1 = np.delete(a, 3, 0 if which == 'row' else 1)
+        else:
+            a1 = np.delete(a, slice(3, 3+p), 0 if which == 'row' else 1)
+
+        # don't overwrite
+        q = q0.copy('F')
+        r = r0.copy('F')
+        q1, r1 = qr_delete(q, r, 3, p, which, False)
+        check_qr(q1, r1, a1, self.rtol, self.atol, assert_sqr)
+        check_qr(q, r, a, self.rtol, self.atol, assert_sqr)
+
+        if test_F:
+            q = q0.copy('F')
+            r = r0.copy('F')
+            q2, r2 = qr_delete(q, r, 3, p, which, True)
+            check_qr(q2, r2, a1, self.rtol, self.atol, assert_sqr)
+            # verify the overwriting
+            assert_allclose(q2, q[qind], rtol=self.rtol, atol=self.atol)
+            assert_allclose(r2, r[rind], rtol=self.rtol, atol=self.atol)
+
+        if test_C:
+            q = q0.copy('C')
+            r = r0.copy('C')
+            q3, r3 = qr_delete(q, r, 3, p, which, True)
+            check_qr(q3, r3, a1, self.rtol, self.atol, assert_sqr)
+            assert_allclose(q3, q[qind], rtol=self.rtol, atol=self.atol)
+            assert_allclose(r3, r[rind], rtol=self.rtol, atol=self.atol)
+
+    def test_overwrite_qr_1_row(self):
+        # any positively strided q and r.
+        self.base_overwrite_qr('row', 1, True, True)
+
+    def test_overwrite_economic_qr_1_row(self):
+        # Any contiguous q and positively strided r.
+        self.base_overwrite_qr('row', 1, True, True, 'economic')
+
+    def test_overwrite_qr_1_col(self):
+        # any positively strided q and r.
+        # full and eco share code paths
+        self.base_overwrite_qr('col', 1, True, True)
+
+    def test_overwrite_qr_p_row(self):
+        # any positively strided q and r.
+        self.base_overwrite_qr('row', 3, True, True)
+
+    def test_overwrite_economic_qr_p_row(self):
+        # any contiguous q and positively strided r
+        self.base_overwrite_qr('row', 3, True, True, 'economic')
+
+    def test_overwrite_qr_p_col(self):
+        # only F ordered q and r can be overwritten for cols
+        # full and eco share code paths
+        self.base_overwrite_qr('col', 3, False, True)
+
+    def test_bad_which(self):
+        a, q, r = self.generate('sqr')
+        assert_raises(ValueError, qr_delete, q, r, 0, which='foo')
+
+    def test_bad_k(self):
+        a, q, r = self.generate('tall')
+        assert_raises(ValueError, qr_delete, q, r, q.shape[0], 1)
+        assert_raises(ValueError, qr_delete, q, r, -q.shape[0]-1, 1)
+        assert_raises(ValueError, qr_delete, q, r, r.shape[0], 1, 'col')
+        assert_raises(ValueError, qr_delete, q, r, -r.shape[0]-1, 1, 'col')
+
+    def test_bad_p(self):
+        a, q, r = self.generate('tall')
+        # p must be positive
+        assert_raises(ValueError, qr_delete, q, r, 0, -1)
+        assert_raises(ValueError, qr_delete, q, r, 0, -1, 'col')
+
+        # and nonzero
+        assert_raises(ValueError, qr_delete, q, r, 0, 0)
+        assert_raises(ValueError, qr_delete, q, r, 0, 0, 'col')
+
+        # must have at least k+p rows or cols, depending.
+        assert_raises(ValueError, qr_delete, q, r, 3, q.shape[0]-2)
+        assert_raises(ValueError, qr_delete, q, r, 3, r.shape[1]-2, 'col')
+
+    def test_empty_q(self):
+        a, q, r = self.generate('tall')
+        # same code path for 'row' and 'col'
+        assert_raises(ValueError, qr_delete, np.array([]), r, 0, 1)
+
+    def test_empty_r(self):
+        a, q, r = self.generate('tall')
+        # same code path for 'row' and 'col'
+        assert_raises(ValueError, qr_delete, q, np.array([]), 0, 1)
+
+    def test_mismatched_q_and_r(self):
+        a, q, r = self.generate('tall')
+        r = r[1:]
+        assert_raises(ValueError, qr_delete, q, r, 0, 1)
+
+    @pytest.mark.filterwarnings("ignore::DeprecationWarning")
+    def test_unsupported_dtypes(self):
+        dts = ['int8', 'int16', 'int32', 'int64',
+               'uint8', 'uint16', 'uint32', 'uint64',
+               'float16', 'longdouble', 'clongdouble',
+               'bool']
+        a, q0, r0 = self.generate('tall')
+        for dtype in dts:
+            q = q0.real.astype(dtype)
+            with np.errstate(invalid="ignore"):
+                r = r0.real.astype(dtype)
+            assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'row')
+            assert_raises(ValueError, qr_delete, q, r0, 0, 2, 'row')
+            assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'col')
+            assert_raises(ValueError, qr_delete, q, r0, 0, 2, 'col')
+
+            assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'row')
+            assert_raises(ValueError, qr_delete, q0, r, 0, 2, 'row')
+            assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'col')
+            assert_raises(ValueError, qr_delete, q0, r, 0, 2, 'col')
+
+    def test_check_finite(self):
+        a0, q0, r0 = self.generate('tall')
+
+        q = q0.copy('F')
+        q[1,1] = np.nan
+        assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'row')
+        assert_raises(ValueError, qr_delete, q, r0, 0, 3, 'row')
+        assert_raises(ValueError, qr_delete, q, r0, 0, 1, 'col')
+        assert_raises(ValueError, qr_delete, q, r0, 0, 3, 'col')
+
+        r = r0.copy('F')
+        r[1,1] = np.nan
+        assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'row')
+        assert_raises(ValueError, qr_delete, q0, r, 0, 3, 'row')
+        assert_raises(ValueError, qr_delete, q0, r, 0, 1, 'col')
+        assert_raises(ValueError, qr_delete, q0, r, 0, 3, 'col')
+
+    def test_qr_scalar(self):
+        a, q, r = self.generate('1x1')
+        assert_raises(ValueError, qr_delete, q[0, 0], r, 0, 1, 'row')
+        assert_raises(ValueError, qr_delete, q, r[0, 0], 0, 1, 'row')
+        assert_raises(ValueError, qr_delete, q[0, 0], r, 0, 1, 'col')
+        assert_raises(ValueError, qr_delete, q, r[0, 0], 0, 1, 'col')
+
+class TestQRdelete_f(BaseQRdelete):
+    dtype = np.dtype('f')
+
+class TestQRdelete_F(BaseQRdelete):
+    dtype = np.dtype('F')
+
+class TestQRdelete_d(BaseQRdelete):
+    dtype = np.dtype('d')
+
+class TestQRdelete_D(BaseQRdelete):
+    dtype = np.dtype('D')
+
+class BaseQRinsert(BaseQRdeltas):
+    def generate(self, type, mode='full', which='row', p=1):
+        a, q, r = super().generate(type, mode)
+
+        assert_(p > 0)
+        rng = np.random.default_rng(1234)
+
+        if which == 'row':
+            if p == 1:
+                u = rng.random(a.shape[1])
+            else:
+                u = rng.random((p, a.shape[1]))
+        elif which == 'col':
+            if p == 1:
+                u = rng.random(a.shape[0])
+            else:
+                u = rng.random((a.shape[0], p))
+        else:
+            raise ValueError('which should be either "row" or "col"')
+
+        if np.iscomplexobj(self.dtype.type(1)):
+            b = rng.random(u.shape)
+            u = u + 1j * b
+
+        u = u.astype(self.dtype)
+        return a, q, r, u
+
+    def test_sqr_1_row(self):
+        a, q, r, u = self.generate('sqr', which='row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_sqr_p_row(self):
+        # sqr + rows --> fat always
+        a, q, r, u = self.generate('sqr', which='row', p=3)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, np.full(3, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_sqr_1_col(self):
+        a, q, r, u = self.generate('sqr', which='col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_sqr_p_col(self):
+        # sqr + cols --> fat always
+        a, q, r, u = self.generate('sqr', which='col', p=3)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(3, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_1_row(self):
+        a, q, r, u = self.generate('tall', which='row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_p_row(self):
+        # tall + rows --> tall always
+        a, q, r, u = self.generate('tall', which='row', p=3)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, np.full(3, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_1_col(self):
+        a, q, r, u = self.generate('tall', which='col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    # for column adds to tall matrices there are three cases to test
+    # tall + pcol --> tall
+    # tall + pcol --> sqr
+    # tall + pcol --> fat
+    def base_tall_p_col_xxx(self, p):
+        a, q, r, u = self.generate('tall', which='col', p=p)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(p, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_p_col_tall(self):
+        # 12x7 + 12x3 = 12x10 --> stays tall
+        self.base_tall_p_col_xxx(3)
+
+    def test_tall_p_col_sqr(self):
+        # 12x7 + 12x5 = 12x12 --> becomes sqr
+        self.base_tall_p_col_xxx(5)
+
+    def test_tall_p_col_fat(self):
+        # 12x7 + 12x7 = 12x14 --> becomes fat
+        self.base_tall_p_col_xxx(7)
+
+    def test_fat_1_row(self):
+        a, q, r, u = self.generate('fat', which='row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    # for row adds to fat matrices there are three cases to test
+    # fat + prow --> fat
+    # fat + prow --> sqr
+    # fat + prow --> tall
+    def base_fat_p_row_xxx(self, p):
+        a, q, r, u = self.generate('fat', which='row', p=p)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, np.full(p, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_p_row_fat(self):
+        # 7x12 + 3x12 = 10x12 --> stays fat
+        self.base_fat_p_row_xxx(3)
+
+    def test_fat_p_row_sqr(self):
+        # 7x12 + 5x12 = 12x12 --> becomes sqr
+        self.base_fat_p_row_xxx(5)
+
+    def test_fat_p_row_tall(self):
+        # 7x12 + 7x12 = 14x12 --> becomes tall
+        self.base_fat_p_row_xxx(7)
+
+    def test_fat_1_col(self):
+        a, q, r, u = self.generate('fat', which='col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_p_col(self):
+        # fat + cols --> fat always
+        a, q, r, u = self.generate('fat', which='col', p=3)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(3, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_economic_1_row(self):
+        a, q, r, u = self.generate('tall', 'economic', 'row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row, overwrite_qru=False)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_economic_p_row(self):
+        # tall + rows --> tall always
+        a, q, r, u = self.generate('tall', 'economic', 'row', 3)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row, overwrite_qru=False)
+            a1 = np.insert(a, np.full(3, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_economic_1_col(self):
+        a, q, r, u = self.generate('tall', 'economic', which='col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u.copy(), col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_economic_1_col_bad_update(self):
+        # When the column to be added lies in the span of Q, the update is
+        # not meaningful.  This is detected, and a LinAlgError is issued.
+        q = np.eye(5, 3, dtype=self.dtype)
+        r = np.eye(3, dtype=self.dtype)
+        u = np.array([1, 0, 0, 0, 0], self.dtype)
+        assert_raises(linalg.LinAlgError, qr_insert, q, r, u, 0, 'col')
+
+    # for column adds to economic matrices there are three cases to test
+    # eco + pcol --> eco
+    # eco + pcol --> sqr
+    # eco + pcol --> fat
+    def base_economic_p_col_xxx(self, p):
+        a, q, r, u = self.generate('tall', 'economic', which='col', p=p)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(p, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_economic_p_col_eco(self):
+        # 12x7 + 12x3 = 12x10 --> stays eco
+        self.base_economic_p_col_xxx(3)
+
+    def test_economic_p_col_sqr(self):
+        # 12x7 + 12x5 = 12x12 --> becomes sqr
+        self.base_economic_p_col_xxx(5)
+
+    def test_economic_p_col_fat(self):
+        # 12x7 + 12x7 = 12x14 --> becomes fat
+        self.base_economic_p_col_xxx(7)
+
+    def test_Mx1_1_row(self):
+        a, q, r, u = self.generate('Mx1', which='row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_p_row(self):
+        a, q, r, u = self.generate('Mx1', which='row', p=3)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, np.full(3, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_1_col(self):
+        a, q, r, u = self.generate('Mx1', which='col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_p_col(self):
+        a, q, r, u = self.generate('Mx1', which='col', p=3)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(3, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_economic_1_row(self):
+        a, q, r, u = self.generate('Mx1', 'economic', 'row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_Mx1_economic_p_row(self):
+        a, q, r, u = self.generate('Mx1', 'economic', 'row', 3)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, np.full(3, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_Mx1_economic_1_col(self):
+        a, q, r, u = self.generate('Mx1', 'economic', 'col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_Mx1_economic_p_col(self):
+        a, q, r, u = self.generate('Mx1', 'economic', 'col', 3)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(3, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_1xN_1_row(self):
+        a, q, r, u = self.generate('1xN', which='row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1xN_p_row(self):
+        a, q, r, u = self.generate('1xN', which='row', p=3)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, np.full(3, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1xN_1_col(self):
+        a, q, r, u = self.generate('1xN', which='col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1xN_p_col(self):
+        a, q, r, u = self.generate('1xN', which='col', p=3)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(3, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_1_row(self):
+        a, q, r, u = self.generate('1x1', which='row')
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, row, u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_p_row(self):
+        a, q, r, u = self.generate('1x1', which='row', p=3)
+        for row in range(r.shape[0] + 1):
+            q1, r1 = qr_insert(q, r, u, row)
+            a1 = np.insert(a, np.full(3, row, np.intp), u, 0)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_1_col(self):
+        a, q, r, u = self.generate('1x1', which='col')
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, col, u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_p_col(self):
+        a, q, r, u = self.generate('1x1', which='col', p=3)
+        for col in range(r.shape[1] + 1):
+            q1, r1 = qr_insert(q, r, u, col, 'col', overwrite_qru=False)
+            a1 = np.insert(a, np.full(3, col, np.intp), u, 1)
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_1_scalar(self):
+        a, q, r, u = self.generate('1x1', which='row')
+        assert_raises(ValueError, qr_insert, q[0, 0], r, u, 0, 'row')
+        assert_raises(ValueError, qr_insert, q, r[0, 0], u, 0, 'row')
+        assert_raises(ValueError, qr_insert, q, r, u[0], 0, 'row')
+
+        assert_raises(ValueError, qr_insert, q[0, 0], r, u, 0, 'col')
+        assert_raises(ValueError, qr_insert, q, r[0, 0], u, 0, 'col')
+        assert_raises(ValueError, qr_insert, q, r, u[0], 0, 'col')
+
+    def base_non_simple_strides(self, adjust_strides, k, p, which):
+        for type in ['sqr', 'tall', 'fat']:
+            a, q0, r0, u0 = self.generate(type, which=which, p=p)
+            qs, rs, us = adjust_strides((q0, r0, u0))
+            if p == 1:
+                ai = np.insert(a, k, u0, 0 if which == 'row' else 1)
+            else:
+                ai = np.insert(a, np.full(p, k, np.intp),
+                        u0 if which == 'row' else u0,
+                        0 if which == 'row' else 1)
+
+            # for each variable, q, r, u we try with it strided and
+            # overwrite=False. Then we try with overwrite=True. Nothing
+            # is checked to see if it can be overwritten, since only
+            # F ordered Q can be overwritten when adding columns.
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            q1, r1 = qr_insert(qs, r, u, k, which, overwrite_qru=False)
+            check_qr(q1, r1, ai, self.rtol, self.atol)
+            q1o, r1o = qr_insert(qs, r, u, k, which, overwrite_qru=True)
+            check_qr(q1o, r1o, ai, self.rtol, self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            q2, r2 = qr_insert(q, rs, u, k, which, overwrite_qru=False)
+            check_qr(q2, r2, ai, self.rtol, self.atol)
+            q2o, r2o = qr_insert(q, rs, u, k, which, overwrite_qru=True)
+            check_qr(q2o, r2o, ai, self.rtol, self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            q3, r3 = qr_insert(q, r, us, k, which, overwrite_qru=False)
+            check_qr(q3, r3, ai, self.rtol, self.atol)
+            q3o, r3o = qr_insert(q, r, us, k, which, overwrite_qru=True)
+            check_qr(q3o, r3o, ai, self.rtol, self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            # since some of these were consumed above
+            qs, rs, us = adjust_strides((q, r, u))
+            q5, r5 = qr_insert(qs, rs, us, k, which, overwrite_qru=False)
+            check_qr(q5, r5, ai, self.rtol, self.atol)
+            q5o, r5o = qr_insert(qs, rs, us, k, which, overwrite_qru=True)
+            check_qr(q5o, r5o, ai, self.rtol, self.atol)
+
+    def test_non_unit_strides_1_row(self):
+        self.base_non_simple_strides(make_strided, 0, 1, 'row')
+
+    def test_non_unit_strides_p_row(self):
+        self.base_non_simple_strides(make_strided, 0, 3, 'row')
+
+    def test_non_unit_strides_1_col(self):
+        self.base_non_simple_strides(make_strided, 0, 1, 'col')
+
+    def test_non_unit_strides_p_col(self):
+        self.base_non_simple_strides(make_strided, 0, 3, 'col')
+
+    def test_neg_strides_1_row(self):
+        self.base_non_simple_strides(negate_strides, 0, 1, 'row')
+
+    def test_neg_strides_p_row(self):
+        self.base_non_simple_strides(negate_strides, 0, 3, 'row')
+
+    def test_neg_strides_1_col(self):
+        self.base_non_simple_strides(negate_strides, 0, 1, 'col')
+
+    def test_neg_strides_p_col(self):
+        self.base_non_simple_strides(negate_strides, 0, 3, 'col')
+
+    def test_non_itemsize_strides_1_row(self):
+        self.base_non_simple_strides(nonitemsize_strides, 0, 1, 'row')
+
+    def test_non_itemsize_strides_p_row(self):
+        self.base_non_simple_strides(nonitemsize_strides, 0, 3, 'row')
+
+    def test_non_itemsize_strides_1_col(self):
+        self.base_non_simple_strides(nonitemsize_strides, 0, 1, 'col')
+
+    def test_non_itemsize_strides_p_col(self):
+        self.base_non_simple_strides(nonitemsize_strides, 0, 3, 'col')
+
+    def test_non_native_byte_order_1_row(self):
+        self.base_non_simple_strides(make_nonnative, 0, 1, 'row')
+
+    def test_non_native_byte_order_p_row(self):
+        self.base_non_simple_strides(make_nonnative, 0, 3, 'row')
+
+    def test_non_native_byte_order_1_col(self):
+        self.base_non_simple_strides(make_nonnative, 0, 1, 'col')
+
+    def test_non_native_byte_order_p_col(self):
+        self.base_non_simple_strides(make_nonnative, 0, 3, 'col')
+
+    def test_overwrite_qu_rank_1(self):
+        # when inserting rows, the size of both Q and R change, so only
+        # column inserts can overwrite q. Only complex column inserts
+        # with C ordered Q overwrite u. Any contiguous Q is overwritten
+        # when inserting 1 column
+        a, q0, r, u, = self.generate('sqr', which='col', p=1)
+        q = q0.copy('C')
+        u0 = u.copy()
+        # don't overwrite
+        q1, r1 = qr_insert(q, r, u, 0, 'col', overwrite_qru=False)
+        a1 = np.insert(a, 0, u0, 1)
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+        check_qr(q, r, a, self.rtol, self.atol)
+
+        # try overwriting
+        q2, r2 = qr_insert(q, r, u, 0, 'col', overwrite_qru=True)
+        check_qr(q2, r2, a1, self.rtol, self.atol)
+        # verify the overwriting
+        assert_allclose(q2, q, rtol=self.rtol, atol=self.atol)
+        assert_allclose(u, u0.conj(), self.rtol, self.atol)
+
+        # now try with a fortran ordered Q
+        qF = q0.copy('F')
+        u1 = u0.copy()
+        q3, r3 = qr_insert(qF, r, u1, 0, 'col', overwrite_qru=False)
+        check_qr(q3, r3, a1, self.rtol, self.atol)
+        check_qr(qF, r, a, self.rtol, self.atol)
+
+        # try overwriting
+        q4, r4 = qr_insert(qF, r, u1, 0, 'col', overwrite_qru=True)
+        check_qr(q4, r4, a1, self.rtol, self.atol)
+        assert_allclose(q4, qF, rtol=self.rtol, atol=self.atol)
+
+    def test_overwrite_qu_rank_p(self):
+        # when inserting rows, the size of both Q and R change, so only
+        # column inserts can potentially overwrite Q.  In practice, only
+        # F ordered Q are overwritten with a rank p update.
+        a, q0, r, u, = self.generate('sqr', which='col', p=3)
+        q = q0.copy('F')
+        a1 = np.insert(a, np.zeros(3, np.intp), u, 1)
+
+        # don't overwrite
+        q1, r1 = qr_insert(q, r, u, 0, 'col', overwrite_qru=False)
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+        check_qr(q, r, a, self.rtol, self.atol)
+
+        # try overwriting
+        q2, r2 = qr_insert(q, r, u, 0, 'col', overwrite_qru=True)
+        check_qr(q2, r2, a1, self.rtol, self.atol)
+        assert_allclose(q2, q, rtol=self.rtol, atol=self.atol)
+
+    def test_empty_inputs(self):
+        a, q, r, u = self.generate('sqr', which='row')
+        assert_raises(ValueError, qr_insert, np.array([]), r, u, 0, 'row')
+        assert_raises(ValueError, qr_insert, q, np.array([]), u, 0, 'row')
+        assert_raises(ValueError, qr_insert, q, r, np.array([]), 0, 'row')
+        assert_raises(ValueError, qr_insert, np.array([]), r, u, 0, 'col')
+        assert_raises(ValueError, qr_insert, q, np.array([]), u, 0, 'col')
+        assert_raises(ValueError, qr_insert, q, r, np.array([]), 0, 'col')
+
+    def test_mismatched_shapes(self):
+        a, q, r, u = self.generate('tall', which='row')
+        assert_raises(ValueError, qr_insert, q, r[1:], u, 0, 'row')
+        assert_raises(ValueError, qr_insert, q[:-2], r, u, 0, 'row')
+        assert_raises(ValueError, qr_insert, q, r, u[1:], 0, 'row')
+        assert_raises(ValueError, qr_insert, q, r[1:], u, 0, 'col')
+        assert_raises(ValueError, qr_insert, q[:-2], r, u, 0, 'col')
+        assert_raises(ValueError, qr_insert, q, r, u[1:], 0, 'col')
+
+    @pytest.mark.filterwarnings("ignore::DeprecationWarning")
+    def test_unsupported_dtypes(self):
+        dts = ['int8', 'int16', 'int32', 'int64',
+               'uint8', 'uint16', 'uint32', 'uint64',
+               'float16', 'longdouble', 'clongdouble',
+               'bool']
+        a, q0, r0, u0 = self.generate('sqr', which='row')
+        for dtype in dts:
+            q = q0.real.astype(dtype)
+            with np.errstate(invalid="ignore"):
+                r = r0.real.astype(dtype)
+            u = u0.real.astype(dtype)
+            assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'row')
+            assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'col')
+            assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'row')
+            assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'col')
+            assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'row')
+            assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'col')
+
+    def test_check_finite(self):
+        a0, q0, r0, u0 = self.generate('sqr', which='row', p=3)
+
+        q = q0.copy('F')
+        q[1,1] = np.nan
+        assert_raises(ValueError, qr_insert, q, r0, u0[:,0], 0, 'row')
+        assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'row')
+        assert_raises(ValueError, qr_insert, q, r0, u0[:,0], 0, 'col')
+        assert_raises(ValueError, qr_insert, q, r0, u0, 0, 'col')
+
+        r = r0.copy('F')
+        r[1,1] = np.nan
+        assert_raises(ValueError, qr_insert, q0, r, u0[:,0], 0, 'row')
+        assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'row')
+        assert_raises(ValueError, qr_insert, q0, r, u0[:,0], 0, 'col')
+        assert_raises(ValueError, qr_insert, q0, r, u0, 0, 'col')
+
+        u = u0.copy('F')
+        u[0,0] = np.nan
+        assert_raises(ValueError, qr_insert, q0, r0, u[:,0], 0, 'row')
+        assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'row')
+        assert_raises(ValueError, qr_insert, q0, r0, u[:,0], 0, 'col')
+        assert_raises(ValueError, qr_insert, q0, r0, u, 0, 'col')
+
+class TestQRinsert_f(BaseQRinsert):
+    dtype = np.dtype('f')
+
+class TestQRinsert_F(BaseQRinsert):
+    dtype = np.dtype('F')
+
+class TestQRinsert_d(BaseQRinsert):
+    dtype = np.dtype('d')
+
+class TestQRinsert_D(BaseQRinsert):
+    dtype = np.dtype('D')
+
+class BaseQRupdate(BaseQRdeltas):
+    def generate(self, type, mode='full', p=1):
+        a, q, r = super().generate(type, mode)
+
+        rng = np.random.default_rng(1234)
+        if p == 1:
+            u = rng.random(q.shape[0])
+            v = rng.random(r.shape[1])
+        else:
+            u = rng.random((q.shape[0], p))
+            v = rng.random((r.shape[1], p))
+
+        if np.iscomplexobj(self.dtype.type(1)):
+            b = rng.random(u.shape)
+            u = u + 1j * b
+
+            c = rng.random(v.shape)
+            v = v + 1j * c
+
+        u = u.astype(self.dtype)
+        v = v.astype(self.dtype)
+        return a, q, r, u, v
+
+    def test_sqr_rank_1(self):
+        a, q, r, u, v = self.generate('sqr')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_sqr_rank_p(self):
+        # test ndim = 2, rank 1 updates here too
+        for p in [1, 2, 3, 5]:
+            a, q, r, u, v = self.generate('sqr', p=p)
+            if p == 1:
+                u = u.reshape(u.size, 1)
+                v = v.reshape(v.size, 1)
+            q1, r1 = qr_update(q, r, u, v, False)
+            a1 = a + np.dot(u, v.T.conj())
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_rank_1(self):
+        a, q, r, u, v = self.generate('tall')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_tall_rank_p(self):
+        for p in [1, 2, 3, 5]:
+            a, q, r, u, v = self.generate('tall', p=p)
+            if p == 1:
+                u = u.reshape(u.size, 1)
+                v = v.reshape(v.size, 1)
+            q1, r1 = qr_update(q, r, u, v, False)
+            a1 = a + np.dot(u, v.T.conj())
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_rank_1(self):
+        a, q, r, u, v = self.generate('fat')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_fat_rank_p(self):
+        for p in [1, 2, 3, 5]:
+            a, q, r, u, v = self.generate('fat', p=p)
+            if p == 1:
+                u = u.reshape(u.size, 1)
+                v = v.reshape(v.size, 1)
+            q1, r1 = qr_update(q, r, u, v, False)
+            a1 = a + np.dot(u, v.T.conj())
+            check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_economic_rank_1(self):
+        a, q, r, u, v = self.generate('tall', 'economic')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_economic_rank_p(self):
+        for p in [1, 2, 3, 5]:
+            a, q, r, u, v = self.generate('tall', 'economic', p)
+            if p == 1:
+                u = u.reshape(u.size, 1)
+                v = v.reshape(v.size, 1)
+            q1, r1 = qr_update(q, r, u, v, False)
+            a1 = a + np.dot(u, v.T.conj())
+            check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_Mx1_rank_1(self):
+        a, q, r, u, v = self.generate('Mx1')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_rank_p(self):
+        # when M or N == 1, only a rank 1 update is allowed. This isn't
+        # fundamental limitation, but the code does not support it.
+        a, q, r, u, v = self.generate('Mx1', p=1)
+        u = u.reshape(u.size, 1)
+        v = v.reshape(v.size, 1)
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.dot(u, v.T.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_Mx1_economic_rank_1(self):
+        a, q, r, u, v = self.generate('Mx1', 'economic')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_Mx1_economic_rank_p(self):
+        # when M or N == 1, only a rank 1 update is allowed. This isn't
+        # fundamental limitation, but the code does not support it.
+        a, q, r, u, v = self.generate('Mx1', 'economic', p=1)
+        u = u.reshape(u.size, 1)
+        v = v.reshape(v.size, 1)
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.dot(u, v.T.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+    def test_1xN_rank_1(self):
+        a, q, r, u, v = self.generate('1xN')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1xN_rank_p(self):
+        # when M or N == 1, only a rank 1 update is allowed. This isn't
+        # fundamental limitation, but the code does not support it.
+        a, q, r, u, v = self.generate('1xN', p=1)
+        u = u.reshape(u.size, 1)
+        v = v.reshape(v.size, 1)
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.dot(u, v.T.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_rank_1(self):
+        a, q, r, u, v = self.generate('1x1')
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_rank_p(self):
+        # when M or N == 1, only a rank 1 update is allowed. This isn't
+        # fundamental limitation, but the code does not support it.
+        a, q, r, u, v = self.generate('1x1', p=1)
+        u = u.reshape(u.size, 1)
+        v = v.reshape(v.size, 1)
+        q1, r1 = qr_update(q, r, u, v, False)
+        a1 = a + np.dot(u, v.T.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+
+    def test_1x1_rank_1_scalar(self):
+        a, q, r, u, v = self.generate('1x1')
+        assert_raises(ValueError, qr_update, q[0, 0], r, u, v)
+        assert_raises(ValueError, qr_update, q, r[0, 0], u, v)
+        assert_raises(ValueError, qr_update, q, r, u[0], v)
+        assert_raises(ValueError, qr_update, q, r, u, v[0])
+
+    def base_non_simple_strides(self, adjust_strides, mode, p, overwriteable):
+        assert_sqr = False if mode == 'economic' else True
+        for type in ['sqr', 'tall', 'fat']:
+            a, q0, r0, u0, v0 = self.generate(type, mode, p)
+            qs, rs, us, vs = adjust_strides((q0, r0, u0, v0))
+            if p == 1:
+                aup = a + np.outer(u0, v0.conj())
+            else:
+                aup = a + np.dot(u0, v0.T.conj())
+
+            # for each variable, q, r, u, v we try with it strided and
+            # overwrite=False. Then we try with overwrite=True, and make
+            # sure that if p == 1, r and v are still overwritten.
+            # a strided q and u must always be copied.
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            v = v0.copy('C')
+            q1, r1 = qr_update(qs, r, u, v, False)
+            check_qr(q1, r1, aup, self.rtol, self.atol, assert_sqr)
+            q1o, r1o = qr_update(qs, r, u, v, True)
+            check_qr(q1o, r1o, aup, self.rtol, self.atol, assert_sqr)
+            if overwriteable:
+                assert_allclose(r1o, r, rtol=self.rtol, atol=self.atol)
+                assert_allclose(v, v0.conj(), rtol=self.rtol, atol=self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            v = v0.copy('C')
+            q2, r2 = qr_update(q, rs, u, v, False)
+            check_qr(q2, r2, aup, self.rtol, self.atol, assert_sqr)
+            q2o, r2o = qr_update(q, rs, u, v, True)
+            check_qr(q2o, r2o, aup, self.rtol, self.atol, assert_sqr)
+            if overwriteable:
+                assert_allclose(r2o, rs, rtol=self.rtol, atol=self.atol)
+                assert_allclose(v, v0.conj(), rtol=self.rtol, atol=self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            v = v0.copy('C')
+            q3, r3 = qr_update(q, r, us, v, False)
+            check_qr(q3, r3, aup, self.rtol, self.atol, assert_sqr)
+            q3o, r3o = qr_update(q, r, us, v, True)
+            check_qr(q3o, r3o, aup, self.rtol, self.atol, assert_sqr)
+            if overwriteable:
+                assert_allclose(r3o, r, rtol=self.rtol, atol=self.atol)
+                assert_allclose(v, v0.conj(), rtol=self.rtol, atol=self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            v = v0.copy('C')
+            q4, r4 = qr_update(q, r, u, vs, False)
+            check_qr(q4, r4, aup, self.rtol, self.atol, assert_sqr)
+            q4o, r4o = qr_update(q, r, u, vs, True)
+            check_qr(q4o, r4o, aup, self.rtol, self.atol, assert_sqr)
+            if overwriteable:
+                assert_allclose(r4o, r, rtol=self.rtol, atol=self.atol)
+                assert_allclose(vs, v0.conj(), rtol=self.rtol, atol=self.atol)
+
+            q = q0.copy('F')
+            r = r0.copy('F')
+            u = u0.copy('F')
+            v = v0.copy('C')
+            # since some of these were consumed above
+            qs, rs, us, vs = adjust_strides((q, r, u, v))
+            q5, r5 = qr_update(qs, rs, us, vs, False)
+            check_qr(q5, r5, aup, self.rtol, self.atol, assert_sqr)
+            q5o, r5o = qr_update(qs, rs, us, vs, True)
+            check_qr(q5o, r5o, aup, self.rtol, self.atol, assert_sqr)
+            if overwriteable:
+                assert_allclose(r5o, rs, rtol=self.rtol, atol=self.atol)
+                assert_allclose(vs, v0.conj(), rtol=self.rtol, atol=self.atol)
+
+    def test_non_unit_strides_rank_1(self):
+        self.base_non_simple_strides(make_strided, 'full', 1, True)
+
+    def test_non_unit_strides_economic_rank_1(self):
+        self.base_non_simple_strides(make_strided, 'economic', 1, True)
+
+    def test_non_unit_strides_rank_p(self):
+        self.base_non_simple_strides(make_strided, 'full', 3, False)
+
+    def test_non_unit_strides_economic_rank_p(self):
+        self.base_non_simple_strides(make_strided, 'economic', 3, False)
+
+    def test_neg_strides_rank_1(self):
+        self.base_non_simple_strides(negate_strides, 'full', 1, False)
+
+    def test_neg_strides_economic_rank_1(self):
+        self.base_non_simple_strides(negate_strides, 'economic', 1, False)
+
+    def test_neg_strides_rank_p(self):
+        self.base_non_simple_strides(negate_strides, 'full', 3, False)
+
+    def test_neg_strides_economic_rank_p(self):
+        self.base_non_simple_strides(negate_strides, 'economic', 3, False)
+
+    def test_non_itemsize_strides_rank_1(self):
+        self.base_non_simple_strides(nonitemsize_strides, 'full', 1, False)
+
+    def test_non_itemsize_strides_economic_rank_1(self):
+        self.base_non_simple_strides(nonitemsize_strides, 'economic', 1, False)
+
+    def test_non_itemsize_strides_rank_p(self):
+        self.base_non_simple_strides(nonitemsize_strides, 'full', 3, False)
+
+    def test_non_itemsize_strides_economic_rank_p(self):
+        self.base_non_simple_strides(nonitemsize_strides, 'economic', 3, False)
+
+    def test_non_native_byte_order_rank_1(self):
+        self.base_non_simple_strides(make_nonnative, 'full', 1, False)
+
+    def test_non_native_byte_order_economic_rank_1(self):
+        self.base_non_simple_strides(make_nonnative, 'economic', 1, False)
+
+    def test_non_native_byte_order_rank_p(self):
+        self.base_non_simple_strides(make_nonnative, 'full', 3, False)
+
+    def test_non_native_byte_order_economic_rank_p(self):
+        self.base_non_simple_strides(make_nonnative, 'economic', 3, False)
+
+    def test_overwrite_qruv_rank_1(self):
+        # Any positive strided q, r, u, and v can be overwritten for a rank 1
+        # update, only checking C and F contiguous.
+        a, q0, r0, u0, v0 = self.generate('sqr')
+        a1 = a + np.outer(u0, v0.conj())
+        q = q0.copy('F')
+        r = r0.copy('F')
+        u = u0.copy('F')
+        v = v0.copy('F')
+
+        # don't overwrite
+        q1, r1 = qr_update(q, r, u, v, False)
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+        check_qr(q, r, a, self.rtol, self.atol)
+
+        q2, r2 = qr_update(q, r, u, v, True)
+        check_qr(q2, r2, a1, self.rtol, self.atol)
+        # verify the overwriting, no good way to check u and v.
+        assert_allclose(q2, q, rtol=self.rtol, atol=self.atol)
+        assert_allclose(r2, r, rtol=self.rtol, atol=self.atol)
+
+        q = q0.copy('C')
+        r = r0.copy('C')
+        u = u0.copy('C')
+        v = v0.copy('C')
+        q3, r3 = qr_update(q, r, u, v, True)
+        check_qr(q3, r3, a1, self.rtol, self.atol)
+        assert_allclose(q3, q, rtol=self.rtol, atol=self.atol)
+        assert_allclose(r3, r, rtol=self.rtol, atol=self.atol)
+
+    def test_overwrite_qruv_rank_1_economic(self):
+        # updating economic decompositions can overwrite any contiguous r,
+        # and positively strided r and u. V is only ever read.
+        # only checking C and F contiguous.
+        a, q0, r0, u0, v0 = self.generate('tall', 'economic')
+        a1 = a + np.outer(u0, v0.conj())
+        q = q0.copy('F')
+        r = r0.copy('F')
+        u = u0.copy('F')
+        v = v0.copy('F')
+
+        # don't overwrite
+        q1, r1 = qr_update(q, r, u, v, False)
+        check_qr(q1, r1, a1, self.rtol, self.atol, False)
+        check_qr(q, r, a, self.rtol, self.atol, False)
+
+        q2, r2 = qr_update(q, r, u, v, True)
+        check_qr(q2, r2, a1, self.rtol, self.atol, False)
+        # verify the overwriting, no good way to check u and v.
+        assert_allclose(q2, q, rtol=self.rtol, atol=self.atol)
+        assert_allclose(r2, r, rtol=self.rtol, atol=self.atol)
+
+        q = q0.copy('C')
+        r = r0.copy('C')
+        u = u0.copy('C')
+        v = v0.copy('C')
+        q3, r3 = qr_update(q, r, u, v, True)
+        check_qr(q3, r3, a1, self.rtol, self.atol, False)
+        assert_allclose(q3, q, rtol=self.rtol, atol=self.atol)
+        assert_allclose(r3, r, rtol=self.rtol, atol=self.atol)
+
+    def test_overwrite_qruv_rank_p(self):
+        # for rank p updates, q r must be F contiguous, v must be C (v.T --> F)
+        # and u can be C or F, but is only overwritten if Q is C and complex
+        a, q0, r0, u0, v0 = self.generate('sqr', p=3)
+        a1 = a + np.dot(u0, v0.T.conj())
+        q = q0.copy('F')
+        r = r0.copy('F')
+        u = u0.copy('F')
+        v = v0.copy('C')
+
+        # don't overwrite
+        q1, r1 = qr_update(q, r, u, v, False)
+        check_qr(q1, r1, a1, self.rtol, self.atol)
+        check_qr(q, r, a, self.rtol, self.atol)
+
+        q2, r2 = qr_update(q, r, u, v, True)
+        check_qr(q2, r2, a1, self.rtol, self.atol)
+        # verify the overwriting, no good way to check u and v.
+        assert_allclose(q2, q, rtol=self.rtol, atol=self.atol)
+        assert_allclose(r2, r, rtol=self.rtol, atol=self.atol)
+
+    def test_empty_inputs(self):
+        a, q, r, u, v = self.generate('tall')
+        assert_raises(ValueError, qr_update, np.array([]), r, u, v)
+        assert_raises(ValueError, qr_update, q, np.array([]), u, v)
+        assert_raises(ValueError, qr_update, q, r, np.array([]), v)
+        assert_raises(ValueError, qr_update, q, r, u, np.array([]))
+
+    def test_mismatched_shapes(self):
+        a, q, r, u, v = self.generate('tall')
+        assert_raises(ValueError, qr_update, q, r[1:], u, v)
+        assert_raises(ValueError, qr_update, q[:-2], r, u, v)
+        assert_raises(ValueError, qr_update, q, r, u[1:], v)
+        assert_raises(ValueError, qr_update, q, r, u, v[1:])
+
+    @pytest.mark.filterwarnings("ignore::DeprecationWarning")
+    def test_unsupported_dtypes(self):
+        dts = ['int8', 'int16', 'int32', 'int64',
+               'uint8', 'uint16', 'uint32', 'uint64',
+               'float16', 'longdouble', 'clongdouble',
+               'bool']
+        a, q0, r0, u0, v0 = self.generate('tall')
+        for dtype in dts:
+            q = q0.real.astype(dtype)
+            with np.errstate(invalid="ignore"):
+                r = r0.real.astype(dtype)
+            u = u0.real.astype(dtype)
+            v = v0.real.astype(dtype)
+            assert_raises(ValueError, qr_update, q, r0, u0, v0)
+            assert_raises(ValueError, qr_update, q0, r, u0, v0)
+            assert_raises(ValueError, qr_update, q0, r0, u, v0)
+            assert_raises(ValueError, qr_update, q0, r0, u0, v)
+
+    def test_integer_input(self):
+        q = np.arange(16).reshape(4, 4)
+        r = q.copy()  # doesn't matter
+        u = q[:, 0].copy()
+        v = r[0, :].copy()
+        assert_raises(ValueError, qr_update, q, r, u, v)
+
+    def test_check_finite(self):
+        a0, q0, r0, u0, v0 = self.generate('tall', p=3)
+
+        q = q0.copy('F')
+        q[1,1] = np.nan
+        assert_raises(ValueError, qr_update, q, r0, u0[:,0], v0[:,0])
+        assert_raises(ValueError, qr_update, q, r0, u0, v0)
+
+        r = r0.copy('F')
+        r[1,1] = np.nan
+        assert_raises(ValueError, qr_update, q0, r, u0[:,0], v0[:,0])
+        assert_raises(ValueError, qr_update, q0, r, u0, v0)
+
+        u = u0.copy('F')
+        u[0,0] = np.nan
+        assert_raises(ValueError, qr_update, q0, r0, u[:,0], v0[:,0])
+        assert_raises(ValueError, qr_update, q0, r0, u, v0)
+
+        v = v0.copy('F')
+        v[0,0] = np.nan
+        assert_raises(ValueError, qr_update, q0, r0, u[:,0], v[:,0])
+        assert_raises(ValueError, qr_update, q0, r0, u, v)
+
+    def test_economic_check_finite(self):
+        a0, q0, r0, u0, v0 = self.generate('tall', mode='economic', p=3)
+
+        q = q0.copy('F')
+        q[1,1] = np.nan
+        assert_raises(ValueError, qr_update, q, r0, u0[:,0], v0[:,0])
+        assert_raises(ValueError, qr_update, q, r0, u0, v0)
+
+        r = r0.copy('F')
+        r[1,1] = np.nan
+        assert_raises(ValueError, qr_update, q0, r, u0[:,0], v0[:,0])
+        assert_raises(ValueError, qr_update, q0, r, u0, v0)
+
+        u = u0.copy('F')
+        u[0,0] = np.nan
+        assert_raises(ValueError, qr_update, q0, r0, u[:,0], v0[:,0])
+        assert_raises(ValueError, qr_update, q0, r0, u, v0)
+
+        v = v0.copy('F')
+        v[0,0] = np.nan
+        assert_raises(ValueError, qr_update, q0, r0, u[:,0], v[:,0])
+        assert_raises(ValueError, qr_update, q0, r0, u, v)
+
+    def test_u_exactly_in_span_q(self):
+        q = np.array([[0, 0], [0, 0], [1, 0], [0, 1]], self.dtype)
+        r = np.array([[1, 0], [0, 1]], self.dtype)
+        u = np.array([0, 0, 0, -1], self.dtype)
+        v = np.array([1, 2], self.dtype)
+        q1, r1 = qr_update(q, r, u, v)
+        a1 = np.dot(q, r) + np.outer(u, v.conj())
+        check_qr(q1, r1, a1, self.rtol, self.atol, False)
+
+class TestQRupdate_f(BaseQRupdate):
+    dtype = np.dtype('f')
+
+class TestQRupdate_F(BaseQRupdate):
+    dtype = np.dtype('F')
+
+class TestQRupdate_d(BaseQRupdate):
+    dtype = np.dtype('d')
+
+class TestQRupdate_D(BaseQRupdate):
+    dtype = np.dtype('D')
+
+def test_form_qTu():
+    # We want to ensure that all of the code paths through this function are
+    # tested. Most of them should be hit with the rest of test suite, but
+    # explicit tests make clear precisely what is being tested.
+    #
+    # This function expects that Q is either C or F contiguous and square.
+    # Economic mode decompositions (Q is (M, N), M != N) do not go through this
+    # function. U may have any positive strides.
+    #
+    # Some of these test are duplicates, since contiguous 1d arrays are both C
+    # and F.
+
+    q_order = ['F', 'C']
+    q_shape = [(8, 8), ]
+    u_order = ['F', 'C', 'A']  # here A means is not F not C
+    u_shape = [1, 3]
+    dtype = ['f', 'd', 'F', 'D']
+
+    for qo, qs, uo, us, d in \
+            itertools.product(q_order, q_shape, u_order, u_shape, dtype):
+        if us == 1:
+            check_form_qTu(qo, qs, uo, us, 1, d)
+            check_form_qTu(qo, qs, uo, us, 2, d)
+        else:
+            check_form_qTu(qo, qs, uo, us, 2, d)
+
+def check_form_qTu(q_order, q_shape, u_order, u_shape, u_ndim, dtype):
+    rng = np.random.default_rng(47)
+    if u_shape == 1 and u_ndim == 1:
+        u_shape = (q_shape[0],)
+    else:
+        u_shape = (q_shape[0], u_shape)
+    dtype = np.dtype(dtype)
+
+    if dtype.char in 'fd':
+        q = rng.random(q_shape)
+        u = rng.random(u_shape)
+    elif dtype.char in 'FD':
+        q = rng.random(q_shape) + 1j*rng.random(q_shape)
+        u = rng.random(u_shape) + 1j*rng.random(u_shape)
+    else:
+        raise ValueError("form_qTu doesn't support this dtype")
+
+    q = np.require(q, dtype, q_order)
+    if u_order != 'A':
+        u = np.require(u, dtype, u_order)
+    else:
+        u, = make_strided((u.astype(dtype),))
+
+    rtol = 10.0 ** -(np.finfo(dtype).precision-2)
+    atol = 2*np.finfo(dtype).eps
+
+    expected = np.dot(q.T.conj(), u)
+    res = _decomp_update._form_qTu(q, u)
+    assert_allclose(res, expected, rtol=rtol, atol=atol)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_deprecations.py

@@ -0,0 +1,223 @@
+'''Test dtype deprecations.
+'''
+import numpy as np
+import scipy.linalg as linalg
+import pytest
+
+from scipy._lib._array_api import SCIPY_ARRAY_API
+
+#
+# When the deprecation expires
+#     1. Change the test from asserting a DeprecationWarning to assert_raises
+#     2. Remove the `_skip_dict` dictionary: it is only used to tell things which
+#        already raise errors from things which now emit DeprecationWarnings
+
+
+# skip these completely
+_skip_names = [
+    # not functions
+    'LinAlgError', 'LinAlgWarning',
+    # do not touch the low-level `scipy.linalg.{blas,lapack}`
+    'get_lapack_funcs', 'get_blas_funcs', 'find_best_blas_type',
+    # qr_update skip for now
+    'qr_update', 'qr_insert', 'qr_delete',
+    # special matrix constructors do not use LAPACK
+    'block_diag',
+    'toeplitz', 'circulant', 'hankel', 'hadamard', 'leslie', 'companion',
+    'helmert', 'hilbert', 'invhilbert', 'pascal', 'invpascal', 'dft',
+    'fiedler', 'fiedler_companion', 'convolution_matrix',
+    # _sketches does not use LAPACK
+    'clarkson_woodruff_transform',
+    # these did not accepted weird dtypes already
+    'expm', 'tanm', 'tanhm', 'sinm', 'cosm', 'sinhm', 'coshm', 'expm_cond',
+    'solve_continuous_are',
+    'rsf2csf',
+]
+
+
+# use this for dtype-specific skips
+# (e.g. if something only accepts real arrays, add complex typecodes here.)
+_skip_dict = {
+    # TypeError: Only real arrays currently supported
+    'eigh_tridiagonal': 'G',
+    'eigvalsh_tridiagonal': 'G',
+    # TypeError: No matching signature found
+    'issymmetric': 'eGmM',
+    'ishermitian': 'eGmM',
+    'bandwidth': 'egGmM',
+    # already raise
+    'det': 'gGmM',
+    'expm': 'gG',
+
+    # datetime specific + float16 specific
+    'cdf2rdf' : 'mM',
+    'expm_frechet' : 'mM' + 'e',
+    'pinv': 'mM',
+    'signm': 'mM',
+    'fractional_matrix_power' : 'mM' + 'e',
+    'khatri_rao' : 'mM',
+    'logm' : 'mM' + 'e',
+    'orthogonal_procrustes' : 'mM' + 'e',
+    'solve_circulant' : 'mM',
+    'solve_continuous_lyapunov': 'M',
+    'solve_discrete_lyapunov': 'mM' + 'e',
+    'solve_lyapunov' : 'M',
+    'solve_sylvester' : 'mM',
+    'solve_discrete_are': 'mM',
+    'matmul_toeplitz': 'mM',
+    'solve_toeplitz': 'mM',
+}
+
+
+# these raise if SCIPY_ARRAY_API env var is defined
+_array_api_skip_dict = {
+    'cho_factor' : 'mM',
+    'cho_solve' : 'mM',
+    'cho_solve_banded' : 'mM',
+    'cholesky' : 'mM',
+    'cholesky_banded' : 'mM',
+    'cossin' : 'mM',
+    'diagsvd' : 'mM',
+    'eig_banded' : 'mM',
+    'eigh' : 'mM',
+    'eigh_tridiagonal' : 'mM',
+    'eigvals_banded' : 'mM',
+    'eigvalsh' : 'mM',
+    'eigvalsh_tridiagonal' : 'mM',
+    'funm' : 'mM',
+    'hessenberg' : 'mM',
+    'ldl' : 'mM',
+    'lu_factor' : 'mM',
+    'lu_solve' : 'mM',
+    'matrix_balance' : 'mM',
+    'null_space' : 'mM',
+    'ordqz' : 'mM',
+    'orth' : 'mM',
+    'pinvh' : 'mM',
+    'polar' : 'mM',
+    'qr' : 'mM',
+    'qr_multiply' : 'mM',
+    'qz' : 'mM',
+    'rq' : 'mM',
+    'schur' : 'mM',
+    'solve_banded' : 'mM',
+    'solve_continuous_lyapunov' : 'mM',
+    'solve_lyapunov' : 'mM',
+    'solve_triangular' : 'mM',
+    'solveh_banded' : 'mM',
+    'subspace_angles' : 'mM',
+}
+
+
+if SCIPY_ARRAY_API:
+    #_skip_dict.update(**_array_api_skip_dict)
+    for key, val in _array_api_skip_dict.items():
+        if key in _skip_dict:
+            _skip_dict[key] += val
+        else:
+            _skip_dict[key] = val
+
+
+# loop over all names
+_names = linalg.__all__
+_objs = {name: getattr(linalg, name)
+    for name in _names
+    if name not in _skip_names
+}
+_objs = {name: obj for name,obj in _objs.items() if callable(obj)}
+
+
+# Some gymnastics to prepare a set of arguments which all functions can accept
+_two_arg_names = set([
+    'cdf2rdf', 'rsf2csf', 'solve', 'solve_triangular', 'lstsq', 'khatri_rao',
+    'subspace_angles', 'qz', 'ordqz', 'orthogonal_procrustes', 'solve_circulant',
+    'solve_discrete_lyapunov', 'solve_continuous_lyapunov', 'solve_lyapunov',
+    'expm_frechet',
+    'matmul_toeplitz', 'solve_toeplitz', 'lu_solve',
+    'eigh_tridiagonal', 'eigvalsh_tridiagonal',
+    'solveh_banded', 'solve_banded', 'solveh_banded', 'qr_multiply',
+])
+
+_three_arg_names = set(['solve_sylvester'])
+
+_four_arg_names = set(['solve_continuous_are', 'solve_discrete_are'])
+
+_arr_and_int_names = set([
+    'clarkson_woodruff_transform', 'convolution_matrix', 'fractional_matrix_power',
+])
+
+_arr_and_two_int_names = set(['diagsvd', 'cossin'])
+
+
+def _patch_args(func_name, args):
+    '''Make sure func(*args) does not raise because *args is wrong.'''
+    if func_name == 'cdf2rdf':
+        args = (args[0][0], args[1])  # cdf2rd(1d, 2d)
+    elif func_name == 'funm':
+        args = (args[0], lambda x: x)
+    elif func_name == 'lu_solve':
+        a = args[0]
+        piv = np.arange(a.shape[0])
+        args = ((a, piv), args[1])
+    elif func_name in ('eigh_tridiagonal', 'eigvalsh_tridiagonal'):       
+        d, e = args
+        args = (d, e[:, :-1])
+    elif func_name == 'cossin':
+        a = args[0]
+        args = (a, a.shape[0]//2, a.shape[1]//2)
+    elif func_name == 'solve_banded':
+        from scipy.linalg._basic import _to_banded
+        args = ((1, 2), _to_banded(1, 2, args[0]), args[0])
+    elif func_name == 'solveh_banded':
+        from scipy.linalg._basic import _to_banded
+        args = (_to_banded(0, 2, args[0])[:, :-1], args[0][0, :-1])
+    elif func_name == 'cholesky_banded':
+        from scipy.linalg._basic import _to_banded
+        args = (_to_banded(0, 2, args[0])[:, :-1],)
+    elif func_name == 'solve_toeplitz':
+        c = args[0][1]
+        args = (c, np.ones_like(c))
+    return args
+
+
+@pytest.mark.skip_xp_backends(np_only=True)
+@pytest.mark.filterwarnings(
+    'ignore:attempting to conjugate non-numeric dtype:DeprecationWarning'
+)
+@pytest.mark.filterwarnings('ignore:overflow encountered in dot:RuntimeWarning')
+@pytest.mark.filterwarnings('ignore:invalid value encountered in dot:RuntimeWarning')
+@pytest.mark.filterwarnings('ignore:logm result may be inaccurate:RuntimeWarning')
+@pytest.mark.filterwarnings('ignore:overflow encountered in multiply:RuntimeWarning')
+@pytest.mark.parametrize('func_name', _objs.keys())
+@pytest.mark.parametrize('tcode', 'e' + 'mMgG')
+def test_deprecations(func_name, tcode):
+    func = _objs[func_name]
+
+    if tcode in _skip_dict.get(func_name, ''):
+        return
+
+    a = np.arange(16).reshape(4, 4) + 8*np.eye(4)
+    a = a.T @ a
+    a = a.astype(tcode)
+
+    if func_name in _two_arg_names:
+        b = a.copy()
+        args = (a, b)
+    elif func_name in _three_arg_names:
+        args = (a, a, a)
+    elif func_name in _four_arg_names:
+        args = (a, a, a, a)
+    elif func_name in _arr_and_int_names:
+        args = (a, a.shape[0])
+    elif func_name in _arr_and_two_int_names:
+        args = (a, a.shape[0], a.shape[1])
+    elif func_name in ['cho_solve', 'cho_solve_banded']:
+        args = ((a, True), a)
+    else:
+        args = (a,)
+
+    args = _patch_args(func_name, args)
+
+    with pytest.warns(DeprecationWarning):
+        func(*args)
+

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_extending.py

@@ -0,0 +1,55 @@
+import os
+import platform
+import sysconfig
+
+import numpy as np
+import pytest
+
+from scipy._lib._testutils import IS_EDITABLE, _test_cython_extension, cython
+from scipy.linalg.blas import get_blas_funcs
+from scipy.linalg.lapack import get_lapack_funcs
+import scipy.linalg.cython_blas as cython_blas
+
+
+@pytest.mark.parallel_threads_limit(4)  # 0.35 GiB per thread RAM usage
+@pytest.mark.fail_slow(120)
+# essential per https://github.com/scipy/scipy/pull/20487#discussion_r1567057247
+@pytest.mark.skipif(IS_EDITABLE,
+                    reason='Editable install cannot find .pxd headers.')
+@pytest.mark.skipif((platform.system() == 'Windows' and
+                     sysconfig.get_config_var('Py_GIL_DISABLED')),
+                    reason='gh-22039')
+@pytest.mark.skipif(platform.machine() in ["wasm32", "wasm64"],
+                    reason="Can't start subprocess")
+@pytest.mark.skipif(cython is None, reason="requires cython")
+def test_cython(tmp_path):
+    srcdir = os.path.dirname(os.path.dirname(__file__))
+    extensions, extensions_cpp = _test_cython_extension(tmp_path, srcdir)
+    # actually test the cython c-extensions
+    a = np.ones(8) * 3
+    b = np.ones(9)
+    c = np.ones(8) * 4
+    x = np.ones(9)
+    dgtsv = get_lapack_funcs('gtsv', dtype=np.float64, ilp64="preferred")
+    _, _, _, x, _ = dgtsv(a, b, c, x)
+    a = np.ones(8) * 3
+    b = np.ones(9)
+    c = np.ones(8) * 4
+    x_c = np.ones(9)
+    extensions.tridiag(a, b, c, x_c)
+    a = np.ones(8) * 3
+    b = np.ones(9)
+    c = np.ones(8) * 4
+    x_cpp = np.ones(9)
+    extensions_cpp.tridiag(a, b, c, x_cpp)
+    np.testing.assert_array_equal(x, x_cpp)
+    cx = np.array([1-1j, 2+2j, 3-3j], dtype=np.complex64)
+    cy = np.array([4+4j, 5-5j, 6+6j], dtype=np.complex64)
+    cdotu = get_blas_funcs('dotu', dtype=np.complex64, ilp64="preferred")
+    np.testing.assert_array_equal(cdotu(cx, cy), extensions.complex_dot(cx, cy))
+    np.testing.assert_array_equal(cdotu(cx, cy), extensions_cpp.complex_dot(cx, cy))
+
+    # Verify blas_int size consistency between installed and JIT-compiled modules
+    expected_size = cython_blas._blas_int_size()
+    assert extensions.get_blas_int_size() == expected_size
+    assert extensions_cpp.get_blas_int_size() == expected_size

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_fblas.py

@@ -0,0 +1,610 @@
+# Test interfaces to fortran blas.
+#
+# The tests are more of interface than they are of the underlying blas.
+# Only very small matrices checked -- N=3 or so.
+#
+# !! Complex calculations really aren't checked that carefully.
+# !! Only real valued complex numbers are used in tests.
+from itertools import product
+import sys
+
+import numpy as np
+import pytest
+from numpy import float32, float64, complex64, complex128, arange, array, \
+                  zeros, shape, transpose, newaxis, common_type, conjugate
+
+from numpy.testing import (
+    assert_array_equal, assert_allclose, assert_array_almost_equal, assert_
+)
+
+import scipy
+
+blas_config = scipy.show_config(mode='dicts')['Build Dependencies']['blas']
+if blas_config['cython blas ilp64']:
+    pytestmark = pytest.skip(allow_module_level=True,
+                             reason="LP64 _fblas wasn't built")
+
+from scipy.linalg import _fblas as fblas  # noqa: E402
+
+
+# decimal accuracy to require between Python and LAPACK/BLAS calculations
+accuracy = 5
+
+# Since numpy.dot likely uses the same blas, use this routine
+# to check.
+
+
+def matrixmultiply(a, b):
+    if len(b.shape) == 1:
+        b_is_vector = True
+        b = b[:, newaxis]
+    else:
+        b_is_vector = False
+    assert_(a.shape[1] == b.shape[0])
+    c = zeros((a.shape[0], b.shape[1]), common_type(a, b))
+    for i in range(a.shape[0]):
+        for j in range(b.shape[1]):
+            s = 0
+            for k in range(a.shape[1]):
+                s += a[i, k] * b[k, j]
+            c[i, j] = s
+    if b_is_vector:
+        c = c.reshape((a.shape[0],))
+    return c
+
+##################################################
+# Test blas ?axpy
+
+
+class BaseAxpy:
+    ''' Mixin class for axpy tests '''
+
+    def test_default_a(self):
+        x = arange(3., dtype=self.dtype)
+        y = arange(3., dtype=x.dtype)
+        real_y = x*1.+y
+        y = self.blas_func(x, y)
+        assert_array_equal(real_y, y)
+
+    def test_simple(self):
+        x = arange(3., dtype=self.dtype)
+        y = arange(3., dtype=x.dtype)
+        real_y = x*3.+y
+        y = self.blas_func(x, y, a=3.)
+        assert_array_equal(real_y, y)
+
+    def test_x_stride(self):
+        x = arange(6., dtype=self.dtype)
+        y = zeros(3, x.dtype)
+        y = arange(3., dtype=x.dtype)
+        real_y = x[::2]*3.+y
+        y = self.blas_func(x, y, a=3., n=3, incx=2)
+        assert_array_equal(real_y, y)
+
+    def test_y_stride(self):
+        x = arange(3., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        real_y = x*3.+y[::2]
+        y = self.blas_func(x, y, a=3., n=3, incy=2)
+        assert_array_equal(real_y, y[::2])
+
+    def test_x_and_y_stride(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        real_y = x[::4]*3.+y[::2]
+        y = self.blas_func(x, y, a=3., n=3, incx=4, incy=2)
+        assert_array_equal(real_y, y[::2])
+
+    def test_x_bad_size(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        with pytest.raises(Exception, match='failed for 1st keyword'):
+            self.blas_func(x, y, n=4, incx=5)
+
+    def test_y_bad_size(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        with pytest.raises(Exception, match='failed for 1st keyword'):
+            self.blas_func(x, y, n=3, incy=5)
+
+
+try:
+    class TestSaxpy(BaseAxpy):
+        blas_func = fblas.saxpy
+        dtype = float32
+except AttributeError:
+    class TestSaxpy:
+        pass
+
+
+class TestDaxpy(BaseAxpy):
+    blas_func = fblas.daxpy
+    dtype = float64
+
+
+try:
+    class TestCaxpy(BaseAxpy):
+        blas_func = fblas.caxpy
+        dtype = complex64
+except AttributeError:
+    class TestCaxpy:
+        pass
+
+
+class TestZaxpy(BaseAxpy):
+    blas_func = fblas.zaxpy
+    dtype = complex128
+
+
+##################################################
+# Test blas ?scal
+
+class BaseScal:
+    ''' Mixin class for scal testing '''
+
+    def test_simple(self):
+        x = arange(3., dtype=self.dtype)
+        real_x = x*3.
+        x = self.blas_func(3., x)
+        assert_array_equal(real_x, x)
+
+    def test_x_stride(self):
+        x = arange(6., dtype=self.dtype)
+        real_x = x.copy()
+        real_x[::2] = x[::2]*array(3., self.dtype)
+        x = self.blas_func(3., x, n=3, incx=2)
+        assert_array_equal(real_x, x)
+
+    def test_x_bad_size(self):
+        x = arange(12., dtype=self.dtype)
+        with pytest.raises(Exception, match='failed for 1st keyword'):
+            self.blas_func(2., x, n=4, incx=5)
+
+
+try:
+    class TestSscal(BaseScal):
+        blas_func = fblas.sscal
+        dtype = float32
+except AttributeError:
+    class TestSscal:
+        pass
+
+
+class TestDscal(BaseScal):
+    blas_func = fblas.dscal
+    dtype = float64
+
+
+try:
+    class TestCscal(BaseScal):
+        blas_func = fblas.cscal
+        dtype = complex64
+except AttributeError:
+    class TestCscal:
+        pass
+
+
+class TestZscal(BaseScal):
+    blas_func = fblas.zscal
+    dtype = complex128
+
+
+##################################################
+# Test blas ?copy
+
+class BaseCopy:
+    ''' Mixin class for copy testing '''
+
+    def test_simple(self):
+        x = arange(3., dtype=self.dtype)
+        y = zeros(shape(x), x.dtype)
+        y = self.blas_func(x, y)
+        assert_array_equal(x, y)
+
+    def test_x_stride(self):
+        x = arange(6., dtype=self.dtype)
+        y = zeros(3, x.dtype)
+        y = self.blas_func(x, y, n=3, incx=2)
+        assert_array_equal(x[::2], y)
+
+    def test_y_stride(self):
+        x = arange(3., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        y = self.blas_func(x, y, n=3, incy=2)
+        assert_array_equal(x, y[::2])
+
+    def test_x_and_y_stride(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        y = self.blas_func(x, y, n=3, incx=4, incy=2)
+        assert_array_equal(x[::4], y[::2])
+
+    def test_x_bad_size(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        with pytest.raises(Exception, match='failed for 1st keyword'):
+            self.blas_func(x, y, n=4, incx=5)
+
+    def test_y_bad_size(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        with pytest.raises(Exception, match='failed for 1st keyword'):
+            self.blas_func(x, y, n=3, incy=5)
+
+    # def test_y_bad_type(self):
+    ##   Hmmm. Should this work?  What should be the output.
+    #    x = arange(3.,dtype=self.dtype)
+    #    y = zeros(shape(x))
+    #    self.blas_func(x,y)
+    #    assert_array_equal(x,y)
+
+
+try:
+    class TestScopy(BaseCopy):
+        blas_func = fblas.scopy
+        dtype = float32
+except AttributeError:
+    class TestScopy:
+        pass
+
+
+class TestDcopy(BaseCopy):
+    blas_func = fblas.dcopy
+    dtype = float64
+
+
+try:
+    class TestCcopy(BaseCopy):
+        blas_func = fblas.ccopy
+        dtype = complex64
+except AttributeError:
+    class TestCcopy:
+        pass
+
+
+class TestZcopy(BaseCopy):
+    blas_func = fblas.zcopy
+    dtype = complex128
+
+
+##################################################
+# Test blas ?swap
+
+class BaseSwap:
+    ''' Mixin class for swap tests '''
+
+    def test_simple(self):
+        x = arange(3., dtype=self.dtype)
+        y = zeros(shape(x), x.dtype)
+        desired_x = y.copy()
+        desired_y = x.copy()
+        x, y = self.blas_func(x, y)
+        assert_array_equal(desired_x, x)
+        assert_array_equal(desired_y, y)
+
+    def test_x_stride(self):
+        x = arange(6., dtype=self.dtype)
+        y = zeros(3, x.dtype)
+        desired_x = y.copy()
+        desired_y = x.copy()[::2]
+        x, y = self.blas_func(x, y, n=3, incx=2)
+        assert_array_equal(desired_x, x[::2])
+        assert_array_equal(desired_y, y)
+
+    def test_y_stride(self):
+        x = arange(3., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        desired_x = y.copy()[::2]
+        desired_y = x.copy()
+        x, y = self.blas_func(x, y, n=3, incy=2)
+        assert_array_equal(desired_x, x)
+        assert_array_equal(desired_y, y[::2])
+
+    def test_x_and_y_stride(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        desired_x = y.copy()[::2]
+        desired_y = x.copy()[::4]
+        x, y = self.blas_func(x, y, n=3, incx=4, incy=2)
+        assert_array_equal(desired_x, x[::4])
+        assert_array_equal(desired_y, y[::2])
+
+    def test_x_bad_size(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        with pytest.raises(Exception, match='failed for 1st keyword'):
+            self.blas_func(x, y, n=4, incx=5)
+
+    def test_y_bad_size(self):
+        x = arange(12., dtype=self.dtype)
+        y = zeros(6, x.dtype)
+        with pytest.raises(Exception, match='failed for 1st keyword'):
+            self.blas_func(x, y, n=3, incy=5)
+
+
+try:
+    class TestSswap(BaseSwap):
+        blas_func = fblas.sswap
+        dtype = float32
+except AttributeError:
+    class TestSswap:
+        pass
+
+
+class TestDswap(BaseSwap):
+    blas_func = fblas.dswap
+    dtype = float64
+
+
+try:
+    class TestCswap(BaseSwap):
+        blas_func = fblas.cswap
+        dtype = complex64
+except AttributeError:
+    class TestCswap:
+        pass
+
+
+class TestZswap(BaseSwap):
+    blas_func = fblas.zswap
+    dtype = complex128
+
+##################################################
+# Test blas ?gemv
+# This will be a mess to test all cases.
+
+
+class BaseGemv:
+    ''' Mixin class for gemv tests '''
+
+    def get_data(self, x_stride=1, y_stride=1):
+        rng = np.random.default_rng(1234)
+        mult = array(1, dtype=self.dtype)
+        if self.dtype in [complex64, complex128]:
+            mult = array(1+1j, dtype=self.dtype)
+        alpha = array(1., dtype=self.dtype) * mult
+        beta = array(1., dtype=self.dtype) * mult
+        a = rng.normal(0., 1., (3, 3)).astype(self.dtype) * mult
+        x = arange(shape(a)[0]*x_stride, dtype=self.dtype) * mult
+        y = arange(shape(a)[1]*y_stride, dtype=self.dtype) * mult
+        return alpha, beta, a, x, y
+
+    def test_simple(self):
+        alpha, beta, a, x, y = self.get_data()
+        desired_y = alpha*matrixmultiply(a, x)+beta*y
+        y = self.blas_func(alpha, a, x, beta, y)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_default_beta_y(self):
+        alpha, beta, a, x, y = self.get_data()
+        desired_y = matrixmultiply(a, x)
+        y = self.blas_func(1, a, x)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_simple_transpose(self):
+        alpha, beta, a, x, y = self.get_data()
+        desired_y = alpha*matrixmultiply(transpose(a), x)+beta*y
+        y = self.blas_func(alpha, a, x, beta, y, trans=1)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_simple_transpose_conj(self):
+        alpha, beta, a, x, y = self.get_data()
+        desired_y = alpha*matrixmultiply(transpose(conjugate(a)), x)+beta*y
+        y = self.blas_func(alpha, a, x, beta, y, trans=2)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_x_stride(self):
+        alpha, beta, a, x, y = self.get_data(x_stride=2)
+        desired_y = alpha*matrixmultiply(a, x[::2])+beta*y
+        y = self.blas_func(alpha, a, x, beta, y, incx=2)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_x_stride_transpose(self):
+        alpha, beta, a, x, y = self.get_data(x_stride=2)
+        desired_y = alpha*matrixmultiply(transpose(a), x[::2])+beta*y
+        y = self.blas_func(alpha, a, x, beta, y, trans=1, incx=2)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_x_stride_assert(self):
+        # What is the use of this test?
+        alpha, beta, a, x, y = self.get_data(x_stride=2)
+        with pytest.raises(Exception, match='failed for 3rd argument'):
+            y = self.blas_func(1, a, x, 1, y, trans=0, incx=3)
+        with pytest.raises(Exception, match='failed for 3rd argument'):
+            y = self.blas_func(1, a, x, 1, y, trans=1, incx=3)
+
+    def test_y_stride(self):
+        alpha, beta, a, x, y = self.get_data(y_stride=2)
+        desired_y = y.copy()
+        desired_y[::2] = alpha*matrixmultiply(a, x)+beta*y[::2]
+        y = self.blas_func(alpha, a, x, beta, y, incy=2)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_y_stride_transpose(self):
+        alpha, beta, a, x, y = self.get_data(y_stride=2)
+        desired_y = y.copy()
+        desired_y[::2] = alpha*matrixmultiply(transpose(a), x)+beta*y[::2]
+        y = self.blas_func(alpha, a, x, beta, y, trans=1, incy=2)
+        assert_array_almost_equal(desired_y, y)
+
+    def test_y_stride_assert(self):
+        # What is the use of this test?
+        alpha, beta, a, x, y = self.get_data(y_stride=2)
+        with pytest.raises(Exception, match='failed for 2nd keyword'):
+            y = self.blas_func(1, a, x, 1, y, trans=0, incy=3)
+        with pytest.raises(Exception, match='failed for 2nd keyword'):
+            y = self.blas_func(1, a, x, 1, y, trans=1, incy=3)
+
+
+try:
+    class TestSgemv(BaseGemv):
+        blas_func = fblas.sgemv
+        dtype = float32
+
+        @pytest.mark.skipif(sys.platform != 'darwin', reason="MacOS specific test")
+        def test_sgemv_on_osx(self):
+            def aligned_array(shape, align, dtype, order='C'):
+                # Make array shape `shape` with aligned at `align` bytes
+                d = dtype()
+                # Make array of correct size with `align` extra bytes
+                N = np.prod(shape)
+                tmp = np.zeros(N * d.nbytes + align, dtype=np.uint8)
+                address = tmp.__array_interface__["data"][0]
+                # Find offset into array giving desired alignment
+                for offset in range(align):
+                    if (address + offset) % align == 0:
+                        break
+                tmp = tmp[offset:offset+N*d.nbytes].view(dtype=dtype)
+                return tmp.reshape(shape, order=order)
+
+            def as_aligned(arr, align, dtype, order='C'):
+                # Copy `arr` into an aligned array with same shape
+                aligned = aligned_array(arr.shape, align, dtype, order)
+                aligned[:] = arr[:]
+                return aligned
+
+            def assert_dot_close(A, X, desired):
+                assert_allclose(self.blas_func(1.0, A, X), desired,
+                                rtol=1e-5, atol=1e-7)
+
+            testdata = product((15, 32), (10000,), (200, 89), ('C', 'F'))
+            rng = np.random.default_rng(1234)
+            for align, m, n, a_order in testdata:
+                A_d = rng.random((m, n))
+                X_d = rng.random(n)
+                desired = np.dot(A_d, X_d)
+                # Calculation with aligned single precision
+                A_f = as_aligned(A_d, align, np.float32, order=a_order)
+                X_f = as_aligned(X_d, align, np.float32, order=a_order)
+                assert_dot_close(A_f, X_f, desired)
+
+except AttributeError:
+    class TestSgemv:
+        pass
+
+
+class TestDgemv(BaseGemv):
+    blas_func = fblas.dgemv
+    dtype = float64
+
+
+try:
+    class TestCgemv(BaseGemv):
+        blas_func = fblas.cgemv
+        dtype = complex64
+except AttributeError:
+    class TestCgemv:
+        pass
+
+
+class TestZgemv(BaseGemv):
+    blas_func = fblas.zgemv
+    dtype = complex128
+
+
+"""
+##################################################
+### Test blas ?ger
+### This will be a mess to test all cases.
+
+class BaseGer:
+    def get_data(self,x_stride=1,y_stride=1):
+        rng = np.random.default_rng(1234)
+        alpha = array(1., dtype = self.dtype)
+        a = rng.normal(0.,1.,(3,3)).astype(self.dtype)
+        x = arange(shape(a)[0]*x_stride,dtype=self.dtype)
+        y = arange(shape(a)[1]*y_stride,dtype=self.dtype)
+        return alpha,a,x,y
+    def test_simple(self):
+        alpha,a,x,y = self.get_data()
+        # transpose takes care of Fortran vs. C(and Python) memory layout
+        desired_a = alpha*transpose(x[:,newaxis]*y) + a
+        self.blas_func(x,y,a)
+        assert_array_almost_equal(desired_a,a)
+    def test_x_stride(self):
+        alpha,a,x,y = self.get_data(x_stride=2)
+        desired_a = alpha*transpose(x[::2,newaxis]*y) + a
+        self.blas_func(x,y,a,incx=2)
+        assert_array_almost_equal(desired_a,a)
+    def test_x_stride_assert(self):
+        alpha,a,x,y = self.get_data(x_stride=2)
+        with pytest.raises(ValueError, match='foo'):
+            self.blas_func(x,y,a,incx=3)
+    def test_y_stride(self):
+        alpha,a,x,y = self.get_data(y_stride=2)
+        desired_a = alpha*transpose(x[:,newaxis]*y[::2]) + a
+        self.blas_func(x,y,a,incy=2)
+        assert_array_almost_equal(desired_a,a)
+
+    def test_y_stride_assert(self):
+        alpha,a,x,y = self.get_data(y_stride=2)
+        with pytest.raises(ValueError, match='foo'):
+            self.blas_func(a,x,y,incy=3)
+
+class TestSger(BaseGer):
+    blas_func = fblas.sger
+    dtype = float32
+class TestDger(BaseGer):
+    blas_func = fblas.dger
+    dtype = float64
+"""
+##################################################
+# Test blas ?gerc
+# This will be a mess to test all cases.
+
+"""
+class BaseGerComplex(BaseGer):
+    def get_data(self,x_stride=1,y_stride=1):
+        rng = np.random.default_rng(1234)
+        alpha = array(1+1j, dtype = self.dtype)
+        a = rng.normal(0.,1.,(3,3)).astype(self.dtype)
+        a = a + rng.normal(0.,1.,(3,3)) * array(1j, dtype = self.dtype)
+        x = rng.normal(0.,1.,shape(a)[0]*x_stride).astype(self.dtype)
+        x = x + x * array(1j, dtype = self.dtype)
+        y = rng.normal(0.,1.,shape(a)[1]*y_stride).astype(self.dtype)
+        y = y + y * array(1j, dtype = self.dtype)
+        return alpha,a,x,y
+    def test_simple(self):
+        alpha,a,x,y = self.get_data()
+        # transpose takes care of Fortran vs. C(and Python) memory layout
+        a = a * array(0.,dtype = self.dtype)
+        #desired_a = alpha*transpose(x[:,newaxis]*self.transform(y)) + a
+        desired_a = alpha*transpose(x[:,newaxis]*y) + a
+        #self.blas_func(x,y,a,alpha = alpha)
+        fblas.cgeru(x,y,a,alpha = alpha)
+        assert_array_almost_equal(desired_a,a)
+
+    #def test_x_stride(self):
+    #    alpha,a,x,y = self.get_data(x_stride=2)
+    #    desired_a = alpha*transpose(x[::2,newaxis]*self.transform(y)) + a
+    #    self.blas_func(x,y,a,incx=2)
+    #    assert_array_almost_equal(desired_a,a)
+    #def test_y_stride(self):
+    #    alpha,a,x,y = self.get_data(y_stride=2)
+    #    desired_a = alpha*transpose(x[:,newaxis]*self.transform(y[::2])) + a
+    #    self.blas_func(x,y,a,incy=2)
+    #    assert_array_almost_equal(desired_a,a)
+
+class TestCgeru(BaseGerComplex):
+    blas_func = fblas.cgeru
+    dtype = complex64
+    def transform(self,x):
+        return x
+class TestZgeru(BaseGerComplex):
+    blas_func = fblas.zgeru
+    dtype = complex128
+    def transform(self,x):
+        return x
+
+class TestCgerc(BaseGerComplex):
+    blas_func = fblas.cgerc
+    dtype = complex64
+    def transform(self,x):
+        return conjugate(x)
+
+class TestZgerc(BaseGerComplex):
+    blas_func = fblas.zgerc
+    dtype = complex128
+    def transform(self,x):
+        return conjugate(x)
+"""

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_interpolative.py

@@ -0,0 +1,232 @@
+#  ******************************************************************************
+#   Copyright (C) 2013 Kenneth L. Ho
+#   Redistribution and use in source and binary forms, with or without
+#   modification, are permitted provided that the following conditions are met:
+#
+#   Redistributions of source code must retain the above copyright notice, this
+#   list of conditions and the following disclaimer. Redistributions in binary
+#   form must reproduce the above copyright notice, this list of conditions and
+#   the following disclaimer in the documentation and/or other materials
+#   provided with the distribution.
+#
+#   None of the names of the copyright holders may be used to endorse or
+#   promote products derived from this software without specific prior written
+#   permission.
+#
+#   THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+#   AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+#   IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+#   ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
+#   LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+#   CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+#   SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+#   INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+#   CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+#   ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+#   POSSIBILITY OF SUCH DAMAGE.
+#  ******************************************************************************
+
+import scipy.linalg.interpolative as pymatrixid
+import numpy as np
+from scipy.linalg import hilbert, svdvals, norm
+from scipy.sparse.linalg import aslinearoperator
+from scipy.linalg.interpolative import interp_decomp
+
+from numpy.testing import (assert_, assert_allclose, assert_equal,
+                           assert_array_equal)
+import pytest
+from pytest import raises as assert_raises
+
+
+@pytest.fixture()
+def eps():
+    yield 1e-12
+
+
+@pytest.fixture()
+def rng():
+    rng = np.random.default_rng(1718313768084012)
+    yield rng
+
+
+@pytest.fixture(params=[np.float64, np.complex128])
+def A(request):
+    # construct Hilbert matrix
+    # set parameters
+    n = 300
+    yield hilbert(n).astype(request.param)
+
+
+@pytest.fixture()
+def L(A):
+    yield aslinearoperator(A)
+
+
+@pytest.fixture()
+def rank(A, eps):
+    S = np.linalg.svd(A, compute_uv=False)
+    try:
+        rank = np.nonzero(S < eps)[0][0]
+    except IndexError:
+        rank = A.shape[0]
+    return rank
+
+
+class TestInterpolativeDecomposition:
+
+    @pytest.mark.parametrize(
+        "rand,lin_op",
+        [(False, False), (True, False), (True, True)])
+    def test_real_id_fixed_precision(self, A, L, eps, rand, lin_op, rng):
+        # Test ID routines on a Hilbert matrix.
+        A_or_L = A if not lin_op else L
+
+        k, idx, proj = pymatrixid.interp_decomp(A_or_L, eps, rand=rand, rng=rng)
+        B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj)
+        assert_allclose(A, B, rtol=eps, atol=1e-08)
+
+    @pytest.mark.parametrize(
+        "rand,lin_op",
+        [(False, False), (True, False), (True, True)])
+    def test_real_id_fixed_rank(self, A, L, eps, rank, rand, lin_op, rng):
+        k = rank
+        A_or_L = A if not lin_op else L
+
+        idx, proj = pymatrixid.interp_decomp(A_or_L, k, rand=rand, rng=rng)
+        B = pymatrixid.reconstruct_matrix_from_id(A[:, idx[:k]], idx, proj)
+        assert_allclose(A, B, rtol=eps, atol=1e-08)
+
+    @pytest.mark.parametrize("rand,lin_op", [(False, False)])
+    def test_real_id_skel_and_interp_matrices(
+            self, A, L, eps, rank, rand, lin_op, rng):
+        k = rank
+        A_or_L = A if not lin_op else L
+
+        idx, proj = pymatrixid.interp_decomp(A_or_L, k, rand=rand, rng=rng)
+        P = pymatrixid.reconstruct_interp_matrix(idx, proj)
+        B = pymatrixid.reconstruct_skel_matrix(A, k, idx)
+        assert_allclose(B, A[:, idx[:k]], rtol=eps, atol=1e-08)
+        assert_allclose(B @ P, A, rtol=eps, atol=1e-08)
+
+    @pytest.mark.parametrize(
+        "rand,lin_op",
+        [(False, False), (True, False), (True, True)])
+    def test_svd_fixed_precision(self, A, L, eps, rand, lin_op, rng):
+        A_or_L = A if not lin_op else L
+
+        U, S, V = pymatrixid.svd(A_or_L, eps, rand=rand, rng=rng)
+        B = U * S @ V.T.conj()
+        assert_allclose(A, B, rtol=eps, atol=1e-08)
+
+    @pytest.mark.parametrize(
+        "rand,lin_op",
+        [(False, False), (True, False), (True, True)])
+    def test_svd_fixed_rank(self, A, L, eps, rank, rand, lin_op, rng):
+        k = rank
+        A_or_L = A if not lin_op else L
+
+        U, S, V = pymatrixid.svd(A_or_L, k, rand=rand, rng=rng)
+        B = U * S @ V.T.conj()
+        assert_allclose(A, B, rtol=eps, atol=1e-08)
+
+    def test_id_to_svd(self, A, eps, rank):
+        k = rank
+
+        idx, proj = pymatrixid.interp_decomp(A, k, rand=False)
+        U, S, V = pymatrixid.id_to_svd(A[:, idx[:k]], idx, proj)
+        B = U * S @ V.T.conj()
+        assert_allclose(A, B, rtol=eps, atol=1e-08)
+
+    def test_estimate_spectral_norm(self, A, rng):
+        s = svdvals(A)
+        norm_2_est = pymatrixid.estimate_spectral_norm(A, rng=rng)
+        assert_allclose(norm_2_est, s[0], rtol=1e-6, atol=1e-8)
+
+    def test_estimate_spectral_norm_diff(self, A, rng):
+        B = A.copy()
+        B[:, 0] *= 1.2
+        s = svdvals(A - B)
+        norm_2_est = pymatrixid.estimate_spectral_norm_diff(A, B, rng=rng)
+        assert_allclose(norm_2_est, s[0], rtol=1e-6, atol=1e-8)
+
+    def test_rank_estimates_array(self, A, rng):
+        B = np.array([[1, 1, 0], [0, 0, 1], [0, 0, 1]], dtype=A.dtype)
+
+        for M in [A, B]:
+            rank_tol = 1e-9
+            rank_np = np.linalg.matrix_rank(M, norm(M, 2) * rank_tol)
+            rank_est = pymatrixid.estimate_rank(M, rank_tol, rng=rng)
+            assert_(rank_est >= rank_np)
+            assert_(rank_est <= rank_np + 10)
+
+    def test_rank_estimates_lin_op(self, A, rng):
+        B = np.array([[1, 1, 0], [0, 0, 1], [0, 0, 1]], dtype=A.dtype)
+
+        for M in [A, B]:
+            ML = aslinearoperator(M)
+            rank_tol = 1e-9
+            rank_np = np.linalg.matrix_rank(M, norm(M, 2) * rank_tol)
+            rank_est = pymatrixid.estimate_rank(ML, rank_tol, rng=rng)
+            assert_(rank_est >= rank_np - 4)
+            assert_(rank_est <= rank_np + 4)
+
+    def test_badcall(self):
+        A = hilbert(5).astype(np.float32)
+        with assert_raises(ValueError):
+            pymatrixid.interp_decomp(A, 1e-6, rand=False)
+
+    def test_rank_too_large(self):
+        # svd(array, k) should not segfault
+        a = np.ones((4, 3))
+        with assert_raises(ValueError):
+            pymatrixid.svd(a, 4)
+
+    def test_full_rank(self):
+        eps = 1.0e-12
+        rng = np.random.default_rng(1234)
+        # fixed precision
+        A = rng.random((16, 8))
+        k, idx, proj = pymatrixid.interp_decomp(A, eps)
+        assert_equal(k, A.shape[1])
+
+        P = pymatrixid.reconstruct_interp_matrix(idx, proj)
+        B = pymatrixid.reconstruct_skel_matrix(A, k, idx)
+        assert_allclose(A, B @ P)
+
+        # fixed rank
+        idx, proj = pymatrixid.interp_decomp(A, k)
+
+        P = pymatrixid.reconstruct_interp_matrix(idx, proj)
+        B = pymatrixid.reconstruct_skel_matrix(A, k, idx)
+        assert_allclose(A, B @ P)
+
+    @pytest.mark.parametrize("dtype", [np.float64, np.complex128])
+    @pytest.mark.parametrize("rand", [True, False])
+    @pytest.mark.parametrize("eps", [1, 0.1])
+    def test_bug_9793(self, dtype, rand, eps):
+        A = np.array([[-1, -1, -1, 0, 0, 0],
+                      [0, 0, 0, 1, 1, 1],
+                      [1, 0, 0, 1, 0, 0],
+                      [0, 1, 0, 0, 1, 0],
+                      [0, 0, 1, 0, 0, 1]],
+                     dtype=dtype, order="C")
+        B = A.copy()
+        interp_decomp(A.T, eps, rand=rand)
+        assert_array_equal(A, B)
+
+    def test_svd_aslinearoperator_shape_check(self):
+        # See gh-issue #22451
+        rng = np.random.default_rng(1744580941832515)
+        x = rng.uniform(size=[7, 5])
+        xl = aslinearoperator(x)
+        u, s, v = pymatrixid.svd(xl, 3)
+        assert_equal(u.shape, (7, 3))
+        assert_equal(s.shape, (3,))
+        assert_equal(v.shape, (5, 3))
+
+        x = rng.uniform(size=[4, 9])
+        xl = aslinearoperator(x)
+        u, s, v = pymatrixid.svd(xl, 2)
+        assert_equal(u.shape, (4, 2))
+        assert_equal(s.shape, (2,))
+        assert_equal(v.shape, (9, 2))

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_matfuncs.py

@@ -0,0 +1,1149 @@
+#
+# Created by: Pearu Peterson, March 2002
+#
+""" Test functions for linalg.matfuncs module
+
+"""
+import functools
+import pytest
+import warnings
+
+import numpy as np
+from numpy import array, identity, sqrt
+from numpy.testing import (assert_array_almost_equal, assert_allclose, assert_,
+                           assert_array_less, assert_array_equal)
+
+import scipy.linalg
+from scipy.linalg import (funm, signm, logm, sqrtm, fractional_matrix_power,
+                          expm, expm_frechet, expm_cond, norm, khatri_rao,
+                          cosm, sinm, tanm, coshm, sinhm, tanhm)
+
+from scipy.linalg import _matfuncs_inv_ssq
+from scipy.linalg._matfuncs_inv_ssq import LogmExactlySingularWarning
+import scipy.linalg._expm_frechet
+from scipy.linalg import LinAlgWarning
+from scipy.optimize import minimize
+
+
+def _get_al_mohy_higham_2012_experiment_1():
+    """
+    Return the test matrix from Experiment (1) of [1]_.
+
+    References
+    ----------
+    .. [1] Awad H. Al-Mohy and Nicholas J. Higham (2012)
+           "Improved Inverse Scaling and Squaring Algorithms
+           for the Matrix Logarithm."
+           SIAM Journal on Scientific Computing, 34 (4). C152-C169.
+           ISSN 1095-7197
+
+    """
+    A = np.array([
+        [3.2346e-1, 3e4, 3e4, 3e4],
+        [0, 3.0089e-1, 3e4, 3e4],
+        [0, 0, 3.2210e-1, 3e4],
+        [0, 0, 0, 3.0744e-1]], dtype=float)
+    return A
+
+
+class TestSignM:
+
+    def test_nils(self):
+        a = array([[29.2, -24.2, 69.5, 49.8, 7.],
+                   [-9.2, 5.2, -18., -16.8, -2.],
+                   [-10., 6., -20., -18., -2.],
+                   [-9.6, 9.6, -25.5, -15.4, -2.],
+                   [9.8, -4.8, 18., 18.2, 2.]])
+        cr = array([[11.94933333,-2.24533333,15.31733333,21.65333333,-2.24533333],
+                    [-3.84266667,0.49866667,-4.59066667,-7.18666667,0.49866667],
+                    [-4.08,0.56,-4.92,-7.6,0.56],
+                    [-4.03466667,1.04266667,-5.59866667,-7.02666667,1.04266667],
+                    [4.15733333,-0.50133333,4.90933333,7.81333333,-0.50133333]])
+        r = signm(a)
+        assert_array_almost_equal(r,cr)
+
+    def test_defective1(self):
+        a = array([[0.0,1,0,0],[1,0,1,0],[0,0,0,1],[0,0,1,0]])
+        signm(a)
+        #XXX: what would be the correct result?
+
+    def test_defective2(self):
+        a = array((
+            [29.2,-24.2,69.5,49.8,7.0],
+            [-9.2,5.2,-18.0,-16.8,-2.0],
+            [-10.0,6.0,-20.0,-18.0,-2.0],
+            [-9.6,9.6,-25.5,-15.4,-2.0],
+            [9.8,-4.8,18.0,18.2,2.0]))
+        signm(a)
+        #XXX: what would be the correct result?
+
+    def test_defective3(self):
+        a = array([[-2., 25., 0., 0., 0., 0., 0.],
+                   [0., -3., 10., 3., 3., 3., 0.],
+                   [0., 0., 2., 15., 3., 3., 0.],
+                   [0., 0., 0., 0., 15., 3., 0.],
+                   [0., 0., 0., 0., 3., 10., 0.],
+                   [0., 0., 0., 0., 0., -2., 25.],
+                   [0., 0., 0., 0., 0., 0., -3.]])
+        signm(a)
+        #XXX: what would be the correct result?
+
+    @pytest.mark.parametrize('dtype', [np.float32, np.float64,
+                                       np.complex64, np.complex128])
+    def test_signm_dtype_preservation(self, dtype):
+        # gh-25657: signm upcast float32/complex64 for defective matrices
+        # (the iterative fall-through branch) while the non-defective branch
+        # preserved dtype. Output dtype must depend on input dtype, not values.
+        non_defective = np.array([[2, 1], [0, 3]], dtype=dtype)  # distinct eigs
+        defective = np.array([[2, 1], [0, 2]], dtype=dtype)      # repeated eig
+        assert signm(non_defective).dtype == dtype
+        assert signm(defective).dtype == dtype
+
+
+class TestLogM:
+    @pytest.mark.filterwarnings("ignore:.*inaccurate.*:RuntimeWarning")
+    def test_nils(self):
+        a = array([[-2., 25., 0., 0., 0., 0., 0.],
+                   [0., -3., 10., 3., 3., 3., 0.],
+                   [0., 0., 2., 15., 3., 3., 0.],
+                   [0., 0., 0., 0., 15., 3., 0.],
+                   [0., 0., 0., 0., 3., 10., 0.],
+                   [0., 0., 0., 0., 0., -2., 25.],
+                   [0., 0., 0., 0., 0., 0., -3.]])
+        m = (identity(7)*3.1+0j)-a
+        logm(m)
+        #XXX: what would be the correct result?
+
+    @pytest.mark.filterwarnings("ignore:.*inaccurate.*:RuntimeWarning")
+    def test_al_mohy_higham_2012_experiment_1_logm(self):
+        # The logm completes the round trip successfully.
+        # Note that the expm leg of the round trip is badly conditioned.
+        A = _get_al_mohy_higham_2012_experiment_1()
+        A_logm = logm(A)
+        A_round_trip = expm(A_logm)
+        assert_allclose(A_round_trip, A, rtol=5e-5, atol=1e-14)
+
+    def test_al_mohy_higham_2012_experiment_1_funm_log(self):
+        # The raw funm with np.log does not complete the round trip.
+        # Note that the expm leg of the round trip is badly conditioned.
+        A = _get_al_mohy_higham_2012_experiment_1()
+        A_funm_log = funm(A, np.log)
+        A_round_trip = expm(A_funm_log)
+        assert_(not np.allclose(A_round_trip, A, rtol=1e-5, atol=1e-14))
+
+    def test_round_trip_random_float(self):
+        rng = np.random.default_rng(1738098768840254)
+        for n in range(1, 6):
+            M_unscaled = rng.uniform(size=(n, n))
+            for scale in np.logspace(-4, 4, 9):
+                M = M_unscaled * scale
+
+                # Eigenvalues are related to the branch cut.
+                W = np.linalg.eigvals(M)
+                err_msg = f'M:{M} eivals:{W}'
+
+                # Check sqrtm round trip because it is used within logm.
+                M_sqrtm = sqrtm(M)
+                M_sqrtm_round_trip = M_sqrtm @ M_sqrtm
+                assert_allclose(M_sqrtm_round_trip, M)
+
+                # Check logm round trip.
+                with warnings.catch_warnings():
+                    warnings.simplefilter("ignore", RuntimeWarning)
+
+                    M_logm = logm(M)
+                    M_logm_round_trip = expm(M_logm)
+                    assert_allclose(M_logm_round_trip, M, err_msg=err_msg)
+
+    def test_round_trip_random_complex(self):
+        rng = np.random.default_rng(1738098768840254)
+        for n in range(1, 6):
+            M_unscaled = (rng.standard_normal((n, n)) +
+                          1j*rng.standard_normal((n, n)))
+            for scale in np.logspace(-4, 4, 9):
+                M = M_unscaled * scale
+                M_logm = logm(M)
+                M_round_trip = expm(M_logm)
+                assert_allclose(M_round_trip, M)
+
+    def test_logm_type_preservation_and_conversion(self):
+        # The logm matrix function should preserve the type of a matrix
+        # whose eigenvalues are positive with zero imaginary part.
+        # Test this preservation for variously structured matrices.
+        complex_dtype_chars = ('F', 'D', 'G')
+        for matrix_as_list in (
+                [[1, 0], [0, 1]],
+                [[1, 0], [1, 1]],
+                [[2, 1], [1, 1]],
+                [[2, 3], [1, 2]]):
+
+            # check that the spectrum has the expected properties
+            W = scipy.linalg.eigvals(matrix_as_list)
+            assert_(not any(w.imag or w.real < 0 for w in W))
+
+            # check float type preservation
+            A = np.array(matrix_as_list, dtype=float)
+            A_logm = logm(A)
+            assert_(A_logm.dtype.char not in complex_dtype_chars)
+
+            # check complex type preservation
+            A = np.array(matrix_as_list, dtype=complex)
+            A_logm = logm(A)
+            assert_(A_logm.dtype.char in complex_dtype_chars)
+
+            # check float->complex type conversion for the matrix negation
+            A = -np.array(matrix_as_list, dtype=float)
+            A_logm = logm(A)
+            assert_(A_logm.dtype.char in complex_dtype_chars)
+
+    def test_complex_spectrum_real_logm(self):
+        # This matrix has complex eigenvalues and real logm.
+        # Its output dtype depends on its input dtype.
+        M = [[1, 1, 2], [2, 1, 1], [1, 2, 1]]
+        for dt in float, complex:
+            X = np.array(M, dtype=dt)
+            w = scipy.linalg.eigvals(X)
+            assert_(1e-2 < np.absolute(w.imag).sum())
+            Y = logm(X)
+            assert_(np.issubdtype(Y.dtype, np.inexact))
+            assert_allclose(expm(Y), X)
+
+    def test_real_mixed_sign_spectrum(self):
+        # These matrices have real eigenvalues with mixed signs.
+        # The output logm dtype is complex, regardless of input dtype.
+        for M in (
+                [[1, 0], [0, -1]],
+                [[0, 1], [1, 0]]):
+            for dt in float, complex:
+                A = np.array(M, dtype=dt)
+                A_logm, info = logm(A)
+                assert_(np.issubdtype(A_logm.dtype, np.complexfloating))
+
+    def test_exactly_singular(self):
+        A = np.array([[0, 0], [1j, 1j]])
+        B = np.asarray([[1, 1], [0, 0]])
+        for M in A, A.T, B, B.T:
+            with pytest.warns(_matfuncs_inv_ssq.LogmExactlySingularWarning):
+                L = logm(M)
+            E = expm(L)
+            assert_allclose(E, M, atol=1e-14)
+
+    def test_nearly_singular(self):
+        M = np.array([[1e-100]])
+        with pytest.warns(_matfuncs_inv_ssq.LogmNearlySingularWarning):
+            L = logm(M)
+        E = expm(L)
+        assert_allclose(E, M, atol=1e-14)
+
+    def test_opposite_sign_complex_eigenvalues(self):
+        # See gh-6113
+        E = [[0, 1], [-1, 0]]
+        L = [[0, np.pi*0.5], [-np.pi*0.5, 0]]
+        assert_allclose(expm(L), E, atol=1e-14)
+        assert_allclose(logm(E), L, atol=1e-14)
+        E = [[1j, 4], [0, -1j]]
+        L = [[1j*np.pi*0.5, 2*np.pi], [0, -1j*np.pi*0.5]]
+        assert_allclose(expm(L), E, atol=1e-14)
+        assert_allclose(logm(E), L, atol=1e-14)
+        E = [[1j, 0], [0, -1j]]
+        L = [[1j*np.pi*0.5, 0], [0, -1j*np.pi*0.5]]
+        assert_allclose(expm(L), E, atol=1e-14)
+        assert_allclose(logm(E), L, atol=1e-14)
+
+    def test_readonly(self):
+        n = 5
+        a = np.ones((n, n)) + np.identity(n)
+        a.flags.writeable = False
+        logm(a)
+
+    @pytest.mark.xfail(reason="ValueError: attempt to get argmax of an empty sequence")
+    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
+    def test_empty(self, dt):
+        a = np.empty((0, 0), dtype=dt)
+        log_a = logm(a)
+        a0 = np.eye(2, dtype=dt)
+        log_a0 = logm(a0)
+
+        assert log_a.shape == (0, 0)
+        assert log_a.dtype == log_a0.dtype
+
+    @pytest.mark.parametrize('dtype', [int, float, np.float32, complex, np.complex64])
+    def test_no_ZeroDivisionError(self, dtype):
+        # gh-17136 reported inconsistent behavior in `logm` depending on input dtype:
+        # sometimes it raised an error, and sometimes it printed a warning message.
+        # check that this is resolved and that the warning is emitted properly.
+        with (pytest.warns(RuntimeWarning, match="logm result may be inaccurate"),
+              pytest.warns(LogmExactlySingularWarning)):
+            logm(np.zeros((2, 2), dtype=dtype))
+
+
+class TestSqrtM:
+
+    def test_round_trip_random_float(self):
+        rng = np.random.default_rng(1738151906092735)
+        for n in range(1, 6):
+            M_unscaled = rng.standard_normal((n, n))
+            for scale in np.logspace(-4, 4, 9):
+                M = M_unscaled * scale
+                M_sqrtm = sqrtm(M)
+                M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm)
+                assert_allclose(M_sqrtm_round_trip, M)
+
+    def test_round_trip_random_complex(self):
+        rng = np.random.default_rng(1738151906092736)
+        for n in range(1, 6):
+            M_unscaled = (rng.standard_normal((n, n)) +
+                          1j * rng.standard_normal((n, n)))
+            for scale in np.logspace(-4, 4, 9):
+                M = M_unscaled * scale
+                M_sqrtm = sqrtm(M)
+                M_sqrtm_round_trip = M_sqrtm.dot(M_sqrtm)
+                assert_allclose(M_sqrtm_round_trip, M)
+
+    def test_bad(self):
+        # See https://web.archive.org/web/20051220232650/http://www.maths.man.ac.uk/~nareports/narep336.ps.gz
+        e = 2**-5
+        se = sqrt(e)
+        a = array([[1.0,0,0,1],
+                   [0,e,0,0],
+                   [0,0,e,0],
+                   [0,0,0,1]])
+        sa = array([[1,0,0,0.5],
+                    [0,se,0,0],
+                    [0,0,se,0],
+                    [0,0,0,1]])
+        assert_array_almost_equal(sa @ sa, a)
+        # Check default sqrtm.
+        esa = sqrtm(a)
+        assert_array_almost_equal(esa @ esa, a)
+
+    def test_sqrtm_type_preservation_and_conversion(self):
+        # The sqrtm matrix function should preserve the type of a matrix
+        # whose eigenvalues are nonnegative with zero imaginary part.
+        # Test this preservation for variously structured matrices.
+        complex_dtype_chars = ('F', 'D', 'G')
+        for matrix_as_list in (
+                [[1, 0], [0, 1]],
+                [[1, 0], [1, 1]],
+                [[2, 1], [1, 1]],
+                [[2, 3], [1, 2]],
+                [[1, 1], [1, 1]]):
+
+            # check that the spectrum has the expected properties
+            W = scipy.linalg.eigvals(matrix_as_list)
+            assert_(not any(w.imag or w.real < 0 for w in W))
+
+            # Last test matrix is singular so suppress the warning
+            with warnings.catch_warnings():
+                warnings.simplefilter("ignore", LinAlgWarning)
+
+                # check float type preservation
+                A = np.array(matrix_as_list, dtype=float)
+                A_sqrtm = sqrtm(A)
+                assert_(A_sqrtm.dtype.char not in complex_dtype_chars)
+
+                # check complex type preservation
+                A = np.array(matrix_as_list, dtype=complex)
+                A_sqrtm = sqrtm(A)
+                assert_(A_sqrtm.dtype.char in complex_dtype_chars)
+
+                # check float->complex type conversion for the matrix negation
+                A = -np.array(matrix_as_list, dtype=float)
+                A_sqrtm = sqrtm(A)
+                assert_(A_sqrtm.dtype.char in complex_dtype_chars)
+
+    def test_sqrtm_type_conversion_mixed_sign_or_complex_spectrum(self):
+        complex_dtype_chars = ('F', 'D', 'G')
+        for matrix_as_list in (
+                [[1, 0], [0, -1]],
+                [[0, 1], [1, 0]],
+                [[0, 1, 0], [0, 0, -1], [1, 0, 0]]):
+
+            # check that the spectrum has the expected properties
+            W = scipy.linalg.eigvals(matrix_as_list)
+            assert_(any(w.imag or w.real < 0 for w in W))
+
+            # check complex->complex
+            A = np.array(matrix_as_list, dtype=complex)
+            A_sqrtm = sqrtm(A)
+            assert_(A_sqrtm.dtype.char in complex_dtype_chars)
+
+            # check float->complex
+            A = np.array(matrix_as_list, dtype=float)
+            A_sqrtm = sqrtm(A)
+            assert_(A_sqrtm.dtype.char in complex_dtype_chars)
+
+    def test_al_mohy_higham_2012_experiment_1(self):
+        # Matrix square root of a tricky upper triangular matrix.
+        A = _get_al_mohy_higham_2012_experiment_1()
+        A_sqrtm = sqrtm(A)
+        A_round_trip = A_sqrtm @ A_sqrtm
+        assert_allclose(A_round_trip, A, rtol=1e-5)
+        assert_allclose(np.tril(A_round_trip), np.tril(A))
+
+    def test_strict_upper_triangular(self):
+        # This matrix has no square root but upper triangular hence upper
+        # triangle will be filled with junk values.
+        for dt in int, float:
+            A = np.array([
+                [0, 3, 0, 0],
+                [0, 0, 3, 0],
+                [0, 0, 0, 3],
+                [0, 0, 0, 0]], dtype=dt)
+            with warnings.catch_warnings():
+                warnings.simplefilter("ignore", LinAlgWarning)
+
+                A_sqrtm = sqrtm(A)
+                assert_allclose(np.tril(A_sqrtm), np.zeros((4, 4)))
+                assert np.isnan(A_sqrtm).any()
+                assert np.isinf(A_sqrtm).any()
+
+    # Future edit: This squareroot is not possible to find algorithmically
+    # with the current methods. Now sqrtm docstring has another example of
+    # such matrix whose squareroot is not a polynomial in it. Hence no need
+    # to test it here.
+    """
+    def test_weird_matrix(self):
+        # The square root of matrix B exists.
+        for dt in int, float:
+            A = np.array([
+                [0, 0, 1],
+                [0, 0, 0],
+                [0, 1, 0]], dtype=dt)
+            B = np.array([
+                [0, 1, 0],
+                [0, 0, 0],
+                [0, 0, 0]], dtype=dt)
+            assert_array_equal(B, A @ A)
+
+            # But scipy sqrtm is not clever enough to find it.
+            B_sqrtm, info = sqrtm(B, disp=False)
+            assert_(np.isnan(B_sqrtm).all())
+    """
+
+    def test_opposite_sign_complex_eigenvalues(self):
+        M = [[2j, 4], [0, -2j]]
+        R = [[1+1j, 2], [0, 1-1j]]
+        assert_allclose(np.dot(R, R), M, atol=1e-14)
+        assert_allclose(sqrtm(M), R, atol=1e-14)
+
+    def test_gh4866(self):
+        M = np.array([[1, 0, 0, 1],
+                      [0, 0, 0, 0],
+                      [0, 0, 0, 0],
+                      [1, 0, 0, 1]])
+        R = np.array([[sqrt(0.5), 0, 0, sqrt(0.5)],
+                      [0, 0, 0, 0],
+                      [0, 0, 0, 0],
+                      [sqrt(0.5), 0, 0, sqrt(0.5)]])
+        assert_allclose(R @ R, M, atol=1e-14)
+        with warnings.catch_warnings():
+            warnings.simplefilter("ignore", LinAlgWarning)
+
+            assert_allclose(sqrtm(M), R, atol=1e-14)
+
+    def test_gh5336(self):
+        M = np.diag([2, 1, 0])
+        R = np.diag([sqrt(2), 1, 0])
+        assert_allclose(R @ R, M, atol=1e-14)
+        with warnings.catch_warnings():
+            warnings.filterwarnings("ignore", category=LinAlgWarning)
+            assert_allclose(sqrtm(M), R, atol=1e-14)
+
+    def test_gh7839(self):
+        M = np.zeros((2, 2))
+        R = np.zeros((2, 2))
+        # Catch and silence LinAlgWarning
+        with warnings.catch_warnings():
+            warnings.filterwarnings("ignore", category=LinAlgWarning)
+
+            assert_allclose(sqrtm(M), R, atol=1e-14)
+
+    def test_gh17918(self):
+        M = np.empty((19, 19))
+        M.fill(0.94)
+        np.fill_diagonal(M, 1)
+        assert np.isrealobj(sqrtm(M))
+
+    def test_gh23278(self):
+        M = np.array([[1., 0., 0.], [0, 1, -1j], [0, 1j, 2]])
+        sq = sqrtm(M)
+        assert_allclose(sq @ sq, M, atol=1e-14)
+        sq = sqrtm(M.astype(np.complex64))
+        assert_allclose(sq @ sq, M, atol=1e-6)
+
+    def test_data_size_preservation_uint_in_float_out(self):
+        M = np.eye(10, dtype=np.uint8)
+        assert sqrtm(M).dtype == np.float64
+        M = np.eye(10, dtype=np.uint16)
+        assert sqrtm(M).dtype == np.float64
+        M = np.eye(10, dtype=np.uint32)
+        assert sqrtm(M).dtype == np.float64
+        M = np.eye(10, dtype=np.uint64)
+        assert sqrtm(M).dtype == np.float64
+
+    def test_data_size_preservation_int_in_float_out(self):
+        M = np.eye(10, dtype=np.int8)
+        assert sqrtm(M).dtype == np.float64
+        M = np.eye(10, dtype=np.int16)
+        assert sqrtm(M).dtype == np.float64
+        M = np.eye(10, dtype=np.int32)
+        assert sqrtm(M).dtype == np.float64
+        M = np.eye(10, dtype=np.int64)
+        assert sqrtm(M).dtype == np.float64
+
+    def test_data_size_preservation_int_in_comp_out(self):
+        M = np.array([[2, 4], [0, -2]], dtype=np.int8)
+        assert sqrtm(M).dtype == np.complex128
+        M = np.array([[2, 4], [0, -2]], dtype=np.int16)
+        assert sqrtm(M).dtype == np.complex128
+        M = np.array([[2, 4], [0, -2]], dtype=np.int32)
+        assert sqrtm(M).dtype == np.complex128
+        M = np.array([[2, 4], [0, -2]], dtype=np.int64)
+        assert sqrtm(M).dtype == np.complex128
+
+    @pytest.mark.filterwarnings("ignore::DeprecationWarning")
+    def test_data_size_preservation_float_in_float_out(self):
+        M = np.eye(10, dtype=np.float16)
+        assert sqrtm(M).dtype == np.float32
+        M = np.eye(10, dtype=np.float32)
+        assert sqrtm(M).dtype == np.float32
+        M = np.eye(10, dtype=np.float64)
+        assert sqrtm(M).dtype == np.float64
+        if hasattr(np, 'float128'):
+            M = np.eye(10, dtype=np.float128)
+            assert sqrtm(M).dtype == np.float64
+
+    @pytest.mark.filterwarnings("ignore::DeprecationWarning")
+    def test_data_size_preservation_float_in_comp_out(self):
+        M = np.array([[2, 4], [0, -2]], dtype=np.float16)
+        assert sqrtm(M).dtype == np.complex64
+        M = np.array([[2, 4], [0, -2]], dtype=np.float32)
+        assert sqrtm(M).dtype == np.complex64
+        M = np.array([[2, 4], [0, -2]], dtype=np.float64)
+        assert sqrtm(M).dtype == np.complex128
+        if hasattr(np, 'float128') and hasattr(np, 'complex256'):
+            M = np.array([[2, 4], [0, -2]], dtype=np.float128)
+            assert sqrtm(M).dtype == np.complex128
+
+    @pytest.mark.filterwarnings("ignore::DeprecationWarning")
+    def test_data_size_preservation_comp_in_comp_out(self):
+        M = np.array([[2j, 4], [0, -2j]], dtype=np.complex64)
+        assert sqrtm(M).dtype == np.complex64
+        M = np.array([[2j, 4], [0, -2j]], dtype=np.complex128)
+        assert sqrtm(M).dtype == np.complex128
+        if hasattr(np, 'complex256'):
+            M = np.array([[2j, 4], [0, -2j]], dtype=np.complex256)
+            assert sqrtm(M).dtype == np.complex128
+
+    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
+    def test_empty(self, dt):
+        a = np.empty((0, 0), dtype=dt)
+        s = sqrtm(a)
+        a0 = np.eye(2, dtype=dt)
+        s0 = sqrtm(a0)
+
+        assert s.shape == (0, 0)
+        assert s.dtype == s0.dtype
+
+    def test_cf_noncontig_nd_inputs(self):
+        # Check that non-contiguous arrays are handled correctly.
+        # Generate an L, U pair for invertible random matrix.
+        rng = np.random.default_rng(1738151906092737)
+        n = 13
+        A = rng.uniform(size=(3, 2*n, 2*n))
+        L, U = np.tril(A, k=-1) + np.eye(2*n), np.triu(A)
+        A = L @ U
+        # Create strided views of 3D array.
+        A_noncontig_c = A[:, ::2, ::2]
+        A_noncontig_f = np.asfortranarray(A)[:, 1::2, 1::2]
+        assert_allclose(sqrtm(A[:, ::2, ::2]), sqrtm(A_noncontig_c))
+        assert_allclose(sqrtm(A[:, 1::2, 1::2]), sqrtm(A_noncontig_f))
+
+    def test_empty_sizes(self):
+        A = np.empty(shape=[4, 0, 5, 5], dtype=float)
+        assert_array_equal(sqrtm(A), A)
+
+    def test_negative_strides(self):
+        rng = np.random.default_rng(1738151906092738)
+        A = rng.uniform(size=(3, 5, 5))
+        A_negneg_orig = A[:, ::-1, ::-1]
+        A_negneg_copy = A[:, ::-1, ::-1].copy()
+        assert_allclose(sqrtm(A_negneg_orig), sqrtm(A_negneg_copy))
+
+        A_posneg_orig = A[:, :, ::-1]
+        A_posneg_copy = A[:, :, ::-1].copy()
+        assert_allclose(sqrtm(A_posneg_orig), sqrtm(A_posneg_copy))
+
+        A_negpos_orig = A[:, ::-1, :]
+        A_negpos_copy = A[:, ::-1, :].copy()
+        assert_allclose(sqrtm(A_negpos_orig), sqrtm(A_negpos_copy))
+
+    @pytest.mark.parametrize('dtyp', [np.float32, np.float64,
+                                      np.complex64, np.complex128])
+    def test_gh24508(self, dtyp):
+        # Check that sqrtm raises a warning when the input is singular
+        # And for this example, the top row is filled with infs.
+        a = np.array([[0, 2, 2], [0, 0, 0], [0, 0, 0]], dtype=dtyp)
+        with pytest.warns(LinAlgWarning, match="Matrix is singular."):
+            res = sqrtm(a)
+            assert np.isinf(res[0, 1:]).all()
+
+
+class TestFractionalMatrixPower:
+    def test_round_trip_random_complex(self):
+        rng = np.random.default_rng(1234)
+        for p in range(1, 5):
+            for n in range(1, 5):
+                M_unscaled = (rng.standard_normal((n, n)) +
+                              1j * rng.standard_normal((n, n)))
+                for scale in np.logspace(-4, 4, 9):
+                    M = M_unscaled * scale
+                    M_root = fractional_matrix_power(M, 1/p)
+                    M_round_trip = np.linalg.matrix_power(M_root, p)
+                    assert_allclose(M_round_trip, M)
+
+    def test_round_trip_random_float(self):
+        # This test is more annoying because it can hit the branch cut;
+        # this happens when the matrix has an eigenvalue
+        # with no imaginary component and with a real negative component,
+        # and it means that the principal branch does not exist.
+        rng = np.random.default_rng(1234)
+        for p in range(1, 5):
+            for n in range(1, 5):
+                M_unscaled = rng.standard_normal((n, n))
+                for scale in np.logspace(-4, 4, 9):
+                    M = M_unscaled * scale
+                    M_root = fractional_matrix_power(M, 1/p)
+                    M_round_trip = np.linalg.matrix_power(M_root, p)
+                    assert_allclose(M_round_trip, M)
+
+    def test_larger_abs_fractional_matrix_powers(self):
+        rng = np.random.default_rng(1234)
+        for n in (2, 3, 5):
+            for i in range(10):
+                M = rng.standard_normal((n, n)) + 1j * rng.standard_normal((n, n))
+                M_one_fifth = fractional_matrix_power(M, 0.2)
+                # Test the round trip.
+                M_round_trip = np.linalg.matrix_power(M_one_fifth, 5)
+                assert_allclose(M, M_round_trip)
+                # Test a large abs fractional power.
+                X = fractional_matrix_power(M, -5.4)
+                Y = np.linalg.matrix_power(M_one_fifth, -27)
+                assert_allclose(X, Y)
+                # Test another large abs fractional power.
+                X = fractional_matrix_power(M, 3.8)
+                Y = np.linalg.matrix_power(M_one_fifth, 19)
+                assert_allclose(X, Y)
+
+    def test_random_matrices_and_powers(self):
+        # Each independent iteration of this fuzz test picks random parameters.
+        # It tries to hit some edge cases.
+        rng = np.random.default_rng(1726500458620605)
+        nsamples = 20
+        for i in range(nsamples):
+            # Sample a matrix size and a random real power.
+            n = rng.integers(1, 5)
+            p = rng.random()
+
+            # Sample a random real or complex matrix.
+            matrix_scale = np.exp(rng.integers(-4, 5))
+            A = rng.random(size=[n, n])
+            if [True, False][rng.choice(2)]:
+                A = A + 1j * rng.random(size=[n, n])
+            A = A * matrix_scale
+
+            # Check a couple of analytically equivalent ways
+            # to compute the fractional matrix power.
+            # These can be compared because they both use the principal branch.
+            A_power = fractional_matrix_power(A, p)
+            A_logm = logm(A)
+            A_power_expm_logm = expm(A_logm * p)
+            assert_allclose(A_power, A_power_expm_logm)
+
+    def test_al_mohy_higham_2012_experiment_1(self):
+        # Fractional powers of a tricky upper triangular matrix.
+        A = _get_al_mohy_higham_2012_experiment_1()
+
+        # Test remainder matrix power.
+        A_funm_sqrt = funm(A, np.sqrt)
+        A_sqrtm = sqrtm(A)
+        A_rem_power = _matfuncs_inv_ssq._remainder_matrix_power(A, 0.5)
+        A_power = fractional_matrix_power(A, 0.5)
+        assert_allclose(A_rem_power, A_power, rtol=1e-11)
+        assert_allclose(A_sqrtm, A_power)
+        assert_allclose(A_sqrtm, A_funm_sqrt)
+
+        # Test more fractional powers.
+        for p in (1/2, 5/3):
+            A_power = fractional_matrix_power(A, p)
+            A_round_trip = fractional_matrix_power(A_power, 1/p)
+            assert_allclose(A_round_trip, A, rtol=1e-2)
+            assert_allclose(np.tril(A_round_trip, 1), np.tril(A, 1))
+
+    def test_briggs_helper_function(self):
+        rng = np.random.default_rng(1234)
+        for a in rng.standard_normal(10) + 1j * rng.standard_normal(10):
+            for k in range(5):
+                x_observed = _matfuncs_inv_ssq._briggs_helper_function(a, k)
+                x_expected = a ** np.exp2(-k) - 1
+                assert_allclose(x_observed, x_expected)
+
+    def test_type_preservation_and_conversion(self):
+        # The fractional_matrix_power matrix function should preserve
+        # the type of a matrix whose eigenvalues
+        # are positive with zero imaginary part.
+        # Test this preservation for variously structured matrices.
+        complex_dtype_chars = ('F', 'D', 'G')
+        for matrix_as_list in (
+                [[1, 0], [0, 1]],
+                [[1, 0], [1, 1]],
+                [[2, 1], [1, 1]],
+                [[2, 3], [1, 2]]):
+
+            # check that the spectrum has the expected properties
+            W = scipy.linalg.eigvals(matrix_as_list)
+            assert_(not any(w.imag or w.real < 0 for w in W))
+
+            # Check various positive and negative powers
+            # with absolute values bigger and smaller than 1.
+            for p in (-2.4, -0.9, 0.2, 3.3):
+
+                # check float type preservation
+                A = np.array(matrix_as_list, dtype=float)
+                A_power = fractional_matrix_power(A, p)
+                assert_(A_power.dtype.char not in complex_dtype_chars)
+
+                # check complex type preservation
+                A = np.array(matrix_as_list, dtype=complex)
+                A_power = fractional_matrix_power(A, p)
+                assert_(A_power.dtype.char in complex_dtype_chars)
+
+                # check float->complex for the matrix negation
+                A = -np.array(matrix_as_list, dtype=float)
+                A_power = fractional_matrix_power(A, p)
+                assert_(A_power.dtype.char in complex_dtype_chars)
+
+    def test_type_conversion_mixed_sign_or_complex_spectrum(self):
+        complex_dtype_chars = ('F', 'D', 'G')
+        for matrix_as_list in (
+                [[1, 0], [0, -1]],
+                [[0, 1], [1, 0]],
+                [[0, 1, 0], [0, 0, 1], [1, 0, 0]]):
+
+            # check that the spectrum has the expected properties
+            W = scipy.linalg.eigvals(matrix_as_list)
+            assert_(any(w.imag or w.real < 0 for w in W))
+
+            # Check various positive and negative powers
+            # with absolute values bigger and smaller than 1.
+            for p in (-2.4, -0.9, 0.2, 3.3):
+
+                # check complex->complex
+                A = np.array(matrix_as_list, dtype=complex)
+                A_power = fractional_matrix_power(A, p)
+                assert_(A_power.dtype.char in complex_dtype_chars)
+
+                # check float->complex
+                A = np.array(matrix_as_list, dtype=float)
+                A_power = fractional_matrix_power(A, p)
+                assert_(A_power.dtype.char in complex_dtype_chars)
+
+    @pytest.mark.xfail(reason='Too unstable across LAPACKs.')
+    def test_singular(self):
+        # Negative fractional powers do not work with singular matrices.
+        for matrix_as_list in (
+                [[0, 0], [0, 0]],
+                [[1, 1], [1, 1]],
+                [[1, 2], [3, 6]],
+                [[0, 0, 0], [0, 1, 1], [0, -1, 1]]):
+
+            # Check fractional powers both for float and for complex types.
+            for newtype in (float, complex):
+                A = np.array(matrix_as_list, dtype=newtype)
+                for p in (-0.7, -0.9, -2.4, -1.3):
+                    A_power = fractional_matrix_power(A, p)
+                    assert_(np.isnan(A_power).all())
+                for p in (0.2, 1.43):
+                    A_power = fractional_matrix_power(A, p)
+                    A_round_trip = fractional_matrix_power(A_power, 1/p)
+                    assert_allclose(A_round_trip, A)
+
+    def test_opposite_sign_complex_eigenvalues(self):
+        M = [[2j, 4], [0, -2j]]
+        R = [[1+1j, 2], [0, 1-1j]]
+        assert_allclose(np.dot(R, R), M, atol=1e-14)
+        assert_allclose(fractional_matrix_power(M, 0.5), R, atol=1e-14)
+
+
+class TestExpM:
+    def test_zero(self):
+        a = array([[0.,0],[0,0]])
+        assert_array_almost_equal(expm(a),[[1,0],[0,1]])
+
+    def test_single_elt(self):
+        elt = expm(1)
+        assert_allclose(elt, np.array([[np.e]]))
+
+    @pytest.mark.parametrize('func', [expm, cosm, sinm, tanm, coshm, sinhm, tanhm])
+    @pytest.mark.parametrize('dt',[int, float, np.float32, complex, np.complex64])
+    @pytest.mark.parametrize('shape', [(0, 0), (1, 1)])
+    def test_small_empty_matrix_input(self, func, dt, shape):
+        # regression test for gh-11082 / gh-20372 - test behavior of expm
+        # and related functions for small and zero-sized arrays.
+        A = np.zeros(shape, dtype=dt)
+        A0 = np.zeros((10, 10), dtype=dt)
+        result = func(A)
+        result0 = func(A0)
+        assert result.shape == shape
+        assert result.dtype == result0.dtype
+
+    def test_2x2_input(self):
+        E = np.e
+        a = array([[1, 4], [1, 1]])
+        aa = (E**4 + 1)/(2*E)
+        bb = (E**4 - 1)/E
+        assert_allclose(expm(a), array([[aa, bb], [bb/4, aa]]))
+        assert expm(a.astype(np.complex64)).dtype.char == 'F'
+        assert expm(a.astype(np.float32)).dtype.char == 'f'
+
+    def test_nx2x2_input(self):
+        E = np.e
+        # These are integer matrices with integer eigenvalues
+        a = np.array([[[1, 4], [1, 1]],
+                      [[1, 3], [1, -1]],
+                      [[1, 3], [4, 5]],
+                      [[1, 3], [5, 3]],
+                      [[4, 5], [-3, -4]]], order='F')
+        # Exact results are computed symbolically
+        a_res = np.array([
+                          [[(E**4+1)/(2*E), (E**4-1)/E],
+                           [(E**4-1)/4/E, (E**4+1)/(2*E)]],
+                          [[1/(4*E**2)+(3*E**2)/4, (3*E**2)/4-3/(4*E**2)],
+                           [E**2/4-1/(4*E**2), 3/(4*E**2)+E**2/4]],
+                          [[3/(4*E)+E**7/4, -3/(8*E)+(3*E**7)/8],
+                           [-1/(2*E)+E**7/2, 1/(4*E)+(3*E**7)/4]],
+                          [[5/(8*E**2)+(3*E**6)/8, -3/(8*E**2)+(3*E**6)/8],
+                           [-5/(8*E**2)+(5*E**6)/8, 3/(8*E**2)+(5*E**6)/8]],
+                          [[-3/(2*E)+(5*E)/2, -5/(2*E)+(5*E)/2],
+                           [3/(2*E)-(3*E)/2, 5/(2*E)-(3*E)/2]]
+                         ])
+        assert_allclose(expm(a), a_res)
+
+    def test_readonly(self):
+        n = 7
+        a = np.ones((n, n))
+        a.flags.writeable = False
+        expm(a)
+
+    @pytest.mark.fail_slow(5)
+    def test_gh18086(self):
+        A = np.zeros((400, 400), dtype=float)
+        rng = np.random.default_rng(100)
+        i = rng.integers(0, 399, 500)
+        j = rng.integers(0, 399, 500)
+        A[i, j] = rng.random(500)
+        # Check that result is accurate
+        first_res = expm(A)
+        np.testing.assert_array_almost_equal(logm(first_res), A)
+        # Check that result is consistent
+        for i in range(5):
+            next_res = expm(A)
+            np.testing.assert_array_almost_equal(first_res, next_res)
+
+    @pytest.mark.parametrize('dt', [np.float32, np.float64,
+                                    np.complex64, np.complex128])
+    def test_gh25924(self, dt):
+        # The scaling exponent of a slice that needed scaling and squaring
+        # leaked into the following slices that did not, squaring their
+        # unscaled results and returning expm(2**s * A) instead of expm(A).
+        rng = np.random.default_rng(1234)
+        n, batch = 6, 4
+        A = rng.standard_normal((batch, n, n))
+        if np.issubdtype(dt, np.complexfloating):
+            A = A + 1j*rng.standard_normal((batch, n, n))
+        A = A.astype(dt)
+        # First slice needs scaling and squaring, the remaining ones do not.
+        A[0] *= 8
+        A[1:] /= np.abs(A[1:]).sum(axis=1).max(axis=-1)[:, None, None]
+
+        one_at_a_time = np.stack([expm(x) for x in A])
+        assert_allclose(expm(A), one_at_a_time,
+                        rtol=10*np.finfo(dt).eps, atol=0)
+
+
+class TestExpmFrechet:
+
+    def test_expm_frechet(self):
+        # a test of the basic functionality
+        M = np.array([
+            [1, 2, 3, 4],
+            [5, 6, 7, 8],
+            [0, 0, 1, 2],
+            [0, 0, 5, 6],
+            ], dtype=float)
+        A = np.array([
+            [1, 2],
+            [5, 6],
+            ], dtype=float)
+        E = np.array([
+            [3, 4],
+            [7, 8],
+            ], dtype=float)
+        expected_expm = scipy.linalg.expm(A)
+        expected_frechet = scipy.linalg.expm(M)[:2, 2:]
+        for kwargs in ({}, {'method':'SPS'}, {'method':'blockEnlarge'}):
+            observed_expm, observed_frechet = expm_frechet(A, E, **kwargs)
+            assert_allclose(expected_expm, observed_expm)
+            assert_allclose(expected_frechet, observed_frechet)
+
+    def test_small_norm_expm_frechet(self):
+        # methodically test matrices with a range of norms, for better coverage
+        M_original = np.array([
+            [1, 2, 3, 4],
+            [5, 6, 7, 8],
+            [0, 0, 1, 2],
+            [0, 0, 5, 6],
+            ], dtype=float)
+        A_original = np.array([
+            [1, 2],
+            [5, 6],
+            ], dtype=float)
+        E_original = np.array([
+            [3, 4],
+            [7, 8],
+            ], dtype=float)
+        A_original_norm_1 = scipy.linalg.norm(A_original, 1)
+        selected_m_list = [1, 3, 5, 7, 9, 11, 13, 15]
+        m_neighbor_pairs = zip(selected_m_list[:-1], selected_m_list[1:])
+        for ma, mb in m_neighbor_pairs:
+            ell_a = scipy.linalg._expm_frechet.ell_table_61[ma]
+            ell_b = scipy.linalg._expm_frechet.ell_table_61[mb]
+            target_norm_1 = 0.5 * (ell_a + ell_b)
+            scale = target_norm_1 / A_original_norm_1
+            M = scale * M_original
+            A = scale * A_original
+            E = scale * E_original
+            expected_expm = scipy.linalg.expm(A)
+            expected_frechet = scipy.linalg.expm(M)[:2, 2:]
+            observed_expm, observed_frechet = expm_frechet(A, E)
+            assert_allclose(expected_expm, observed_expm)
+            assert_allclose(expected_frechet, observed_frechet)
+
+    def test_fuzz(self):
+        rng = np.random.default_rng(1726500908359153)
+        # try a bunch of crazy inputs
+        rfuncs = (
+                rng.uniform,
+                rng.normal,
+                rng.standard_cauchy,
+                rng.exponential)
+        ntests = 100
+        for i in range(ntests):
+            rfunc = rfuncs[rng.choice(4)]
+            target_norm_1 = rng.exponential()
+            n = rng.integers(2, 16)
+            A_original = rfunc(size=(n,n))
+            E_original = rfunc(size=(n,n))
+            A_original_norm_1 = scipy.linalg.norm(A_original, 1)
+            scale = target_norm_1 / A_original_norm_1
+            A = scale * A_original
+            E = scale * E_original
+            M = np.vstack([
+                np.hstack([A, E]),
+                np.hstack([np.zeros_like(A), A])])
+            expected_expm = scipy.linalg.expm(A)
+            expected_frechet = scipy.linalg.expm(M)[:n, n:]
+            observed_expm, observed_frechet = expm_frechet(A, E)
+            assert_allclose(expected_expm, observed_expm, atol=5e-8)
+            assert_allclose(expected_frechet, observed_frechet, atol=1e-7)
+
+    def test_problematic_matrix(self):
+        # this test case uncovered a bug which has since been fixed
+        A = np.array([
+                [1.50591997, 1.93537998],
+                [0.41203263, 0.23443516],
+                ], dtype=float)
+        E = np.array([
+                [1.87864034, 2.07055038],
+                [1.34102727, 0.67341123],
+                ], dtype=float)
+        scipy.linalg.norm(A, 1)
+        sps_expm, sps_frechet = expm_frechet(
+                A, E, method='SPS')
+        blockEnlarge_expm, blockEnlarge_frechet = expm_frechet(
+                A, E, method='blockEnlarge')
+        assert_allclose(sps_expm, blockEnlarge_expm)
+        assert_allclose(sps_frechet, blockEnlarge_frechet)
+
+    @pytest.mark.slow
+    @pytest.mark.skip(reason='this test is deliberately slow')
+    def test_medium_matrix(self):
+        # profile this to see the speed difference
+        n = 1000
+        rng = np.random.default_rng(1234)
+        A = rng.exponential(size=(n, n))
+        E = rng.exponential(size=(n, n))
+        sps_expm, sps_frechet = expm_frechet(
+                A, E, method='SPS')
+        blockEnlarge_expm, blockEnlarge_frechet = expm_frechet(
+                A, E, method='blockEnlarge')
+        assert_allclose(sps_expm, blockEnlarge_expm)
+        assert_allclose(sps_frechet, blockEnlarge_frechet)
+
+
+def _help_expm_cond_search(A, A_norm, X, X_norm, eps, p):
+    p = np.reshape(p, A.shape)
+    p_norm = norm(p)
+    perturbation = eps * p * (A_norm / p_norm)
+    X_prime = expm(A + perturbation)
+    scaled_relative_error = norm(X_prime - X) / (X_norm * eps)
+    return -scaled_relative_error
+
+
+def _normalized_like(A, B):
+    return A * (scipy.linalg.norm(B) / scipy.linalg.norm(A))
+
+
+def _relative_error(f, A, perturbation):
+    X = f(A)
+    X_prime = f(A + perturbation)
+    return norm(X_prime - X) / norm(X)
+
+
+class TestExpmConditionNumber:
+    def test_expm_cond_smoke(self):
+        rng = np.random.default_rng(1234)
+        for n in range(1, 4):
+            A = rng.standard_normal((n, n))
+            kappa = expm_cond(A)
+            assert_array_less(0, kappa)
+
+    def test_expm_bad_condition_number(self):
+        A = np.array([
+            [-1.128679820, 9.614183771e4, -4.524855739e9, 2.924969411e14],
+            [0, -1.201010529, 9.634696872e4, -4.681048289e9],
+            [0, 0, -1.132893222, 9.532491830e4],
+            [0, 0, 0, -1.179475332],
+            ])
+        kappa = expm_cond(A)
+        assert_array_less(1e36, kappa)
+
+    def test_univariate(self):
+        rng = np.random.default_rng(1234)
+        for x in np.linspace(-5, 5, num=11):
+            A = np.array([[x]])
+            assert_allclose(expm_cond(A), abs(x))
+        for x in np.logspace(-2, 2, num=11):
+            A = np.array([[x]])
+            assert_allclose(expm_cond(A), abs(x))
+        for i in range(10):
+            A = rng.standard_normal((1, 1))
+            assert_allclose(expm_cond(A), np.absolute(A)[0, 0])
+
+    @pytest.mark.slow
+    def test_expm_cond_fuzz(self):
+        rng = np.random.RandomState(12345)
+        eps = 1e-5
+        nsamples = 10
+        for i in range(nsamples):
+            n = rng.randint(2, 5)
+            A = rng.randn(n, n)
+            A_norm = scipy.linalg.norm(A)
+            X = expm(A)
+            X_norm = scipy.linalg.norm(X)
+            kappa = expm_cond(A)
+
+            # Look for the small perturbation that gives the greatest
+            # relative error.
+            f = functools.partial(_help_expm_cond_search,
+                    A, A_norm, X, X_norm, eps)
+            guess = np.ones(n*n)
+            out = minimize(f, guess, method='L-BFGS-B')
+            xopt = out.x
+            yopt = f(xopt)
+            p_best = eps * _normalized_like(np.reshape(xopt, A.shape), A)
+            p_best_relerr = _relative_error(expm, A, p_best)
+            assert_allclose(p_best_relerr, -yopt * eps)
+
+            # Check that the identified perturbation indeed gives greater
+            # relative error than random perturbations with similar norms.
+            for j in range(5):
+                p_rand = eps * _normalized_like(rng.randn(*A.shape), A)
+                assert_allclose(norm(p_best), norm(p_rand))
+                p_rand_relerr = _relative_error(expm, A, p_rand)
+                assert_array_less(p_rand_relerr, p_best_relerr)
+
+            # The greatest relative error should not be much greater than
+            # eps times the condition number kappa.
+            # In the limit as eps approaches zero it should never be greater.
+            assert_array_less(p_best_relerr, (1 + 2*eps) * eps * kappa)
+
+
+class TestKhatriRao:
+
+    def test_basic(self):
+        a = khatri_rao(array([[1, 2], [3, 4]]),
+                       array([[5, 6], [7, 8]]))
+
+        assert_array_equal(a, array([[5, 12],
+                                     [7, 16],
+                                     [15, 24],
+                                     [21, 32]]))
+
+        b = khatri_rao(np.empty([2, 2]), np.empty([2, 2]))
+        assert_array_equal(b.shape, (4, 2))
+
+    def test_number_of_columns_equality(self):
+        with pytest.raises(ValueError):
+            a = array([[1, 2, 3],
+                       [4, 5, 6]])
+            b = array([[1, 2],
+                       [3, 4]])
+            khatri_rao(a, b)
+
+    def test_to_assure_2d_array(self):
+        with pytest.raises(ValueError):
+            # both arrays are 1-D
+            a = array([1, 2, 3])
+            b = array([4, 5, 6])
+            khatri_rao(a, b)
+
+        with pytest.raises(ValueError):
+            # first array is 1-D
+            a = array([1, 2, 3])
+            b = array([
+                [1, 2, 3],
+                [4, 5, 6]
+            ])
+            khatri_rao(a, b)
+
+        with pytest.raises(ValueError):
+            # second array is 1-D
+            a = array([
+                [1, 2, 3],
+                [7, 8, 9]
+            ])
+            b = array([4, 5, 6])
+            khatri_rao(a, b)
+
+    def test_equality_of_two_equations(self):
+        a = array([[1, 2], [3, 4]])
+        b = array([[5, 6], [7, 8]])
+
+        res1 = khatri_rao(a, b)
+        res2 = np.vstack([np.kron(a[:, k], b[:, k])
+                          for k in range(b.shape[1])]).T
+
+        assert_array_equal(res1, res2)
+
+    def test_empty(self):
+        a = np.empty((0, 2))
+        b = np.empty((3, 2))
+        res = khatri_rao(a, b)
+        assert_allclose(res, np.empty((0, 2)))
+
+        a = np.empty((3, 0))
+        b = np.empty((5, 0))
+        res = khatri_rao(a, b)
+        assert_allclose(res, np.empty((15, 0)))

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_matmul_toeplitz.py

@@ -0,0 +1,136 @@
+"""Test functions for linalg.matmul_toeplitz function
+"""
+
+import numpy as np
+from scipy.linalg import toeplitz, matmul_toeplitz
+
+from pytest import raises as assert_raises
+from numpy.testing import assert_allclose
+
+
+class TestMatmulToeplitz:
+
+    def setup_method(self):
+        self.rng = np.random.RandomState(42)
+        self.tolerance = 1.5e-13
+
+    def test_real(self):
+        cases = []
+
+        n = 1
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n)
+        x = self.rng.normal(size=(n, 1))
+        cases.append((x, c, r, False))
+
+        n = 2
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n)
+        x = self.rng.normal(size=(n, 1))
+        cases.append((x, c, r, False))
+
+        n = 101
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n)
+        x = self.rng.normal(size=(n, 1))
+        cases.append((x, c, r, True))
+
+        n = 1000
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n)
+        x = self.rng.normal(size=(n, 1))
+        cases.append((x, c, r, False))
+
+        n = 100
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n)
+        x = self.rng.normal(size=(n, self.rng.randint(1, 10)))
+        cases.append((x, c, r, False))
+
+        n = 100
+        c = self.rng.normal(size=(n, 1))
+        r = self.rng.normal(size=(n, 1))
+        x = self.rng.normal(size=(n, self.rng.randint(1, 10)))
+        cases.append((x, c, r, True))
+
+        n = 100
+        c = self.rng.normal(size=(n, 1))
+        r = None
+        x = self.rng.normal(size=(n, self.rng.randint(1, 10)))
+        cases.append((x, c, r, True, -1))
+
+        n = 100
+        c = self.rng.normal(size=(n, 1))
+        r = None
+        x = self.rng.normal(size=n)
+        cases.append((x, c, r, False))
+
+        n = 101
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n-27)
+        x = self.rng.normal(size=(n-27, 1))
+        cases.append((x, c, r, True))
+
+        n = 100
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n//4)
+        x = self.rng.normal(size=(n//4, self.rng.randint(1, 10)))
+        cases.append((x, c, r, True))
+
+        [self.do(*i) for i in cases]
+
+    def test_complex(self):
+        n = 127
+        c = self.rng.normal(size=(n, 1)) + self.rng.normal(size=(n, 1))*1j
+        r = self.rng.normal(size=(n, 1)) + self.rng.normal(size=(n, 1))*1j
+        x = self.rng.normal(size=(n, 3)) + self.rng.normal(size=(n, 3))*1j
+        self.do(x, c, r, False)
+
+        n = 100
+        c = self.rng.normal(size=(n, 1)) + self.rng.normal(size=(n, 1))*1j
+        r = self.rng.normal(size=(n//2, 1)) +\
+            self.rng.normal(size=(n//2, 1))*1j
+        x = self.rng.normal(size=(n//2, 3)) +\
+            self.rng.normal(size=(n//2, 3))*1j
+        self.do(x, c, r, False)
+
+    def test_empty(self):
+        c = []
+        r = []
+        x = []
+        self.do(x, c, r, False)
+
+        x = np.empty((0, 0))
+        self.do(x, c, r, False)
+
+    def test_exceptions(self):
+
+        n = 100
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=2*n)
+        x = self.rng.normal(size=n)
+        assert_raises(ValueError, matmul_toeplitz, (c, r), x, True)
+
+        n = 100
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n)
+        x = self.rng.normal(size=n-1)
+        assert_raises(ValueError, matmul_toeplitz, (c, r), x, True)
+
+        n = 100
+        c = self.rng.normal(size=n)
+        r = self.rng.normal(size=n//2)
+        x = self.rng.normal(size=n//2-1)
+        assert_raises(ValueError, matmul_toeplitz, (c, r), x, True)
+
+    # For toeplitz matrices, matmul_toeplitz() should be equivalent to @.
+    def do(self, x, c, r=None, check_finite=False, workers=None):
+        c = np.ravel(c)
+        if r is None:
+            actual = matmul_toeplitz(c, x, check_finite, workers)
+        else:
+            r = np.ravel(r)
+            actual = matmul_toeplitz((c, r), x, check_finite)
+        desired = toeplitz(c, r) @ x
+        assert_allclose(actual, desired,
+            rtol=self.tolerance, atol=self.tolerance)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_procrustes.py

@@ -0,0 +1,209 @@
+from itertools import product, permutations
+
+import numpy as np
+import pytest
+from numpy.testing import assert_allclose
+from pytest import raises as assert_raises
+
+from scipy.linalg import orthogonal_procrustes
+from scipy.sparse._sputils import matrix
+from scipy._lib._array_api import make_xp_test_case, xp_assert_close
+from scipy.conftest import skip_xp_invalid_arg
+
+def _centered(A, xp):
+    mu = xp.mean(A, axis=0)
+    return A - mu, mu
+
+@make_xp_test_case(orthogonal_procrustes)
+class TestOrthogonalProcrustes:
+    def test_orthogonal_procrustes_ndim_too_small(self, xp):
+        rng = np.random.RandomState(1234)
+        A = xp.asarray(rng.randn(3))
+        B = xp.asarray(rng.randn(3))
+        assert_raises(ValueError, orthogonal_procrustes, A, B)
+
+    def test_orthogonal_procrustes_shape_mismatch(self, xp):
+        rng = np.random.RandomState(1234)
+        shapes = ((3, 3), (3, 4), (4, 3), (4, 4))
+        for a, b in permutations(shapes, 2):
+            A = xp.asarray(rng.randn(*a))
+            B = xp.asarray(rng.randn(*b))
+            assert_raises(ValueError, orthogonal_procrustes, A, B)
+
+    def test_orthogonal_procrustes_checkfinite_exception(self, xp):
+        rng = np.random.RandomState(1234)
+        m, n = 2, 3
+        A_good = rng.randn(m, n)
+        B_good = rng.randn(m, n)
+        for bad_value in np.inf, -np.inf, np.nan:
+            A_bad = A_good.copy()
+            A_bad[1, 2] = bad_value
+            B_bad = B_good.copy()
+            B_bad[1, 2] = bad_value
+            for A, B in ((A_good, B_bad), (A_bad, B_good), (A_bad, B_bad)):
+                assert_raises(ValueError, orthogonal_procrustes, xp.asarray(A),
+                              xp.asarray(B))
+
+    def test_orthogonal_procrustes_scale_invariance(self, xp):
+        rng = np.random.RandomState(1234)
+        m, n = 4, 3
+        for i in range(3):
+            A_orig = xp.asarray(rng.randn(m, n))
+            B_orig = xp.asarray(rng.randn(m, n))
+            R_orig, s = orthogonal_procrustes(A_orig, B_orig)
+            for A_scale in np.square(rng.randn(3)):
+                for B_scale in np.square(rng.randn(3)):
+                    R, s = orthogonal_procrustes(A_orig * xp.asarray(A_scale),
+                                                 B_orig * xp.asarray(B_scale))
+                    xp_assert_close(R, R_orig)
+
+    @skip_xp_invalid_arg()
+    def test_orthogonal_procrustes_array_conversion(self):
+        rng = np.random.RandomState(1234)
+        for m, n in ((6, 4), (4, 4), (4, 6)):
+            A_arr = rng.randn(m, n)
+            B_arr = rng.randn(m, n)
+            As = (A_arr, A_arr.tolist(), matrix(A_arr))
+            Bs = (B_arr, B_arr.tolist(), matrix(B_arr))
+            R_arr, s = orthogonal_procrustes(A_arr, B_arr)
+            AR_arr = A_arr.dot(R_arr)
+            for A, B in product(As, Bs):
+                R, s = orthogonal_procrustes(A, B)
+                AR = A_arr.dot(R)
+                assert_allclose(AR, AR_arr)
+
+    def test_orthogonal_procrustes(self, xp):
+        rng = np.random.RandomState(1234)
+        for m, n in ((6, 4), (4, 4), (4, 6)):
+            # Sample a random target matrix.
+            B = xp.asarray(rng.randn(m, n))
+            # Sample a random orthogonal matrix
+            # by computing eigh of a sampled symmetric matrix.
+            X = xp.asarray(rng.randn(n, n))
+            w, V = xp.linalg.eigh(X.T + X)
+            xp_assert_close(xp.linalg.inv(V), V.T)
+            # Compute a matrix with a known orthogonal transformation that gives B.
+            A = B @ V.T
+            # Check that an orthogonal transformation from A to B can be recovered.
+            R, s = orthogonal_procrustes(A, B)
+            xp_assert_close(xp.linalg.inv(R), R.T)
+            xp_assert_close(A @ R, B)
+            # Create a perturbed input matrix.
+            A_perturbed = A + 1e-2 * xp.asarray(rng.randn(m, n))
+            # Check that the orthogonal procrustes function can find an orthogonal
+            # transformation that is better than the orthogonal transformation
+            # computed from the original input matrix.
+            R_prime, s = orthogonal_procrustes(A_perturbed, B)
+            xp_assert_close(xp.linalg.inv(R_prime), R_prime.T)
+            # Compute the naive and optimal transformations of the perturbed input.
+            naive_approx = A_perturbed @ R
+            optim_approx = A_perturbed @ R_prime
+            # Compute the Frobenius norm errors of the matrix approximations.
+            naive_approx_error = xp.linalg.matrix_norm(naive_approx - B, ord='fro')
+            optim_approx_error = xp.linalg.matrix_norm(optim_approx - B, ord='fro')
+            # Check that the orthogonal Procrustes approximation is better.
+            assert xp.all(optim_approx_error < naive_approx_error)
+
+    def test_orthogonal_procrustes_exact_example(self, xp):
+        # Check a small application.
+        # It uses translation, scaling, reflection, and rotation.
+        #
+        #         |
+        #   a  b  |
+        #         |
+        #   d  c  |        w
+        #         |
+        # --------+--- x ----- z ---
+        #         |
+        #         |        y
+        #         |
+        #
+        A_orig = xp.asarray([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=xp.float64)
+        B_orig = xp.asarray([[3, 2], [1, 0], [3, -2], [5, 0]], dtype=xp.float64)
+        A, A_mu = _centered(A_orig, xp)
+        B, B_mu = _centered(B_orig, xp)
+        R, s = orthogonal_procrustes(A, B)
+        scale = s / xp.linalg.matrix_norm(A)**2
+        B_approx = scale * A @ R + B_mu
+        xp_assert_close(B_approx, B_orig, atol=1e-8)
+
+    def test_orthogonal_procrustes_stretched_example(self, xp):
+        # Try again with a target with a stretched y axis.
+        A_orig = xp.asarray([[-3, 3], [-2, 3], [-2, 2], [-3, 2]], dtype=xp.float64)
+        B_orig = xp.asarray([[3, 40], [1, 0], [3, -40], [5, 0]], dtype=xp.float64)
+        A, A_mu = _centered(A_orig, xp)
+        B, B_mu = _centered(B_orig, xp)
+        R, s = orthogonal_procrustes(A, B)
+        scale = s / xp.linalg.matrix_norm(A)**2
+        B_approx = scale * A @ R + B_mu
+        expected = xp.asarray([[3, 21], [-18, 0], [3, -21], [24, 0]], dtype=xp.float64)
+        xp_assert_close(B_approx, expected, atol=1e-8)
+        # Check disparity symmetry.
+        expected_disparity = xp.asarray(0.4501246882793018, dtype=xp.float64)[()]
+        AB_disparity = (xp.linalg.matrix_norm(B_approx - B_orig)
+                        / xp.linalg.matrix_norm(B))**2
+        xp_assert_close(AB_disparity, expected_disparity)
+        R, s = orthogonal_procrustes(B, A)
+        scale = s / xp.linalg.matrix_norm(B)**2
+        A_approx = scale * B @ R + A_mu
+        BA_disparity = (xp.linalg.matrix_norm(A_approx - A_orig)
+                        / xp.linalg.matrix_norm(A))**2
+        xp_assert_close(BA_disparity, expected_disparity)
+
+    def test_orthogonal_procrustes_skbio_example(self, xp):
+        # This transformation is also exact.
+        # It uses translation, scaling, and reflection.
+        #
+        #   |
+        #   | a
+        #   | b
+        #   | c d
+        # --+---------
+        #   |
+        #   |       w
+        #   |
+        #   |       x
+        #   |
+        #   |   z   y
+        #   |
+        #
+        A_orig = xp.asarray([[4, -2], [4, -4], [4, -6], [2, -6]], dtype=xp.float64)
+        B_orig = xp.asarray([[1, 3], [1, 2], [1, 1], [2, 1]], dtype=xp.float64)
+        B_standardized = xp.asarray([[-0.13363062, 0.6681531],
+                                     [-0.13363062, 0.13363062],
+                                     [-0.13363062, -0.40089186],
+                                     [0.40089186, -0.40089186]], dtype=xp.float64)
+        A, A_mu = _centered(A_orig, xp)
+        B, B_mu = _centered(B_orig, xp)
+        R, s = orthogonal_procrustes(A, B)
+        scale = s / xp.linalg.matrix_norm(A)**2
+        B_approx = scale * A @ R + B_mu
+        xp_assert_close(B_approx, B_orig)
+        xp_assert_close(B / xp.linalg.matrix_norm(B), B_standardized)
+
+    def test_empty(self, xp):
+        a = xp.empty((0, 0))
+        r, s = orthogonal_procrustes(a, a)
+        xp_assert_close(r, xp.empty((0, 0)))
+
+        a = xp.empty((0, 3))
+        r, s = orthogonal_procrustes(a, a)
+        xp_assert_close(r, xp.eye(3))
+
+    @pytest.mark.parametrize('shape', [(4, 5), (5, 5), (5, 4)])
+    def test_unitary(self, shape, xp):
+        # gh-12071 added support for unitary matrices; check that it
+        # works as intended.
+        m, n = shape
+        rng = np.random.default_rng(589234981235)
+        A = xp.asarray(rng.random(shape) + rng.random(shape) * 1j)
+        Q = xp.asarray(rng.random((n, n)) + rng.random((n, n)) * 1j)
+        Q, _ = xp.linalg.qr(Q)
+        B = A @ Q
+        R, scale = orthogonal_procrustes(A, B)
+        xp_assert_close(R @ xp.conj(R).T, xp.eye(n, dtype=xp.complex128), atol=1e-14)
+        xp_assert_close(A @ Q, B)
+        if shape != (4, 5):  # solution is unique
+            xp_assert_close(R, Q)
+        _, s, _ = xp.linalg.svd(xp.conj(A).T @ B)
+        xp_assert_close(scale, xp.sum(s))

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_sketches.py

@@ -0,0 +1,112 @@
+"""Tests for _sketches.py."""
+
+import numpy as np
+from numpy.testing import assert_, assert_equal
+from scipy.linalg import clarkson_woodruff_transform
+from scipy.linalg._sketches import cwt_matrix
+from scipy.sparse import issparse, random_array
+from scipy.sparse.linalg import norm
+
+
+class TestClarksonWoodruffTransform:
+    """
+    Testing the Clarkson Woodruff Transform
+    """
+    # set seed for generating test matrices
+    rng = np.random.default_rng(1179103485)
+
+    # Test matrix parameters
+    n_rows = 2000
+    n_cols = 100
+    density = 0.1
+
+    # Sketch matrix dimensions
+    n_sketch_rows = 200
+
+    # Seeds to test with
+    seeds = [1755490010, 934377150, 1391612830, 1752708722, 2008891431,
+             1302443994, 1521083269, 1501189312, 1126232505, 1533465685]
+
+    shape = (n_rows, n_cols)
+    A_dense = rng.random(shape)
+    A_csc = random_array(shape, density=density, format='csc', rng=rng)
+    A_csr = random_array(shape, density=density, format='csr', rng=rng)
+    A_coo = random_array(shape, density=density, format='coo', rng=rng)
+
+    # Collect the test matrices
+    test_matrices = [A_dense, A_csc, A_csr, A_coo]
+
+    # Test vector with norm ~1
+    x = rng.random((n_rows, 1)) / np.sqrt(n_rows)
+    del rng  # Not deterministic in pytest-run-parallel
+
+    def test_sketch_dimensions(self):
+        for A in self.test_matrices:
+            for seed in self.seeds:
+                # seed to ensure backwards compatibility post SPEC7
+                sketch = clarkson_woodruff_transform(
+                    A, self.n_sketch_rows, seed=seed
+                )
+                assert_(sketch.shape == (self.n_sketch_rows, self.n_cols))
+
+    def test_seed_returns_identical_transform_matrix(self):
+        for seed in self.seeds:
+            S1 = cwt_matrix(
+                self.n_sketch_rows, self.n_rows, rng=seed
+            ).toarray()
+            S2 = cwt_matrix(
+                self.n_sketch_rows, self.n_rows, rng=seed
+            ).toarray()
+            assert_equal(S1, S2)
+
+    def test_seed_returns_identically(self):
+        for A in self.test_matrices:
+            for seed in self.seeds:
+                sketch1 = clarkson_woodruff_transform(
+                    A, self.n_sketch_rows, rng=seed
+                )
+                sketch2 = clarkson_woodruff_transform(
+                    A, self.n_sketch_rows, rng=seed
+                )
+                if issparse(sketch1):
+                    sketch1 = sketch1.toarray()
+                if issparse(sketch2):
+                    sketch2 = sketch2.toarray()
+                assert_equal(sketch1, sketch2)
+
+    def test_sketch_preserves_frobenius_norm(self):
+        # Given the probabilistic nature of the sketches
+        # we run the test multiple times and check that
+        # we pass all/almost all the tries.
+        n_errors = 0
+        for A in self.test_matrices:
+            if issparse(A):
+                true_norm = norm(A)
+            else:
+                true_norm = np.linalg.norm(A)
+            for seed in self.seeds:
+                sketch = clarkson_woodruff_transform(
+                    A, self.n_sketch_rows, rng=seed,
+                )
+                if issparse(sketch):
+                    sketch_norm = norm(sketch)
+                else:
+                    sketch_norm = np.linalg.norm(sketch)
+
+                if np.abs(true_norm - sketch_norm) > 0.1 * true_norm:
+                    n_errors += 1
+        assert_(n_errors == 0)
+
+    def test_sketch_preserves_vector_norm(self):
+        n_errors = 0
+        n_sketch_rows = int(np.ceil(2. / (0.01 * 0.5**2)))
+        true_norm = np.linalg.norm(self.x)
+        for seed in self.seeds:
+            sketch = clarkson_woodruff_transform(
+                self.x, n_sketch_rows, rng=seed,
+            )
+            sketch_norm = np.linalg.norm(sketch)
+
+            if np.abs(true_norm - sketch_norm) > 0.5 * true_norm:
+                n_errors += 1
+        assert_(n_errors == 0)

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_solve_toeplitz.py

@@ -0,0 +1,137 @@
+"""Test functions for linalg._solve_toeplitz module
+"""
+import numpy as np
+from scipy.linalg._solve_toeplitz import levinson
+from scipy.linalg import solve, toeplitz, solve_toeplitz
+from numpy.testing import assert_equal, assert_allclose
+
+import pytest
+from pytest import raises as assert_raises
+
+
+def test_solve_equivalence():
+    # For toeplitz matrices, solve_toeplitz() should be equivalent to solve().
+    random = np.random.RandomState(1234)
+    for n in (1, 2, 3, 10):
+        c = random.randn(n)
+        if random.rand() < 0.5:
+            c = c + 1j * random.randn(n)
+        r = random.randn(n)
+        if random.rand() < 0.5:
+            r = r + 1j * random.randn(n)
+        y = random.randn(n)
+        if random.rand() < 0.5:
+            y = y + 1j * random.randn(n)
+
+        # Check equivalence when both the column and row are provided.
+        actual = solve_toeplitz((c,r), y)
+        desired = solve(toeplitz(c, r=r), y)
+        assert_allclose(actual, desired)
+
+        # Check equivalence when the column is provided but not the row.
+        actual = solve_toeplitz(c, b=y)
+        desired = solve(toeplitz(c), y)
+        assert_allclose(actual, desired)
+
+
+def test_multiple_rhs():
+    random = np.random.RandomState(1234)
+    c = random.randn(4)
+    r = random.randn(4)
+    for offset in [0, 1j]:
+        for yshape in ((4,), (4, 3)):
+            y = random.randn(*yshape) + offset
+            actual = solve_toeplitz((c,r), b=y)
+            desired = solve(toeplitz(c, r=r), y)
+            assert_equal(actual.shape, yshape)
+            assert_equal(desired.shape, yshape)
+            assert_allclose(actual, desired)
+
+
+def test_native_list_arguments():
+    c = [1,2,4,7]
+    r = [1,3,9,12]
+    y = [5,1,4,2]
+    actual = solve_toeplitz((c,r), y)
+    desired = solve(toeplitz(c, r=r), y)
+    assert_allclose(actual, desired)
+
+
+def test_zero_diag_error():
+    # The Levinson-Durbin implementation fails when the diagonal is zero.
+    random = np.random.RandomState(1234)
+    n = 4
+    c = random.randn(n)
+    r = random.randn(n)
+    y = random.randn(n)
+    c[0] = 0
+    assert_raises(np.linalg.LinAlgError,
+        solve_toeplitz, (c, r), b=y)
+
+
+def test_wikipedia_counterexample():
+    # The Levinson-Durbin implementation also fails in other cases.
+    # This example is from the talk page of the wikipedia article.
+    random = np.random.RandomState(1234)
+    c = [2, 2, 1]
+    y = random.randn(3)
+    assert_raises(np.linalg.LinAlgError, solve_toeplitz, c, b=y)
+
+
+def test_reflection_coeffs():
+    # check that the partial solutions are given by the reflection
+    # coefficients
+
+    random = np.random.RandomState(1234)
+    y_d = random.randn(10)
+    y_z = random.randn(10) + 1j
+    reflection_coeffs_d = [1]
+    reflection_coeffs_z = [1]
+    for i in range(2, 10):
+        reflection_coeffs_d.append(solve_toeplitz(y_d[:(i-1)], b=y_d[1:i])[-1])
+        reflection_coeffs_z.append(solve_toeplitz(y_z[:(i-1)], b=y_z[1:i])[-1])
+
+    y_d_concat = np.concatenate((y_d[-2:0:-1], y_d[:-1]))
+    y_z_concat = np.concatenate((y_z[-2:0:-1].conj(), y_z[:-1]))
+    _, ref_d = levinson(y_d_concat, b=y_d[1:])
+    _, ref_z = levinson(y_z_concat, b=y_z[1:])
+
+    assert_allclose(reflection_coeffs_d, ref_d[:-1])
+    assert_allclose(reflection_coeffs_z, ref_z[:-1])
+
+
+@pytest.mark.xfail(reason='Instability of Levinson iteration')
+def test_unstable():
+    # this is a "Gaussian Toeplitz matrix", as mentioned in Example 2 of
+    # I. Gohbert, T. Kailath and V. Olshevsky "Fast Gaussian Elimination with
+    # Partial Pivoting for Matrices with Displacement Structure"
+    # Mathematics of Computation, 64, 212 (1995), pp 1557-1576
+    # which can be unstable for levinson recursion.
+
+    # other fast toeplitz solvers such as GKO or Burg should be better.
+    random = np.random.RandomState(1234)
+    n = 100
+    c = 0.9 ** (np.arange(n)**2)
+    y = random.randn(n)
+
+    solution1 = solve_toeplitz(c, b=y)
+    solution2 = solve(toeplitz(c), y)
+
+    assert_allclose(solution1, solution2)
+
+
+@pytest.mark.parametrize('dt_c', [int, float, np.float32, complex, np.complex64])
+@pytest.mark.parametrize('dt_b', [int, float, np.float32, complex, np.complex64])
+def test_empty(dt_c, dt_b):
+    c = np.array([], dtype=dt_c)
+    b = np.array([], dtype=dt_b)
+    x = solve_toeplitz(c, b)
+    assert x.shape == (0,)
+    assert x.dtype == solve_toeplitz(np.array([2, 1], dtype=dt_c),
+                                      np.ones(2, dtype=dt_b)).dtype
+
+    b = np.empty((0, 0), dtype=dt_b)
+    x1 = solve_toeplitz(c, b)
+    assert x1.shape == (0, 0)
+    assert x1.dtype == x.dtype
+

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_solvers.py

@@ -0,0 +1,862 @@
+import os
+import numpy as np
+
+from numpy.testing import assert_array_almost_equal, assert_allclose
+import pytest
+from pytest import raises as assert_raises
+
+from scipy.linalg import solve_sylvester
+from scipy.linalg import solve_continuous_lyapunov, solve_discrete_lyapunov
+from scipy.linalg import solve_continuous_are, solve_discrete_are
+from scipy.linalg import block_diag, solve, LinAlgError
+from scipy.sparse._sputils import matrix
+from scipy.conftest import skip_xp_invalid_arg
+
+
+# dtypes for testing size-0 case following precedent set in gh-20295
+dtypes = [int, float, np.float32, complex, np.complex64]
+
+
+def _load_data(name):
+    """
+    Load npz data file under data/
+    Returns a copy of the data, rather than keeping the npz file open.
+    """
+    filename = os.path.join(os.path.abspath(os.path.dirname(__file__)),
+                            'data', name)
+    with np.load(filename) as f:
+        return dict(f.items())
+
+
+class TestSolveLyapunov:
+
+    cases = [
+        # empty case
+        (np.empty((0, 0)),
+         np.empty((0, 0))),
+        (np.array([[1, 2], [3, 4]]),
+         np.array([[9, 10], [11, 12]])),
+        # a, q all complex.
+        (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]),
+         np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])),
+        # a real; q complex.
+        (np.array([[1.0, 2.0], [3.0, 5.0]]),
+         np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])),
+        # a complex; q real.
+        (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]),
+         np.array([[2.0, 2.0], [-1.0, 2.0]])),
+        # An example from Kitagawa, 1977
+        (np.array([[3, 9, 5, 1, 4], [1, 2, 3, 8, 4], [4, 6, 6, 6, 3],
+                   [1, 5, 2, 0, 7], [5, 3, 3, 1, 5]]),
+         np.array([[2, 4, 1, 0, 1], [4, 1, 0, 2, 0], [1, 0, 3, 0, 3],
+                   [0, 2, 0, 1, 0], [1, 0, 3, 0, 4]])),
+        # Companion matrix example. a complex; q real; a.shape[0] = 11
+        (np.array([[0.100+0.j, 0.091+0.j, 0.082+0.j, 0.073+0.j, 0.064+0.j,
+                    0.055+0.j, 0.046+0.j, 0.037+0.j, 0.028+0.j, 0.019+0.j,
+                    0.010+0.j],
+                   [1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j,
+                    0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j,
+                    0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j, 0.000+0.j,
+                    0.000+0.j],
+                   [0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j,
+                    0.000+0.j, 0.000+0.j, 0.000+0.j, 0.000+0.j, 1.000+0.j,
+                    0.000+0.j]]),
+         np.eye(11)),
+        # https://github.com/scipy/scipy/issues/4176
+        (matrix([[0, 1], [-1/2, -1]]),
+         (matrix([0, 3]).T @ matrix([0, 3]).T.T)),
+        # https://github.com/scipy/scipy/issues/4176
+        (matrix([[0, 1], [-1/2, -1]]),
+         (np.array(matrix([0, 3]).T @ matrix([0, 3]).T.T))),
+        ]
+
+    def test_continuous_squareness_and_shape(self):
+        nsq = np.ones((3, 2))
+        sq = np.eye(3)
+        assert_raises(ValueError, solve_continuous_lyapunov, nsq, sq)
+        assert_raises(ValueError, solve_continuous_lyapunov, sq, nsq)
+        assert_raises(ValueError, solve_continuous_lyapunov, sq, np.eye(2))
+
+    def check_continuous_case(self, a, q):
+        x = solve_continuous_lyapunov(a, q)
+        assert_array_almost_equal(
+                          np.dot(a, x) + np.dot(x, a.conj().transpose()), q)
+
+    def check_discrete_case(self, a, q, method=None):
+        x = solve_discrete_lyapunov(a, q, method=method)
+        assert_array_almost_equal(
+                      np.dot(np.dot(a, x), a.conj().transpose()) - x, -1.0*q)
+
+    @skip_xp_invalid_arg
+    def test_cases(self):
+        for case in self.cases:
+            self.check_continuous_case(case[0], case[1])
+            self.check_discrete_case(case[0], case[1])
+            self.check_discrete_case(case[0], case[1], method='direct')
+            self.check_discrete_case(case[0], case[1], method='bilinear')
+
+    @pytest.mark.parametrize("dtype_a", dtypes)
+    @pytest.mark.parametrize("dtype_q", dtypes)
+    def test_size_0(self, dtype_a, dtype_q):
+        rng = np.random.default_rng(234598235)
+
+        a = np.zeros((0, 0), dtype=dtype_a)
+        q = np.zeros((0, 0), dtype=dtype_q)
+        res = solve_continuous_lyapunov(a, q)
+
+        a = (rng.random((5, 5))*100).astype(dtype_a)
+        q = (rng.random((5, 5))*100).astype(dtype_q)
+        ref = solve_continuous_lyapunov(a, q)
+
+        assert res.shape == (0, 0)
+        assert res.dtype == ref.dtype
+
+
+class TestSolveContinuousAre:
+    mat6 = _load_data('carex_6_data.npz')
+    mat15 = _load_data('carex_15_data.npz')
+    mat18 = _load_data('carex_18_data.npz')
+    mat19 = _load_data('carex_19_data.npz')
+    mat20 = _load_data('carex_20_data.npz')
+    cases = [
+        # Carex examples taken from (with default parameters):
+        # [1] P.BENNER, A.J. LAUB, V. MEHRMANN: 'A Collection of Benchmark
+        #     Examples for the Numerical Solution of Algebraic Riccati
+        #     Equations II: Continuous-Time Case', Tech. Report SPC 95_23,
+        #     Fak. f. Mathematik, TU Chemnitz-Zwickau (Germany), 1995.
+        #
+        # The format of the data is (a, b, q, r, knownfailure), where
+        # knownfailure is None if the test passes or a string
+        # indicating the reason for failure.
+        #
+        # Test Case 0: carex #1
+        (np.diag([1.], 1),
+         np.array([[0], [1]]),
+         block_diag(1., 2.),
+         1,
+         None),
+        # Test Case 1: carex #2
+        (np.array([[4, 3], [-4.5, -3.5]]),
+         np.array([[1], [-1]]),
+         np.array([[9, 6], [6, 4.]]),
+         1,
+         None),
+        # Test Case 2: carex #3
+        (np.array([[0, 1, 0, 0],
+                   [0, -1.89, 0.39, -5.53],
+                   [0, -0.034, -2.98, 2.43],
+                   [0.034, -0.0011, -0.99, -0.21]]),
+         np.array([[0, 0], [0.36, -1.6], [-0.95, -0.032], [0.03, 0]]),
+         np.array([[2.313, 2.727, 0.688, 0.023],
+                   [2.727, 4.271, 1.148, 0.323],
+                   [0.688, 1.148, 0.313, 0.102],
+                   [0.023, 0.323, 0.102, 0.083]]),
+         np.eye(2),
+         None),
+        # Test Case 3: carex #4
+        (np.array([[-0.991, 0.529, 0, 0, 0, 0, 0, 0],
+                   [0.522, -1.051, 0.596, 0, 0, 0, 0, 0],
+                   [0, 0.522, -1.118, 0.596, 0, 0, 0, 0],
+                   [0, 0, 0.522, -1.548, 0.718, 0, 0, 0],
+                   [0, 0, 0, 0.922, -1.64, 0.799, 0, 0],
+                   [0, 0, 0, 0, 0.922, -1.721, 0.901, 0],
+                   [0, 0, 0, 0, 0, 0.922, -1.823, 1.021],
+                   [0, 0, 0, 0, 0, 0, 0.922, -1.943]]),
+         np.array([[3.84, 4.00, 37.60, 3.08, 2.36, 2.88, 3.08, 3.00],
+                   [-2.88, -3.04, -2.80, -2.32, -3.32, -3.82, -4.12, -3.96]]
+                  ).T * 0.001,
+         np.array([[1.0, 0.0, 0.0, 0.0, 0.5, 0.0, 0.0, 0.1],
+                   [0.0, 1.0, 0.0, 0.0, 0.1, 0.0, 0.0, 0.0],
+                   [0.0, 0.0, 1.0, 0.0, 0.0, 0.5, 0.0, 0.0],
+                   [0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0],
+                   [0.5, 0.1, 0.0, 0.0, 0.1, 0.0, 0.0, 0.0],
+                   [0.0, 0.0, 0.5, 0.0, 0.0, 0.1, 0.0, 0.0],
+                   [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.1, 0.0],
+                   [0.1, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.1]]),
+         np.eye(2),
+         None),
+        # Test Case 4: carex #5
+        (np.array(
+          [[-4.019, 5.120, 0., 0., -2.082, 0., 0., 0., 0.870],
+           [-0.346, 0.986, 0., 0., -2.340, 0., 0., 0., 0.970],
+           [-7.909, 15.407, -4.069, 0., -6.450, 0., 0., 0., 2.680],
+           [-21.816, 35.606, -0.339, -3.870, -17.800, 0., 0., 0., 7.390],
+           [-60.196, 98.188, -7.907, 0.340, -53.008, 0., 0., 0., 20.400],
+           [0, 0, 0, 0, 94.000, -147.200, 0., 53.200, 0.],
+           [0, 0, 0, 0, 0, 94.000, -147.200, 0, 0],
+           [0, 0, 0, 0, 0, 12.800, 0.000, -31.600, 0],
+           [0, 0, 0, 0, 12.800, 0.000, 0.000, 18.800, -31.600]]),
+         np.array([[0.010, -0.011, -0.151],
+                   [0.003, -0.021, 0.000],
+                   [0.009, -0.059, 0.000],
+                   [0.024, -0.162, 0.000],
+                   [0.068, -0.445, 0.000],
+                   [0.000, 0.000, 0.000],
+                   [0.000, 0.000, 0.000],
+                   [0.000, 0.000, 0.000],
+                   [0.000, 0.000, 0.000]]),
+         np.eye(9),
+         np.eye(3),
+         None),
+        # Test Case 5: carex #6
+        (mat6['A'], mat6['B'], mat6['Q'], mat6['R'], None),
+        # Test Case 6: carex #7
+        (np.array([[1, 0], [0, -2.]]),
+         np.array([[1e-6], [0]]),
+         np.ones((2, 2)),
+         1.,
+         'Bad residual accuracy'),
+        # Test Case 7: carex #8
+        (block_diag(-0.1, -0.02),
+         np.array([[0.100, 0.000], [0.001, 0.010]]),
+         np.array([[100, 1000], [1000, 10000]]),
+         np.ones((2, 2)) + block_diag(1e-6, 0),
+         None),
+        # Test Case 8: carex #9
+        (np.array([[0, 1e6], [0, 0]]),
+         np.array([[0], [1.]]),
+         np.eye(2),
+         1.,
+         None),
+        # Test Case 9: carex #10
+        (np.array([[1.0000001, 1], [1., 1.0000001]]),
+         np.eye(2),
+         np.eye(2),
+         np.eye(2),
+         None),
+        # Test Case 10: carex #11
+        (np.array([[3, 1.], [4, 2]]),
+         np.array([[1], [1]]),
+         np.array([[-11, -5], [-5, -2.]]),
+         1.,
+         None),
+        # Test Case 11: carex #12
+        (np.array([[7000000., 2000000., -0.],
+                   [2000000., 6000000., -2000000.],
+                   [0., -2000000., 5000000.]]) / 3,
+         np.eye(3),
+         np.array([[1., -2., -2.], [-2., 1., -2.], [-2., -2., 1.]]).dot(
+                np.diag([1e-6, 1, 1e6])).dot(
+            np.array([[1., -2., -2.], [-2., 1., -2.], [-2., -2., 1.]])) / 9,
+         np.eye(3) * 1e6,
+         'Bad Residual Accuracy'),
+        # Test Case 12: carex #13
+        (np.array([[0, 0.4, 0, 0],
+                   [0, 0, 0.345, 0],
+                   [0, -0.524e6, -0.465e6, 0.262e6],
+                   [0, 0, 0, -1e6]]),
+         np.array([[0, 0, 0, 1e6]]).T,
+         np.diag([1, 0, 1, 0]),
+         1.,
+         None),
+        # Test Case 13: carex #14
+        (np.array([[-1e-6, 1, 0, 0],
+                   [-1, -1e-6, 0, 0],
+                   [0, 0, 1e-6, 1],
+                   [0, 0, -1, 1e-6]]),
+         np.ones((4, 1)),
+         np.ones((4, 4)),
+         1.,
+         None),
+        # Test Case 14: carex #15
+        (mat15['A'], mat15['B'], mat15['Q'], mat15['R'], None),
+        # Test Case 15: carex #16
+        (np.eye(64, 64, k=-1) + np.eye(64, 64)*(-2.) + np.rot90(
+                 block_diag(1, np.zeros((62, 62)), 1)) + np.eye(64, 64, k=1),
+         np.eye(64),
+         np.eye(64),
+         np.eye(64),
+         None),
+        # Test Case 16: carex #17
+        (np.diag(np.ones((20, )), 1),
+         np.flipud(np.eye(21, 1)),
+         np.eye(21, 1) * np.eye(21, 1).T,
+         1,
+         'Bad Residual Accuracy'),
+        # Test Case 17: carex #18
+        (mat18['A'], mat18['B'], mat18['Q'], mat18['R'], None),
+        # Test Case 18: carex #19
+        (mat19['A'], mat19['B'], mat19['Q'], mat19['R'],
+         'Bad Residual Accuracy'),
+        # Test Case 19: carex #20
+        (mat20['A'], mat20['B'], mat20['Q'], mat20['R'],
+         'Bad Residual Accuracy')
+        ]
+    # Makes the minimum precision requirements customized to the test.
+    # Here numbers represent the number of decimals that agrees with zero
+    # matrix when the solution x is plugged in to the equation.
+    #
+    # res = array([[8e-3,1e-16],[1e-16,1e-20]]) --> min_decimal[k] = 2
+    #
+    # If the test is failing use "None" for that entry.
+    #
+    min_decimal = (14, 12, 13, 14, 11, 6, None, 5, 7, 14, 14,
+                   None, 9, 14, 13, 14, None, 12, None, None)
+
+    @pytest.mark.parametrize("j, case", list(enumerate(cases)))
+    def test_solve_continuous_are(self, j, case):
+        """Checks if 0 = XA + A'X - XB(R)^{-1} B'X + Q is true"""
+        a, b, q, r, knownfailure = case
+        if knownfailure:
+            pytest.xfail(reason=knownfailure)
+
+        dec = self.min_decimal[j]
+        x = solve_continuous_are(a, b, q, r)
+        res = x @ a + a.conj().T @ x + q
+        out_fact = x @ b
+        res -= out_fact @ solve(np.atleast_2d(r), out_fact.conj().T)
+        assert_array_almost_equal(res, np.zeros_like(res), decimal=dec)
+
+
+class TestSolveDiscreteAre:
+    cases = [
+        # Darex examples taken from (with default parameters):
+        # [1] P.BENNER, A.J. LAUB, V. MEHRMANN: 'A Collection of Benchmark
+        #     Examples for the Numerical Solution of Algebraic Riccati
+        #     Equations II: Discrete-Time Case', Tech. Report SPC 95_23,
+        #     Fak. f. Mathematik, TU Chemnitz-Zwickau (Germany), 1995.
+        # [2] T. GUDMUNDSSON, C. KENNEY, A.J. LAUB: 'Scaling of the
+        #     Discrete-Time Algebraic Riccati Equation to Enhance Stability
+        #     of the Schur Solution Method', IEEE Trans.Aut.Cont., vol.37(4)
+        #
+        # The format of the data is (a, b, q, r, knownfailure), where
+        # knownfailure is None if the test passes or a string
+        # indicating the reason for failure.
+        #
+        # TEST CASE 0 : Complex a; real b, q, r
+        (np.array([[2, 1-2j], [0, -3j]]),
+         np.array([[0], [1]]),
+         np.array([[1, 0], [0, 2]]),
+         np.array([[1]]),
+         None),
+        # TEST CASE 1 :Real a, q, r; complex b
+        (np.array([[2, 1], [0, -1]]),
+         np.array([[-2j], [1j]]),
+         np.array([[1, 0], [0, 2]]),
+         np.array([[1]]),
+         None),
+        # TEST CASE 2 : Real a, b; complex q, r
+        (np.array([[3, 1], [0, -1]]),
+         np.array([[1, 2], [1, 3]]),
+         np.array([[1, 1+1j], [1-1j, 2]]),
+         np.array([[2, -2j], [2j, 3]]),
+         None),
+        # TEST CASE 3 : User-reported gh-2251 (Trac #1732)
+        (np.array([[0.63399379, 0.54906824, 0.76253406],
+                   [0.5404729, 0.53745766, 0.08731853],
+                   [0.27524045, 0.84922129, 0.4681622]]),
+         np.array([[0.96861695], [0.05532739], [0.78934047]]),
+         np.eye(3),
+         np.eye(1),
+         None),
+        # TEST CASE 4 : darex #1
+        (np.array([[4, 3], [-4.5, -3.5]]),
+         np.array([[1], [-1]]),
+         np.array([[9, 6], [6, 4]]),
+         np.array([[1]]),
+         None),
+        # TEST CASE 5 : darex #2
+        (np.array([[0.9512, 0], [0, 0.9048]]),
+         np.array([[4.877, 4.877], [-1.1895, 3.569]]),
+         np.array([[0.005, 0], [0, 0.02]]),
+         np.array([[1/3, 0], [0, 3]]),
+         None),
+        # TEST CASE 6 : darex #3
+        (np.array([[2, -1], [1, 0]]),
+         np.array([[1], [0]]),
+         np.array([[0, 0], [0, 1]]),
+         np.array([[0]]),
+         None),
+        # TEST CASE 7 : darex #4 (skipped the gen. Ric. term S)
+        (np.array([[0, 1], [0, -1]]),
+         np.array([[1, 0], [2, 1]]),
+         np.array([[-4, -4], [-4, 7]]) * (1/11),
+         np.array([[9, 3], [3, 1]]),
+         None),
+        # TEST CASE 8 : darex #5
+        (np.array([[0, 1], [0, 0]]),
+         np.array([[0], [1]]),
+         np.array([[1, 2], [2, 4]]),
+         np.array([[1]]),
+         None),
+        # TEST CASE 9 : darex #6
+        (np.array([[0.998, 0.067, 0, 0],
+                   [-.067, 0.998, 0, 0],
+                   [0, 0, 0.998, 0.153],
+                   [0, 0, -.153, 0.998]]),
+         np.array([[0.0033, 0.0200],
+                   [0.1000, -.0007],
+                   [0.0400, 0.0073],
+                   [-.0028, 0.1000]]),
+         np.array([[1.87, 0, 0, -0.244],
+                   [0, 0.744, 0.205, 0],
+                   [0, 0.205, 0.589, 0],
+                   [-0.244, 0, 0, 1.048]]),
+         np.eye(2),
+         None),
+        # TEST CASE 10 : darex #7
+        (np.array([[0.984750, -.079903, 0.0009054, -.0010765],
+                   [0.041588, 0.998990, -.0358550, 0.0126840],
+                   [-.546620, 0.044916, -.3299100, 0.1931800],
+                   [2.662400, -.100450, -.9245500, -.2632500]]),
+         np.array([[0.0037112, 0.0007361],
+                   [-.0870510, 9.3411e-6],
+                   [-1.198440, -4.1378e-4],
+                   [-3.192700, 9.2535e-4]]),
+         np.eye(4)*1e-2,
+         np.eye(2),
+         None),
+        # TEST CASE 11 : darex #8
+        (np.array([[-0.6000000, -2.2000000, -3.6000000, -5.4000180],
+                   [1.0000000, 0.6000000, 0.8000000, 3.3999820],
+                   [0.0000000, 1.0000000, 1.8000000, 3.7999820],
+                   [0.0000000, 0.0000000, 0.0000000, -0.9999820]]),
+         np.array([[1.0, -1.0, -1.0, -1.0],
+                   [0.0, 1.0, -1.0, -1.0],
+                   [0.0, 0.0, 1.0, -1.0],
+                   [0.0, 0.0, 0.0, 1.0]]),
+         np.array([[2, 1, 3, 6],
+                   [1, 2, 2, 5],
+                   [3, 2, 6, 11],
+                   [6, 5, 11, 22]]),
+         np.eye(4),
+         None),
+        # TEST CASE 12 : darex #9
+        (np.array([[95.4070, 1.9643, 0.3597, 0.0673, 0.0190],
+                   [40.8490, 41.3170, 16.0840, 4.4679, 1.1971],
+                   [12.2170, 26.3260, 36.1490, 15.9300, 12.3830],
+                   [4.1118, 12.8580, 27.2090, 21.4420, 40.9760],
+                   [0.1305, 0.5808, 1.8750, 3.6162, 94.2800]]) * 0.01,
+         np.array([[0.0434, -0.0122],
+                   [2.6606, -1.0453],
+                   [3.7530, -5.5100],
+                   [3.6076, -6.6000],
+                   [0.4617, -0.9148]]) * 0.01,
+         np.eye(5),
+         np.eye(2),
+         None),
+        # TEST CASE 13 : darex #10
+        (np.kron(np.eye(2), np.diag([1, 1], k=1)),
+         np.kron(np.eye(2), np.array([[0], [0], [1]])),
+         np.array([[1, 1, 0, 0, 0, 0],
+                   [1, 1, 0, 0, 0, 0],
+                   [0, 0, 0, 0, 0, 0],
+                   [0, 0, 0, 1, -1, 0],
+                   [0, 0, 0, -1, 1, 0],
+                   [0, 0, 0, 0, 0, 0]]),
+         np.array([[3, 0], [0, 1]]),
+         None),
+        # TEST CASE 14 : darex #11
+        (0.001 * np.array(
+         [[870.1, 135.0, 11.59, .5014, -37.22, .3484, 0, 4.242, 7.249],
+          [76.55, 897.4, 12.72, 0.5504, -40.16, .3743, 0, 4.53, 7.499],
+          [-127.2, 357.5, 817, 1.455, -102.8, .987, 0, 11.85, 18.72],
+          [-363.5, 633.9, 74.91, 796.6, -273.5, 2.653, 0, 31.72, 48.82],
+          [-960, 1645.9, -128.9, -5.597, 71.42, 7.108, 0, 84.52, 125.9],
+          [-664.4, 112.96, -88.89, -3.854, 84.47, 13.6, 0, 144.3, 101.6],
+          [-410.2, 693, -54.71, -2.371, 66.49, 12.49, .1063, 99.97, 69.67],
+          [-179.9, 301.7, -23.93, -1.035, 60.59, 22.16, 0, 213.9, 35.54],
+          [-345.1, 580.4, -45.96, -1.989, 105.6, 19.86, 0, 219.1, 215.2]]),
+         np.array([[4.7600, -0.5701, -83.6800],
+                   [0.8790, -4.7730, -2.7300],
+                   [1.4820, -13.1200, 8.8760],
+                   [3.8920, -35.1300, 24.8000],
+                   [10.3400, -92.7500, 66.8000],
+                   [7.2030, -61.5900, 38.3400],
+                   [4.4540, -36.8300, 20.2900],
+                   [1.9710, -15.5400, 6.9370],
+                   [3.7730, -30.2800, 14.6900]]) * 0.001,
+         np.diag([50, 0, 0, 0, 50, 0, 0, 0, 0]),
+         np.eye(3),
+         None),
+        # TEST CASE 15 : darex #12 - numerically least accurate example
+        (np.array([[0, 1e6], [0, 0]]),
+         np.array([[0], [1]]),
+         np.eye(2),
+         np.array([[1]]),
+        None),
+        # TEST CASE 16 : darex #13
+        (np.array([[16, 10, -2],
+                  [10, 13, -8],
+                  [-2, -8, 7]]) * (1/9),
+         np.eye(3),
+         1e6 * np.eye(3),
+         1e6 * np.eye(3),
+        None),
+        # TEST CASE 17 : darex #14
+        (np.array([[1 - 1/1e8, 0, 0, 0],
+                  [1, 0, 0, 0],
+                  [0, 1, 0, 0],
+                  [0, 0, 1, 0]]),
+         np.array([[1e-08], [0], [0], [0]]),
+         np.diag([0, 0, 0, 1]),
+         np.array([[0.25]]),
+         None),
+        # TEST CASE 18 : darex #15
+        (np.eye(100, k=1),
+         np.flipud(np.eye(100, 1)),
+         np.eye(100),
+         np.array([[1]]),
+         None)
+        ]
+
+    # Makes the minimum precision requirements customized to the test.
+    # Here numbers represent the number of decimals that agrees with zero
+    # matrix when the solution x is plugged in to the equation.
+    #
+    # res = array([[8e-3,1e-16],[1e-16,1e-20]]) --> min_decimal[k] = 2
+    #
+    # If the test is failing use "None" for that entry.
+    #
+    min_decimal = (12, 14, 13, 14, 13, 16, 18, 14, 14, 13,
+                   14, 13, 13, 14, 12, 2, 4, 6, 10)
+    max_tol = [1.5 * 10**-ind for ind in min_decimal]
+    # relaxed tolerance in gh-18012 after bump to OpenBLAS
+    max_tol[11] = 2.5e-13
+
+    # relaxed tolerance in gh-20335 for linux-aarch64 build on Cirrus
+    # with OpenBLAS from ubuntu jammy
+    max_tol[15] = 2.0e-2
+
+    # relaxed tolerance in gh-20335 for OpenBLAS 3.20 on ubuntu jammy
+    # bump not needed for OpenBLAS 3.26
+    max_tol[16] = 2.0e-4
+
+    @pytest.mark.parametrize("j, case", list(enumerate(cases)))
+    def test_solve_discrete_are(self, j, case):
+        """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true"""
+        a, b, q, r, knownfailure = case
+        if knownfailure:
+            pytest.xfail(reason=knownfailure)
+
+        atol = self.max_tol[j]
+
+        x = solve_discrete_are(a, b, q, r)
+        bH = b.conj().T
+        xa, xb = x @ a, x @ b
+
+        res = a.conj().T @ xa - x + q
+        res -= a.conj().T @ xb @ (solve(r + bH @ xb, bH) @ xa)
+
+        # changed from
+        # assert_array_almost_equal(res, np.zeros_like(res), decimal=dec)
+        # in gh-18012 as it's easier to relax a tolerance and allclose is
+        # preferred
+        assert_allclose(res, np.zeros_like(res), atol=atol)
+
+    def test_infeasible(self):
+        # An infeasible example taken from https://arxiv.org/abs/1505.04861v1
+        A = np.triu(np.ones((3, 3)))
+        A[0, 1] = -1
+        B = np.array([[1, 1, 0], [0, 0, 1]]).T
+        Q = np.full_like(A, -2) + np.diag([8, -1, -1.9])
+        R = np.diag([-10, 0.1])
+        assert_raises(LinAlgError, solve_continuous_are, A, B, Q, R)
+
+
+class TestSolveCommonAre:
+    @pytest.mark.parametrize("solver", [solve_continuous_are, solve_discrete_are])
+    def test_with_skipped_array_argument_gh23336(self, solver):
+        # gh-23336 reported a failure when optional argument `e` was skipped
+        A = np.array([[-0.9, 0.25], [0, -1.1]])
+        B = np.array([[0.23], [0.45]])
+        Q = np.eye(2)
+        R = np.atleast_2d(0.45)
+        E = np.eye(2)
+        S = np.array([[0.1], [0.2]])
+
+        res = solver(A, B, Q, R, s=S)
+        ref = solver(A, B, Q, R, E, S)
+        np.testing.assert_allclose(res, ref)
+
+
+def test_solve_generalized_continuous_are():
+    cases = [
+        # Two random examples differ by s term
+        # in the absence of any literature for demanding examples.
+        (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01],
+                   [4.617139e-02, 6.948286e-01, 3.444608e-02],
+                   [9.713178e-02, 3.170995e-01, 4.387444e-01]]),
+         np.array([[3.815585e-01, 1.868726e-01],
+                   [7.655168e-01, 4.897644e-01],
+                   [7.951999e-01, 4.455862e-01]]),
+         np.eye(3),
+         np.eye(2),
+         np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01],
+                   [7.093648e-01, 6.797027e-01, 1.189977e-01],
+                   [7.546867e-01, 6.550980e-01, 4.983641e-01]]),
+         np.zeros((3, 2)),
+         None),
+        (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01],
+                   [4.617139e-02, 6.948286e-01, 3.444608e-02],
+                   [9.713178e-02, 3.170995e-01, 4.387444e-01]]),
+         np.array([[3.815585e-01, 1.868726e-01],
+                   [7.655168e-01, 4.897644e-01],
+                   [7.951999e-01, 4.455862e-01]]),
+         np.eye(3),
+         np.eye(2),
+         np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01],
+                   [7.093648e-01, 6.797027e-01, 1.189977e-01],
+                   [7.546867e-01, 6.550980e-01, 4.983641e-01]]),
+         np.ones((3, 2)),
+         None)
+        ]
+
+    min_decimal = (10, 10)
+
+    def _test_factory(case, dec):
+        """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true"""
+        a, b, q, r, e, s, knownfailure = case
+        if knownfailure:
+            pytest.xfail(reason=knownfailure)
+
+        x = solve_continuous_are(a, b, q, r, e, s)
+        res = a.conj().T.dot(x.dot(e)) + e.conj().T.dot(x.dot(a)) + q
+        out_fact = e.conj().T.dot(x).dot(b) + s
+        res -= out_fact.dot(solve(np.atleast_2d(r), out_fact.conj().T))
+        assert_array_almost_equal(res, np.zeros_like(res), decimal=dec)
+
+    for ind, case in enumerate(cases):
+        _test_factory(case, min_decimal[ind])
+
+
+def test_solve_generalized_discrete_are():
+    mat20170120 = _load_data('gendare_20170120_data.npz')
+
+    cases = [
+        # Two random examples differ by s term
+        # in the absence of any literature for demanding examples.
+        (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01],
+                   [4.617139e-02, 6.948286e-01, 3.444608e-02],
+                   [9.713178e-02, 3.170995e-01, 4.387444e-01]]),
+         np.array([[3.815585e-01, 1.868726e-01],
+                   [7.655168e-01, 4.897644e-01],
+                   [7.951999e-01, 4.455862e-01]]),
+         np.eye(3),
+         np.eye(2),
+         np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01],
+                   [7.093648e-01, 6.797027e-01, 1.189977e-01],
+                   [7.546867e-01, 6.550980e-01, 4.983641e-01]]),
+         np.zeros((3, 2)),
+         None),
+        (np.array([[2.769230e-01, 8.234578e-01, 9.502220e-01],
+                   [4.617139e-02, 6.948286e-01, 3.444608e-02],
+                   [9.713178e-02, 3.170995e-01, 4.387444e-01]]),
+         np.array([[3.815585e-01, 1.868726e-01],
+                   [7.655168e-01, 4.897644e-01],
+                   [7.951999e-01, 4.455862e-01]]),
+         np.eye(3),
+         np.eye(2),
+         np.array([[6.463130e-01, 2.760251e-01, 1.626117e-01],
+                   [7.093648e-01, 6.797027e-01, 1.189977e-01],
+                   [7.546867e-01, 6.550980e-01, 4.983641e-01]]),
+         np.ones((3, 2)),
+         None),
+        # user-reported (under PR-6616) 20-Jan-2017
+        # tests against the case where E is None but S is provided
+        (mat20170120['A'],
+         mat20170120['B'],
+         mat20170120['Q'],
+         mat20170120['R'],
+         None,
+         mat20170120['S'],
+         None),
+        ]
+
+    max_atol = (1.5e-11, 1.5e-11, 3.5e-16)
+
+    def _test_factory(case, atol):
+        """Checks if X = A'XA-(A'XB)(R+B'XB)^-1(B'XA)+Q) is true"""
+        a, b, q, r, e, s, knownfailure = case
+        if knownfailure:
+            pytest.xfail(reason=knownfailure)
+
+        x = solve_discrete_are(a, b, q, r, e, s)
+        if e is None:
+            e = np.eye(a.shape[0])
+        if s is None:
+            s = np.zeros_like(b)
+        res = a.conj().T.dot(x.dot(a)) - e.conj().T.dot(x.dot(e)) + q
+        res -= (a.conj().T.dot(x.dot(b)) + s).dot(
+                    solve(r+b.conj().T.dot(x.dot(b)),
+                          (b.conj().T.dot(x.dot(a)) + s.conj().T)
+                          )
+                )
+        # changed from:
+        # assert_array_almost_equal(res, np.zeros_like(res), decimal=dec)
+        # in gh-17950 because of a Linux 32 bit fail.
+        assert_allclose(res, np.zeros_like(res), atol=atol)
+
+    for ind, case in enumerate(cases):
+        _test_factory(case, max_atol[ind])
+
+
+def test_are_validate_args():
+
+    def test_square_shape():
+        nsq = np.ones((3, 2))
+        sq = np.eye(3)
+        for x in (solve_continuous_are, solve_discrete_are):
+            assert_raises(ValueError, x, nsq, 1, 1, 1)
+            assert_raises(ValueError, x, sq, sq, nsq, 1)
+            assert_raises(ValueError, x, sq, sq, sq, nsq)
+            assert_raises(ValueError, x, sq, sq, sq, sq, nsq)
+
+    def test_compatible_sizes():
+        nsq = np.ones((3, 2))
+        sq = np.eye(4)
+        for x in (solve_continuous_are, solve_discrete_are):
+            assert_raises(ValueError, x, sq, nsq, 1, 1)
+            assert_raises(ValueError, x, sq, sq, sq, sq, sq, nsq)
+            assert_raises(ValueError, x, sq, sq, np.eye(3), sq)
+            assert_raises(ValueError, x, sq, sq, sq, np.eye(3))
+            assert_raises(ValueError, x, sq, sq, sq, sq, np.eye(3))
+
+    def test_symmetry():
+        nsym = np.arange(9).reshape(3, 3)
+        sym = np.eye(3)
+        for x in (solve_continuous_are, solve_discrete_are):
+            assert_raises(ValueError, x, sym, sym, nsym, sym)
+            assert_raises(ValueError, x, sym, sym, sym, nsym)
+
+    def test_singularity():
+        sing = np.full((3, 3), 1e12)
+        sing[2, 2] -= 1
+        sq = np.eye(3)
+        for x in (solve_continuous_are, solve_discrete_are):
+            assert_raises(ValueError, x, sq, sq, sq, sq, sing)
+
+        assert_raises(ValueError, solve_continuous_are, sq, sq, sq, sing)
+
+    def test_finiteness():
+        nm = np.full((2, 2), np.nan)
+        sq = np.eye(2)
+        for x in (solve_continuous_are, solve_discrete_are):
+            assert_raises(ValueError, x, nm, sq, sq, sq)
+            assert_raises(ValueError, x, sq, nm, sq, sq)
+            assert_raises(ValueError, x, sq, sq, nm, sq)
+            assert_raises(ValueError, x, sq, sq, sq, nm)
+            assert_raises(ValueError, x, sq, sq, sq, sq, nm)
+            assert_raises(ValueError, x, sq, sq, sq, sq, sq, nm)
+
+
+class TestSolveSylvester:
+    cases = [
+        # empty cases
+        (np.empty((0, 0)),
+         np.empty((0, 0)),
+         np.empty((0, 0))),
+         (np.empty((0, 0)),
+         np.empty((2, 2)),
+         np.empty((0, 2))),
+         (np.empty((2, 2)),
+         np.empty((0, 0)),
+         np.empty((2, 0))),
+        # a, b, c all real.
+        (np.array([[1, 2], [0, 4]]),
+         np.array([[5, 6], [0, 8]]),
+         np.array([[9, 10], [11, 12]])),
+        # a, b, c all real, 4x4. a and b have non-trivial 2x2 blocks in their
+        # quasi-triangular form.
+        (np.array([[1.0, 0, 0, 0],
+                   [0, 1.0, 2.0, 0.0],
+                   [0, 0, 3.0, -4],
+                   [0, 0, 2, 5]]),
+         np.array([[2.0, 0, 0, 1.0],
+                   [0, 1.0, 0.0, 0.0],
+                   [0, 0, 1.0, -1],
+                   [0, 0, 1, 1]]),
+         np.array([[1.0, 0, 0, 0],
+                   [0, 1.0, 0, 0],
+                   [0, 0, 1.0, 0],
+                   [0, 0, 0, 1.0]])),
+        # a, b, c all complex.
+        (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]),
+         np.array([[-1.0, 2j], [3.0, 4.0]]),
+         np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])),
+        # a and b real; c complex.
+        (np.array([[1.0, 2.0], [3.0, 5.0]]),
+         np.array([[-1.0, 0], [3.0, 4.0]]),
+         np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])),
+        # a and c complex; b real.
+        (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]),
+         np.array([[-1.0, 0], [3.0, 4.0]]),
+         np.array([[2.0-2j, 2.0+2j], [-1.0-1j, 2.0]])),
+        # a complex; b and c real.
+        (np.array([[1.0+1j, 2.0], [3.0-4.0j, 5.0]]),
+         np.array([[-1.0, 0], [3.0, 4.0]]),
+         np.array([[2.0, 2.0], [-1.0, 2.0]])),
+        # not square matrices, real
+        (np.array([[8, 1, 6], [3, 5, 7], [4, 9, 2]]),
+         np.array([[2, 3], [4, 5]]),
+         np.array([[1, 2], [3, 4], [5, 6]])),
+        # not square matrices, complex
+        (np.array([[8, 1j, 6+2j], [3, 5, 7], [4, 9, 2]]),
+         np.array([[2, 3], [4, 5-1j]]),
+         np.array([[1, 2j], [3, 4j], [5j, 6+7j]])),
+    ]
+
+    def check_case(self, a, b, c):
+        x = solve_sylvester(a, b, c)
+        assert_array_almost_equal(np.dot(a, x) + np.dot(x, b), c)
+
+    def test_cases(self):
+        for case in self.cases:
+            self.check_case(case[0], case[1], case[2])
+
+    def test_trivial(self):
+        a = np.array([[1.0, 0.0], [0.0, 1.0]])
+        b = np.array([[1.0]])
+        c = np.array([2.0, 2.0]).reshape(-1, 1)
+        x = solve_sylvester(a, b, c)
+        assert_array_almost_equal(x, np.array([1.0, 1.0]).reshape(-1, 1))
+
+    # Feel free to adjust this to test fewer dtypes or random selections rather than
+    # the Cartesian product. It doesn't take very long to test all combinations,
+    # though, so we'll start there and trim it down as we see fit.
+    @pytest.mark.parametrize("dtype_a", dtypes)
+    @pytest.mark.parametrize("dtype_b", dtypes)
+    @pytest.mark.parametrize("dtype_q", dtypes)
+    @pytest.mark.parametrize("m", [0, 3])
+    @pytest.mark.parametrize("n", [0, 3])
+    def test_size_0(self, m, n, dtype_a, dtype_b, dtype_q):
+        if m == n != 0:
+            pytest.skip('m = n != 0 is not a case that needs to be tested here.')
+
+        rng = np.random.default_rng(598435298262546)
+
+        a = np.zeros((m, m), dtype=dtype_a)
+        b = np.zeros((n, n), dtype=dtype_b)
+        q = np.zeros((m, n), dtype=dtype_q)
+        res = solve_sylvester(a, b, q)
+
+        a = (rng.random((5, 5))*100).astype(dtype_a)
+        b = (rng.random((6, 6))*100).astype(dtype_b)
+        q = (rng.random((5, 6))*100).astype(dtype_q)
+        ref = solve_sylvester(a, b, q)
+
+        assert res.shape == (m, n)
+        assert res.dtype == ref.dtype

.venv/lib/python3.12/site-packages/scipy/linalg/tests/test_special_matrices.py

@@ -0,0 +1,619 @@
+import pytest
+import numpy as np
+from numpy import arange, array, eye, copy, sqrt
+from numpy.testing import (assert_equal, assert_array_equal,
+                           assert_array_almost_equal, assert_allclose)
+from pytest import raises as assert_raises
+
+from scipy.fft import fft
+from scipy.special import comb
+from scipy.linalg import (toeplitz, hankel, circulant, hadamard, leslie, dft,
+                          companion, block_diag,
+                          helmert, hilbert, invhilbert, pascal, invpascal,
+                          fiedler, fiedler_companion, eigvals,
+                          convolution_matrix)
+from numpy.linalg import cond
+from scipy._lib._array_api import (make_xp_test_case, xp_assert_equal, xp_size,
+                                   xp_default_dtype)
+
+
+class TestToeplitz:
+
+    def test_basic(self):
+        y = toeplitz([1, 2, 3])
+        assert_array_equal(y, [[1, 2, 3], [2, 1, 2], [3, 2, 1]])
+        y = toeplitz([1, 2, 3], [1, 4, 5])
+        assert_array_equal(y, [[1, 4, 5], [2, 1, 4], [3, 2, 1]])
+
+    def test_complex_01(self):
+        data = (1.0 + arange(3.0)) * (1.0 + 1.0j)
+        x = copy(data)
+        t = toeplitz(x)
+        # Calling toeplitz should not change x.
+        assert_array_equal(x, data)
+        # According to the docstring, x should be the first column of t.
+        col0 = t[:, 0]
+        assert_array_equal(col0, data)
+        assert_array_equal(t[0, 1:], data[1:].conj())
+
+    def test_scalar_00(self):
+        """Scalar arguments still produce a 2D array."""
+        t = toeplitz(10)
+        assert_array_equal(t, [[10]])
+        t = toeplitz(10, 20)
+        assert_array_equal(t, [[10]])
+
+    def test_scalar_01(self):
+        c = array([1, 2, 3])
+        t = toeplitz(c, 1)
+        assert_array_equal(t, [[1], [2], [3]])
+
+    def test_scalar_02(self):
+        c = array([1, 2, 3])
+        t = toeplitz(c, array(1))
+        assert_array_equal(t, [[1], [2], [3]])
+
+    def test_scalar_03(self):
+        c = array([1, 2, 3])
+        t = toeplitz(c, array([1]))
+        assert_array_equal(t, [[1], [2], [3]])
+
+    def test_scalar_04(self):
+        r = array([10, 2, 3])
+        t = toeplitz(1, r)
+        assert_array_equal(t, [[1, 2, 3]])
+
+
+class TestHankel:
+    def test_basic(self):
+        y = hankel([1, 2, 3])
+        assert_array_equal(y, [[1, 2, 3], [2, 3, 0], [3, 0, 0]])
+        y = hankel([1, 2, 3], [3, 4, 5])
+        assert_array_equal(y, [[1, 2, 3], [2, 3, 4], [3, 4, 5]])
+
+
+class TestCirculant:
+    def test_basic(self):
+        y = circulant([1, 2, 3])
+        assert_array_equal(y, [[1, 3, 2], [2, 1, 3], [3, 2, 1]])
+
+
+class TestHadamard:
+
+    def test_basic(self):
+
+        y = hadamard(1)
+        assert_array_equal(y, [[1]])
+
+        y = hadamard(2, dtype=float)
+        assert_array_equal(y, [[1.0, 1.0], [1.0, -1.0]])
+
+        y = hadamard(4)
+        assert_array_equal(y, [[1, 1, 1, 1],
+                               [1, -1, 1, -1],
+                               [1, 1, -1, -1],
+                               [1, -1, -1, 1]])
+
+        assert_raises(ValueError, hadamard, 0)
+        assert_raises(ValueError, hadamard, 5)
+
+
+class TestLeslie:
+
+    def test_bad_shapes(self):
+        assert_raises(ValueError, leslie, [[1, 1], [2, 2]], [3, 4, 5])
+        assert_raises(ValueError, leslie, [1, 2], [1, 2])
+        assert_raises(ValueError, leslie, [1], [])
+
+    def test_basic(self):
+        a = leslie([1, 2, 3], [0.25, 0.5])
+        expected = array([[1.0, 2.0, 3.0],
+                          [0.25, 0.0, 0.0],
+                          [0.0, 0.5, 0.0]])
+        assert_array_equal(a, expected)
+
+
+class TestCompanion:
+
+    def test_bad_shapes(self):
+        assert_raises(ValueError, companion, [0, 4, 5])
+        assert_raises(ValueError, companion, [1])
+        assert_raises(ValueError, companion, [])
+
+    def test_basic(self):
+        c = companion([1, 2, 3])
+        expected = array([
+            [-2.0, -3.0],
+            [1.0, 0.0]])
+        assert_array_equal(c, expected)
+
+        c = companion([2.0, 5.0, -10.0])
+        expected = array([
+            [-2.5, 5.0],
+            [1.0, 0.0]])
+        assert_array_equal(c, expected)
+
+        c = companion([(1.0, 2.0, 3.0),
+                       (4.0, 5.0, 6.0)])
+        expected = array([
+            ([-2.00, -3.00],
+             [+1.00, +0.00]),
+            ([-1.25, -1.50],
+             [+1.00, +0.00])
+        ])
+        assert_array_equal(c, expected)
+
+
+@make_xp_test_case(block_diag)
+class TestBlockDiag:
+    def test_basic(self, xp):
+        dtype = xp.asarray(1).dtype
+        x = block_diag(xp.eye(2, dtype=dtype), xp.asarray([[1, 2], [3, 4], [5, 6]]),
+                       xp.asarray([[1, 2, 3]]))
+        xp_assert_equal(x, xp.asarray([[1, 0, 0, 0, 0, 0, 0],
+                                       [0, 1, 0, 0, 0, 0, 0],
+                                       [0, 0, 1, 2, 0, 0, 0],
+                                       [0, 0, 3, 4, 0, 0, 0],
+                                       [0, 0, 5, 6, 0, 0, 0],
+                                       [0, 0, 0, 0, 1, 2, 3]]))
+
+    def test_dtype(self, xp):
+        x = block_diag(xp.asarray([[1.5]]))
+        assert x.dtype == xp_default_dtype(xp)
+
+        x = block_diag(xp.asarray([[True]]))
+        assert x.dtype == xp.bool
+
+    def test_mixed_dtypes(self, xp):
+        actual = block_diag(xp.asarray([[1.]]), xp.asarray([[1j]]))
+        desired = xp.asarray([[1, 0], [0, 1j]])
+        xp_assert_equal(actual, desired)
+
+    def test_scalar_and_1d_args(self, xp):
+        a = block_diag(xp.asarray(1))
+        assert a.shape == (1, 1)
+        xp_assert_equal(a, xp.asarray([[1]]))
+
+        a = block_diag(xp.asarray([2, 3]), xp.asarray(4))
+        xp_assert_equal(a, xp.asarray([[2, 3, 0], [0, 0, 4]]))
+
+    def test_no_args(self):
+        a = block_diag()
+        assert a.ndim == 2
+        assert a.nbytes == 0
+
+    def test_empty_matrix_arg(self, xp):
+        # regression test for gh-4596: check the shape of the result
+        # for empty matrix inputs. Empty matrices are no longer ignored
+        # (gh-4908) it is viewed as a shape (1, 0) matrix.
+        dtype = xp.asarray(1).dtype
+        a = block_diag(xp.asarray([[1, 0], [0, 1]]),
+                       xp.asarray([], dtype=dtype),
+                       xp.asarray([[2, 3], [4, 5], [6, 7]]))
+        xp_assert_equal(a, xp.asarray([[1, 0, 0, 0],
+                                       [0, 1, 0, 0],
+                                       [0, 0, 0, 0],
+                                       [0, 0, 2, 3],
+                                       [0, 0, 4, 5],
+                                       [0, 0, 6, 7]]))
+
+    @pytest.mark.skip_xp_backends("dask.array", reason="dask/dask#11800")
+    def test_zerosized_matrix_arg(self, xp):
+        # test for gh-4908: check the shape of the result for
+        # zero-sized matrix inputs, i.e. matrices with shape (0,n) or (n,0).
+        # note that [[]] takes shape (1,0)
+        dtype = xp.asarray(1).dtype
+        a = block_diag(xp.asarray([[1, 0], [0, 1]]),
+                       xp.asarray([[]], dtype=dtype),
+                       xp.asarray([[2, 3], [4, 5], [6, 7]]),
+                       xp.zeros([0, 2], dtype=dtype))
+        xp_assert_equal(a, xp.asarray([[1, 0, 0, 0, 0, 0],
+                                       [0, 1, 0, 0, 0, 0],
+                                       [0, 0, 0, 0, 0, 0],
+                                       [0, 0, 2, 3, 0, 0],
+                                       [0, 0, 4, 5, 0, 0],
+                                       [0, 0, 6, 7, 0, 0]]))
+
+
+class TestHelmert:
+
+    def test_orthogonality(self):
+        for n in range(1, 7):
+            H = helmert(n, full=True)
+            Id = np.eye(n)
+            assert_allclose(H.dot(H.T), Id, atol=1e-12)
+            assert_allclose(H.T.dot(H), Id, atol=1e-12)
+
+    def test_subspace(self):
+        for n in range(2, 7):
+            H_full = helmert(n, full=True)
+            H_partial = helmert(n)
+            for U in H_full[1:, :].T, H_partial.T:
+                C = np.eye(n) - np.full((n, n), 1 / n)
+                assert_allclose(U.dot(U.T), C)
+                assert_allclose(U.T.dot(U), np.eye(n-1), atol=1e-12)
+
+
+class TestHilbert:
+
+    def test_basic(self):
+        h3 = array([[1.0, 1/2., 1/3.],
+                    [1/2., 1/3., 1/4.],
+                    [1/3., 1/4., 1/5.]])
+        assert_array_almost_equal(hilbert(3), h3)
+
+        assert_array_equal(hilbert(1), [[1.0]])
+
+        h0 = hilbert(0)
+        assert_equal(h0.shape, (0, 0))
+
+
+class TestInvHilbert:
+
+    def test_basic(self):
+        invh1 = array([[1]])
+        assert_array_equal(invhilbert(1, exact=True), invh1)
+        assert_array_equal(invhilbert(1), invh1)
+
+        invh2 = array([[4, -6],
+                       [-6, 12]])
+        assert_array_equal(invhilbert(2, exact=True), invh2)
+        assert_array_almost_equal(invhilbert(2), invh2)
+
+        invh3 = array([[9, -36, 30],
+                       [-36, 192, -180],
+                       [30, -180, 180]])
+        assert_array_equal(invhilbert(3, exact=True), invh3)
+        assert_array_almost_equal(invhilbert(3), invh3)
+
+        invh4 = array([[16, -120, 240, -140],
+                       [-120, 1200, -2700, 1680],
+                       [240, -2700, 6480, -4200],
+                       [-140, 1680, -4200, 2800]])
+        assert_array_equal(invhilbert(4, exact=True), invh4)
+        assert_array_almost_equal(invhilbert(4), invh4)
+
+        invh5 = array([[25, -300, 1050, -1400, 630],
+                       [-300, 4800, -18900, 26880, -12600],
+                       [1050, -18900, 79380, -117600, 56700],
+                       [-1400, 26880, -117600, 179200, -88200],
+                       [630, -12600, 56700, -88200, 44100]])
+        assert_array_equal(invhilbert(5, exact=True), invh5)
+        assert_array_almost_equal(invhilbert(5), invh5)
+
+        invh17 = array([
+            [289, -41616, 1976760, -46124400, 629598060, -5540462928,
+             33374693352, -143034400080, 446982500250, -1033026222800,
+             1774926873720, -2258997839280, 2099709530100, -1384423866000,
+             613101997800, -163493866080, 19835652870],
+            [-41616, 7990272, -426980160, 10627061760, -151103534400,
+             1367702848512, -8410422724704, 36616806420480, -115857864064800,
+             270465047424000, -468580694662080, 600545887119360,
+             -561522320049600, 372133135180800, -165537539406000,
+             44316454993920, -5395297580640],
+            [1976760, -426980160, 24337869120, -630981792000, 9228108708000,
+             -85267724461920, 532660105897920, -2348052711713280,
+             7504429831470000, -17664748409880000, 30818191841236800,
+             -39732544853164800, 37341234283298400, -24857330514030000,
+             11100752642520000, -2982128117299200, 364182586693200],
+            [-46124400, 10627061760, -630981792000, 16826181120000,
+             -251209625940000, 2358021022156800, -14914482965141760,
+             66409571644416000, -214015221119700000, 507295338950400000,
+             -890303319857952000, 1153715376477081600, -1089119333262870000,
+             727848632044800000, -326170262829600000, 87894302404608000,
+             -10763618673376800],
+            [629598060, -151103534400, 9228108708000,
+             -251209625940000, 3810012660090000, -36210360321495360,
+             231343968720664800, -1038687206500944000, 3370739732635275000,
+             -8037460526495400000, 14178080368737885600, -18454939322943942000,
+             17489975175339030000, -11728977435138600000, 5272370630081100000,
+             -1424711708039692800, 174908803442373000],
+            [-5540462928, 1367702848512, -85267724461920, 2358021022156800,
+             -36210360321495360, 347619459086355456, -2239409617216035264,
+             10124803292907663360, -33052510749726468000,
+             79217210949138662400, -140362995650505067440,
+             183420385176741672960, -174433352415381259200,
+             117339159519533952000, -52892422160973595200,
+             14328529177999196160, -1763080738699119840],
+            [33374693352, -8410422724704, 532660105897920,
+             -14914482965141760, 231343968720664800, -2239409617216035264,
+             14527452132196331328, -66072377044391477760,
+             216799987176909536400, -521925895055522958000,
+             928414062734059661760, -1217424500995626443520,
+             1161358898976091015200, -783401860847777371200,
+             354015418167362952000, -96120549902411274240,
+             11851820521255194480],
+            [-143034400080, 36616806420480, -2348052711713280,
+             66409571644416000, -1038687206500944000, 10124803292907663360,
+             -66072377044391477760, 302045152202932469760,
+             -995510145200094810000, 2405996923185123840000,
+             -4294704507885446054400, 5649058909023744614400,
+             -5403874060541811254400, 3654352703663101440000,
+             -1655137020003255360000, 450325202737117593600,
+             -55630994283442749600],
+            [446982500250, -115857864064800, 7504429831470000,
+             -214015221119700000, 3370739732635275000, -33052510749726468000,
+             216799987176909536400, -995510145200094810000,
+             3293967392206196062500, -7988661659013106500000,
+             14303908928401362270000, -18866974090684772052000,
+             18093328327706957325000, -12263364009096700500000,
+             5565847995255512250000, -1517208935002984080000,
+             187754605706619279900],
+            [-1033026222800, 270465047424000, -17664748409880000,
+             507295338950400000, -8037460526495400000, 79217210949138662400,
+             -521925895055522958000, 2405996923185123840000,
+             -7988661659013106500000, 19434404971634224000000,
+             -34894474126569249192000, 46141453390504792320000,
+             -44349976506971935800000, 30121928988527376000000,
+             -13697025107665828500000, 3740200989399948902400,
+             -463591619028689580000],
+            [1774926873720, -468580694662080,
+             30818191841236800, -890303319857952000, 14178080368737885600,
+             -140362995650505067440, 928414062734059661760,
+             -4294704507885446054400, 14303908928401362270000,
+             -34894474126569249192000, 62810053427824648545600,
+             -83243376594051600326400, 80177044485212743068000,
+             -54558343880470209780000, 24851882355348879230400,
+             -6797096028813368678400, 843736746632215035600],
+            [-2258997839280, 600545887119360, -39732544853164800,
+             1153715376477081600, -18454939322943942000, 183420385176741672960,
+             -1217424500995626443520, 5649058909023744614400,
+             -18866974090684772052000, 46141453390504792320000,
+             -83243376594051600326400, 110552468520163390156800,
+             -106681852579497947388000, 72720410752415168870400,
+             -33177973900974346080000, 9087761081682520473600,
+             -1129631016152221783200],
+            [2099709530100, -561522320049600, 37341234283298400,
+             -1089119333262870000, 17489975175339030000,
+             -174433352415381259200, 1161358898976091015200,
+             -5403874060541811254400, 18093328327706957325000,
+             -44349976506971935800000, 80177044485212743068000,
+             -106681852579497947388000, 103125790826848015808400,
+             -70409051543137015800000, 32171029219823375700000,
+             -8824053728865840192000, 1098252376814660067000],
+            [-1384423866000, 372133135180800,
+             -24857330514030000, 727848632044800000, -11728977435138600000,
+             117339159519533952000, -783401860847777371200,
+             3654352703663101440000, -12263364009096700500000,
+             30121928988527376000000, -54558343880470209780000,
+             72720410752415168870400, -70409051543137015800000,
+             48142941226076592000000, -22027500987368499000000,
+             6049545098753157120000, -753830033789944188000],
+            [613101997800, -165537539406000,
+             11100752642520000, -326170262829600000, 5272370630081100000,
+             -52892422160973595200, 354015418167362952000,
+             -1655137020003255360000, 5565847995255512250000,
+             -13697025107665828500000, 24851882355348879230400,
+             -33177973900974346080000, 32171029219823375700000,
+             -22027500987368499000000, 10091416708498869000000,
+             -2774765838662800128000, 346146444087219270000],
+            [-163493866080, 44316454993920, -2982128117299200,
+             87894302404608000, -1424711708039692800,
+             14328529177999196160, -96120549902411274240,
+             450325202737117593600, -1517208935002984080000,
+             3740200989399948902400, -6797096028813368678400,
+             9087761081682520473600, -8824053728865840192000,
+             6049545098753157120000, -2774765838662800128000,
+             763806510427609497600, -95382575704033754400],
+            [19835652870, -5395297580640, 364182586693200, -10763618673376800,
+             174908803442373000, -1763080738699119840, 11851820521255194480,
+             -55630994283442749600, 187754605706619279900,
+             -463591619028689580000, 843736746632215035600,
+             -1129631016152221783200, 1098252376814660067000,
+             -753830033789944188000, 346146444087219270000,
+             -95382575704033754400, 11922821963004219300]
+        ])
+        assert_array_equal(invhilbert(17, exact=True), invh17)
+        assert_allclose(invhilbert(17), invh17.astype(float), rtol=1e-12)
+
+    def test_inverse(self):
+        for n in range(1, 10):
+            a = hilbert(n)
+            b = invhilbert(n)
+            # The Hilbert matrix is increasingly badly conditioned,
+            # so take that into account in the test
+            c = cond(a)
+            assert_allclose(a.dot(b), eye(n), atol=1e-15*c, rtol=1e-15*c)
+
+
+class TestPascal:
+
+    cases = [
+        (1, array([[1]]), array([[1]])),
+        (2, array([[1, 1],
+                   [1, 2]]),
+            array([[1, 0],
+                   [1, 1]])),
+        (3, array([[1, 1, 1],
+                   [1, 2, 3],
+                   [1, 3, 6]]),
+            array([[1, 0, 0],
+                   [1, 1, 0],
+                   [1, 2, 1]])),
+        (4, array([[1, 1, 1, 1],
+                   [1, 2, 3, 4],
+                   [1, 3, 6, 10],
+                   [1, 4, 10, 20]]),
+            array([[1, 0, 0, 0],
+                   [1, 1, 0, 0],
+                   [1, 2, 1, 0],
+                   [1, 3, 3, 1]])),
+    ]
+
+    def check_case(self, n, sym, low):
+        assert_array_equal(pascal(n), sym)
+        assert_array_equal(pascal(n, kind='lower'), low)
+        assert_array_equal(pascal(n, kind='upper'), low.T)
+        assert_array_almost_equal(pascal(n, exact=False), sym)
+        assert_array_almost_equal(pascal(n, exact=False, kind='lower'), low)
+        assert_array_almost_equal(pascal(n, exact=False, kind='upper'), low.T)
+
+    def test_cases(self):
+        for n, sym, low in self.cases:
+            self.check_case(n, sym, low)
+
+    def test_big(self):
+        p = pascal(50)
+        assert p[-1, -1] == comb(98, 49, exact=True)
+
+    def test_threshold(self):
+        # Regression test.  An early version of `pascal` returned an
+        # array of type np.uint64 for n=35, but that data type is too small
+        # to hold p[-1, -1].  The second assert_equal below would fail
+        # because p[-1, -1] overflowed.
+        p = pascal(34)
+        assert_equal(2*p.item(-1, -2), p.item(-1, -1), err_msg="n = 34")
+        p = pascal(35)
+        assert_equal(2.*p.item(-1, -2), 1.*p.item(-1, -1), err_msg="n = 35")
+
+
+def test_invpascal():
+
+    def check_invpascal(n, kind, exact):
+        ip = invpascal(n, kind=kind, exact=exact)
+        p = pascal(n, kind=kind, exact=exact)
+        # Matrix-multiply ip and p, and check that we get the identity matrix.
+        # We can't use the simple expression e = ip.dot(p), because when
+        # n < 35 and exact is True, p.dtype is np.uint64 and ip.dtype is
+        # np.int64. The product of those dtypes is np.float64, which loses
+        # precision when n is greater than 18.  Instead we'll cast both to
+        # object arrays, and then multiply.
+        e = ip.astype(object).dot(p.astype(object))
+        assert_array_equal(e, eye(n), err_msg=f"n={n}  kind={kind!r} exact={exact!r}")
+
+    kinds = ['symmetric', 'lower', 'upper']
+
+    ns = [1, 2, 5, 18]
+    for n in ns:
+        for kind in kinds:
+            for exact in [True, False]:
+                check_invpascal(n, kind, exact)
+
+    ns = [19, 34, 35, 50]
+    for n in ns:
+        for kind in kinds:
+            check_invpascal(n, kind, True)
+
+
+def test_dft():
+    m = dft(2)
+    expected = array([[1.0, 1.0], [1.0, -1.0]])
+    assert_array_almost_equal(m, expected)
+    m = dft(2, scale='n')
+    assert_array_almost_equal(m, expected/2.0)
+    m = dft(2, scale='sqrtn')
+    assert_array_almost_equal(m, expected/sqrt(2.0))
+
+    x = array([0, 1, 2, 3, 4, 5, 0, 1])
+    m = dft(8)
+    mx = m.dot(x)
+    fx = fft(x)
+    assert_array_almost_equal(mx, fx)
+
+
+@make_xp_test_case(fiedler)
+def test_fiedler(xp):
+    f = fiedler(xp.asarray([]))
+    assert xp_size(f) == 0
+    assert f.shape == (0, 0)
+
+    f = fiedler(xp.asarray([123.]))
+    xp_assert_equal(f, xp.asarray([[0.]]))
+
+    f = fiedler(xp.arange(1, 7))
+    des = xp.asarray([[0, 1, 2, 3, 4, 5],
+                      [1, 0, 1, 2, 3, 4],
+                      [2, 1, 0, 1, 2, 3],
+                      [3, 2, 1, 0, 1, 2],
+                      [4, 3, 2, 1, 0, 1],
+                      [5, 4, 3, 2, 1, 0]])
+    xp_assert_equal(f, des)
+
+
+def test_fiedler_companion():
+    fc = fiedler_companion([])
+    assert_equal(fc.size, 0)
+    fc = fiedler_companion([1.])
+    assert_equal(fc.size, 0)
+    assert_equal(fc.shape, (0, 0))
+    fc = fiedler_companion([1., 2.])
+    assert_array_equal(fc, np.array([[-2.]]))
+    fc = fiedler_companion([1e-12, 2., 3.])
+    assert_array_almost_equal(fc, companion([1e-12, 2., 3.]))
+    with assert_raises(ValueError):
+        fiedler_companion([0, 1, 2])
+    fc = fiedler_companion([1., -16., 86., -176., 105.])
+    assert_array_almost_equal(eigvals(fc),
+                              np.array([7., 5., 3., 1.]))
+
+
+class TestConvolutionMatrix:
+    """
+    Test convolution_matrix vs. numpy.convolve for various parameters.
+    """
+
+    def create_vector(self, n, cpx):
+        """Make a complex or real test vector of length n."""
+        x = np.linspace(-2.5, 2.2, n)
+        if cpx:
+            x = x + 1j*np.linspace(-1.5, 3.1, n)
+        return x
+
+    def test_bad_n(self):
+        # n must be a positive integer
+        with pytest.raises(ValueError, match='n must be a positive integer'):
+            convolution_matrix([1, 2, 3], 0)
+
+    def test_empty_first_arg(self):
+        # first arg must have at least one value
+        with pytest.raises(ValueError, match=r'len\(a\)'):
+            convolution_matrix([], 4)
+
+    def test_bad_mode(self):
+        # mode must be in ('full', 'valid', 'same')
+        with pytest.raises(ValueError, match='mode.*must be one of'):
+            convolution_matrix((1, 1), 4, mode='invalid argument')
+
+    @pytest.mark.parametrize('cpx', [False, True])
+    @pytest.mark.parametrize('na', [1, 2, 9])
+    @pytest.mark.parametrize('nv', [1, 2, 9])
+    @pytest.mark.parametrize('mode', [None, 'full', 'valid', 'same'])
+    def test_against_numpy_convolve(self, cpx, na, nv, mode):
+        a = self.create_vector(na, cpx)
+        v = self.create_vector(nv, cpx)
+        if mode is None:
+            y1 = np.convolve(v, a)
+            A = convolution_matrix(a, nv)
+        else:
+            y1 = np.convolve(v, a, mode)
+            A = convolution_matrix(a, nv, mode)
+        y2 = A @ v
+        assert_array_almost_equal(y1, y2)
+
+
+@pytest.mark.fail_slow(5)  # `leslie` has an import in the function
+@pytest.mark.parametrize('f, args', [(circulant, ()),
+                                     (companion, ()),
+                                     (convolution_matrix, (5, 'same')),
+                                     (fiedler, ()),
+                                     (fiedler_companion, ()),
+                                     (hankel, (np.arange(9),)),
+                                     (leslie, (np.arange(9),)),
+                                     (toeplitz, (np.arange(9),)),
+                                     ])
+def test_batch(f, args):
+    rng = np.random.default_rng(283592436523456)
+    batch_shape = (2, 3)
+    m = 10
+    A = rng.random(batch_shape + (m,))
+
+    if f in {hankel}:
+        message = "Beginning in SciPy 1.19, multidimensional input will be..."
+        with pytest.warns(FutureWarning, match=message):
+            f(A, *args)
+        return
+
+    res = f(A, *args)
+    ref = np.asarray([f(a, *args) for a in A.reshape(-1, m)])
+    ref = ref.reshape(A.shape[:-1] + ref.shape[-2:])
+    assert_allclose(res, ref)

.venv/lib/python3.12/site-packages/scipy/misc/__init__.py

@@ -0,0 +1,6 @@
+import warnings
+warnings.warn(
+    "scipy.misc is deprecated and will be removed in 2.0.0",
+    DeprecationWarning,
+    stacklevel=2
+)

.venv/lib/python3.12/site-packages/scipy/misc/common.py

@@ -0,0 +1,6 @@
+import warnings
+warnings.warn(
+    "scipy.misc.common is deprecated and will be removed in 2.0.0",
+    DeprecationWarning,
+    stacklevel=2
+)

.venv/lib/python3.12/site-packages/scipy/misc/doccer.py

@@ -0,0 +1,6 @@
+import warnings
+warnings.warn(
+    "scipy.misc.doccer is deprecated and will be removed in 2.0.0",
+    DeprecationWarning,
+    stacklevel=2
+)

.venv/lib/python3.12/site-packages/scipy/ndimage/__init__.py

@@ -0,0 +1,174 @@
+"""
+=========================================================
+Multidimensional image processing (:mod:`scipy.ndimage`)
+=========================================================
+
+.. currentmodule:: scipy.ndimage
+
+This package contains various functions for multidimensional image
+processing.
+
+
+Filters
+=======
+
+.. autosummary::
+   :toctree: generated/
+
+   convolve - Multidimensional convolution
+   convolve1d - 1-D convolution along the given axis
+   correlate - Multidimensional correlation
+   correlate1d - 1-D correlation along the given axis
+   gaussian_filter
+   gaussian_filter1d
+   gaussian_gradient_magnitude
+   gaussian_laplace
+   generic_filter - Multidimensional filter using a given function
+   generic_filter1d - 1-D generic filter along the given axis
+   generic_gradient_magnitude
+   generic_laplace
+   laplace - N-D Laplace filter based on approximate second derivatives
+   maximum_filter
+   maximum_filter1d
+   median_filter - Calculates a multidimensional median filter
+   minimum_filter
+   minimum_filter1d
+   percentile_filter - Calculates a multidimensional percentile filter
+   prewitt
+   rank_filter - Calculates a multidimensional rank filter
+   sobel
+   uniform_filter - Multidimensional uniform filter
+   uniform_filter1d - 1-D uniform filter along the given axis
+   vectorized_filter
+
+Fourier filters
+===============
+
+.. autosummary::
+   :toctree: generated/
+
+   fourier_ellipsoid
+   fourier_gaussian
+   fourier_shift
+   fourier_uniform
+
+Interpolation
+=============
+
+.. autosummary::
+   :toctree: generated/
+
+   affine_transform - Apply an affine transformation
+   geometric_transform - Apply an arbitrary geometric transform
+   map_coordinates - Map input array to new coordinates by interpolation
+   rotate - Rotate an array
+   shift - Shift an array
+   spline_filter
+   spline_filter1d
+   zoom - Zoom an array
+
+Measurements
+============
+
+.. autosummary::
+   :toctree: generated/
+
+   center_of_mass - The center of mass of the values of an array at labels
+   extrema - Min's and max's of an array at labels, with their positions
+   find_objects - Find objects in a labeled array
+   histogram - Histogram of the values of an array, optionally at labels
+   label - Label features in an array
+   labeled_comprehension
+   maximum
+   maximum_position
+   mean - Mean of the values of an array at labels
+   median
+   minimum
+   minimum_position
+   standard_deviation - Standard deviation of an N-D image array
+   sum_labels - Sum of the values of the array
+   value_indices - Find indices of each distinct value in given array
+   variance - Variance of the values of an N-D image array
+   watershed_ift
+
+Morphology
+==========
+
+.. autosummary::
+   :toctree: generated/
+
+   binary_closing
+   binary_dilation
+   binary_erosion
+   binary_fill_holes
+   binary_hit_or_miss
+   binary_opening
+   binary_propagation
+   black_tophat
+   distance_transform_bf
+   distance_transform_cdt
+   distance_transform_edt
+   generate_binary_structure
+   grey_closing
+   grey_dilation
+   grey_erosion
+   grey_opening
+   iterate_structure
+   morphological_gradient
+   morphological_laplace
+   white_tophat
+
+"""
+
+# Copyright (C) 2003-2005 Peter J. Verveer
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above
+#    copyright notice, this list of conditions and the following
+#    disclaimer in the documentation and/or other materials provided
+#    with the distribution.
+#
+# 3. The name of the author may not be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS
+# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
+# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+# bring in the public functionality from private namespaces
+
+# mypy: ignore-errors
+
+from ._support_alternative_backends import *
+
+# adjust __all__ and do not leak implementation details
+from . import _support_alternative_backends
+__all__ = _support_alternative_backends.__all__
+del _support_alternative_backends, _ndimage_api, _delegators  # noqa: F821  # pyrefly:ignore[unbound-name]
+
+
+# Deprecated namespaces, to be removed in v2.0.0
+from . import filters
+from . import fourier
+from . import interpolation
+from . import measurements
+from . import morphology
+
+from scipy._lib._testutils import PytestTester
+test = PytestTester(__name__)
+del PytestTester

.venv/lib/python3.12/site-packages/scipy/ndimage/_delegators.py

@@ -0,0 +1,303 @@
+"""Delegators for alternative backends in scipy.ndimage.
+
+The signature of `func_signature` must match the signature of ndimage.func.
+The job of a `func_signature` is to know which arguments of `ndimage.func`
+are arrays.
+
+* signatures are generated by
+
+--------------
+import inspect
+from scipy import ndimage
+
+names = [x for x in dir(ndimage) if not x.startswith('_')]
+objs = [getattr(ndimage, name) for name in names]
+funcs = [obj for obj in objs if inspect.isroutine(obj)]
+
+for func in funcs:
+    sig = inspect.signature(func)
+    print(f"def {func.__name__}_signature{sig}:\n\tpass\n\n")
+---------------
+
+* which arguments to delegate on: manually trawled the documentation for
+  array-like and array arguments
+
+"""
+import numpy as np
+from scipy._lib._array_api import array_namespace
+from scipy.ndimage._ni_support import _skip_if_dtype
+
+
+def affine_transform_signature(
+    input, matrix, offset=0.0, output_shape=None, output=None, *args, **kwds
+):
+    return array_namespace(input, matrix, _skip_if_dtype(output))
+
+
+def binary_closing_signature(
+    input, structure=None, iterations=1, output=None, *args, **kwds
+):
+    return array_namespace(input, structure, _skip_if_dtype(output))
+
+binary_opening_signature = binary_closing_signature
+
+
+def binary_dilation_signature(
+    input, structure=None, iterations=1, mask=None, output=None, *args, **kwds
+):
+    return array_namespace(input, structure, _skip_if_dtype(output), mask)
+
+binary_erosion_signature = binary_dilation_signature
+
+
+def binary_fill_holes_signature(
+    input, structure=None, output=None, origin=0, *args, **kwargs
+):
+    return array_namespace(input, structure, _skip_if_dtype(output))
+
+
+def label_signature(input, structure=None, output=None, origin=0):
+    return array_namespace(input, structure, _skip_if_dtype(output))
+
+
+def binary_hit_or_miss_signature(
+    input, structure1=None, structure2=None, output=None, *args, **kwds
+):
+    return array_namespace(input, structure1, structure2, _skip_if_dtype(output))
+
+
+def binary_propagation_signature(
+    input, structure=None, mask=None, output=None, *args, **kwds
+):
+    return array_namespace(input, structure, mask, _skip_if_dtype(output))
+
+
+def convolve_signature(input, weights, output=None, *args, **kwds):
+    return array_namespace(input, weights, _skip_if_dtype(output))
+
+correlate_signature = convolve_signature
+
+
+def convolve1d_signature(input, weights, axis=-1, output=None, *args, **kwds):
+    return array_namespace(input, weights, _skip_if_dtype(output))
+
+correlate1d_signature = convolve1d_signature
+
+
+def distance_transform_bf_signature(
+    input, metric='euclidean', sampling=None, return_distances=True,
+    return_indices=False, distances=None, indices=None
+):
+    return array_namespace(input, distances, indices)
+
+
+def distance_transform_cdt_signature(
+    input, metric='chessboard', return_distances=True, return_indices=False,
+    distances=None, indices=None
+):
+    return array_namespace(input, distances, indices)
+
+
+def distance_transform_edt_signature(
+    input, sampling=None, return_distances=True, return_indices=False,
+    distances=None, indices=None
+):
+    return array_namespace(input, distances, indices)
+
+
+def find_objects_signature(input, max_label=0):
+    return array_namespace(input)
+
+
+def fourier_ellipsoid_signature(input, size, n=-1, axis=-1, output=None):
+    return array_namespace(input, _skip_if_dtype(output))
+
+fourier_uniform_signature = fourier_ellipsoid_signature
+
+
+def fourier_gaussian_signature(input, sigma, n=-1, axis=-1, output=None):
+    return array_namespace(input, _skip_if_dtype(output))
+
+def fourier_shift_signature(input, shift, n=-1, axis=-1, output=None):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def gaussian_filter_signature(input, sigma, order=0, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def gaussian_filter1d_signature(
+    input, sigma, axis=-1, order=0, output=None, *args, **kwds
+):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def gaussian_gradient_magnitude_signature(input, sigma, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+gaussian_laplace_signature = gaussian_gradient_magnitude_signature
+
+
+def generate_binary_structure_signature(rank, connectivity):
+    # XXX: no input arrays; always return numpy
+    return np
+
+
+def generic_filter_signature(
+    input, function, size=None, footprint=None, output=None, *args, **kwds
+):
+    # XXX: function LowLevelCallable w/backends
+    return array_namespace(input, footprint, _skip_if_dtype(output))
+
+
+def generic_filter1d_signature(
+    input, function, filter_size, axis=-1, output=None, *args, **kwds
+):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def generic_gradient_magnitude_signature(
+    input, derivative, output=None, *args, **kwds
+):
+    # XXX: function LowLevelCallable w/backends
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def generic_laplace_signature(input, derivative2, output=None, *args, **kwds):
+    # XXX: function LowLevelCallable w/backends
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def geometric_transform_signature(
+    input, mapping, output_shape=None, output=None, *args, **kwds
+):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def histogram_signature(input, min, max, bins, labels=None, index=None):
+    return array_namespace(input, labels)
+
+
+def iterate_structure_signature(structure, iterations, origin=None):
+    return array_namespace(structure)
+
+
+def labeled_comprehension_signature(input, labels, *args, **kwds):
+    return array_namespace(input, labels)
+
+
+def laplace_signature(input, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def map_coordinates_signature(input, coordinates, output=None, *args, **kwds):
+    return array_namespace(input, coordinates, _skip_if_dtype(output))
+
+
+def maximum_filter1d_signature(input, size, axis=-1, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+minimum_filter1d_signature = maximum_filter1d_signature
+uniform_filter1d_signature = maximum_filter1d_signature
+
+
+def maximum_signature(input, labels=None, index=None):
+    return array_namespace(input, labels, index)
+
+minimum_signature = maximum_signature
+median_signature = maximum_signature
+mean_signature = maximum_signature
+variance_signature = maximum_signature
+standard_deviation_signature = maximum_signature
+sum_labels_signature = maximum_signature
+sum_signature = maximum_signature  # ndimage.sum is sum_labels
+
+maximum_position_signature = maximum_signature
+minimum_position_signature = maximum_signature
+
+extrema_signature = maximum_signature
+center_of_mass_signature = extrema_signature
+
+
+def median_filter_signature(
+    input, size=None, footprint=None, output=None, *args, **kwds
+):
+    return array_namespace(input, footprint, _skip_if_dtype(output))
+
+minimum_filter_signature = median_filter_signature
+maximum_filter_signature = median_filter_signature
+
+
+def morphological_gradient_signature(
+    input, size=None, footprint=None, structure=None, output=None, *args, **kwds
+):
+    return array_namespace(input, footprint, structure, _skip_if_dtype(output))
+
+morphological_laplace_signature = morphological_gradient_signature
+white_tophat_signature = morphological_gradient_signature
+black_tophat_signature = morphological_gradient_signature
+grey_closing_signature = morphological_gradient_signature
+grey_dilation_signature = morphological_gradient_signature
+grey_erosion_signature = morphological_gradient_signature
+grey_opening_signature = morphological_gradient_signature
+
+
+def percentile_filter_signature(
+    input, percentile, size=None, footprint=None, output=None, *args, **kwds
+):
+    return array_namespace(input, footprint, _skip_if_dtype(output))
+
+
+def prewitt_signature(input, axis=-1, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+sobel_signature = prewitt_signature
+
+
+def rank_filter_signature(
+    input, rank, size=None, footprint=None, output=None, *args, **kwds
+):
+    return array_namespace(input, footprint, _skip_if_dtype(output))
+
+
+def rotate_signature(
+    input, angle, axes=(1, 0), reshape=True, output=None , *args, **kwds
+):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def shift_signature(input, shift, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def spline_filter_signature(input, order=3, output=np.float64, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def spline_filter1d_signature(
+    input, order=3, axis=-1, output=np.float64, *args, **kwds
+):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def uniform_filter_signature(input, size=3, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+
+
+def value_indices_signature(arr, *args, **kwds):
+    return array_namespace(arr)
+
+
+def vectorized_filter_signature(
+    input, function, size=None, footprint=None, output=None, *args, **kwds
+):
+    return array_namespace(input, footprint, _skip_if_dtype(output))
+
+
+def watershed_ift_signature(input, markers, structure=None, output=None):
+    return array_namespace(input, markers, structure, _skip_if_dtype(output))
+
+
+def zoom_signature(input, zoom, output=None, *args, **kwds):
+    return array_namespace(input, _skip_if_dtype(output))
+

.venv/lib/python3.12/site-packages/scipy/ndimage/_filters.py

@@ -0,0 +1,2446 @@
+# Copyright (C) 2003-2005 Peter J. Verveer
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above
+#    copyright notice, this list of conditions and the following
+#    disclaimer in the documentation and/or other materials provided
+#    with the distribution.
+#
+# 3. The name of the author may not be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS
+# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
+# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+from collections.abc import Iterable
+import numbers
+import warnings
+import numpy as np
+import operator
+import math
+
+from scipy._lib._util import normalize_axis_index
+from scipy._lib._array_api import array_namespace, is_cupy, xp_size
+from . import _ni_support
+from . import _nd_image
+from . import _ni_docstrings
+from . import _rank_filter_1d
+
+__all__ = ['correlate1d', 'convolve1d', 'gaussian_filter1d', 'gaussian_filter',
+           'prewitt', 'sobel', 'generic_laplace', 'laplace',
+           'gaussian_laplace', 'generic_gradient_magnitude',
+           'gaussian_gradient_magnitude', 'correlate', 'convolve',
+           'uniform_filter1d', 'uniform_filter', 'minimum_filter1d',
+           'maximum_filter1d', 'minimum_filter', 'maximum_filter',
+           'rank_filter', 'median_filter', 'percentile_filter',
+           'generic_filter1d', 'generic_filter', 'vectorized_filter']
+
+
+def _vectorized_filter_iv(input, function, size, footprint, output, mode, cval, origin,
+                          axes, batch_memory):
+    xp = array_namespace(input, footprint, output)
+
+    # vectorized_filter input validation and standardization
+    input = xp.asarray(input)
+
+    if not callable(function):
+        raise ValueError("`function` must be a callable.")
+
+    if size is None and footprint is None:
+        raise ValueError("Either `size` or `footprint` must be provided.")
+
+    if size is not None and footprint is not None:
+        raise ValueError("Either `size` or `footprint` may be provided, not both.")
+
+    if axes is None:
+        axes = tuple(range(-input.ndim, 0))
+    elif np.isscalar(axes):
+        axes = (axes,)
+    n_axes = len(axes)
+    n_batch = input.ndim - n_axes
+
+    if n_axes > input.ndim:
+        message = ("The length of `axes` may not exceed the dimensionality of `input`"
+                   "(`input.ndim`).")
+        raise ValueError(message)
+
+    # Either footprint or size must be provided
+    footprinted_function = function
+    if size is not None:
+        # If provided, size must be an integer or tuple of integers.
+        size = (size,)*n_axes if np.isscalar(size) else tuple(size)
+        valid = [xp.isdtype(xp.asarray(i).dtype, 'integral') and i > 0 for i in size]
+        if not all(valid):
+            raise ValueError("All elements of `size` must be positive integers.")
+    else:
+        # If provided, `footprint` must be array-like
+        footprint = xp.asarray(footprint, dtype=xp.bool)
+        size = footprint.shape
+        def footprinted_function(input, *args, axis=-1, **kwargs):
+            return function(input[..., footprint], *args, axis=-1, **kwargs)
+
+    # And by now, the dimensionality of the footprint must equal the number of axes
+    if n_axes != len(size):
+        message = ("`axes` must be compatible with the dimensionality "
+                   "of the window specified by `size` or `footprint`.")
+        raise ValueError(message)
+
+    # If this is not *equal* to the dimensionality of `input`, then `axes`
+    # must be a provided tuple, and its length must equal the core dimensionality.
+    elif n_axes < input.ndim:
+        if axes is None:
+            message = ("`axes` must be provided if the dimensionality of the window "
+                       "(`len(size)` or `footprint.ndim`) does not equal the number "
+                       "of axes of `input` (`input.ndim`).")
+            raise ValueError(message)
+    else:
+        axes = tuple(range(-n_axes, 0)) if axes is None else axes
+
+    axes = (axes,) if np.isscalar(axes) else axes
+
+    # If `origin` is provided, then it must be "broadcastable" to a tuple with length
+    # equal to the core dimensionality.
+    if origin is None:
+        origin = (0,) * n_axes
+    else:
+        origin = (origin,)*n_axes if np.isscalar(origin) else tuple(origin)
+        integral = [xp.isdtype(xp.asarray(i).dtype, 'integral') for i in origin]
+        if not all(integral):
+            raise ValueError("All elements of `origin` must be integers.")
+        if not len(origin) == n_axes:
+            message = ("`origin` must be an integer or tuple of integers with length "
+                       "equal to the number of axes.")
+            raise ValueError(message)
+
+    # mode must be one of the allowed strings, and we should convert it to the
+    # value required by `np.pad`/`cp.pad` here.
+    valid_modes = {'reflect', 'constant', 'nearest', 'mirror', 'wrap',
+                   'grid-mirror', 'grid-constant', 'grid-wrap', 'valid'}
+    if mode not in valid_modes:
+        raise ValueError(f"`mode` must be one of {valid_modes}.")
+    mode_map = {'nearest': 'edge', 'reflect': 'symmetric', 'mirror': 'reflect',
+                'grid-mirror': 'reflect', 'grid-constant': 'constant',
+                'grid-wrap': 'wrap'}
+    mode = mode_map.get(mode, mode)
+
+    if mode == 'valid' and any(origin):
+        raise ValueError("`mode='valid'` is incompatible with use of `origin`.")
+
+    if cval is None:
+        cval = 0.0
+    elif mode != 'constant':
+        raise ValueError("Use of `cval` is compatible only with `mode='constant'`.")
+
+    # `cval` must be a scalar or "broadcastable" to a tuple with the same
+    # dimensionality of `input`. (Full input validation done by `np.pad`/`cp.pad`.)
+    if not xp.isdtype(xp.asarray(cval).dtype, 'numeric'):
+        raise ValueError("`cval` must include only numbers.")
+
+    # `batch_memory` must be a positive number.
+    temp = xp.asarray(batch_memory)
+    if temp.ndim != 0 or (not xp.isdtype(temp.dtype, 'numeric')) or temp <= 0:
+        raise ValueError("`batch_memory` must be positive number.")
+
+    # For simplicity, work with `axes` at the end.
+    working_axes = tuple(range(-n_axes, 0))
+    input = xp.moveaxis(input, axes, working_axes)
+    output = (xp.moveaxis(output, axes, working_axes)
+              if output is not None else output)
+
+    # Wrap the function to limit maximum memory usage, deal with `footprint`,
+    # and populate `output`. The latter requires some verbosity because we
+    # don't know the output dtype.
+    def wrapped_function(view, output=output):
+        kwargs = {'axis': working_axes}
+
+        if working_axes == ():
+            return footprinted_function(xp.asarray(view), **kwargs)
+
+        # for now, assume we only have to iterate over zeroth axis
+        chunk_size = math.prod(view.shape[1:]) * view.dtype.itemsize
+        slices_per_batch = min(view.shape[0], batch_memory // chunk_size)
+        if slices_per_batch < 1:
+            raise ValueError("`batch_memory` is insufficient for minimum chunk size.")
+
+        elif slices_per_batch == view.shape[0]:
+            if output is None:
+                return footprinted_function(xp.asarray(view), **kwargs)
+            else:
+                output[...] = footprinted_function(xp.asarray(view), **kwargs)
+                return output
+
+        for i in range(0, view.shape[0], slices_per_batch):
+            i2 = min(i + slices_per_batch, view.shape[0])
+            if output is None:
+                # Look at the dtype before allocating the array. (In a follow-up, we
+                # can also look at the shape to support non-scalar elements.)
+                temp = footprinted_function(xp.asarray(view[i:i2]), **kwargs)
+                output = xp.empty(view.shape[:-n_axes], dtype=temp.dtype)
+                output[i:i2, ...] = temp
+            else:
+                output[i:i2, ...] = footprinted_function(xp.asarray(view[i:i2]),
+                                                         **kwargs)
+        return output
+
+    return (input, wrapped_function, size, mode, cval, origin,
+            working_axes, axes, n_axes, n_batch, xp)
+
+
+@_ni_docstrings.docfiller
+def vectorized_filter(input, function, *, size=None, footprint=None, output=None,
+                      mode='reflect', cval=None, origin=None, axes=None,
+                      batch_memory=2**30):
+    """Filter an array with a vectorized Python callable as the kernel.
+
+    Parameters
+    ----------
+    %(input)s
+    function : callable
+        Kernel to apply over a window centered at each element of `input`.
+        Callable must have signature::
+
+            function(window: ndarray, *, axis: int | tuple) -> scalar
+
+        where ``axis`` specifies the axis (or axes) of ``window`` along which
+        the filter function is evaluated.
+    size : scalar or tuple, optional
+        See `footprint` below. Ignored if `footprint` is given.
+    footprint : array, optional
+        Either `size` or `footprint` must be defined. `size` gives
+        the shape that is taken from the input array, at every element
+        position, to define the input to the filter function.
+        `footprint` is a boolean array that specifies (implicitly) a
+        shape, but also which of the elements within this shape will get
+        passed to the filter function. Thus ``size=(n, m)`` is equivalent
+        to ``footprint=np.ones((n, m))``.
+        We adjust `size` to the number of dimensions indicated by `axes`.
+        For instance, if `axes` is ``(0, 2, 1)`` and ``n`` is passed for ``size``,
+        then the effective `size` is ``(n, n, n)``.
+    output : array, optional
+        The array in which to place the output. By default, an array of the dtype
+        returned by `function` will be created.
+    mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional
+        The `mode` parameter determines how the input array is extended
+        beyond its boundaries. Default is 'reflect'. Behavior for each valid
+        value is as follows:
+
+        'reflect' (`d c b a | a b c d | d c b a`)
+            The input is extended by reflecting about the edge of the last
+            pixel. This mode is also sometimes referred to as half-sample
+            symmetric.
+
+        'constant' (`k k k k | a b c d | k k k k`)
+            The input is extended by filling all values beyond the edge with
+            the same constant value, defined by the `cval` parameter.
+
+        'nearest' (`a a a a | a b c d | d d d d`)
+            The input is extended by replicating the last pixel.
+
+        'mirror' (`d c b | a b c d | c b a`)
+            The input is extended by reflecting about the center of the last
+            pixel. This mode is also sometimes referred to as whole-sample
+            symmetric.
+
+        'wrap' (`a b c d | a b c d | a b c d`)
+            The input is extended by wrapping around to the opposite edge.
+
+        'valid' (`| a b c d |`)
+            The input is not extended; rather, the output shape is reduced depending
+            on the window size according to the following calculation::
+
+                window_size = np.asarray(size if size is not None else footprint.shape)
+                output_shape = np.asarray(input.shape)
+                output_shape[np.asarray(axes)] -= (window_size - 1)
+
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int, optional
+        If None, `input` is filtered along all axes. Otherwise, `input` is filtered
+        along the specified axes. When `axes` is specified, the dimensionality of
+        `footprint` and the length of any tuples used for `size` or `origin` must
+        match the length of `axes`. The ith axis of `footprint` and the ith element
+        in these tuples corresponds to the ith element of `axes`.
+    batch_memory : int, default: 2**30
+        The maximum number of bytes occupied by data in the ``window``
+        array passed to ``function``.
+
+    Returns
+    -------
+    output : ndarray
+        Filtered array. The dtype is the output dtype of `function`. If `function` is
+        scalar-valued when applied to a single window, the shape of the output is that
+        of `input` (unless ``mode=='valid'``; see `mode` documentation). If `function`
+        is multi-valued when applied to a single window, the placement of the
+        corresponding dimensions within the output shape depends entirely on the
+        behavior of `function`; see Examples.
+
+    See Also
+    --------
+    scipy.ndimage.generic_filter
+
+    Notes
+    -----
+    This function works by padding `input` according to `mode`, then calling the
+    provided `function` on chunks of a sliding window view over the padded array.
+    This approach is very simple and flexible, and so the function has many features
+    not offered by some other filter functions (e.g. memory control, ``float16``
+    and complex dtype support, and any NaN-handling features provided by the
+    `function` argument).
+
+    However, this brute-force approach may perform considerable redundant work.
+    Use a specialized filter (e.g. `minimum_filter` instead of this function with
+    `numpy.min` as the callable; `uniform_filter` instead of this function with
+    `numpy.mean` as the callable) when possible, as it may use a more efficient
+    algorithm.
+
+    When a specialized filter is not available, this function is ideal when `function`
+    is a vectorized, pure-Python callable. Even better performance may be possible
+    by passing a `scipy.LowLevelCallable` to `generic_filter`. `generic_filter` may
+    also be preferred for expensive callables with large filter footprints and
+    callables that are not vectorized (i.e. those without ``axis`` support).
+
+    This function does not provide the ``extra_arguments`` or ``extra_keywords``
+    arguments provided by some `ndimage` functions. There are two reasons:
+
+    - The passthrough functionality can be achieved by the user: simply wrap the
+      original callable in another function that provides the required arguments;
+      e.g., ``function=lambda input, axis: function(input, *extra_arguments, axis=axis, **extra_keywords)``.
+    - There are use cases for `function` to be passed additional *sliding-window data*
+      to `function` besides `input`. This is not yet implemented, but we reserve
+      these argument names for such a feature, which would add capability rather than
+      providing a duplicate interface to existing capability.
+
+    Examples
+    --------
+    Suppose we wish to perform a median filter with even window size on a ``float16``
+    image. Furthermore, the image has NaNs that we wish to be ignored (and effectively
+    removed by the filter). `median_filter` does not support ``float16`` data, its
+    behavior when NaNs are present is not defined, and for even window sizes, it does
+    not return the usual sample median - the average of the two middle elements. This
+    would be an excellent use case for `vectorized_filter` with
+    ``function=np.nanmedian``, which supports the required interface: it accepts a
+    data array of any shape as the first positional argument, and tuple of axes as
+    keyword argument ``axis``.
+
+    >>> import numpy as np
+    >>> from scipy import datasets, ndimage
+    >>> from scipy.ndimage import vectorized_filter
+    >>> import matplotlib.pyplot as plt
+    >>> ascent = ndimage.zoom(datasets.ascent(), 0.5).astype(np.float16)
+    >>> ascent[::16, ::16] = np.nan
+    >>> result = vectorized_filter(ascent, function=np.nanmedian, size=4)
+
+    Plot the original and filtered images.
+
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> fig.tight_layout()
+    >>> plt.show()
+
+    Another need satisfied by `vectorized_filter` is to perform multi-output
+    filters. For instance, suppose we wish to filter an image according to the 25th
+    and 75th percentiles in addition to the median. We could perform the three
+    filters separately.
+
+    >>> ascent = ndimage.zoom(datasets.ascent(), 0.5)
+    >>> def get_quantile_fun(p):
+    ...     return lambda x, axis: np.quantile(x, p, axis=axis)
+    >>> ref1 = vectorized_filter(ascent, get_quantile_fun(0.25), size=4)
+    >>> ref2 = vectorized_filter(ascent, get_quantile_fun(0.50), size=4)
+    >>> ref3 = vectorized_filter(ascent, get_quantile_fun(0.75), size=4)
+    >>> ref = np.stack([ref1, ref2, ref3])
+
+    However, `vectorized_filter` also supports filters that return multiple outputs
+    as long as `output` is unspecified and `batch_memory` is sufficiently high to
+    perform the calculation in a single chunk.
+
+    >>> def quartiles(x, axis):
+    ...     return np.quantile(x, [0.25, 0.50, 0.75], axis=axis)
+    >>> res = vectorized_filter(ascent, quartiles, size=4, batch_memory=np.inf)
+    >>> np.all(np.isclose(res, ref))
+    np.True_
+
+    The placement of the additional dimension(s) corresponding with multiple outputs
+    is at the discretion of `function`. `quartiles` happens to prepend one dimension
+    corresponding with the three outputs simply because that is the behavior of
+    `np.quantile`:
+
+    >>> res.shape == (3,) + ascent.shape
+    True
+
+    If we wished for this dimension to be appended:
+
+    >>> def quartiles(x, axis):
+    ...     return np.moveaxis(np.quantile(x, [0.25, 0.50, 0.75], axis=axis), 0, -1)
+    >>> res = vectorized_filter(ascent, quartiles, size=4, batch_memory=np.inf)
+    >>> res.shape == ascent.shape + (3,)
+    True
+
+    Suppose we wish to implment a "mode" filter - a filter that selects the most
+    frequently occuring value within the window. A simple (but rather slow)
+    approach is to use `generic_filter` with `scipy.stats.mode`.
+
+    >>> from scipy import stats
+    >>> rng = np.random.default_rng(3195824598724609246)
+    >>> input = rng.integers(255, size=(50, 50)).astype(np.uint8)
+    >>> def simple_mode(input):
+    ...     return stats.mode(input, axis=None).mode
+    >>> ref = ndimage.generic_filter(input, simple_mode, size=5)
+
+    If speed is important, `vectorized_filter` can take advantage of the performance
+    benefit of a vectorized callable.
+
+    >>> def vectorized_mode(x, axis=(-1,)):
+    ...     n_axes = 1 if np.isscalar(axis) else len(axis)
+    ...     x = np.moveaxis(x, axis, tuple(range(-n_axes, 0)))
+    ...     x = np.reshape(x, x.shape[:-n_axes] + (-1,))
+    ...     y = np.sort(x, axis=-1)
+    ...     i = np.concatenate([np.ones(y.shape[:-1] + (1,), dtype=bool),
+    ...                         y[..., :-1] != y[..., 1:]], axis=-1)
+    ...     indices = np.arange(y.size)[i.ravel()]
+    ...     counts = np.diff(indices, append=y.size)
+    ...     counts = np.reshape(np.repeat(counts, counts), y.shape)
+    ...     k = np.argmax(counts, axis=-1, keepdims=True)
+    ...     return np.take_along_axis(y, k, axis=-1)[..., 0]
+    >>> res = vectorized_filter(input, vectorized_mode, size=5)
+    >>> np.all(res == ref)
+    np.True_
+
+    Depending on the machine, the `vectorized_filter` version may be as much as
+    100x faster.
+
+    """  # noqa: E501
+
+    (input, function, size, mode, cval, origin, working_axes, axes, n_axes, n_batch, xp
+     ) = _vectorized_filter_iv(input, function, size, footprint, output, mode, cval,
+        origin, axes, batch_memory)
+
+    # `np.pad`/`cp.pad` raises with these sorts of cases, but the best result is
+    # probably to return the original array. It could be argued that we should call
+    # the function on the empty array with `axis=None` just to determine the output
+    # dtype, but I can also see rationale against that.
+    if xp_size(input) == 0:
+        return xp.asarray(input)
+
+    # This seems to be defined.
+    if input.ndim == 0 and size == ():
+        return xp.asarray(function(input) if footprint is None
+                          else function(input[footprint]))
+
+    if is_cupy(xp):
+        # CuPy is the only GPU backend that has `pad` (with all modes)
+        # and `sliding_window_view`. An enhancement would be to use
+        # no-copy conversion to CuPy whenever the data is on the GPU.
+        cp = xp  # let there be no ambiguity!
+        swv = cp.lib.stride_tricks.sliding_window_view
+        pad = cp.pad
+    else:
+        # Try to perform no-copy conversion to NumPy for padding and
+        # `sliding_window_view`. (If that fails, fine - for now, the only
+        # GPU backend we support is CuPy.)
+        swv = np.lib.stride_tricks.sliding_window_view
+        pad = np.pad
+        input = np.asarray(input)
+        cval = np.asarray(cval)[()] if mode == 'constant' else None
+
+    # Border the image according to `mode` and `offset`.
+    if mode != 'valid':
+        kwargs = {'constant_values': cval} if mode == 'constant' else {}
+        borders = tuple((i//2 + j, (i-1)//2 - j) for i, j in zip(size, origin))
+        bordered_input = pad(input, ((0, 0),)*n_batch + borders, mode=mode, **kwargs)
+    else:
+        bordered_input = input
+
+    # Evaluate function with sliding window view. Function is already wrapped to
+    # manage memory, deal with `footprint`, populate `output`, etc.
+    view = swv(bordered_input, size, working_axes)
+    res = function(view)
+
+    # move working_axes back to original positions
+    return xp.moveaxis(res, working_axes, axes)
+
+
+def _invalid_origin(origin, lenw):
+    return (origin < -(lenw // 2)) or (origin > (lenw - 1) // 2)
+
+
+def _complex_via_real_components(func, input, weights, output, cval, **kwargs):
+    """Complex convolution via a linear combination of real convolutions."""
+    complex_input = input.dtype.kind == 'c'
+    complex_weights = weights.dtype.kind == 'c'
+    if complex_input and complex_weights:
+        # real component of the output
+        func(input.real, weights.real, output=output.real,
+             cval=np.real(cval), **kwargs)
+        output.real -= func(input.imag, weights.imag, output=None,
+                            cval=np.imag(cval), **kwargs)
+        # imaginary component of the output
+        func(input.real, weights.imag, output=output.imag,
+             cval=np.real(cval), **kwargs)
+        output.imag += func(input.imag, weights.real, output=None,
+                            cval=np.imag(cval), **kwargs)
+    elif complex_input:
+        func(input.real, weights, output=output.real, cval=np.real(cval),
+             **kwargs)
+        func(input.imag, weights, output=output.imag, cval=np.imag(cval),
+             **kwargs)
+    else:
+        if np.iscomplexobj(cval):
+            raise ValueError("Cannot provide a complex-valued cval when the "
+                             "input is real.")
+        func(input, weights.real, output=output.real, cval=cval, **kwargs)
+        func(input, weights.imag, output=output.imag, cval=cval, **kwargs)
+    return output
+
+
+def _expand_origin(ndim_image, axes, origin):
+    num_axes = len(axes)
+    origins = _ni_support._normalize_sequence(origin, num_axes)
+    if num_axes < ndim_image:
+        # set origin = 0 for any axes not being filtered
+        origins_temp = [0,] * ndim_image
+        for o, ax in zip(origins, axes):
+            origins_temp[ax] = o
+        origins = origins_temp
+    return origins
+
+
+def _expand_footprint(ndim_image, axes, footprint,
+                      footprint_name="footprint"):
+    num_axes = len(axes)
+    if num_axes < ndim_image:
+        if footprint.ndim != num_axes:
+            raise RuntimeError(f"{footprint_name}.ndim ({footprint.ndim}) "
+                               f"must match len(axes) ({num_axes})")
+
+        footprint = np.expand_dims(
+            footprint,
+            tuple(ax for ax in range(ndim_image) if ax not in axes)
+        )
+    return footprint
+
+
+def _expand_mode(ndim_image, axes, mode):
+    num_axes = len(axes)
+    if not isinstance(mode, str) and isinstance(mode, Iterable):
+        # set mode = 'constant' for any axes not being filtered
+        modes = _ni_support._normalize_sequence(mode, num_axes)
+        modes_temp = ['constant'] * ndim_image
+        for m, ax in zip(modes, axes):
+            modes_temp[ax] = m
+        mode = modes_temp
+    return mode
+
+
+@_ni_docstrings.docfiller
+def correlate1d(input, weights, axis=-1, output=None, mode="reflect",
+                cval=0.0, origin=0):
+    """Calculate a 1-D correlation along the given axis.
+
+    The lines of the array along the given axis are correlated with the
+    given weights.
+
+    Parameters
+    ----------
+    %(input)s
+    weights : array
+        1-D sequence of numbers.
+    %(axis)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin)s
+
+    Returns
+    -------
+    result : ndarray
+        Correlation result. Has the same shape as `input`.
+
+    Examples
+    --------
+    >>> from scipy.ndimage import correlate1d
+    >>> correlate1d([2, 8, 0, 4, 1, 9, 9, 0], weights=[1, 3])
+    array([ 8, 26,  8, 12,  7, 28, 36,  9])
+    """
+    input = np.asarray(input)
+    weights = np.asarray(weights)
+    complex_input = input.dtype.kind == 'c'
+    complex_weights = weights.dtype.kind == 'c'
+    if complex_input or complex_weights:
+        if complex_weights:
+            weights = weights.conj()
+            weights = weights.astype(np.complex128, copy=False)
+        kwargs = dict(axis=axis, mode=mode, origin=origin)
+        output = _ni_support._get_output(output, input, complex_output=True)
+        return _complex_via_real_components(correlate1d, input, weights,
+                                            output, cval, **kwargs)
+
+    output = _ni_support._get_output(output, input)
+    weights = np.asarray(weights, dtype=np.float64)
+    if weights.ndim != 1 or weights.shape[0] < 1:
+        raise RuntimeError('no filter weights given')
+    if not weights.flags.contiguous:
+        weights = weights.copy()
+    axis = normalize_axis_index(axis, input.ndim)
+    if _invalid_origin(origin, len(weights)):
+        raise ValueError('Invalid origin; origin must satisfy '
+                         '-(len(weights) // 2) <= origin <= '
+                         '(len(weights)-1) // 2')
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.correlate1d(input, weights, axis, output, mode, cval,
+                          origin)
+    return output
+
+
+@_ni_docstrings.docfiller
+def convolve1d(input, weights, axis=-1, output=None, mode="reflect",
+               cval=0.0, origin=0):
+    """Calculate a 1-D convolution along the given axis.
+
+    The lines of the array along the given axis are convolved with the
+    given weights.
+
+    Parameters
+    ----------
+    %(input)s
+    weights : ndarray
+        1-D sequence of numbers.
+    %(axis)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin)s
+
+    Returns
+    -------
+    convolve1d : ndarray
+        Convolved array with same shape as input
+
+    Examples
+    --------
+    >>> from scipy.ndimage import convolve1d
+    >>> convolve1d([2, 8, 0, 4, 1, 9, 9, 0], weights=[1, 3])
+    array([14, 24,  4, 13, 12, 36, 27,  0])
+    """
+    weights = np.asarray(weights)
+    weights = weights[::-1]
+    origin = -origin
+    if not weights.shape[0] & 1:
+        origin -= 1
+    if weights.dtype.kind == 'c':
+        # pre-conjugate here to counteract the conjugation in correlate1d
+        weights = weights.conj()
+    return correlate1d(input, weights, axis, output, mode, cval, origin)
+
+
+def _gaussian_kernel1d(sigma, order, radius):
+    """
+    Computes a 1-D Gaussian convolution kernel.
+    """
+    if order < 0:
+        raise ValueError('order must be non-negative')
+    exponent_range = np.arange(order + 1)
+    sigma2 = sigma * sigma
+    x = np.arange(-radius, radius+1)
+    phi_x = np.exp(-0.5 / sigma2 * x ** 2)
+    phi_x = phi_x / phi_x.sum()
+
+    if order == 0:
+        return phi_x
+    else:
+        # f(x) = q(x) * phi(x) = q(x) * exp(p(x))
+        # f'(x) = (q'(x) + q(x) * p'(x)) * phi(x)
+        # p'(x) = -1 / sigma ** 2
+        # Implement q'(x) + q(x) * p'(x) as a matrix operator and apply to the
+        # coefficients of q(x)
+        q = np.zeros(order + 1)
+        q[0] = 1
+        D = np.diag(exponent_range[1:], 1)  # D @ q(x) = q'(x)
+        P = np.diag(np.ones(order)/-sigma2, -1)  # P @ q(x) = q(x) * p'(x)
+        Q_deriv = D + P
+        for _ in range(order):
+            q = Q_deriv.dot(q)
+        q = (x[:, None] ** exponent_range).dot(q)
+        return q * phi_x
+
+
+@_ni_docstrings.docfiller
+def gaussian_filter1d(input, sigma, axis=-1, order=0, output=None,
+                      mode="reflect", cval=0.0, truncate=4.0, *, radius=None):
+    """1-D Gaussian filter.
+
+    Parameters
+    ----------
+    %(input)s
+    sigma : scalar
+        standard deviation for Gaussian kernel
+    %(axis)s
+    order : int, optional
+        An order of 0 corresponds to convolution with a Gaussian
+        kernel. A positive order corresponds to convolution with
+        that derivative of a Gaussian.
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    truncate : float, optional
+        Truncate the filter at this many standard deviations.
+        Default is 4.0.
+    radius : None or int, optional
+        Radius of the Gaussian kernel. If specified, the size of
+        the kernel will be ``2*radius + 1``, and `truncate` is ignored.
+        Default is None.
+
+    Returns
+    -------
+    gaussian_filter1d : ndarray
+        Returned array of same shape as `input`.
+
+    Notes
+    -----
+    The Gaussian kernel will have size ``2*radius + 1`` along each axis. If
+    `radius` is None, a default ``radius = round(truncate * sigma)`` will be
+    used.
+
+    Examples
+    --------
+    >>> from scipy.ndimage import gaussian_filter1d
+    >>> import numpy as np
+    >>> gaussian_filter1d([1.0, 2.0, 3.0, 4.0, 5.0], 1)
+    array([ 1.42704095,  2.06782203,  3.        ,  3.93217797,  4.57295905])
+    >>> gaussian_filter1d([1.0, 2.0, 3.0, 4.0, 5.0], 4)
+    array([ 2.91948343,  2.95023502,  3.        ,  3.04976498,  3.08051657])
+    >>> import matplotlib.pyplot as plt
+    >>> rng = np.random.default_rng()
+    >>> x = rng.standard_normal(101).cumsum()
+    >>> y3 = gaussian_filter1d(x, 3)
+    >>> y6 = gaussian_filter1d(x, 6)
+    >>> plt.plot(x, 'k', label='original data')
+    >>> plt.plot(y3, '--', label='filtered, sigma=3')
+    >>> plt.plot(y6, ':', label='filtered, sigma=6')
+    >>> plt.legend()
+    >>> plt.grid()
+    >>> plt.show()
+
+    """
+    sd = float(sigma)
+    # make the radius of the filter equal to truncate standard deviations
+    lw = int(truncate * sd + 0.5)
+    if radius is not None:
+        lw = radius
+    if not isinstance(lw, numbers.Integral) or lw < 0:
+        raise ValueError('Radius must be a nonnegative integer.')
+    # Since we are calling correlate, not convolve, revert the kernel
+    weights = _gaussian_kernel1d(sigma, order, lw)[::-1]
+    return correlate1d(input, weights, axis, output, mode, cval, 0)
+
+
+@_ni_docstrings.docfiller
+def gaussian_filter(input, sigma, order=0, output=None,
+                    mode="reflect", cval=0.0, truncate=4.0, *, radius=None,
+                    axes=None):
+    """Multidimensional Gaussian filter.
+
+    Parameters
+    ----------
+    %(input)s
+    sigma : scalar or sequence of scalars
+        Standard deviation for Gaussian kernel. The standard
+        deviations of the Gaussian filter are given for each axis as a
+        sequence, or as a single number, in which case it is equal for
+        all axes.
+    order : int or sequence of ints, optional
+        The order of the filter along each axis is given as a sequence
+        of integers, or as a single number. An order of 0 corresponds
+        to convolution with a Gaussian kernel. A positive order
+        corresponds to convolution with that derivative of a Gaussian.
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    truncate : float, optional
+        Truncate the filter at this many standard deviations.
+        Default is 4.0.
+    radius : None or int or sequence of ints, optional
+        Radius of the Gaussian kernel. The radius are given for each axis
+        as a sequence, or as a single number, in which case it is equal
+        for all axes. If specified, the size of the kernel along each axis
+        will be ``2*radius + 1``, and `truncate` is ignored.
+        Default is None.
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `sigma`, `order`, `mode` and/or `radius`
+        must match the length of `axes`. The ith entry in any of these tuples
+        corresponds to the ith entry in `axes`.
+
+    Returns
+    -------
+    gaussian_filter : ndarray
+        Returned array of same shape as `input`.
+
+    Notes
+    -----
+    The multidimensional filter is implemented as a sequence of
+    1-D convolution filters. The intermediate arrays are
+    stored in the same data type as the output. Therefore, for output
+    types with a limited precision, the results may be imprecise
+    because intermediate results may be stored with insufficient
+    precision.
+
+    The Gaussian kernel will have size ``2*radius + 1`` along each axis. If
+    `radius` is None, the default ``radius = round(truncate * sigma)`` will be
+    used.
+
+    Examples
+    --------
+    >>> from scipy.ndimage import gaussian_filter
+    >>> import numpy as np
+    >>> a = np.arange(50, step=2).reshape((5,5))
+    >>> a
+    array([[ 0,  2,  4,  6,  8],
+           [10, 12, 14, 16, 18],
+           [20, 22, 24, 26, 28],
+           [30, 32, 34, 36, 38],
+           [40, 42, 44, 46, 48]])
+    >>> gaussian_filter(a, sigma=1)
+    array([[ 4,  6,  8,  9, 11],
+           [10, 12, 14, 15, 17],
+           [20, 22, 24, 25, 27],
+           [29, 31, 33, 34, 36],
+           [35, 37, 39, 40, 42]])
+
+    >>> from scipy import datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = gaussian_filter(ascent, sigma=5)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+    output = _ni_support._get_output(output, input)
+
+    axes = _ni_support._check_axes(axes, input.ndim)
+    num_axes = len(axes)
+    orders = _ni_support._normalize_sequence(order, num_axes)
+    sigmas = _ni_support._normalize_sequence(sigma, num_axes)
+    modes = _ni_support._normalize_sequence(mode, num_axes)
+    radiuses = _ni_support._normalize_sequence(radius, num_axes)
+    axes = [(axes[ii], sigmas[ii], orders[ii], modes[ii], radiuses[ii])
+            for ii in range(num_axes) if sigmas[ii] > 1e-15]
+    if len(axes) > 0:
+        for axis, sigma, order, mode, radius in axes:
+            gaussian_filter1d(input, sigma, axis, order, output,
+                              mode, cval, truncate, radius=radius)
+            input = output
+    else:
+        output[...] = input[...]
+    return output
+
+
+@_ni_docstrings.docfiller
+def prewitt(input, axis=-1, output=None, mode="reflect", cval=0.0):
+    """Calculate a Prewitt filter.
+
+    Parameters
+    ----------
+    %(input)s
+    %(axis)s
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+
+    Returns
+    -------
+    prewitt : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    See Also
+    --------
+    sobel: Sobel filter
+
+    Notes
+    -----
+    This function computes the one-dimensional Prewitt filter.
+    Horizontal edges are emphasised with the horizontal transform (axis=0),
+    vertical edges with the vertical transform (axis=1), and so on for higher
+    dimensions. These can be combined to give the magnitude.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> import numpy as np
+    >>> ascent = datasets.ascent()
+    >>> prewitt_h = ndimage.prewitt(ascent, axis=0)
+    >>> prewitt_v = ndimage.prewitt(ascent, axis=1)
+    >>> magnitude = np.sqrt(prewitt_h ** 2 + prewitt_v ** 2)
+    >>> magnitude *= 255 / np.max(magnitude) # Normalization
+    >>> fig, axes = plt.subplots(2, 2, figsize = (8, 8))
+    >>> plt.gray()
+    >>> axes[0, 0].imshow(ascent)
+    >>> axes[0, 1].imshow(prewitt_h)
+    >>> axes[1, 0].imshow(prewitt_v)
+    >>> axes[1, 1].imshow(magnitude)
+    >>> titles = ["original", "horizontal", "vertical", "magnitude"]
+    >>> for i, ax in enumerate(axes.ravel()):
+    ...     ax.set_title(titles[i])
+    ...     ax.axis("off")
+    >>> plt.show()
+
+    """
+    input = np.asarray(input)
+    axis = normalize_axis_index(axis, input.ndim)
+    output = _ni_support._get_output(output, input)
+    modes = _ni_support._normalize_sequence(mode, input.ndim)
+    correlate1d(input, [-1, 0, 1], axis, output, modes[axis], cval, 0)
+    axes = [ii for ii in range(input.ndim) if ii != axis]
+    for ii in axes:
+        correlate1d(output, [1, 1, 1], ii, output, modes[ii], cval, 0,)
+    return output
+
+
+@_ni_docstrings.docfiller
+def sobel(input, axis=-1, output=None, mode="reflect", cval=0.0):
+    """Calculate a Sobel filter.
+
+    Parameters
+    ----------
+    %(input)s
+    %(axis)s
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+
+    Returns
+    -------
+    sobel : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Notes
+    -----
+    This function computes the axis-specific Sobel gradient.
+    The horizontal edges can be emphasised with the horizontal transform (axis=0),
+    the vertical edges with the vertical transform (axis=1) and so on for higher
+    dimensions. These can be combined to give the magnitude.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> import numpy as np
+    >>> ascent = datasets.ascent().astype('int32')
+    >>> sobel_h = ndimage.sobel(ascent, 0)  # horizontal gradient
+    >>> sobel_v = ndimage.sobel(ascent, 1)  # vertical gradient
+    >>> magnitude = np.sqrt(sobel_h**2 + sobel_v**2)
+    >>> magnitude *= 255.0 / np.max(magnitude)  # normalization
+    >>> fig, axs = plt.subplots(2, 2, figsize=(8, 8))
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> axs[0, 0].imshow(ascent)
+    >>> axs[0, 1].imshow(sobel_h)
+    >>> axs[1, 0].imshow(sobel_v)
+    >>> axs[1, 1].imshow(magnitude)
+    >>> titles = ["original", "horizontal", "vertical", "magnitude"]
+    >>> for i, ax in enumerate(axs.ravel()):
+    ...     ax.set_title(titles[i])
+    ...     ax.axis("off")
+    >>> plt.show()
+
+    """
+    input = np.asarray(input)
+    axis = normalize_axis_index(axis, input.ndim)
+    output = _ni_support._get_output(output, input)
+    modes = _ni_support._normalize_sequence(mode, input.ndim)
+    correlate1d(input, [-1, 0, 1], axis, output, modes[axis], cval, 0)
+    axes = [ii for ii in range(input.ndim) if ii != axis]
+    for ii in axes:
+        correlate1d(output, [1, 2, 1], ii, output, modes[ii], cval, 0)
+    return output
+
+
+@_ni_docstrings.docfiller
+def generic_laplace(input, derivative2, output=None, mode="reflect",
+                    cval=0.0,
+                    extra_arguments=(),
+                    extra_keywords=None,
+                    *, axes=None):
+    """
+    N-D Laplace filter using a provided second derivative function.
+
+    Parameters
+    ----------
+    %(input)s
+    derivative2 : callable
+        Callable with the following signature::
+
+            derivative2(input, axis, output, mode, cval,
+                        *extra_arguments, **extra_keywords)
+
+        See `extra_arguments`, `extra_keywords` below.
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    %(extra_arguments)s
+    %(extra_keywords)s
+    axes : tuple of int or None
+        The axes over which to apply the filter. If a `mode` tuple is
+        provided, its length must match the number of axes.
+
+    Returns
+    -------
+    generic_laplace : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    """
+    if extra_keywords is None:
+        extra_keywords = {}
+    input = np.asarray(input)
+    output = _ni_support._get_output(output, input)
+    axes = _ni_support._check_axes(axes, input.ndim)
+    if len(axes) > 0:
+        modes = _ni_support._normalize_sequence(mode, len(axes))
+        derivative2(input, axes[0], output, modes[0], cval,
+                    *extra_arguments, **extra_keywords)
+        for ii in range(1, len(axes)):
+            tmp = derivative2(input, axes[ii], output.dtype, modes[ii], cval,
+                              *extra_arguments, **extra_keywords)
+            output += tmp
+    else:
+        output[...] = input[...]
+    return output
+
+
+@_ni_docstrings.docfiller
+def laplace(input, output=None, mode="reflect", cval=0.0, *, axes=None):
+    """N-D Laplace filter based on approximate second derivatives.
+
+    Parameters
+    ----------
+    %(input)s
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    axes : tuple of int or None
+        The axes over which to apply the filter. If a `mode` tuple is
+        provided, its length must match the number of axes.
+
+    Returns
+    -------
+    laplace : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.laplace(ascent)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    def derivative2(input, axis, output, mode, cval):
+        return correlate1d(input, [1, -2, 1], axis, output, mode, cval, 0)
+    return generic_laplace(input, derivative2, output, mode, cval, axes=axes)
+
+
+@_ni_docstrings.docfiller
+def gaussian_laplace(input, sigma, output=None, mode="reflect",
+                     cval=0.0, *, axes=None, **kwargs):
+    """Multidimensional Laplace filter using Gaussian second derivatives.
+
+    Parameters
+    ----------
+    %(input)s
+    sigma : scalar or sequence of scalars
+        The standard deviations of the Gaussian filter are given for
+        each axis as a sequence, or as a single number, in which case
+        it is equal for all axes.
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    axes : tuple of int or None
+        The axes over which to apply the filter. If `sigma` or `mode` tuples
+        are provided, their length must match the number of axes.
+    **kwargs
+        Extra keyword arguments will be passed to `gaussian_filter`.
+
+    Returns
+    -------
+    gaussian_laplace : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> ascent = datasets.ascent()
+
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+
+    >>> result = ndimage.gaussian_laplace(ascent, sigma=1)
+    >>> ax1.imshow(result)
+
+    >>> result = ndimage.gaussian_laplace(ascent, sigma=3)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+
+    def derivative2(input, axis, output, mode, cval, sigma, **kwargs):
+        order = [0] * input.ndim
+        order[axis] = 2
+        return gaussian_filter(input, sigma, order, output, mode, cval,
+                               **kwargs)
+
+    axes = _ni_support._check_axes(axes, input.ndim)
+    num_axes = len(axes)
+    sigma = _ni_support._normalize_sequence(sigma, num_axes)
+    if num_axes < input.ndim:
+        # set sigma = 0 for any axes not being filtered
+        sigma_temp = [0,] * input.ndim
+        for s, ax in zip(sigma, axes):
+            sigma_temp[ax] = s
+        sigma = sigma_temp
+
+    return generic_laplace(input, derivative2, output, mode, cval,
+                           extra_arguments=(sigma,),
+                           extra_keywords=kwargs,
+                           axes=axes)
+
+
+@_ni_docstrings.docfiller
+def generic_gradient_magnitude(input, derivative, output=None,
+                               mode="reflect", cval=0.0,
+                               extra_arguments=(), extra_keywords=None,
+                               *, axes=None):
+    """Gradient magnitude using a provided gradient function.
+
+    Parameters
+    ----------
+    %(input)s
+    derivative : callable
+        Callable with the following signature::
+
+            derivative(input, axis, output, mode, cval,
+                       *extra_arguments, **extra_keywords)
+
+        See `extra_arguments`, `extra_keywords` below.
+        `derivative` can assume that `input` and `output` are ndarrays.
+        Note that the output from `derivative` is modified inplace;
+        be careful to copy important inputs before returning them.
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    %(extra_arguments)s
+    %(extra_keywords)s
+    axes : tuple of int or None
+        The axes over which to apply the filter. If a `mode` tuple is
+        provided, its length must match the number of axes.
+
+    Returns
+    -------
+    generic_gradient_magnitude : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    """
+    if extra_keywords is None:
+        extra_keywords = {}
+    input = np.asarray(input)
+    output = _ni_support._get_output(output, input)
+    axes = _ni_support._check_axes(axes, input.ndim)
+    if len(axes) > 0:
+        modes = _ni_support._normalize_sequence(mode, len(axes))
+        derivative(input, axes[0], output, modes[0], cval,
+                   *extra_arguments, **extra_keywords)
+        np.multiply(output, output, output)
+        for ii in range(1, len(axes)):
+            tmp = derivative(input, axes[ii], output.dtype, modes[ii], cval,
+                             *extra_arguments, **extra_keywords)
+            np.multiply(tmp, tmp, tmp)
+            output += tmp
+        # This allows the sqrt to work with a different default casting
+        np.sqrt(output, output, casting='unsafe')
+    else:
+        output[...] = input[...]
+    return output
+
+
+@_ni_docstrings.docfiller
+def gaussian_gradient_magnitude(input, sigma, output=None,
+                                mode="reflect", cval=0.0, *, axes=None,
+                                **kwargs):
+    """Multidimensional gradient magnitude using Gaussian derivatives.
+
+    Parameters
+    ----------
+    %(input)s
+    sigma : scalar or sequence of scalars
+        The standard deviations of the Gaussian filter are given for
+        each axis as a sequence, or as a single number, in which case
+        it is equal for all axes.
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    axes : tuple of int or None
+        The axes over which to apply the filter. If `sigma` or `mode` tuples
+        are provided, their length must match the number of axes.
+    **kwargs
+        Extra keyword arguments will be passed to `gaussian_filter`.
+
+    Returns
+    -------
+    gaussian_gradient_magnitude : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.gaussian_gradient_magnitude(ascent, sigma=5)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+
+    def derivative(input, axis, output, mode, cval, sigma, **kwargs):
+        order = [0] * input.ndim
+        order[axis] = 1
+        return gaussian_filter(input, sigma, order, output, mode,
+                               cval, **kwargs)
+
+    return generic_gradient_magnitude(input, derivative, output, mode,
+                                      cval, extra_arguments=(sigma,),
+                                      extra_keywords=kwargs, axes=axes)
+
+
+def _correlate_or_convolve(input, weights, output, mode, cval, origin,
+                           convolution, axes):
+    input = np.asarray(input)
+    weights = np.asarray(weights)
+    complex_input = input.dtype.kind == 'c'
+    complex_weights = weights.dtype.kind == 'c'
+    if complex_input or complex_weights:
+        if complex_weights and not convolution:
+            # As for np.correlate, conjugate weights rather than input.
+            weights = weights.conj()
+        kwargs = dict(
+            mode=mode, origin=origin, convolution=convolution, axes=axes
+        )
+        output = _ni_support._get_output(output, input, complex_output=True)
+
+        return _complex_via_real_components(_correlate_or_convolve, input,
+                                            weights, output, cval, **kwargs)
+
+    axes = _ni_support._check_axes(axes, input.ndim)
+    weights = np.asarray(weights, dtype=np.float64)
+
+    # expand weights and origins if num_axes < input.ndim
+    weights = _expand_footprint(input.ndim, axes, weights, "weights")
+    origins = _expand_origin(input.ndim, axes, origin)
+
+    wshape = [ii for ii in weights.shape if ii > 0]
+    if len(wshape) != input.ndim:
+        raise RuntimeError(f"weights.ndim ({len(wshape)}) must match "
+                           f"len(axes) ({len(axes)})")
+    if convolution:
+        weights = weights[tuple([slice(None, None, -1)] * weights.ndim)]
+        for ii in range(len(origins)):
+            origins[ii] = -origins[ii]
+            if not weights.shape[ii] & 1:
+                origins[ii] -= 1
+    for origin, lenw in zip(origins, wshape):
+        if _invalid_origin(origin, lenw):
+            raise ValueError('Invalid origin; origin must satisfy '
+                             '-(weights.shape[k] // 2) <= origin[k] <= '
+                             '(weights.shape[k]-1) // 2')
+
+    if not weights.flags.contiguous:
+        weights = weights.copy()
+    output = _ni_support._get_output(output, input)
+    temp_needed = np.may_share_memory(input, output)
+    if temp_needed:
+        # input and output arrays cannot share memory
+        temp = output
+        output = _ni_support._get_output(output.dtype, input)
+    if not isinstance(mode, str) and isinstance(mode, Iterable):
+        raise RuntimeError("A sequence of modes is not supported")
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.correlate(input, weights, output, mode, cval, origins)
+    if temp_needed:
+        temp[...] = output
+        output = temp
+    return output
+
+
+@_ni_docstrings.docfiller
+def correlate(input, weights, output=None, mode='reflect', cval=0.0,
+              origin=0, *, axes=None):
+    """
+    Multidimensional correlation.
+
+    The array is correlated with the given kernel.
+
+    Parameters
+    ----------
+    %(input)s
+    weights : ndarray
+        array of weights, same number of dimensions as input
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `mode` or `origin` must match the length
+        of `axes`. The ith entry in any of these tuples corresponds to the ith
+        entry in `axes`.
+
+    Returns
+    -------
+    result : ndarray
+        The result of correlation of `input` with `weights`.
+
+    See Also
+    --------
+    convolve : Convolve an image with a kernel.
+
+    Examples
+    --------
+    Correlation is the process of moving a filter mask often referred to
+    as kernel over the image and computing the sum of products at each location.
+
+    >>> from scipy.ndimage import correlate
+    >>> import numpy as np
+    >>> input_img = np.arange(25).reshape(5,5)
+    >>> print(input_img)
+    [[ 0  1  2  3  4]
+    [ 5  6  7  8  9]
+    [10 11 12 13 14]
+    [15 16 17 18 19]
+    [20 21 22 23 24]]
+
+    Define a kernel (weights) for correlation. In this example, it is for sum of
+    center and up, down, left and right next elements.
+
+    >>> weights = [[0, 1, 0],
+    ...            [1, 1, 1],
+    ...            [0, 1, 0]]
+
+    We can calculate a correlation result:
+    For example, element ``[2,2]`` is ``7 + 11 + 12 + 13 + 17 = 60``.
+
+    >>> correlate(input_img, weights)
+    array([[  6,  10,  15,  20,  24],
+        [ 26,  30,  35,  40,  44],
+        [ 51,  55,  60,  65,  69],
+        [ 76,  80,  85,  90,  94],
+        [ 96, 100, 105, 110, 114]])
+
+    """
+    return _correlate_or_convolve(input, weights, output, mode, cval,
+                                  origin, False, axes)
+
+
+@_ni_docstrings.docfiller
+def convolve(input, weights, output=None, mode='reflect', cval=0.0,
+             origin=0, *, axes=None):
+    """
+    Multidimensional convolution.
+
+    The array is convolved with the given kernel.
+
+    Parameters
+    ----------
+    %(input)s
+    weights : array_like
+        Array of weights, same number of dimensions as input
+    %(output)s
+    %(mode_reflect)s
+    cval : scalar, optional
+        Value to fill past edges of input if `mode` is 'constant'. Default
+        is 0.0
+    origin : int or sequence, optional
+        Controls the placement of the filter on the input array's pixels.
+        A value of 0 (the default) centers the filter over the pixel, with
+        positive values shifting the filter to the right, and negative ones
+        to the left. By passing a sequence of origins with length equal to
+        the number of dimensions of the input array, different shifts can
+        be specified along each axis.
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `mode` or `origin` must match the length
+        of `axes`. The ith entry in any of these tuples corresponds to the ith
+        entry in `axes`.
+
+    Returns
+    -------
+    result : ndarray
+        The result of convolution of `input` with `weights`.
+
+    See Also
+    --------
+    correlate : Correlate an image with a kernel.
+
+    Notes
+    -----
+    Each value in result is :math:`C_i = \\sum_j{I_{i+k-j} W_j}`, where
+    W is the `weights` kernel,
+    j is the N-D spatial index over :math:`W`,
+    I is the `input` and k is the coordinate of the center of
+    W, specified by `origin` in the input parameters.
+
+    Examples
+    --------
+    Perhaps the simplest case to understand is ``mode='constant', cval=0.0``,
+    because in this case borders (i.e., where the `weights` kernel, centered
+    on any one value, extends beyond an edge of `input`) are treated as zeros.
+
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> k = np.array([[1,1,1],[1,1,0],[1,0,0]])
+    >>> from scipy import ndimage
+    >>> ndimage.convolve(a, k, mode='constant', cval=0.0)
+    array([[11, 10,  7,  4],
+           [10,  3, 11, 11],
+           [15, 12, 14,  7],
+           [12,  3,  7,  0]])
+
+    Setting ``cval=1.0`` is equivalent to padding the outer edge of `input`
+    with 1.0's (and then extracting only the original region of the result).
+
+    >>> ndimage.convolve(a, k, mode='constant', cval=1.0)
+    array([[13, 11,  8,  7],
+           [11,  3, 11, 14],
+           [16, 12, 14, 10],
+           [15,  6, 10,  5]])
+
+    With ``mode='reflect'`` (the default), outer values are reflected at the
+    edge of `input` to fill in missing values.
+
+    >>> b = np.array([[2, 0, 0],
+    ...               [1, 0, 0],
+    ...               [0, 0, 0]])
+    >>> k = np.array([[0,1,0], [0,1,0], [0,1,0]])
+    >>> ndimage.convolve(b, k, mode='reflect')
+    array([[5, 0, 0],
+           [3, 0, 0],
+           [1, 0, 0]])
+
+    This includes diagonally at the corners.
+
+    >>> k = np.array([[1,0,0],[0,1,0],[0,0,1]])
+    >>> ndimage.convolve(b, k)
+    array([[4, 2, 0],
+           [3, 2, 0],
+           [1, 1, 0]])
+
+    With ``mode='nearest'``, the single nearest value in to an edge in
+    `input` is repeated as many times as needed to match the overlapping
+    `weights`.
+
+    >>> c = np.array([[2, 0, 1],
+    ...               [1, 0, 0],
+    ...               [0, 0, 0]])
+    >>> k = np.array([[0, 1, 0],
+    ...               [0, 1, 0],
+    ...               [0, 1, 0],
+    ...               [0, 1, 0],
+    ...               [0, 1, 0]])
+    >>> ndimage.convolve(c, k, mode='nearest')
+    array([[7, 0, 3],
+           [5, 0, 2],
+           [3, 0, 1]])
+
+    """
+    return _correlate_or_convolve(input, weights, output, mode, cval,
+                                  origin, True, axes)
+
+
+@_ni_docstrings.docfiller
+def uniform_filter1d(input, size, axis=-1, output=None,
+                     mode="reflect", cval=0.0, origin=0):
+    """Calculate a 1-D uniform filter along the given axis.
+
+    The lines of the array along the given axis are filtered with a
+    uniform filter of given size.
+
+    Parameters
+    ----------
+    %(input)s
+    size : int
+        length of uniform filter
+    %(axis)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin)s
+
+    Returns
+    -------
+    result : ndarray
+        Filtered array. Has same shape as `input`.
+
+    Examples
+    --------
+    >>> from scipy.ndimage import uniform_filter1d
+    >>> uniform_filter1d([2, 8, 0, 4, 1, 9, 9, 0], size=3)
+    array([4, 3, 4, 1, 4, 6, 6, 3])
+    """
+    input = np.asarray(input)
+    axis = normalize_axis_index(axis, input.ndim)
+    if size < 1:
+        raise RuntimeError('incorrect filter size')
+    complex_output = input.dtype.kind == 'c'
+    output = _ni_support._get_output(output, input,
+                                     complex_output=complex_output)
+    if (size // 2 + origin < 0) or (size // 2 + origin >= size):
+        raise ValueError('invalid origin')
+    mode = _ni_support._extend_mode_to_code(mode)
+    if not complex_output:
+        _nd_image.uniform_filter1d(input, size, axis, output, mode, cval,
+                                   origin)
+    else:
+        _nd_image.uniform_filter1d(input.real, size, axis, output.real, mode,
+                                   np.real(cval), origin)
+        _nd_image.uniform_filter1d(input.imag, size, axis, output.imag, mode,
+                                   np.imag(cval), origin)
+    return output
+
+
+@_ni_docstrings.docfiller
+def uniform_filter(input, size=3, output=None, mode="reflect",
+                   cval=0.0, origin=0, *, axes=None):
+    """Multidimensional uniform filter.
+
+    Parameters
+    ----------
+    %(input)s
+    size : int or sequence of ints, optional
+        The sizes of the uniform filter are given for each axis as a
+        sequence, or as a single number, in which case the size is
+        equal for all axes.
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `size`, `origin`, and/or `mode`
+        must match the length of `axes`. The ith entry in any of these tuples
+        corresponds to the ith entry in `axes`.
+
+    Returns
+    -------
+    uniform_filter : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Notes
+    -----
+    The multidimensional filter is implemented as a sequence of
+    1-D uniform filters. The intermediate arrays are stored
+    in the same data type as the output. Therefore, for output types
+    with a limited precision, the results may be imprecise because
+    intermediate results may be stored with insufficient precision.
+
+    %(nan)s
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.uniform_filter(ascent, size=20)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+    output = _ni_support._get_output(output, input,
+                                     complex_output=input.dtype.kind == 'c')
+    axes = _ni_support._check_axes(axes, input.ndim)
+    num_axes = len(axes)
+    sizes = _ni_support._normalize_sequence(size, num_axes)
+    origins = _ni_support._normalize_sequence(origin, num_axes)
+    modes = _ni_support._normalize_sequence(mode, num_axes)
+    axes = [(axes[ii], sizes[ii], origins[ii], modes[ii])
+            for ii in range(num_axes) if sizes[ii] > 1]
+    if len(axes) > 0:
+        for axis, size, origin, mode in axes:
+            uniform_filter1d(input, int(size), axis, output, mode,
+                             cval, origin)
+            input = output
+    else:
+        output[...] = input[...]
+    return output
+
+
+@_ni_docstrings.docfiller
+def minimum_filter1d(input, size, axis=-1, output=None,
+                     mode="reflect", cval=0.0, origin=0):
+    """Calculate a 1-D minimum filter along the given axis.
+
+    The lines of the array along the given axis are filtered with a
+    minimum filter of given size.
+
+    Parameters
+    ----------
+    %(input)s
+    size : int
+        length along which to calculate 1D minimum
+    %(axis)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin)s
+
+    Returns
+    -------
+    result : ndarray.
+        Filtered image. Has the same shape as `input`.
+
+    Notes
+    -----
+    This function implements the MINLIST algorithm [1]_, as described by
+    Richard Harter [2]_, and has a guaranteed O(n) performance, `n` being
+    the `input` length, regardless of filter size.
+
+    References
+    ----------
+    .. [1] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.2777.
+    .. [2] http://www.richardhartersworld.com/cri/2001/slidingmin.html
+
+
+    Examples
+    --------
+    >>> from scipy.ndimage import minimum_filter1d
+    >>> minimum_filter1d([2, 8, 0, 4, 1, 9, 9, 0], size=3)
+    array([2, 0, 0, 0, 1, 1, 0, 0])
+    """
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError('Complex type not supported')
+    axis = normalize_axis_index(axis, input.ndim)
+    if size < 1:
+        raise RuntimeError('incorrect filter size')
+    output = _ni_support._get_output(output, input)
+    if (size // 2 + origin < 0) or (size // 2 + origin >= size):
+        raise ValueError('invalid origin')
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.min_or_max_filter1d(input, size, axis, output, mode, cval,
+                                  origin, 1)
+    return output
+
+
+@_ni_docstrings.docfiller
+def maximum_filter1d(input, size, axis=-1, output=None,
+                     mode="reflect", cval=0.0, origin=0):
+    """Calculate a 1-D maximum filter along the given axis.
+
+    The lines of the array along the given axis are filtered with a
+    maximum filter of given size.
+
+    Parameters
+    ----------
+    %(input)s
+    size : int
+        Length along which to calculate the 1-D maximum.
+    %(axis)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin)s
+
+    Returns
+    -------
+    maximum1d : ndarray, None
+        Maximum-filtered array with same shape as input.
+        None if `output` is not None
+
+    Notes
+    -----
+    This function implements the MAXLIST algorithm [1]_, as described by
+    Richard Harter [2]_, and has a guaranteed O(n) performance, `n` being
+    the `input` length, regardless of filter size.
+
+    References
+    ----------
+    .. [1] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.42.2777.
+    .. [2] http://www.richardhartersworld.com/cri/2001/slidingmin.html
+
+    Examples
+    --------
+    >>> from scipy.ndimage import maximum_filter1d
+    >>> maximum_filter1d([2, 8, 0, 4, 1, 9, 9, 0], size=3)
+    array([8, 8, 8, 4, 9, 9, 9, 9])
+    """
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError('Complex type not supported')
+    axis = normalize_axis_index(axis, input.ndim)
+    if size < 1:
+        raise RuntimeError('incorrect filter size')
+    output = _ni_support._get_output(output, input)
+    if (size // 2 + origin < 0) or (size // 2 + origin >= size):
+        raise ValueError('invalid origin')
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.min_or_max_filter1d(input, size, axis, output, mode, cval,
+                                  origin, 0)
+    return output
+
+
+def _min_or_max_filter(input, size, footprint, structure, output, mode,
+                       cval, origin, minimum, axes=None):
+    if (size is not None) and (footprint is not None):
+        warnings.warn("ignoring size because footprint is set",
+                      UserWarning, stacklevel=3)
+    if structure is None:
+        if footprint is None:
+            if size is None:
+                raise RuntimeError("no footprint provided")
+            separable = True
+        else:
+            footprint = np.asarray(footprint, dtype=bool)
+            if not footprint.any():
+                raise ValueError("All-zero footprint is not supported.")
+            if footprint.all():
+                size = footprint.shape
+                footprint = None
+                separable = True
+            else:
+                separable = False
+    else:
+        structure = np.asarray(structure, dtype=np.float64)
+        separable = False
+        if footprint is None:
+            footprint = np.ones(structure.shape, bool)
+        else:
+            footprint = np.asarray(footprint, dtype=bool)
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError("Complex type not supported")
+    output = _ni_support._get_output(output, input)
+    temp_needed = np.may_share_memory(input, output)
+    if temp_needed:
+        # input and output arrays cannot share memory
+        temp = output
+        output = _ni_support._get_output(output.dtype, input)
+    axes = _ni_support._check_axes(axes, input.ndim)
+    num_axes = len(axes)
+    if separable:
+        origins = _ni_support._normalize_sequence(origin, num_axes)
+        sizes = _ni_support._normalize_sequence(size, num_axes)
+        modes = _ni_support._normalize_sequence(mode, num_axes)
+        axes = [(axes[ii], sizes[ii], origins[ii], modes[ii])
+                for ii in range(len(axes)) if sizes[ii] > 1]
+        if minimum:
+            filter_ = minimum_filter1d
+        else:
+            filter_ = maximum_filter1d
+        if len(axes) > 0:
+            for axis, size, origin, mode in axes:
+                filter_(input, int(size), axis, output, mode, cval, origin)
+                input = output
+        else:
+            output[...] = input[...]
+    else:
+        # expand origins and footprint if num_axes < input.ndim
+        footprint = _expand_footprint(input.ndim, axes, footprint)
+        origins = _expand_origin(input.ndim, axes, origin)
+
+        fshape = [ii for ii in footprint.shape if ii > 0]
+        if len(fshape) != input.ndim:
+            raise RuntimeError(f"footprint.ndim ({footprint.ndim}) must match "
+                               f"len(axes) ({len(axes)})")
+        for origin, lenf in zip(origins, fshape):
+            if (lenf // 2 + origin < 0) or (lenf // 2 + origin >= lenf):
+                raise ValueError("invalid origin")
+        if not footprint.flags.contiguous:
+            footprint = footprint.copy()
+        if structure is not None:
+            if len(structure.shape) != num_axes:
+                raise RuntimeError("structure array has incorrect shape")
+            if num_axes != structure.ndim:
+                structure = np.expand_dims(
+                    structure,
+                    tuple(ax for ax in range(structure.ndim) if ax not in axes)
+                )
+            if not structure.flags.contiguous:
+                structure = structure.copy()
+        if not isinstance(mode, str) and isinstance(mode, Iterable):
+            raise RuntimeError(
+                "A sequence of modes is not supported for non-separable "
+                "footprints")
+        mode = _ni_support._extend_mode_to_code(mode)
+        _nd_image.min_or_max_filter(input, footprint, structure, output,
+                                    mode, cval, origins, minimum)
+    if temp_needed:
+        temp[...] = output
+        output = temp
+    return output
+
+
+@_ni_docstrings.docfiller
+def minimum_filter(input, size=None, footprint=None, output=None,
+                   mode="reflect", cval=0.0, origin=0, *, axes=None):
+    """Calculate a multidimensional minimum filter.
+
+    Parameters
+    ----------
+    %(input)s
+    %(size_foot)s
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `size`, `origin`, and/or `mode`
+        must match the length of `axes`. The ith entry in any of these tuples
+        corresponds to the ith entry in `axes`.
+
+    Returns
+    -------
+    minimum_filter : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Notes
+    -----
+    A sequence of modes (one per axis) is only supported when the footprint is
+    separable. Otherwise, a single mode string must be provided.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.minimum_filter(ascent, size=20)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    return _min_or_max_filter(input, size, footprint, None, output, mode,
+                              cval, origin, 1, axes)
+
+
+@_ni_docstrings.docfiller
+def maximum_filter(input, size=None, footprint=None, output=None,
+                   mode="reflect", cval=0.0, origin=0, *, axes=None):
+    """Calculate a multidimensional maximum filter.
+
+    Parameters
+    ----------
+    %(input)s
+    %(size_foot)s
+    %(output)s
+    %(mode_multiple)s
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `size`, `origin`, and/or `mode`
+        must match the length of `axes`. The ith entry in any of these tuples
+        corresponds to the ith entry in `axes`.
+
+    Returns
+    -------
+    maximum_filter : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Notes
+    -----
+    A sequence of modes (one per axis) is only supported when the footprint is
+    separable. Otherwise, a single mode string must be provided.
+
+    %(nan)s
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.maximum_filter(ascent, size=20)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    return _min_or_max_filter(input, size, footprint, None, output, mode,
+                              cval, origin, 0, axes)
+
+
+@_ni_docstrings.docfiller
+def _rank_filter(input, rank, size=None, footprint=None, output=None,
+                 mode="reflect", cval=0.0, origin=0, operation='rank',
+                 axes=None):
+    if (size is not None) and (footprint is not None):
+        warnings.warn("ignoring size because footprint is set",
+                      UserWarning, stacklevel=3)
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError('Complex type not supported')
+    axes = _ni_support._check_axes(axes, input.ndim)
+    num_axes = len(axes)
+    if footprint is None:
+        if size is None:
+            raise RuntimeError("no footprint or filter size provided")
+        sizes = _ni_support._normalize_sequence(size, num_axes)
+        footprint = np.ones(sizes, dtype=bool)
+    else:
+        footprint = np.asarray(footprint, dtype=bool)
+    # expand origins, footprint and modes if num_axes < input.ndim
+    footprint = _expand_footprint(input.ndim, axes, footprint)
+    origins = _expand_origin(input.ndim, axes, origin)
+    mode = _expand_mode(input.ndim, axes, mode)
+
+    fshape = [ii for ii in footprint.shape if ii > 0]
+    if len(fshape) != input.ndim:
+        raise RuntimeError(f"footprint.ndim ({footprint.ndim}) must match "
+                           f"len(axes) ({len(axes)})")
+    for origin, lenf in zip(origins, fshape):
+        if (lenf // 2 + origin < 0) or (lenf // 2 + origin >= lenf):
+            raise ValueError('invalid origin')
+    if not footprint.flags.contiguous:
+        footprint = footprint.copy()
+    filter_size = np.where(footprint, 1, 0).sum()
+    if operation == 'median':
+        rank = filter_size // 2
+    elif operation == 'percentile':
+        percentile = rank
+        if percentile < 0.0:
+            percentile += 100.0
+        if percentile < 0 or percentile > 100:
+            raise RuntimeError('invalid percentile')
+        if percentile == 100.0:
+            rank = filter_size - 1
+        else:
+            rank = int(float(filter_size) * percentile / 100.0)
+    if rank < 0:
+        rank += filter_size
+    if rank < 0 or rank >= filter_size:
+        raise RuntimeError('rank not within filter footprint size')
+    if rank == 0:
+        return minimum_filter(input, None, footprint, output, mode, cval,
+                              origins, axes=None)
+    elif rank == filter_size - 1:
+        return maximum_filter(input, None, footprint, output, mode, cval,
+                              origins, axes=None)
+    else:
+        output = _ni_support._get_output(output, input)
+        temp_needed = np.may_share_memory(input, output)
+        if temp_needed:
+            # input and output arrays cannot share memory
+            temp = output
+            output = _ni_support._get_output(output.dtype, input)
+        if not isinstance(mode, str) and isinstance(mode, Iterable):
+            raise RuntimeError(
+                "A sequence of modes is not supported by non-separable rank "
+                "filters")
+        mode = _ni_support._extend_mode_to_code(mode, is_filter=True)
+        # Some corner cases are currently not allowed to use the
+        # "new"/fast 1D rank filter code, including when the
+        # footprint is large compared to the array size.
+        # See discussion in gh-23293; longer-term it may be possible
+        # to allow the fast path for these corner cases as well,
+        # if algorithmic fixes are found.
+        lim2 = input.size - ((footprint.size - 1) // 2 - origin)
+        if input.ndim == 1 and ((lim2 >= 0) or (input.size == 1)):
+            if input.dtype in (np.int64, np.float64, np.float32):
+                x = input
+                x_out = output
+            elif input.dtype == np.float16:
+                x = input.astype('float32')
+                x_out = np.empty(x.shape, dtype='float32')
+            elif np.result_type(input, np.int64) == np.int64:
+                x = input.astype('int64')
+                x_out = np.empty(x.shape, dtype='int64')
+            elif input.dtype.kind in 'biu':
+                # cast any other boolean, integer or unsigned type to int64
+                x = input.astype('int64')
+                x_out = np.empty(x.shape, dtype='int64')
+            else:
+                raise RuntimeError('Unsupported array type')
+            cval = x.dtype.type(cval)
+            _rank_filter_1d.rank_filter(x, rank, footprint.size, x_out, mode, cval,
+                                        origin)
+            if input.dtype not in (np.int64, np.float64, np.float32):
+                np.copyto(output, x_out, casting='unsafe')
+        else:
+            _nd_image.rank_filter(input, rank, footprint, output, mode, cval, origins)
+        if temp_needed:
+            temp[...] = output
+            output = temp
+        return output
+
+
+@_ni_docstrings.docfiller
+def rank_filter(input, rank, size=None, footprint=None, output=None,
+                mode="reflect", cval=0.0, origin=0, *, axes=None):
+    """Calculate a multidimensional rank filter.
+
+    Parameters
+    ----------
+    %(input)s
+    rank : int
+        The rank parameter may be less than zero, i.e., rank = -1
+        indicates the largest element.
+    %(size_foot)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `size`, `origin`, and/or `mode`
+        must match the length of `axes`. The ith entry in any of these tuples
+        corresponds to the ith entry in `axes`.
+
+    Returns
+    -------
+    rank_filter : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.rank_filter(ascent, rank=42, size=20)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    rank = operator.index(rank)
+    return _rank_filter(input, rank, size, footprint, output, mode, cval,
+                        origin, 'rank', axes=axes)
+
+
+@_ni_docstrings.docfiller
+def median_filter(input, size=None, footprint=None, output=None,
+                  mode="reflect", cval=0.0, origin=0, *, axes=None):
+    """
+    Calculate a multidimensional median filter.
+
+    Parameters
+    ----------
+    %(input)s
+    %(size_foot)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `size`, `origin`, and/or `mode`
+        must match the length of `axes`. The ith entry in any of these tuples
+        corresponds to the ith entry in `axes`.
+
+    Returns
+    -------
+    median_filter : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    See Also
+    --------
+    scipy.signal.medfilt2d
+
+    Notes
+    -----
+    For 2-dimensional images with ``uint8``, ``float32`` or ``float64`` dtypes
+    the specialised function `scipy.signal.medfilt2d` may be faster. It is
+    however limited to constant mode with ``cval=0``.
+
+    The filter always returns the argument that would appear at index ``n // 2`` in
+    a sorted array, where ``n`` is the number of elements in the footprint of the
+    filter. Note that this differs from the conventional definition of the median
+    when ``n`` is even. Also, this function does not support the ``float16`` dtype,
+    behavior in the presence of NaNs is undefined, and memory consumption scales with
+    ``n**4``. For ``float16`` support, greater control over the definition of the
+    filter, and to limit memory usage, consider using `vectorized_filter` with
+    NumPy functions `np.median` or `np.nanmedian`.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.median_filter(ascent, size=20)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    return _rank_filter(input, 0, size, footprint, output, mode, cval,
+                        origin, 'median', axes=axes)
+
+
+@_ni_docstrings.docfiller
+def percentile_filter(input, percentile, size=None, footprint=None,
+                      output=None, mode="reflect", cval=0.0, origin=0, *,
+                      axes=None):
+    """Calculate a multidimensional percentile filter.
+
+    Parameters
+    ----------
+    %(input)s
+    percentile : scalar
+        The percentile parameter may be less than zero, i.e.,
+        percentile = -20 equals percentile = 80
+    %(size_foot)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin_multiple)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `size`, `origin`, and/or `mode`
+        must match the length of `axes`. The ith entry in any of these tuples
+        corresponds to the ith entry in `axes`.
+
+    Returns
+    -------
+    percentile_filter : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure()
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.percentile_filter(ascent, percentile=20, size=20)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result)
+    >>> plt.show()
+    """
+    return _rank_filter(input, percentile, size, footprint, output, mode,
+                        cval, origin, 'percentile', axes=axes)
+
+
+@_ni_docstrings.docfiller
+def generic_filter1d(input, function, filter_size, axis=-1,
+                     output=None, mode="reflect", cval=0.0, origin=0,
+                     extra_arguments=(), extra_keywords=None):
+    """Calculate a 1-D filter along the given axis.
+
+    `generic_filter1d` iterates over the lines of the array, calling the
+    given function at each line. The arguments of the line are the
+    input line, and the output line. The input and output lines are 1-D
+    double arrays. The input line is extended appropriately according
+    to the filter size and origin. The output line must be modified
+    in-place with the result.
+
+    Parameters
+    ----------
+    %(input)s
+    function : {callable, scipy.LowLevelCallable}
+        Function to apply along given axis.
+    filter_size : scalar
+        Length of the filter.
+    %(axis)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin)s
+    %(extra_arguments)s
+    %(extra_keywords)s
+
+    Returns
+    -------
+    generic_filter1d : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    Notes
+    -----
+    This function also accepts low-level callback functions with one of
+    the following signatures and wrapped in `scipy.LowLevelCallable`:
+
+    .. code:: c
+
+       int function(double *input_line, npy_intp input_length,
+                    double *output_line, npy_intp output_length,
+                    void *user_data)
+       int function(double *input_line, intptr_t input_length,
+                    double *output_line, intptr_t output_length,
+                    void *user_data)
+
+    The calling function iterates over the lines of the input and output
+    arrays, calling the callback function at each line. The current line
+    is extended according to the border conditions set by the calling
+    function, and the result is copied into the array that is passed
+    through ``input_line``. The length of the input line (after extension)
+    is passed through ``input_length``. The callback function should apply
+    the filter and store the result in the array passed through
+    ``output_line``. The length of the output line is passed through
+    ``output_length``. ``user_data`` is the data pointer provided
+    to `scipy.LowLevelCallable` as-is.
+
+    The callback function must return an integer error status that is zero
+    if something went wrong and one otherwise. If an error occurs, you should
+    normally set the python error status with an informative message
+    before returning, otherwise a default error message is set by the
+    calling function.
+
+    In addition, some other low-level function pointer specifications
+    are accepted, but these are for backward compatibility only and should
+    not be used in new code.
+
+    Examples
+    --------
+    This example defines and applies a custom callback function that computes
+    the range (maximum minus minimum) within a sliding window of size 3.
+    It utilizes `numpy.lib.stride_tricks.sliding_window_view` and demonstrates the
+    required in-place modification of the ``output_line`` array.
+
+    >>> import numpy as np
+    >>> from scipy.ndimage import generic_filter1d
+    >>> from numpy.lib.stride_tricks import sliding_window_view
+
+    >>> def local_range(input_line, output_line):
+    ...     # input_line includes padded values according to `filter_size` and `mode`
+    ...     v = sliding_window_view(input_line, 3)
+    ...     # modify `output_line` in-place rather than returning the result
+    ...     output_line[:] = v.max(axis=1) - v.min(axis=1)
+
+    >>> x = np.array([1, 2, 3, 4, 5], dtype=np.float64)
+    >>> generic_filter1d(x, local_range, filter_size=3)
+    array([1., 2., 2., 2., 1.])
+
+    """
+    if extra_keywords is None:
+        extra_keywords = {}
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError('Complex type not supported')
+    output = _ni_support._get_output(output, input)
+    if filter_size < 1:
+        raise RuntimeError('invalid filter size')
+    axis = normalize_axis_index(axis, input.ndim)
+    if (filter_size // 2 + origin < 0) or (filter_size // 2 + origin >=
+                                           filter_size):
+        raise ValueError('invalid origin')
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.generic_filter1d(input, function, filter_size, axis, output,
+                               mode, cval, origin, extra_arguments,
+                               extra_keywords)
+    return output
+
+
+@_ni_docstrings.docfiller
+def generic_filter(input, function, size=None, footprint=None,
+                   output=None, mode="reflect", cval=0.0, origin=0,
+                   extra_arguments=(), extra_keywords=None, *, axes=None):
+    """Calculate a multidimensional filter using the given function.
+
+    At each element the provided function is called. The input values
+    within the filter footprint at that element are passed to the function
+    as a 1-D array of double values.
+
+    Parameters
+    ----------
+    %(input)s
+    function : {callable, scipy.LowLevelCallable}
+        Function to apply at each element.
+    %(size_foot)s
+    %(output)s
+    %(mode_reflect)s
+    %(cval)s
+    %(origin_multiple)s
+    %(extra_arguments)s
+    %(extra_keywords)s
+    axes : tuple of int or None, optional
+        If None, `input` is filtered along all axes. Otherwise,
+        `input` is filtered along the specified axes. When `axes` is
+        specified, any tuples used for `size` or `origin` must match the length
+        of `axes`. The ith entry in any of these tuples corresponds to the ith
+        entry in `axes`.
+
+    Returns
+    -------
+    output : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    See Also
+    --------
+    vectorized_filter : similar functionality, but optimized for vectorized callables
+
+    Notes
+    -----
+    This function is ideal for use with instances of `scipy.LowLevelCallable`;
+    for vectorized, pure-Python callables, consider `vectorized_filter` for improved
+    performance.
+
+    Low-level callback functions must have one of the following signatures:
+
+    .. code:: c
+
+       int callback(double *buffer, npy_intp filter_size,
+                    double *return_value, void *user_data)
+       int callback(double *buffer, intptr_t filter_size,
+                    double *return_value, void *user_data)
+
+    The calling function iterates over the elements of the input and
+    output arrays, calling the callback function at each element. The
+    elements within the footprint of the filter at the current element are
+    passed through the ``buffer`` parameter, and the number of elements
+    within the footprint through ``filter_size``. The calculated value is
+    returned in ``return_value``. ``user_data`` is the data pointer provided
+    to `scipy.LowLevelCallable` as-is.
+
+    The callback function must return an integer error status that is zero
+    if something went wrong and one otherwise. If an error occurs, you should
+    normally set the python error status with an informative message
+    before returning, otherwise a default error message is set by the
+    calling function.
+
+    In addition, some other low-level function pointer specifications
+    are accepted, but these are for backward compatibility only and should
+    not be used in new code.
+
+    Examples
+    --------
+    Import the necessary modules and load the example image used for
+    filtering.
+
+    >>> import numpy as np
+    >>> from scipy import datasets
+    >>> from scipy.ndimage import zoom, generic_filter
+    >>> import matplotlib.pyplot as plt
+    >>> ascent = zoom(datasets.ascent(), 0.5)
+
+    Compute a maximum filter with kernel size 5 by passing a simple NumPy
+    aggregation function as argument to `function`.
+
+    >>> maximum_filter_result = generic_filter(ascent, np.amax, [5, 5])
+
+    While a maximum filter could also directly be obtained using
+    `maximum_filter`, `generic_filter` allows generic Python function or
+    `scipy.LowLevelCallable` to be used as a filter. Here, we compute the
+    range between maximum and minimum value as an example for a kernel size
+    of 5.
+
+    >>> def custom_filter(image):
+    ...     return np.amax(image) - np.amin(image)
+    >>> custom_filter_result = generic_filter(ascent, custom_filter, [5, 5])
+
+    Plot the original and filtered images.
+
+    >>> fig, axes = plt.subplots(3, 1, figsize=(3, 9))
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> top, middle, bottom = axes
+    >>> for ax in axes:
+    ...     ax.set_axis_off()  # remove coordinate system
+    >>> top.imshow(ascent)
+    >>> top.set_title("Original image")
+    >>> middle.imshow(maximum_filter_result)
+    >>> middle.set_title("Maximum filter, Kernel: 5x5")
+    >>> bottom.imshow(custom_filter_result)
+    >>> bottom.set_title("Custom filter, Kernel: 5x5")
+    >>> fig.tight_layout()
+
+    """
+    if (size is not None) and (footprint is not None):
+        warnings.warn("ignoring size because footprint is set",
+                      UserWarning, stacklevel=2)
+    if extra_keywords is None:
+        extra_keywords = {}
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError('Complex type not supported')
+    axes = _ni_support._check_axes(axes, input.ndim)
+    num_axes = len(axes)
+    if footprint is None:
+        if size is None:
+            raise RuntimeError("no footprint or filter size provided")
+        sizes = _ni_support._normalize_sequence(size, num_axes)
+        footprint = np.ones(sizes, dtype=bool)
+    else:
+        footprint = np.asarray(footprint, dtype=bool)
+
+    # expand origins, footprint if num_axes < input.ndim
+    footprint = _expand_footprint(input.ndim, axes, footprint)
+    origins = _expand_origin(input.ndim, axes, origin)
+
+    fshape = [ii for ii in footprint.shape if ii > 0]
+    if len(fshape) != input.ndim:
+        raise RuntimeError(f"footprint.ndim ({footprint.ndim}) "
+                           f"must match len(axes) ({num_axes})")
+    for origin, lenf in zip(origins, fshape):
+        if (lenf // 2 + origin < 0) or (lenf // 2 + origin >= lenf):
+            raise ValueError('invalid origin')
+    if not footprint.flags.contiguous:
+        footprint = footprint.copy()
+    output = _ni_support._get_output(output, input)
+
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.generic_filter(input, function, footprint, output, mode,
+                             cval, origins, extra_arguments, extra_keywords)
+    return output

.venv/lib/python3.12/site-packages/scipy/ndimage/_fourier.py

@@ -0,0 +1,306 @@
+# Copyright (C) 2003-2005 Peter J. Verveer
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above
+#    copyright notice, this list of conditions and the following
+#    disclaimer in the documentation and/or other materials provided
+#    with the distribution.
+#
+# 3. The name of the author may not be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS
+# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
+# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+import numpy as np
+from scipy._lib._util import normalize_axis_index
+from . import _ni_support
+from . import _nd_image
+
+__all__ = ['fourier_gaussian', 'fourier_uniform', 'fourier_ellipsoid',
+           'fourier_shift']
+
+
+def _get_output_fourier(output, input):
+    if output is None:
+        if input.dtype.type in [np.complex64, np.complex128, np.float32]:
+            output = np.zeros(input.shape, dtype=input.dtype)
+        else:
+            output = np.zeros(input.shape, dtype=np.float64)
+    elif type(output) is type:
+        if output not in [np.complex64, np.complex128,
+                          np.float32, np.float64]:
+            raise RuntimeError("output type not supported")
+        output = np.zeros(input.shape, dtype=output)
+    elif output.shape != input.shape:
+        raise RuntimeError("output shape not correct")
+    return output
+
+
+def _get_output_fourier_complex(output, input):
+    if output is None:
+        if input.dtype.type in [np.complex64, np.complex128]:
+            output = np.zeros(input.shape, dtype=input.dtype)
+        else:
+            output = np.zeros(input.shape, dtype=np.complex128)
+    elif type(output) is type:
+        if output not in [np.complex64, np.complex128]:
+            raise RuntimeError("output type not supported")
+        output = np.zeros(input.shape, dtype=output)
+    elif output.shape != input.shape:
+        raise RuntimeError("output shape not correct")
+    return output
+
+
+def fourier_gaussian(input, sigma, n=-1, axis=-1, output=None):
+    """
+    Multidimensional Gaussian fourier filter.
+
+    The array is multiplied with the fourier transform of a Gaussian
+    kernel.
+
+    Parameters
+    ----------
+    input : array_like
+        The input array.
+    sigma : float or sequence
+        The sigma of the Gaussian kernel. If a float, `sigma` is the same for
+        all axes. If a sequence, `sigma` has to contain one value for each
+        axis.
+    n : int, optional
+        If `n` is negative (default), then the input is assumed to be the
+        result of a complex fft.
+        If `n` is larger than or equal to zero, the input is assumed to be the
+        result of a real fft, and `n` gives the length of the array before
+        transformation along the real transform direction.
+    axis : int, optional
+        The axis of the real transform.
+    output : ndarray, optional
+        If given, the result of filtering the input is placed in this array.
+
+    Returns
+    -------
+    fourier_gaussian : ndarray
+        The filtered input.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import numpy.fft
+    >>> import matplotlib.pyplot as plt
+    >>> fig, (ax1, ax2) = plt.subplots(1, 2)
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ascent = datasets.ascent()
+    >>> input_ = numpy.fft.fft2(ascent)
+    >>> result = ndimage.fourier_gaussian(input_, sigma=4)
+    >>> result = numpy.fft.ifft2(result)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result.real)  # the imaginary part is an artifact
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+    output = _get_output_fourier(output, input)
+    axis = normalize_axis_index(axis, input.ndim)
+    sigmas = _ni_support._normalize_sequence(sigma, input.ndim)
+    sigmas = np.asarray(sigmas, dtype=np.float64)
+    if not sigmas.flags.contiguous:
+        sigmas = sigmas.copy()
+
+    _nd_image.fourier_filter(input, sigmas, n, axis, output, 0)
+    return output
+
+
+def fourier_uniform(input, size, n=-1, axis=-1, output=None):
+    """
+    Multidimensional uniform fourier filter.
+
+    The array is multiplied with the Fourier transform of a box of given
+    size.
+
+    Parameters
+    ----------
+    input : array_like
+        The input array.
+    size : float or sequence
+        The size of the box used for filtering.
+        If a float, `size` is the same for all axes. If a sequence, `size` has
+        to contain one value for each axis.
+    n : int, optional
+        If `n` is negative (default), then the input is assumed to be the
+        result of a complex fft.
+        If `n` is larger than or equal to zero, the input is assumed to be the
+        result of a real fft, and `n` gives the length of the array before
+        transformation along the real transform direction.
+    axis : int, optional
+        The axis of the real transform.
+    output : ndarray, optional
+        If given, the result of filtering the input is placed in this array.
+
+    Returns
+    -------
+    fourier_uniform : ndarray
+        The filtered input.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import numpy.fft
+    >>> import matplotlib.pyplot as plt
+    >>> fig, (ax1, ax2) = plt.subplots(1, 2)
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ascent = datasets.ascent()
+    >>> input_ = numpy.fft.fft2(ascent)
+    >>> result = ndimage.fourier_uniform(input_, size=20)
+    >>> result = numpy.fft.ifft2(result)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result.real)  # the imaginary part is an artifact
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+    output = _get_output_fourier(output, input)
+    axis = normalize_axis_index(axis, input.ndim)
+    sizes = _ni_support._normalize_sequence(size, input.ndim)
+    sizes = np.asarray(sizes, dtype=np.float64)
+    if not sizes.flags.contiguous:
+        sizes = sizes.copy()
+    _nd_image.fourier_filter(input, sizes, n, axis, output, 1)
+    return output
+
+
+def fourier_ellipsoid(input, size, n=-1, axis=-1, output=None):
+    """
+    Multidimensional ellipsoid Fourier filter.
+
+    The array is multiplied with the fourier transform of an ellipsoid of
+    given sizes.
+
+    Parameters
+    ----------
+    input : array_like
+        The input array.
+    size : float or sequence
+        The size of the box used for filtering.
+        If a float, `size` is the same for all axes. If a sequence, `size` has
+        to contain one value for each axis.
+    n : int, optional
+        If `n` is negative (default), then the input is assumed to be the
+        result of a complex fft.
+        If `n` is larger than or equal to zero, the input is assumed to be the
+        result of a real fft, and `n` gives the length of the array before
+        transformation along the real transform direction.
+    axis : int, optional
+        The axis of the real transform.
+    output : ndarray, optional
+        If given, the result of filtering the input is placed in this array.
+
+    Returns
+    -------
+    fourier_ellipsoid : ndarray
+        The filtered input.
+
+    Notes
+    -----
+    This function is implemented for arrays of rank 1, 2, or 3.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import numpy.fft
+    >>> import matplotlib.pyplot as plt
+    >>> fig, (ax1, ax2) = plt.subplots(1, 2)
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ascent = datasets.ascent()
+    >>> input_ = numpy.fft.fft2(ascent)
+    >>> result = ndimage.fourier_ellipsoid(input_, size=20)
+    >>> result = numpy.fft.ifft2(result)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result.real)  # the imaginary part is an artifact
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+    if input.ndim > 3:
+        raise NotImplementedError("Only 1d, 2d and 3d inputs are supported")
+    output = _get_output_fourier(output, input)
+    if output.size == 0:
+        # The C code has a bug that can result in a segfault with arrays
+        # that have size 0 (gh-17270), so check here.
+        return output
+    axis = normalize_axis_index(axis, input.ndim)
+    sizes = _ni_support._normalize_sequence(size, input.ndim)
+    sizes = np.asarray(sizes, dtype=np.float64)
+    if not sizes.flags.contiguous:
+        sizes = sizes.copy()
+    _nd_image.fourier_filter(input, sizes, n, axis, output, 2)
+    return output
+
+
+def fourier_shift(input, shift, n=-1, axis=-1, output=None):
+    """
+    Multidimensional Fourier shift filter.
+
+    The array is multiplied with the Fourier transform of a shift operation.
+
+    Parameters
+    ----------
+    input : array_like
+        The input array.
+    shift : float or sequence
+        The size of the box used for filtering.
+        If a float, `shift` is the same for all axes. If a sequence, `shift`
+        has to contain one value for each axis.
+    n : int, optional
+        If `n` is negative (default), then the input is assumed to be the
+        result of a complex fft.
+        If `n` is larger than or equal to zero, the input is assumed to be the
+        result of a real fft, and `n` gives the length of the array before
+        transformation along the real transform direction.
+    axis : int, optional
+        The axis of the real transform.
+    output : ndarray, optional
+        If given, the result of shifting the input is placed in this array.
+
+    Returns
+    -------
+    fourier_shift : ndarray
+        The shifted input.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> import numpy.fft
+    >>> fig, (ax1, ax2) = plt.subplots(1, 2)
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> ascent = datasets.ascent()
+    >>> input_ = numpy.fft.fft2(ascent)
+    >>> result = ndimage.fourier_shift(input_, shift=200)
+    >>> result = numpy.fft.ifft2(result)
+    >>> ax1.imshow(ascent)
+    >>> ax2.imshow(result.real)  # the imaginary part is an artifact
+    >>> plt.show()
+    """
+    input = np.asarray(input)
+    output = _get_output_fourier_complex(output, input)
+    axis = normalize_axis_index(axis, input.ndim)
+    shifts = _ni_support._normalize_sequence(shift, input.ndim)
+    shifts = np.asarray(shifts, dtype=np.float64)
+    if not shifts.flags.contiguous:
+        shifts = shifts.copy()
+    _nd_image.fourier_shift(input, shifts, n, axis, output)
+    return output

.venv/lib/python3.12/site-packages/scipy/ndimage/_interpolation.py

@@ -0,0 +1,1033 @@
+# Copyright (C) 2003-2005 Peter J. Verveer
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above
+#    copyright notice, this list of conditions and the following
+#    disclaimer in the documentation and/or other materials provided
+#    with the distribution.
+#
+# 3. The name of the author may not be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS
+# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
+# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+import itertools
+import warnings
+
+import numpy as np
+from scipy._lib._util import normalize_axis_index
+from scipy._external import array_api_extra as xpx
+
+from scipy import special
+from . import _ni_support
+from . import _nd_image
+from ._ni_docstrings import docfiller
+
+
+__all__ = ['spline_filter1d', 'spline_filter', 'geometric_transform',
+           'map_coordinates', 'affine_transform', 'shift', 'zoom', 'rotate']
+
+
+@docfiller
+def spline_filter1d(input, order=3, axis=-1, output=np.float64,
+                    mode='mirror'):
+    """
+    Calculate a 1-D spline filter along the given axis.
+
+    The lines of the array along the given axis are filtered by a
+    spline filter. The order of the spline must be >= 2 and <= 5.
+
+    Parameters
+    ----------
+    %(input)s
+    order : int, optional
+        The order of the spline, default is 3.
+    axis : int, optional
+        The axis along which the spline filter is applied. Default is the last
+        axis.
+    output : ndarray or dtype, optional
+        The array in which to place the output, or the dtype of the returned
+        array. Default is ``numpy.float64``.
+    %(mode_interp_mirror)s
+
+    Returns
+    -------
+    spline_filter1d : ndarray
+        The filtered input.
+
+    See Also
+    --------
+    spline_filter : Multidimensional spline filter.
+
+    Notes
+    -----
+    All of the interpolation functions in `ndimage` do spline interpolation of
+    the input image. If using B-splines of `order > 1`, the input image
+    values have to be converted to B-spline coefficients first, which is
+    done by applying this 1-D filter sequentially along all
+    axes of the input. All functions that require B-spline coefficients
+    will automatically filter their inputs, a behavior controllable with
+    the `prefilter` keyword argument. For functions that accept a `mode`
+    parameter, the result will only be correct if it matches the `mode`
+    used when filtering.
+
+    For complex-valued `input`, this function processes the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    Examples
+    --------
+    We can filter an image using 1-D spline along the given axis:
+
+    >>> from scipy.ndimage import spline_filter1d
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> orig_img = np.eye(20)  # create an image
+    >>> orig_img[10, :] = 1.0
+    >>> sp_filter_axis_0 = spline_filter1d(orig_img, axis=0)
+    >>> sp_filter_axis_1 = spline_filter1d(orig_img, axis=1)
+    >>> f, ax = plt.subplots(1, 3, sharex=True)
+    >>> for ind, data in enumerate([[orig_img, "original image"],
+    ...             [sp_filter_axis_0, "spline filter (axis=0)"],
+    ...             [sp_filter_axis_1, "spline filter (axis=1)"]]):
+    ...     ax[ind].imshow(data[0], cmap='gray_r')
+    ...     ax[ind].set_title(data[1])
+    >>> plt.tight_layout()
+    >>> plt.show()
+
+    """
+    if order < 0 or order > 5:
+        raise RuntimeError('spline order not supported')
+    input = np.asarray(input)
+    complex_output = np.iscomplexobj(input)
+    output = _ni_support._get_output(output, input,
+                                     complex_output=complex_output)
+    if complex_output:
+        spline_filter1d(input.real, order, axis, output.real, mode)
+        spline_filter1d(input.imag, order, axis, output.imag, mode)
+        return output
+    if order in [0, 1]:
+        output[...] = np.array(input)
+    else:
+        mode = _ni_support._extend_mode_to_code(mode)
+        axis = normalize_axis_index(axis, input.ndim)
+        _nd_image.spline_filter1d(input, order, axis, output, mode)
+    return output
+
+@docfiller
+def spline_filter(input, order=3, output=np.float64, mode='mirror'):
+    """
+    Multidimensional spline filter.
+
+    Parameters
+    ----------
+    %(input)s
+    order : int, optional
+        The order of the spline, default is 3.
+    output : ndarray or dtype, optional
+        The array in which to place the output, or the dtype of the returned
+        array. Default is ``numpy.float64``.
+    %(mode_interp_mirror)s
+
+    Returns
+    -------
+    spline_filter : ndarray
+        Filtered array. Has the same shape as `input`.
+
+    See Also
+    --------
+    spline_filter1d : Calculate a 1-D spline filter along the given axis.
+
+    Notes
+    -----
+    The multidimensional filter is implemented as a sequence of
+    1-D spline filters. The intermediate arrays are stored
+    in the same data type as the output. Therefore, for output types
+    with a limited precision, the results may be imprecise because
+    intermediate results may be stored with insufficient precision.
+
+    For complex-valued `input`, this function processes the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    Examples
+    --------
+    We can filter an image using multidimensional splines:
+
+    >>> from scipy.ndimage import spline_filter
+    >>> import numpy as np
+    >>> import matplotlib.pyplot as plt
+    >>> orig_img = np.eye(20)  # create an image
+    >>> orig_img[10, :] = 1.0
+    >>> sp_filter = spline_filter(orig_img, order=3)
+    >>> f, ax = plt.subplots(1, 2, sharex=True)
+    >>> for ind, data in enumerate([[orig_img, "original image"],
+    ...                             [sp_filter, "spline filter"]]):
+    ...     ax[ind].imshow(data[0], cmap='gray_r')
+    ...     ax[ind].set_title(data[1])
+    >>> plt.tight_layout()
+    >>> plt.show()
+
+    """
+    if order < 2 or order > 5:
+        raise RuntimeError('spline order not supported')
+    input = np.asarray(input)
+    complex_output = np.iscomplexobj(input)
+    output = _ni_support._get_output(output, input,
+                                     complex_output=complex_output)
+    if complex_output:
+        spline_filter(input.real, order, output.real, mode)
+        spline_filter(input.imag, order, output.imag, mode)
+        return output
+    if order not in [0, 1] and input.ndim > 0:
+        for axis in range(input.ndim):
+            spline_filter1d(input, order, axis, output=output, mode=mode)
+            input = output
+    else:
+        output[...] = input[...]
+    return output
+
+
+def _prepad_for_spline_filter(input, mode, cval):
+    if mode in ['nearest', 'grid-constant']:
+        npad = 12
+        if mode == 'grid-constant':
+            padded = np.pad(input, npad, mode='constant',
+                               constant_values=cval)
+        elif mode == 'nearest':
+            padded = np.pad(input, npad, mode='edge')
+    else:
+        # other modes have exact boundary conditions implemented so
+        # no prepadding is needed
+        npad = 0
+        padded = input
+    return padded, npad
+
+
+@docfiller
+def geometric_transform(input, mapping, output_shape=None,
+                        output=None, order=3,
+                        mode='constant', cval=0.0, prefilter=True,
+                        extra_arguments=(), extra_keywords=None):
+    """
+    Apply an arbitrary geometric transform.
+
+    The given mapping function is used to find, for each point in the
+    output, the corresponding coordinates in the input. The value of the
+    input at those coordinates is determined by spline interpolation of
+    the requested order.
+
+    Parameters
+    ----------
+    %(input)s
+    mapping : {callable, scipy.LowLevelCallable}
+        A callable object that accepts a tuple of length equal to the output
+        array rank, and returns the corresponding input coordinates as a tuple
+        of length equal to the input array rank.
+    output_shape : tuple of ints, optional
+        Shape tuple.
+    %(output)s
+    order : int, optional
+        The order of the spline interpolation, default is 3.
+        The order has to be in the range 0-5.
+    %(mode_interp_constant)s
+    %(cval)s
+    %(prefilter)s
+    extra_arguments : tuple, optional
+        Extra arguments passed to `mapping`.
+    extra_keywords : dict, optional
+        Extra keywords passed to `mapping`.
+
+    Returns
+    -------
+    output : ndarray
+        The filtered input.
+
+    See Also
+    --------
+    map_coordinates, affine_transform, spline_filter1d
+
+
+    Notes
+    -----
+    This function also accepts low-level callback functions with one
+    the following signatures and wrapped in `scipy.LowLevelCallable`:
+
+    .. code:: c
+
+       int mapping(npy_intp *output_coordinates, double *input_coordinates,
+                   int output_rank, int input_rank, void *user_data)
+       int mapping(intptr_t *output_coordinates, double *input_coordinates,
+                   int output_rank, int input_rank, void *user_data)
+
+    The calling function iterates over the elements of the output array,
+    calling the callback function at each element. The coordinates of the
+    current output element are passed through ``output_coordinates``. The
+    callback function must return the coordinates at which the input must
+    be interpolated in ``input_coordinates``. The rank of the input and
+    output arrays are given by ``input_rank`` and ``output_rank``
+    respectively. ``user_data`` is the data pointer provided
+    to `scipy.LowLevelCallable` as-is.
+
+    The callback function must return an integer error status that is zero
+    if something went wrong and one otherwise. If an error occurs, you should
+    normally set the Python error status with an informative message
+    before returning, otherwise a default error message is set by the
+    calling function.
+
+    In addition, some other low-level function pointer specifications
+    are accepted, but these are for backward compatibility only and should
+    not be used in new code.
+
+    For complex-valued `input`, this function transforms the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy.ndimage import geometric_transform
+    >>> a = np.arange(12.).reshape((4, 3))
+    >>> def shift_func(output_coords):
+    ...     return (output_coords[0] - 0.5, output_coords[1] - 0.5)
+    ...
+    >>> geometric_transform(a, shift_func)
+    array([[ 0.   ,  0.   ,  0.   ],
+           [ 0.   ,  1.362,  2.738],
+           [ 0.   ,  4.812,  6.187],
+           [ 0.   ,  8.263,  9.637]])
+
+    >>> b = [1, 2, 3, 4, 5]
+    >>> def shift_func(output_coords):
+    ...     return (output_coords[0] - 3,)
+    ...
+    >>> geometric_transform(b, shift_func, mode='constant')
+    array([0, 0, 0, 1, 2])
+    >>> geometric_transform(b, shift_func, mode='nearest')
+    array([1, 1, 1, 1, 2])
+    >>> geometric_transform(b, shift_func, mode='reflect')
+    array([3, 2, 1, 1, 2])
+    >>> geometric_transform(b, shift_func, mode='wrap')
+    array([2, 3, 4, 1, 2])
+
+    """
+    if extra_keywords is None:
+        extra_keywords = {}
+    if order < 0 or order > 5:
+        raise RuntimeError('spline order not supported')
+    input = np.asarray(input)
+    if output_shape is None:
+        output_shape = input.shape
+    if input.ndim < 1 or len(output_shape) < 1:
+        raise RuntimeError('input and output rank must be > 0')
+    complex_output = np.iscomplexobj(input)
+    output = _ni_support._get_output(output, input, shape=output_shape,
+                                     complex_output=complex_output)
+    if complex_output:
+        kwargs = dict(order=order, mode=mode, prefilter=prefilter,
+                      output_shape=output_shape,
+                      extra_arguments=extra_arguments,
+                      extra_keywords=extra_keywords)
+        geometric_transform(input.real, mapping, output=output.real,
+                            cval=np.real(cval), **kwargs)
+        geometric_transform(input.imag, mapping, output=output.imag,
+                            cval=np.imag(cval), **kwargs)
+        return output
+
+    if prefilter and order > 1:
+        padded, npad = _prepad_for_spline_filter(input, mode, cval)
+        filtered = spline_filter(padded, order, output=np.float64,
+                                 mode=mode)
+    else:
+        npad = 0
+        filtered = input
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.geometric_transform(filtered, mapping, None, None, None, output,
+                                  order, mode, cval, npad, extra_arguments,
+                                  extra_keywords)
+    return output
+
+
+@docfiller
+def map_coordinates(input, coordinates, output=None, order=3,
+                    mode='constant', cval=0.0, prefilter=True):
+    """
+    Map the input array to new coordinates by interpolation.
+
+    The array of coordinates is used to find, for each point in the output,
+    the corresponding coordinates in the input. The value of the input at
+    those coordinates is determined by spline interpolation of the
+    requested order.
+
+    The shape of the output is derived from that of the coordinate
+    array by dropping the first axis. The values of the array along
+    the first axis are the coordinates in the input array at which the
+    output value is found.
+
+    Parameters
+    ----------
+    %(input)s
+    coordinates : array_like
+        The coordinates at which `input` is evaluated.
+    %(output)s
+    order : int, optional
+        The order of the spline interpolation, default is 3.
+        The order has to be in the range 0-5.
+    %(mode_interp_constant)s
+    %(cval)s
+    %(prefilter)s
+
+    Returns
+    -------
+    map_coordinates : ndarray
+        The result of transforming the input. The shape of the output is
+        derived from that of `coordinates` by dropping the first axis.
+
+    See Also
+    --------
+    spline_filter, geometric_transform, scipy.interpolate
+
+    Notes
+    -----
+    For complex-valued `input`, this function maps the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    Examples
+    --------
+    >>> from scipy import ndimage
+    >>> import numpy as np
+    >>> a = np.arange(12.).reshape((4, 3))
+    >>> a
+    array([[  0.,   1.,   2.],
+           [  3.,   4.,   5.],
+           [  6.,   7.,   8.],
+           [  9.,  10.,  11.]])
+    >>> ndimage.map_coordinates(a, [[0.5, 2], [0.5, 1]], order=1)
+    array([ 2.,  7.])
+
+    Above, the interpolated value of a[0.5, 0.5] gives output[0], while
+    a[2, 1] is output[1].
+
+    >>> inds = np.array([[0.5, 2], [0.5, 4]])
+    >>> ndimage.map_coordinates(a, inds, order=1, cval=-33.3)
+    array([  2. , -33.3])
+    >>> ndimage.map_coordinates(a, inds, order=1, mode='nearest')
+    array([ 2.,  8.])
+    >>> ndimage.map_coordinates(a, inds, order=1, cval=0, output=bool)
+    array([ True, False], dtype=bool)
+
+    """
+    if order < 0 or order > 5:
+        raise RuntimeError('spline order not supported')
+    input = np.asarray(input)
+    coordinates = np.asarray(coordinates)
+    if np.iscomplexobj(coordinates):
+        raise TypeError('Complex type not supported')
+    output_shape = coordinates.shape[1:]
+    if input.ndim < 1 or len(output_shape) < 1:
+        raise RuntimeError('input and output rank must be > 0')
+    if coordinates.shape[0] != input.ndim:
+        raise RuntimeError('invalid shape for coordinate array')
+    complex_output = np.iscomplexobj(input)
+    output = _ni_support._get_output(output, input, shape=output_shape,
+                                     complex_output=complex_output)
+    if complex_output:
+        kwargs = dict(order=order, mode=mode, prefilter=prefilter)
+        map_coordinates(input.real, coordinates, output=output.real,
+                        cval=np.real(cval), **kwargs)
+        map_coordinates(input.imag, coordinates, output=output.imag,
+                        cval=np.imag(cval), **kwargs)
+        return output
+    if prefilter and order > 1:
+        padded, npad = _prepad_for_spline_filter(input, mode, cval)
+        filtered = spline_filter(padded, order, output=np.float64, mode=mode)
+    else:
+        npad = 0
+        filtered = input
+    mode = _ni_support._extend_mode_to_code(mode)
+    _nd_image.geometric_transform(filtered, None, coordinates, None, None,
+                                  output, order, mode, cval, npad, None, None)
+    return output
+
+
+@docfiller
+def affine_transform(input, matrix, offset=0.0, output_shape=None,
+                     output=None, order=3,
+                     mode='constant', cval=0.0, prefilter=True):
+    """
+    Apply an affine transformation.
+
+    Given an output image pixel index vector ``o``, the pixel value
+    is determined from the input image at position
+    ``np.dot(matrix, o) + offset``.
+
+    This does 'pull' (or 'backward') resampling, transforming the output space
+    to the input to locate data. Affine transformations are often described in
+    the 'push' (or 'forward') direction, transforming input to output. If you
+    have a matrix for the 'push' transformation, use its inverse
+    (:func:`numpy.linalg.inv`) in this function.
+
+    Parameters
+    ----------
+    %(input)s
+    matrix : ndarray
+        The inverse coordinate transformation matrix, mapping output
+        coordinates to input coordinates. If ``ndim`` is the number of
+        dimensions of ``input``, the given matrix must have one of the
+        following shapes:
+
+        - ``(ndim, ndim)``: the linear transformation matrix for each
+          output coordinate.
+        - ``(ndim,)``: assume that the 2-D transformation matrix is
+          diagonal, with the diagonal specified by the given value. A more
+          efficient algorithm is then used that exploits the separability
+          of the problem.
+        - ``(ndim + 1, ndim + 1)``: assume that the transformation is
+          specified using homogeneous coordinates [1]_. In this case, any
+          value passed to ``offset`` is ignored.
+        - ``(ndim, ndim + 1)``: as above, but the bottom row of a
+          homogeneous transformation matrix is always ``[0, 0, ..., 1]``,
+          and may be omitted.
+
+    offset : float or sequence, optional
+        The offset into the array where the transform is applied. If a float,
+        `offset` is the same for each axis. If a sequence, `offset` should
+        contain one value for each axis.
+    output_shape : tuple of ints, optional
+        Shape tuple.
+    %(output)s
+    order : int, optional
+        The order of the spline interpolation, default is 3.
+        The order has to be in the range 0-5.
+    %(mode_interp_constant)s
+    %(cval)s
+    %(prefilter)s
+
+    Returns
+    -------
+    affine_transform : ndarray
+        The transformed input.
+
+    Examples
+    --------
+    Use `affine_transform` to stretch an image::
+
+    >>> from scipy.ndimage import affine_transform
+    >>> from scipy.datasets import face
+    >>> from matplotlib import pyplot as plt
+    >>> import numpy as np
+    >>> im = face(gray=True)
+    >>> matrix = (0.5, 2)
+    >>> im2 = affine_transform(im, matrix)
+    >>> plt.imshow(im2)
+    >>> plt.show()
+
+    Rotate an image by 90 degrees and project it onto an expanded canvas::
+
+    >>> matrix = ((0, 1), (1, 0))
+    >>> im3 = affine_transform(im, matrix, output_shape=(1024, 1024))
+    >>> plt.imshow(im3)
+    >>> plt.show()
+
+    Offset the rotation so that the image is centred::
+
+    >>> output_shape = (1200, 1200)
+    >>> offset = (np.array(im.shape) - output_shape) / 2
+    >>> im4 = affine_transform(im, matrix, offset=offset, output_shape=output_shape)
+    >>> plt.imshow(im4)
+    >>> plt.show()
+
+    Notes
+    -----
+    The given matrix and offset are used to find for each point in the
+    output the corresponding coordinates in the input by an affine
+    transformation. The value of the input at those coordinates is
+    determined by spline interpolation of the requested order. Points
+    outside the boundaries of the input are filled according to the given
+    mode.
+
+    .. versionchanged:: 0.18.0
+        Previously, the exact interpretation of the affine transformation
+        depended on whether the matrix was supplied as a 1-D or a
+        2-D array. If a 1-D array was supplied
+        to the matrix parameter, the output pixel value at index ``o``
+        was determined from the input image at position
+        ``matrix * (o + offset)``.
+
+    For complex-valued `input`, this function transforms the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    References
+    ----------
+    .. [1] https://en.wikipedia.org/wiki/Homogeneous_coordinates
+    """
+    if order < 0 or order > 5:
+        raise RuntimeError('spline order not supported')
+    input = np.asarray(input)
+    if output_shape is None:
+        if isinstance(output, np.ndarray):
+            output_shape = output.shape
+        else:
+            output_shape = input.shape
+    if input.ndim < 1 or len(output_shape) < 1:
+        raise RuntimeError('input and output rank must be > 0')
+    complex_output = np.iscomplexobj(input)
+    output = _ni_support._get_output(output, input, shape=output_shape,
+                                     complex_output=complex_output)
+    if complex_output:
+        kwargs = dict(offset=offset, output_shape=output_shape, order=order,
+                      mode=mode, prefilter=prefilter)
+        affine_transform(input.real, matrix, output=output.real,
+                         cval=np.real(cval), **kwargs)
+        affine_transform(input.imag, matrix, output=output.imag,
+                         cval=np.imag(cval), **kwargs)
+        return output
+    if prefilter and order > 1:
+        padded, npad = _prepad_for_spline_filter(input, mode, cval)
+        filtered = spline_filter(padded, order, output=np.float64, mode=mode)
+    else:
+        npad = 0
+        filtered = input
+    mode = _ni_support._extend_mode_to_code(mode)
+    matrix = np.asarray(matrix, dtype=np.float64)
+    if matrix.ndim not in [1, 2] or matrix.shape[0] < 1:
+        raise RuntimeError('no proper affine matrix provided')
+    if (matrix.ndim == 2 and matrix.shape[1] == input.ndim + 1 and
+            (matrix.shape[0] in [input.ndim, input.ndim + 1])):
+        if matrix.shape[0] == input.ndim + 1:
+            exptd = [0] * input.ndim + [1]
+            if not np.all(matrix[input.ndim] == exptd):
+                msg = (f'Expected homogeneous transformation matrix with '
+                       f'shape {matrix.shape} for image shape {input.shape}, '
+                       f'but bottom row was not equal to {exptd}')
+                raise ValueError(msg)
+        # assume input is homogeneous coordinate transformation matrix
+        offset = matrix[:input.ndim, input.ndim]
+        matrix = matrix[:input.ndim, :input.ndim]
+    if matrix.shape[0] != input.ndim:
+        raise RuntimeError('affine matrix has wrong number of rows')
+    if matrix.ndim == 2 and matrix.shape[1] != output.ndim:
+        raise RuntimeError('affine matrix has wrong number of columns')
+    if not matrix.flags.contiguous:
+        matrix = matrix.copy()
+    offset = _ni_support._normalize_sequence(offset, input.ndim)
+    offset = np.asarray(offset, dtype=np.float64)
+    if offset.ndim != 1 or offset.shape[0] < 1:
+        raise RuntimeError('no proper offset provided')
+    if not offset.flags.contiguous:
+        offset = offset.copy()
+    if matrix.ndim == 1:
+        _nd_image.zoom_shift(filtered, matrix, offset/matrix, output, order,
+                             mode, cval, npad, False)
+    else:
+        _nd_image.geometric_transform(filtered, None, None, matrix, offset,
+                                      output, order, mode, cval, npad, None,
+                                      None)
+    return output
+
+
+@docfiller
+def shift(input, shift, output=None, order=3, mode='constant', cval=0.0,
+          prefilter=True):
+    """
+    Shift an array.
+
+    The array is shifted using spline interpolation of the requested order.
+    Points outside the boundaries of the input are filled according to the
+    given mode.
+
+    Parameters
+    ----------
+    %(input)s
+    shift : float or sequence
+        The shift along the axes. If a float, `shift` is the same for each
+        axis. If a sequence, `shift` should contain one value for each axis.
+    %(output)s
+    order : int, optional
+        The order of the spline interpolation, default is 3.
+        The order has to be in the range 0-5.
+    %(mode_interp_constant)s
+    %(cval)s
+    %(prefilter)s
+
+    Returns
+    -------
+    shift : ndarray
+        The shifted input.
+
+    See Also
+    --------
+    affine_transform : Affine transformations
+
+    Notes
+    -----
+    For complex-valued `input`, this function shifts the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    Examples
+    --------
+    Import the necessary modules and an exemplary image.
+
+    >>> from scipy.ndimage import shift
+    >>> import matplotlib.pyplot as plt
+    >>> from scipy import datasets
+    >>> image = datasets.ascent()
+
+    Shift the image vertically by 20 pixels.
+
+    >>> image_shifted_vertically = shift(image, (20, 0))
+
+    Shift the image vertically by -200 pixels and horizontally by 100 pixels.
+
+    >>> image_shifted_both_directions = shift(image, (-200, 100))
+
+    Plot the original and the shifted images.
+
+    >>> fig, axes = plt.subplots(3, 1, figsize=(4, 12))
+    >>> plt.gray()  # show the filtered result in grayscale
+    >>> top, middle, bottom = axes
+    >>> for ax in axes:
+    ...     ax.set_axis_off()  # remove coordinate system
+    >>> top.imshow(image)
+    >>> top.set_title("Original image")
+    >>> middle.imshow(image_shifted_vertically)
+    >>> middle.set_title("Vertically shifted image")
+    >>> bottom.imshow(image_shifted_both_directions)
+    >>> bottom.set_title("Image shifted in both directions")
+    >>> fig.tight_layout()
+    """
+    if order < 0 or order > 5:
+        raise RuntimeError('spline order not supported')
+    input = np.asarray(input)
+    if input.ndim < 1:
+        raise RuntimeError('input and output rank must be > 0')
+    complex_output = np.iscomplexobj(input)
+    output = _ni_support._get_output(output, input, complex_output=complex_output)
+    if complex_output:
+        # import under different name to avoid confusion with shift parameter
+        from scipy.ndimage._interpolation import shift as _shift
+
+        kwargs = dict(order=order, mode=mode, prefilter=prefilter)
+        _shift(input.real, shift, output=output.real, cval=np.real(cval), **kwargs)
+        _shift(input.imag, shift, output=output.imag, cval=np.imag(cval), **kwargs)
+        return output
+    if prefilter and order > 1:
+        padded, npad = _prepad_for_spline_filter(input, mode, cval)
+        filtered = spline_filter(padded, order, output=np.float64, mode=mode)
+    else:
+        npad = 0
+        filtered = input
+    mode = _ni_support._extend_mode_to_code(mode)
+    shift = _ni_support._normalize_sequence(shift, input.ndim)
+    shift = [-ii for ii in shift]
+    shift = np.asarray(shift, dtype=np.float64)
+    if not shift.flags.contiguous:
+        shift = shift.copy()
+    _nd_image.zoom_shift(filtered, None, shift, output, order, mode, cval,
+                         npad, False)
+    return output
+
+
+@docfiller
+def zoom(input, zoom, output=None, order=3, mode='constant', cval=0.0,
+         prefilter=True, *, grid_mode=False):
+    """
+    Zoom an array.
+
+    The array is zoomed using spline interpolation of the requested order.
+
+    Parameters
+    ----------
+    %(input)s
+    zoom : float or sequence
+        The zoom factor along the axes. If a float, `zoom` is the same for each
+        axis. If a sequence, `zoom` should contain one value for each axis.
+    %(output)s
+    order : int, optional
+        The order of the spline interpolation, default is 3.
+        The order has to be in the range 0-5.
+    %(mode_interp_constant)s
+    %(cval)s
+    %(prefilter)s
+    grid_mode : bool, optional
+        If False, the distance from the pixel centers is zoomed. Otherwise, the
+        distance including the full pixel extent is used. For example, a 1d
+        signal of length 5 is considered to have length 4 when `grid_mode` is
+        False, but length 5 when `grid_mode` is True. See the following
+        visual illustration:
+
+        .. code-block:: text
+
+                | pixel 1 | pixel 2 | pixel 3 | pixel 4 | pixel 5 |
+                     |<-------------------------------------->|
+                                        vs.
+                |<----------------------------------------------->|
+
+        The starting point of the arrow in the diagram above corresponds to
+        coordinate location 0 in each mode.
+
+    Returns
+    -------
+    zoom : ndarray
+        The zoomed input.
+
+    Notes
+    -----
+    For complex-valued `input`, this function zooms the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+
+    >>> fig = plt.figure()
+    >>> ax1 = fig.add_subplot(121)  # left side
+    >>> ax2 = fig.add_subplot(122)  # right side
+    >>> ascent = datasets.ascent()
+    >>> result = ndimage.zoom(ascent, 3.0)
+    >>> ax1.imshow(ascent, vmin=0, vmax=255)
+    >>> ax2.imshow(result, vmin=0, vmax=255)
+    >>> plt.show()
+
+    >>> print(ascent.shape)
+    (512, 512)
+
+    >>> print(result.shape)
+    (1536, 1536)
+    """
+    if order < 0 or order > 5:
+        raise RuntimeError('spline order not supported')
+    input = np.asarray(input)
+    if input.ndim < 1:
+        raise RuntimeError('input and output rank must be > 0')
+    zoom = _ni_support._normalize_sequence(zoom, input.ndim)
+    output_shape = tuple(
+            [int(round(ii * jj)) for ii, jj in zip(input.shape, zoom)])
+    complex_output = np.iscomplexobj(input)
+    output = _ni_support._get_output(output, input, shape=output_shape,
+                                     complex_output=complex_output)
+    if all(z == 1 for z in zoom) and prefilter:  # early exit for gh-20999
+        # zoom 1 means "return original image". If `prefilter=False`,
+        # `input` is *not* the original image; processing is still needed
+        # to undo the filter. So we only early exit if `prefilter`.
+        output = xpx.at(output)[...].set(input)
+        return output
+    if complex_output:
+        # import under different name to avoid confusion with zoom parameter
+        from scipy.ndimage._interpolation import zoom as _zoom
+
+        kwargs = dict(order=order, mode=mode, prefilter=prefilter)
+        _zoom(input.real, zoom, output=output.real, cval=np.real(cval), **kwargs)
+        _zoom(input.imag, zoom, output=output.imag, cval=np.imag(cval), **kwargs)
+        return output
+    if prefilter and order > 1:
+        padded, npad = _prepad_for_spline_filter(input, mode, cval)
+        filtered = spline_filter(padded, order, output=np.float64, mode=mode)
+    else:
+        npad = 0
+        filtered = input
+    if grid_mode:
+        # warn about modes that may have surprising behavior
+        suggest_mode = None
+        if mode == 'constant':
+            suggest_mode = 'grid-constant'
+        elif mode == 'wrap':
+            suggest_mode = 'grid-wrap'
+        if suggest_mode is not None:
+            warnings.warn(
+                (f"It is recommended to use mode = {suggest_mode} instead of {mode} "
+                 f"when grid_mode is True."),
+                stacklevel=2
+            )
+    mode = _ni_support._extend_mode_to_code(mode)
+
+    zoom_div = np.array(output_shape)
+    zoom_nominator = np.array(input.shape)
+    if not grid_mode:
+        zoom_div -= 1
+        zoom_nominator -= 1
+
+    # Zooming to infinite values is unpredictable, so just choose
+    # zoom factor 1 instead
+    zoom = np.divide(zoom_nominator, zoom_div,
+                     out=np.ones_like(input.shape, dtype=np.float64),
+                     where=zoom_div != 0)
+    zoom = np.ascontiguousarray(zoom)
+    _nd_image.zoom_shift(filtered, zoom, None, output, order, mode, cval, npad,
+                         grid_mode)
+    return output
+
+
+@docfiller
+def rotate(input, angle, axes=(1, 0), reshape=True, output=None, order=3,
+           mode='constant', cval=0.0, prefilter=True):
+    """
+    Rotate an array.
+
+    The array is rotated in the plane defined by the two axes given by the
+    `axes` parameter using spline interpolation of the requested order.
+
+    Parameters
+    ----------
+    %(input)s
+    angle : float
+        The rotation angle in degrees.
+    axes : tuple of 2 ints, optional
+        The two axes that define the plane of rotation. Default is the first
+        two axes.
+    reshape : bool, optional
+        If `reshape` is true, the output shape is adapted so that the input
+        array is contained completely in the output. Default is True.
+    %(output)s
+    order : int, optional
+        The order of the spline interpolation, default is 3.
+        The order has to be in the range 0-5.
+    %(mode_interp_constant)s
+    %(cval)s
+    %(prefilter)s
+
+    Returns
+    -------
+    rotate : ndarray
+        The rotated input.
+
+    Notes
+    -----
+    For complex-valued `input`, this function rotates the real and imaginary
+    components independently.
+
+    .. versionadded:: 1.6.0
+        Complex-valued support added.
+
+    Examples
+    --------
+    >>> from scipy import ndimage, datasets
+    >>> import matplotlib.pyplot as plt
+    >>> fig = plt.figure(figsize=(10, 3))
+    >>> ax1, ax2, ax3 = fig.subplots(1, 3)
+    >>> img = datasets.ascent()
+    >>> img_45 = ndimage.rotate(img, 45, reshape=False)
+    >>> full_img_45 = ndimage.rotate(img, 45, reshape=True)
+    >>> ax1.imshow(img, cmap='gray')
+    >>> ax1.set_axis_off()
+    >>> ax2.imshow(img_45, cmap='gray')
+    >>> ax2.set_axis_off()
+    >>> ax3.imshow(full_img_45, cmap='gray')
+    >>> ax3.set_axis_off()
+    >>> fig.set_layout_engine('tight')
+    >>> plt.show()
+    >>> print(img.shape)
+    (512, 512)
+    >>> print(img_45.shape)
+    (512, 512)
+    >>> print(full_img_45.shape)
+    (724, 724)
+
+    """
+    input_arr = np.asarray(input)
+    ndim = input_arr.ndim
+
+    if ndim < 2:
+        raise ValueError('input array should be at least 2D')
+
+    axes = list(axes)
+
+    if len(axes) != 2:
+        raise ValueError('axes should contain exactly two values')
+
+    if not all([float(ax).is_integer() for ax in axes]):
+        raise ValueError('axes should contain only integer values')
+
+    if axes[0] < 0:
+        axes[0] += ndim
+    if axes[1] < 0:
+        axes[1] += ndim
+    if axes[0] < 0 or axes[1] < 0 or axes[0] >= ndim or axes[1] >= ndim:
+        raise ValueError('invalid rotation plane specified')
+
+    axes.sort()
+
+    c, s = special.cosdg(angle), special.sindg(angle)
+
+    rot_matrix = np.array([[c, s],
+                           [-s, c]])
+
+    img_shape = np.asarray(input_arr.shape)
+    in_plane_shape = img_shape[axes]
+    if reshape:
+        # Compute transformed input bounds
+        iy, ix = in_plane_shape
+        out_bounds = rot_matrix @ [[0, 0, iy, iy],
+                                   [0, ix, 0, ix]]
+        # Compute the shape of the transformed input plane
+        out_plane_shape = (np.ptp(out_bounds, axis=1) + 0.5).astype(int)
+    else:
+        out_plane_shape = img_shape[axes]
+
+    out_center = rot_matrix @ ((out_plane_shape - 1) / 2)
+    in_center = (in_plane_shape - 1) / 2
+    offset = in_center - out_center
+
+    output_shape = img_shape
+    output_shape[axes] = out_plane_shape
+    output_shape = tuple(output_shape)
+
+    complex_output = np.iscomplexobj(input_arr)
+    output = _ni_support._get_output(output, input_arr, shape=output_shape,
+                                     complex_output=complex_output)
+
+    if ndim <= 2:
+        affine_transform(input_arr, rot_matrix, offset, output_shape, output,
+                         order, mode, cval, prefilter)
+    else:
+        # If ndim > 2, the rotation is applied over all the planes
+        # parallel to axes
+        planes_coord = itertools.product(
+            *[[slice(None)] if ax in axes else range(img_shape[ax])
+              for ax in range(ndim)])
+
+        out_plane_shape = tuple(out_plane_shape)
+
+        for coordinates in planes_coord:
+            ia = input_arr[coordinates]
+            oa = output[coordinates]
+            affine_transform(ia, rot_matrix, offset, out_plane_shape,
+                             oa, order, mode, cval, prefilter)
+
+    return output

.venv/lib/python3.12/site-packages/scipy/ndimage/_measurements.py

@@ -0,0 +1,1689 @@
+# Copyright (C) 2003-2005 Peter J. Verveer
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above
+#    copyright notice, this list of conditions and the following
+#    disclaimer in the documentation and/or other materials provided
+#    with the distribution.
+#
+# 3. The name of the author may not be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS
+# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
+# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+import numpy as np
+from . import _ni_support
+from . import _ni_label
+from . import _nd_image
+from . import _morphology
+
+__all__ = ['label', 'find_objects', 'labeled_comprehension', 'sum', 'mean',
+           'variance', 'standard_deviation', 'minimum', 'maximum', 'median',
+           'minimum_position', 'maximum_position', 'extrema', 'center_of_mass',
+           'histogram', 'watershed_ift', 'sum_labels', 'value_indices']
+
+
+def label(input, structure=None, output=None):
+    """
+    Label features in an array.
+
+    Parameters
+    ----------
+    input : array_like
+        An array-like object to be labeled. Any non-zero values in `input` are
+        counted as features and zero values are considered the background.
+    structure : array_like, optional
+        A structuring element that defines feature connections.
+        `structure` must be centrosymmetric
+        (see Notes).
+        If no structuring element is provided,
+        one is automatically generated with a squared connectivity equal to
+        one.  That is, for a 2-D `input` array, the default structuring element
+        is::
+
+            [[0,1,0],
+             [1,1,1],
+             [0,1,0]]
+
+    output : (None, data-type, array_like), optional
+        If `output` is a data type, it specifies the type of the resulting
+        labeled feature array.
+        If `output` is an array-like object, then `output` will be updated
+        with the labeled features from this function.  This function can
+        operate in-place, by passing output=input.
+        Note that the output must be able to store the largest label, or this
+        function will raise an Exception.
+
+    Returns
+    -------
+    label : ndarray or int
+        An integer ndarray where each unique feature in `input` has a unique
+        label in the returned array.
+    num_features : int
+        How many objects were found.
+
+        If `output` is None, this function returns a tuple of
+        (`labeled_array`, `num_features`).
+
+        If `output` is an ndarray, then it will be updated with values in
+        `labeled_array` and only `num_features` will be returned by this
+        function.
+
+    See Also
+    --------
+    find_objects : generate a list of slices for the labeled features (or
+                   objects); useful for finding features' position or
+                   dimensions
+
+    Notes
+    -----
+    A centrosymmetric matrix is a matrix that is symmetric about the center.
+    See [1]_ for more information.
+
+    The `structure` matrix must be centrosymmetric to ensure
+    two-way connections.
+    For instance, if the `structure` matrix is not centrosymmetric
+    and is defined as::
+
+        [[0,1,0],
+         [1,1,0],
+         [0,0,0]]
+
+    and the `input` is::
+
+        [[1,2],
+         [0,3]]
+
+    then the structure matrix would indicate the
+    entry 2 in the input is connected to 1,
+    but 1 is not connected to 2.
+
+    References
+    ----------
+    .. [1] James R. Weaver, "Centrosymmetric (cross-symmetric)
+       matrices, their basic properties, eigenvalues, and
+       eigenvectors." The American Mathematical Monthly 92.10
+       (1985): 711-717.
+
+    Examples
+    --------
+    Create an image with some features, then label it using the default
+    (cross-shaped) structuring element:
+
+    >>> from scipy.ndimage import label, generate_binary_structure
+    >>> import numpy as np
+    >>> a = np.array([[0,0,1,1,0,0],
+    ...               [0,0,0,1,0,0],
+    ...               [1,1,0,0,1,0],
+    ...               [0,0,0,1,0,0]])
+    >>> labeled_array, num_features = label(a)
+
+    Each of the 4 features are labeled with a different integer:
+
+    >>> num_features
+    4
+    >>> labeled_array
+    array([[0, 0, 1, 1, 0, 0],
+           [0, 0, 0, 1, 0, 0],
+           [2, 2, 0, 0, 3, 0],
+           [0, 0, 0, 4, 0, 0]], dtype=int32)
+
+    Generate a structuring element that will consider features connected even
+    if they touch diagonally:
+
+    >>> s = generate_binary_structure(2,2)
+
+    or,
+
+    >>> s = [[1,1,1],
+    ...      [1,1,1],
+    ...      [1,1,1]]
+
+    Label the image using the new structuring element:
+
+    >>> labeled_array, num_features = label(a, structure=s)
+
+    Show the 2 labeled features (note that features 1, 3, and 4 from above are
+    now considered a single feature):
+
+    >>> num_features
+    2
+    >>> labeled_array
+    array([[0, 0, 1, 1, 0, 0],
+           [0, 0, 0, 1, 0, 0],
+           [2, 2, 0, 0, 1, 0],
+           [0, 0, 0, 1, 0, 0]], dtype=int32)
+
+    """
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError('Complex type not supported')
+    if structure is None:
+        structure = _morphology.generate_binary_structure(input.ndim, 1)
+    structure = np.asarray(structure, dtype=bool)
+    if structure.ndim != input.ndim:
+        raise RuntimeError('structure and input must have equal rank')
+    for ii in structure.shape:
+        if ii != 3:
+            raise ValueError('structure dimensions must be equal to 3')
+
+    # Use 32 bits if it's large enough for this image.
+    # _ni_label.label() needs two entries for background and
+    # foreground tracking
+    need_64bits = input.size >= (2**31 - 2)
+
+    if isinstance(output, np.ndarray):
+        if output.shape != input.shape:
+            raise ValueError("output shape not correct")
+        caller_provided_output = True
+    else:
+        caller_provided_output = False
+        if output is None:
+            output = np.empty(input.shape, np.intp if need_64bits else np.int32)
+        else:
+            output = np.empty(input.shape, output)
+
+    # handle scalars, 0-D arrays
+    if input.ndim == 0 or input.size == 0:
+        if input.ndim == 0:
+            # scalar
+            maxlabel = 1 if (input != 0) else 0
+            output[...] = maxlabel
+        else:
+            # 0-D
+            maxlabel = 0
+        if caller_provided_output:
+            return maxlabel
+        else:
+            return output, maxlabel
+
+    try:
+        max_label = _ni_label._label(input, structure, output)
+    except _ni_label.NeedMoreBits as e:
+        # Make another attempt with enough bits, then try to cast to the
+        # new type.
+        tmp_output = np.empty(input.shape, np.intp if need_64bits else np.int32)
+        max_label = _ni_label._label(input, structure, tmp_output)
+        output[...] = tmp_output[...]
+        if not np.all(output == tmp_output):
+            # refuse to return bad results
+            raise RuntimeError(
+                "insufficient bit-depth in requested output type"
+            ) from e
+
+    if caller_provided_output:
+        # result was written in-place
+        return max_label
+    else:
+        return output, max_label
+
+
+def find_objects(input, max_label=0):
+    """
+    Find objects in a labeled array.
+
+    Parameters
+    ----------
+    input : ndarray of ints
+        Array containing objects defined by different labels. Labels with
+        value 0 are ignored.
+    max_label : int, optional
+        Maximum label to be searched for in `input`. If max_label is not
+        given, the positions of all objects are returned.
+
+    Returns
+    -------
+    object_slices : list of tuples
+        A list of tuples, with each tuple containing N slices (with N the
+        dimension of the input array). Slices correspond to the minimal
+        parallelepiped that contains the object. If a number is missing,
+        None is returned instead of a slice. The label ``l`` corresponds to
+        the index ``l-1`` in the returned list.
+
+    See Also
+    --------
+    label, center_of_mass
+
+    Notes
+    -----
+    This function is very useful for isolating a volume of interest inside
+    a 3-D array, that cannot be "seen through".
+
+    Examples
+    --------
+    >>> from scipy import ndimage
+    >>> import numpy as np
+    >>> a = np.zeros((6,6), dtype=int)
+    >>> a[2:4, 2:4] = 1
+    >>> a[4, 4] = 1
+    >>> a[:2, :3] = 2
+    >>> a[0, 5] = 3
+    >>> a
+    array([[2, 2, 2, 0, 0, 3],
+           [2, 2, 2, 0, 0, 0],
+           [0, 0, 1, 1, 0, 0],
+           [0, 0, 1, 1, 0, 0],
+           [0, 0, 0, 0, 1, 0],
+           [0, 0, 0, 0, 0, 0]])
+    >>> ndimage.find_objects(a)
+    [(slice(2, 5, None), slice(2, 5, None)),
+     (slice(0, 2, None), slice(0, 3, None)),
+     (slice(0, 1, None), slice(5, 6, None))]
+    >>> ndimage.find_objects(a, max_label=2)
+    [(slice(2, 5, None), slice(2, 5, None)), (slice(0, 2, None), slice(0, 3, None))]
+    >>> ndimage.find_objects(a == 1, max_label=2)
+    [(slice(2, 5, None), slice(2, 5, None)), None]
+
+    >>> loc = ndimage.find_objects(a)[0]
+    >>> a[loc]
+    array([[1, 1, 0],
+           [1, 1, 0],
+           [0, 0, 1]])
+
+    """
+    input = np.asarray(input)
+    if np.iscomplexobj(input):
+        raise TypeError('Complex type not supported')
+
+    if max_label < 1:
+        max_label = input.max()
+
+    return _nd_image.find_objects(input, max_label)
+
+
+def value_indices(arr, *, ignore_value=None):
+    """
+    Find indices of each distinct value in given array.
+
+    Parameters
+    ----------
+    arr : ndarray of ints
+        Array containing integer values.
+    ignore_value : int, optional
+        This value will be ignored in searching the `arr` array. If not
+        given, all values found will be included in output. Default
+        is None.
+
+    Returns
+    -------
+    indices : dictionary
+        A Python dictionary of array indices for each distinct value. The
+        dictionary is keyed by the distinct values, the entries are array
+        index tuples covering all occurrences of the value within the
+        array.
+
+        This dictionary can occupy significant memory, usually several times
+        the size of the input array.
+
+    See Also
+    --------
+    label, maximum, median, minimum_position, extrema, sum, mean, variance,
+    standard_deviation, numpy.where, numpy.unique
+
+    Notes
+    -----
+    For a small array with few distinct values, one might use
+    `numpy.unique()` to find all possible values, and ``(arr == val)`` to
+    locate each value within that array. However, for large arrays,
+    with many distinct values, this can become extremely inefficient,
+    as locating each value would require a new search through the entire
+    array. Using this function, there is essentially one search, with
+    the indices saved for all distinct values.
+
+    This is useful when matching a categorical image (e.g. a segmentation
+    or classification) to an associated image of other data, allowing
+    any per-class statistic(s) to then be calculated. Provides a
+    more flexible alternative to functions like ``scipy.ndimage.mean()``
+    and ``scipy.ndimage.variance()``.
+
+    Some other closely related functionality, with different strengths and
+    weaknesses, can also be found in ``scipy.stats.binned_statistic()`` and
+    the `scikit-image <https://scikit-image.org/>`_ function
+    ``skimage.measure.regionprops()``.
+
+    Note for IDL users: this provides functionality equivalent to IDL's
+    REVERSE_INDICES option (as per the IDL documentation for the
+    `HISTOGRAM <https://www.l3harrisgeospatial.com/docs/histogram.html>`_
+    function).
+
+    .. versionadded:: 1.10.0
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from scipy import ndimage
+    >>> a = np.zeros((6, 6), dtype=int)
+    >>> a[2:4, 2:4] = 1
+    >>> a[4, 4] = 1
+    >>> a[:2, :3] = 2
+    >>> a[0, 5] = 3
+    >>> a
+    array([[2, 2, 2, 0, 0, 3],
+           [2, 2, 2, 0, 0, 0],
+           [0, 0, 1, 1, 0, 0],
+           [0, 0, 1, 1, 0, 0],
+           [0, 0, 0, 0, 1, 0],
+           [0, 0, 0, 0, 0, 0]])
+    >>> val_indices = ndimage.value_indices(a)
+
+    The dictionary `val_indices` will have an entry for each distinct
+    value in the input array.
+
+    >>> val_indices.keys()
+    dict_keys([np.int64(0), np.int64(1), np.int64(2), np.int64(3)])
+
+    The entry for each value is an index tuple, locating the elements
+    with that value.
+
+    >>> ndx1 = val_indices[1]
+    >>> ndx1
+    (array([2, 2, 3, 3, 4]), array([2, 3, 2, 3, 4]))
+
+    This can be used to index into the original array, or any other
+    array with the same shape.
+
+    >>> a[ndx1]
+    array([1, 1, 1, 1, 1])
+
+    If the zeros were to be ignored, then the resulting dictionary
+    would no longer have an entry for zero.
+
+    >>> val_indices = ndimage.value_indices(a, ignore_value=0)
+    >>> val_indices.keys()
+    dict_keys([np.int64(1), np.int64(2), np.int64(3)])
+
+    """
+    # Cope with ignore_value being None, without too much extra complexity
+    # in the C code. If not None, the value is passed in as a numpy array
+    # with the same dtype as arr.
+    arr = np.asarray(arr)
+    ignore_value_arr = np.zeros((1,), dtype=arr.dtype)
+    ignoreIsNone = (ignore_value is None)
+    if not ignoreIsNone:
+        ignore_value_arr[0] = ignore_value_arr.dtype.type(ignore_value)
+
+    val_indices = _nd_image.value_indices(arr, ignoreIsNone, ignore_value_arr)
+    return val_indices
+
+
+def labeled_comprehension(input, labels, index, func, out_dtype, default,
+                          pass_positions=False):
+    """
+    Roughly equivalent to [func(input[labels == i]) for i in index].
+
+    Sequentially applies an arbitrary function (that works on array_like input)
+    to subsets of an N-D image array specified by `labels` and `index`.
+    The option exists to provide the function with positional parameters as the
+    second argument.
+
+    Parameters
+    ----------
+    input : array_like
+        Data from which to select `labels` to process.
+    labels : array_like or None
+        Labels to objects in `input`.
+        If not None, array must be same shape as `input`.
+        If None, `func` is applied to raveled `input`.
+    index : int, sequence of ints or None
+        Subset of `labels` to which to apply `func`.
+        If a scalar, a single value is returned.
+        If None, `func` is applied to all non-zero values of `labels`.
+    func : callable
+        Python function to apply to `labels` from `input`.
+    out_dtype : dtype
+        Dtype to use for `result`.
+    default : int, float or None
+        Default return value when an element of `index` does not exist
+        in `labels`.
+    pass_positions : bool, optional
+        If True, pass linear indices to `func` as a second argument.
+        Default is False.
+
+    Returns
+    -------
+    result : ndarray
+        Result of applying `func` to each of `labels` to `input` in `index`.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> from scipy import ndimage
+    >>> lbl, nlbl = ndimage.label(a)
+    >>> lbls = np.arange(1, nlbl+1)
+    >>> ndimage.labeled_comprehension(a, lbl, lbls, np.mean, float, 0)
+    array([ 2.75,  5.5 ,  6.  ])
+
+    Falling back to `default`:
+
+    >>> lbls = np.arange(1, nlbl+2)
+    >>> ndimage.labeled_comprehension(a, lbl, lbls, np.mean, float, -1)
+    array([ 2.75,  5.5 ,  6.  , -1.  ])
+
+    Passing positions:
+
+    >>> def fn(val, pos):
+    ...     print("fn says: %s : %s" % (val, pos))
+    ...     return (val.sum()) if (pos.sum() % 2 == 0) else (-val.sum())
+    ...
+    >>> ndimage.labeled_comprehension(a, lbl, lbls, fn, float, 0, True)
+    fn says: [1 2 5 3] : [0 1 4 5]
+    fn says: [4 7] : [ 7 11]
+    fn says: [9 3] : [12 13]
+    array([ 11.,  11., -12.,   0.])
+
+    """
+
+    as_scalar = np.isscalar(index)
+    input = np.asarray(input)
+
+    if pass_positions:
+        positions = np.arange(input.size).reshape(input.shape)
+
+    if labels is None:
+        if index is not None:
+            raise ValueError("index without defined labels")
+        if not pass_positions:
+            return func(input.ravel())
+        else:
+            return func(input.ravel(), positions.ravel())
+
+    labels = np.asarray(labels)
+
+    try:
+        input, labels = np.broadcast_arrays(input, labels)
+    except ValueError as e:
+        raise ValueError("input and labels must have the same shape "
+                            "(excepting dimensions with width 1)") from e
+
+    if index is None:
+        if not pass_positions:
+            return func(input[labels > 0])
+        else:
+            return func(input[labels > 0], positions[labels > 0])
+
+    index = np.atleast_1d(index)
+    if np.any(index.astype(labels.dtype).astype(index.dtype) != index):
+        raise ValueError(f"Cannot convert index values from <{index.dtype}> to "
+                         f"<{labels.dtype}> (labels' type) without loss of precision")
+
+    index = index.astype(labels.dtype)
+
+    # optimization: find min/max in index,
+    # and select those parts of labels, input, and positions
+    lo = index.min()
+    hi = index.max()
+    mask = (labels >= lo) & (labels <= hi)
+
+    # this also ravels the arrays
+    labels = labels[mask]
+    input = input[mask]
+    if pass_positions:
+        positions = positions[mask]
+
+    # sort everything by labels
+    label_order = labels.argsort()
+    labels = labels[label_order]
+    input = input[label_order]
+    if pass_positions:
+        positions = positions[label_order]
+
+    index_order = index.argsort()
+    sorted_index = index[index_order]
+
+    def do_map(inputs, output):
+        """labels must be sorted"""
+        nidx = sorted_index.size
+
+        # Find boundaries for each stretch of constant labels
+        # This could be faster, but we already paid N log N to sort labels.
+        lo = np.searchsorted(labels, sorted_index, side='left')
+        hi = np.searchsorted(labels, sorted_index, side='right')
+
+        for i, l, h in zip(range(nidx), lo, hi):
+            if l == h:
+                continue
+            output[i] = func(*[inp[l:h] for inp in inputs])
+
+    temp = np.empty(index.shape, out_dtype)
+    temp[:] = default
+    if not pass_positions:
+        do_map([input], temp)
+    else:
+        do_map([input, positions], temp)
+
+    output = np.zeros(index.shape, out_dtype)
+    output[index_order] = temp
+    if as_scalar:
+        output = output[0]
+
+    return output
+
+
+def _safely_castable_to_int(dt):
+    """Test whether the NumPy data type `dt` can be safely cast to an int."""
+    int_size = np.dtype(int).itemsize
+    safe = ((np.issubdtype(dt, np.signedinteger) and dt.itemsize <= int_size) or
+            (np.issubdtype(dt, np.unsignedinteger) and dt.itemsize < int_size))
+    return safe
+
+
+def _stats(input, labels=None, index=None, centered=False):
+    """Count, sum, and optionally compute (sum - centre)^2 of input by label
+
+    Parameters
+    ----------
+    input : array_like, N-D
+        The input data to be analyzed.
+    labels : array_like (N-D), optional
+        The labels of the data in `input`. This array must be broadcast
+        compatible with `input`; typically, it is the same shape as `input`.
+        If `labels` is None, all nonzero values in `input` are treated as
+        the single labeled group.
+    index : label or sequence of labels, optional
+        These are the labels of the groups for which the stats are computed.
+        If `index` is None, the stats are computed for the single group where
+        `labels` is greater than 0.
+    centered : bool, optional
+        If True, the centered sum of squares for each labeled group is
+        also returned. Default is False.
+
+    Returns
+    -------
+    counts : int or ndarray of ints
+        The number of elements in each labeled group.
+    sums : scalar or ndarray of scalars
+        The sums of the values in each labeled group.
+    sums_c : scalar or ndarray of scalars, optional
+        The sums of mean-centered squares of the values in each labeled group.
+        This is only returned if `centered` is True.
+
+    """
+    def single_group(vals):
+        if centered:
+            vals_c = vals - vals.mean()
+            return vals.size, vals.sum(), (vals_c * vals_c.conjugate()).sum()
+        else:
+            return vals.size, vals.sum()
+
+    input = np.asarray(input)
+    if labels is None:
+        return single_group(input)
+
+    labels = np.asarray(labels)
+    # ensure input and labels match sizes
+    input, labels = np.broadcast_arrays(input, labels)
+
+    if index is None:
+        return single_group(input[labels > 0])
+
+    if np.isscalar(index):
+        return single_group(input[labels == index])
+
+    def _sum_centered(labels):
+        # `labels` is expected to be an ndarray with the same shape as `input`.
+        # It must contain the label indices (which are not necessarily the labels
+        # themselves).
+        means = sums / counts
+        centered_input = input - means[labels]
+        # bincount expects 1-D inputs, so we ravel the arguments.
+        bc = np.bincount(labels.ravel(),
+                              weights=(centered_input *
+                                       centered_input.conjugate()).ravel())
+        return bc
+
+    # Remap labels to unique integers if necessary, or if the largest
+    # label is larger than the number of values.
+
+    if (not _safely_castable_to_int(labels.dtype) or
+            labels.min() < 0 or labels.max() > labels.size):
+        # Use np.unique to generate the label indices.  `new_labels` will
+        # be 1-D, but it should be interpreted as the flattened N-D array of
+        # label indices.
+        unique_labels, new_labels = np.unique(labels, return_inverse=True)
+        new_labels = np.reshape(new_labels, (-1,))  # flatten, since it may be >1-D
+        counts = np.bincount(new_labels)
+        sums = np.bincount(new_labels, weights=input.ravel())
+        if centered:
+            # Compute the sum of the mean-centered squares.
+            # We must reshape new_labels to the N-D shape of `input` before
+            # passing it _sum_centered.
+            sums_c = _sum_centered(new_labels.reshape(labels.shape))
+        idxs = np.searchsorted(unique_labels, index)
+        # make all of idxs valid
+        idxs[idxs >= unique_labels.size] = 0
+        found = (unique_labels[idxs] == index)
+    else:
+        # labels are an integer type allowed by bincount, and there aren't too
+        # many, so call bincount directly.
+        counts = np.bincount(labels.ravel())
+        sums = np.bincount(labels.ravel(), weights=input.ravel())
+        if centered:
+            sums_c = _sum_centered(labels)
+        # make sure all index values are valid
+        idxs = np.asanyarray(index, np.int_).copy()
+        found = (idxs >= 0) & (idxs < counts.size)
+        idxs[~found] = 0
+
+    counts = counts[idxs]
+    counts[~found] = 0
+    sums = sums[idxs]
+    sums[~found] = 0
+
+    if not centered:
+        return (counts, sums)
+    else:
+        sums_c = sums_c[idxs]
+        sums_c[~found] = 0
+        return (counts, sums, sums_c)
+
+
+def sum(input, labels=None, index=None):
+    """
+    Calculate the sum of the values of the array.
+
+    Notes
+    -----
+    This is an alias for `ndimage.sum_labels` kept for backwards compatibility
+    reasons, for new code please prefer `sum_labels`.  See the `sum_labels`
+    docstring for more details.
+
+    """  # numpydoc ignore=RT01
+    return sum_labels(input, labels, index)
+
+
+def sum_labels(input, labels=None, index=None):
+    """
+    Calculate the sum of the values of the array.
+
+    Parameters
+    ----------
+    input : array_like
+        Values of `input` inside the regions defined by `labels`
+        are summed together.
+    labels : array_like of ints, optional
+        Assign labels to the values of the array. Has to have the same shape as
+        `input`.
+    index : array_like, optional
+        A single label number or a sequence of label numbers of
+        the objects to be measured.
+
+    Returns
+    -------
+    sum : ndarray or scalar
+        An array of the sums of values of `input` inside the regions defined
+        by `labels` with the same shape as `index`. If 'index' is None or scalar,
+        a scalar is returned.
+
+    See Also
+    --------
+    mean, median
+
+    Examples
+    --------
+    >>> from scipy import ndimage
+    >>> input =  [0,1,2,3]
+    >>> labels = [1,1,2,2]
+    >>> ndimage.sum_labels(input, labels, index=[1,2])
+    [1.0, 5.0]
+    >>> ndimage.sum_labels(input, labels, index=1)
+    1
+    >>> ndimage.sum_labels(input, labels)
+    6
+
+
+    """
+    count, sum = _stats(input, labels, index)
+    return sum
+
+
+def mean(input, labels=None, index=None):
+    """
+    Calculate the mean of the values of an array at labels.
+
+    Parameters
+    ----------
+    input : array_like
+        Array on which to compute the mean of elements over distinct
+        regions.
+    labels : array_like, optional
+        Array of labels of same shape, or broadcastable to the same shape as
+        `input`. All elements sharing the same label form one region over
+        which the mean of the elements is computed.
+    index : int or sequence of ints, optional
+        Labels of the objects over which the mean is to be computed.
+        Default is None, in which case the mean for all values where label is
+        greater than 0 is calculated.
+
+    Returns
+    -------
+    out : list
+        Sequence of same length as `index`, with the mean of the different
+        regions labeled by the labels in `index`.
+
+    See Also
+    --------
+    variance, standard_deviation, minimum, maximum, sum, label
+
+    Examples
+    --------
+    >>> from scipy import ndimage
+    >>> import numpy as np
+    >>> a = np.arange(25).reshape((5,5))
+    >>> labels = np.zeros_like(a)
+    >>> labels[3:5,3:5] = 1
+    >>> index = np.unique(labels)
+    >>> labels
+    array([[0, 0, 0, 0, 0],
+           [0, 0, 0, 0, 0],
+           [0, 0, 0, 0, 0],
+           [0, 0, 0, 1, 1],
+           [0, 0, 0, 1, 1]])
+    >>> index
+    array([0, 1])
+    >>> ndimage.mean(a, labels=labels, index=index)
+    [10.285714285714286, 21.0]
+
+    """
+
+    count, sum = _stats(input, labels, index)
+    return sum / np.asanyarray(count).astype(np.float64)
+
+
+def variance(input, labels=None, index=None):
+    """
+    Calculate the variance of the values of an N-D image array, optionally at
+    specified sub-regions.
+
+    Parameters
+    ----------
+    input : array_like
+        Nd-image data to process.
+    labels : array_like, optional
+        Labels defining sub-regions in `input`.
+        If not None, must be same shape as `input`.
+    index : int or sequence of ints, optional
+        `labels` to include in output.  If None (default), all values where
+        `labels` is non-zero are used.
+
+    Returns
+    -------
+    variance : float or ndarray
+        Values of variance, for each sub-region if `labels` and `index` are
+        specified.
+
+    See Also
+    --------
+    label, standard_deviation, maximum, minimum, extrema
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> from scipy import ndimage
+    >>> ndimage.variance(a)
+    7.609375
+
+    Features to process can be specified using `labels` and `index`:
+
+    >>> lbl, nlbl = ndimage.label(a)
+    >>> ndimage.variance(a, lbl, index=np.arange(1, nlbl+1))
+    array([ 2.1875,  2.25  ,  9.    ])
+
+    If no index is given, all non-zero `labels` are processed:
+
+    >>> ndimage.variance(a, lbl)
+    6.1875
+
+    """
+    count, sum, sum_c_sq = _stats(input, labels, index, centered=True)
+    return sum_c_sq / np.asanyarray(count).astype(float)
+
+
+def standard_deviation(input, labels=None, index=None):
+    """
+    Calculate the standard deviation of the values of an N-D image array,
+    optionally at specified sub-regions.
+
+    Parameters
+    ----------
+    input : array_like
+        N-D image data to process.
+    labels : array_like, optional
+        Labels to identify sub-regions in `input`.
+        If not None, must be same shape as `input`.
+    index : int or sequence of ints, optional
+        `labels` to include in output. If None (default), all values where
+        `labels` is non-zero are used.
+
+    Returns
+    -------
+    standard_deviation : float or ndarray
+        Values of standard deviation, for each sub-region if `labels` and
+        `index` are specified.
+
+    See Also
+    --------
+    label, variance, maximum, minimum, extrema
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> from scipy import ndimage
+    >>> ndimage.standard_deviation(a)
+    2.7585095613392387
+
+    Features to process can be specified using `labels` and `index`:
+
+    >>> lbl, nlbl = ndimage.label(a)
+    >>> ndimage.standard_deviation(a, lbl, index=np.arange(1, nlbl+1))
+    array([ 1.479,  1.5  ,  3.   ])
+
+    If no index is given, non-zero `labels` are processed:
+
+    >>> ndimage.standard_deviation(a, lbl)
+    2.4874685927665499
+
+    """
+    return np.sqrt(variance(input, labels, index))
+
+
+def _select(input, labels=None, index=None, find_min=False, find_max=False,
+            find_min_positions=False, find_max_positions=False,
+            find_median=False):
+    """Returns min, max, or both, plus their positions (if requested), and
+    median."""
+
+    input = np.asanyarray(input)
+
+    find_positions = find_min_positions or find_max_positions
+    positions = None
+    if find_positions:
+        positions = np.arange(input.size).reshape(input.shape)
+
+    def single_group(vals, positions):
+        result = []
+        if find_min:
+            result += [vals.min()]
+        if find_min_positions:
+            result += [positions[vals == vals.min()][0]]
+        if find_max:
+            result += [vals.max()]
+        if find_max_positions:
+            result += [positions[vals == vals.max()][0]]
+        if find_median:
+            result += [np.median(vals)]
+        return result
+
+    if labels is None:
+        return single_group(input, positions)
+
+    labels = np.asarray(labels)
+    # ensure input and labels match sizes
+    input, labels = np.broadcast_arrays(input, labels)
+
+    if index is None:
+        mask = (labels > 0)
+        masked_positions = None
+        if find_positions:
+            masked_positions = positions[mask]
+        return single_group(input[mask], masked_positions)
+
+    if np.isscalar(index):
+        mask = (labels == index)
+        masked_positions = None
+        if find_positions:
+            masked_positions = positions[mask]
+        return single_group(input[mask], masked_positions)
+
+    index = np.asarray(index)
+
+    # remap labels to unique integers if necessary, or if the largest
+    # label is larger than the number of values.
+    if (not _safely_castable_to_int(labels.dtype) or
+            labels.min() < 0 or labels.max() > labels.size):
+        # remap labels, and indexes
+        unique_labels, labels = np.unique(labels, return_inverse=True)
+        idxs = np.searchsorted(unique_labels, index)
+
+        # make all of idxs valid
+        idxs[idxs >= unique_labels.size] = 0
+        found = (unique_labels[idxs] == index)
+    else:
+        # labels are an integer type, and there aren't too many
+        idxs = np.asanyarray(index, np.int_).copy()
+        found = (idxs >= 0) & (idxs <= labels.max())
+
+    idxs[~ found] = labels.max() + 1
+
+    if find_median:
+        order = np.lexsort((input.ravel(), labels.ravel()))
+    else:
+        order = input.ravel().argsort()
+    input = input.ravel()[order]
+    labels = labels.ravel()[order]
+    if find_positions:
+        positions = positions.ravel()[order]
+
+    result = []
+    if find_min:
+        mins = np.zeros(labels.max() + 2, input.dtype)
+        mins[labels[::-1]] = input[::-1]
+        result += [mins[idxs]]
+    if find_min_positions:
+        minpos = np.zeros(labels.max() + 2, int)
+        minpos[labels[::-1]] = positions[::-1]
+        result += [minpos[idxs]]
+    if find_max:
+        maxs = np.zeros(labels.max() + 2, input.dtype)
+        maxs[labels] = input
+        result += [maxs[idxs]]
+    if find_max_positions:
+        maxpos = np.zeros(labels.max() + 2, int)
+        maxpos[labels] = positions
+        result += [maxpos[idxs]]
+    if find_median:
+        locs = np.arange(len(labels))
+        lo = np.zeros(labels.max() + 2, np.int_)
+        lo[labels[::-1]] = locs[::-1]
+        hi = np.zeros(labels.max() + 2, np.int_)
+        hi[labels] = locs
+        lo = lo[idxs]
+        hi = hi[idxs]
+        # lo is an index to the lowest value in input for each label,
+        # hi is an index to the largest value.
+        # move them to be either the same ((hi - lo) % 2 == 0) or next
+        # to each other ((hi - lo) % 2 == 1), then average.
+        step = (hi - lo) // 2
+        lo += step
+        hi -= step
+        if (np.issubdtype(input.dtype, np.integer)
+                or np.issubdtype(input.dtype, np.bool_)):
+            # avoid integer overflow or boolean addition (gh-12836)
+            result += [(input[lo].astype('d') + input[hi].astype('d')) / 2.0]
+        else:
+            result += [(input[lo] + input[hi]) / 2.0]
+
+    return result
+
+
+def minimum(input, labels=None, index=None):
+    """
+    Calculate the minimum of the values of an array over labeled regions.
+
+    Parameters
+    ----------
+    input : array_like
+        Array_like of values. For each region specified by `labels`, the
+        minimal values of `input` over the region is computed.
+    labels : array_like, optional
+        An array_like of integers marking different regions over which the
+        minimum value of `input` is to be computed. `labels` must have the
+        same shape as `input`. If `labels` is not specified, the minimum
+        over the whole array is returned.
+    index : array_like, optional
+        A list of region labels that are taken into account for computing the
+        minima. If index is None, the minimum over all elements where `labels`
+        is non-zero is returned.
+
+    Returns
+    -------
+    output : a scalar or list of integers or floats based on input type.
+        List of minima of `input` over the regions determined by `labels` and
+        whose index is in `index`. If `index` or `labels` are not specified, a
+        float is returned: the minimal value of `input` if `labels` is None,
+        and the minimal value of elements where `labels` is greater than zero
+        if `index` is None.
+
+    See Also
+    --------
+    label, maximum, median, minimum_position, extrema, sum, mean, variance,
+    standard_deviation
+
+    Notes
+    -----
+    The function returns a Python list and not a NumPy array, use
+    `np.array` to convert the list to an array.
+
+    Examples
+    --------
+    >>> from scipy import ndimage
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> labels, labels_nb = ndimage.label(a)
+    >>> labels
+    array([[1, 1, 0, 0],
+           [1, 1, 0, 2],
+           [0, 0, 0, 2],
+           [3, 3, 0, 0]], dtype=int32)
+    >>> ndimage.minimum(a, labels=labels, index=np.arange(1, labels_nb + 1))
+    [1, 4, 3]
+    >>> ndimage.minimum(a)
+    0
+    >>> ndimage.minimum(a, labels=labels)
+    1
+
+    """
+    return _select(input, labels, index, find_min=True)[0]
+
+
+def maximum(input, labels=None, index=None):
+    """
+    Calculate the maximum of the values of an array over labeled regions.
+
+    Parameters
+    ----------
+    input : array_like
+        Array_like of values. For each region specified by `labels`, the
+        maximal values of `input` over the region is computed.
+    labels : array_like, optional
+        An array of integers marking different regions over which the
+        maximum value of `input` is to be computed. `labels` must have the
+        same shape as `input`. If `labels` is not specified, the maximum
+        over the whole array is returned.
+    index : array_like, optional
+        A list of region labels that are taken into account for computing the
+        maxima. If index is None, the maximum over all elements where `labels`
+        is non-zero is returned.
+
+    Returns
+    -------
+    output : a scalar or list of integers or floats based on input type.
+        List of maxima of `input` over the regions determined by `labels` and
+        whose index is in `index`. If `index` or `labels` are not specified, a
+        float is returned: the maximal value of `input` if `labels` is None,
+        and the maximal value of elements where `labels` is greater than zero
+        if `index` is None.
+
+    See Also
+    --------
+    label, minimum, median, maximum_position, extrema, sum, mean, variance,
+    standard_deviation
+
+    Notes
+    -----
+    The function returns a Python list and not a NumPy array, use
+    `np.array` to convert the list to an array.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.arange(16).reshape((4,4))
+    >>> a
+    array([[ 0,  1,  2,  3],
+           [ 4,  5,  6,  7],
+           [ 8,  9, 10, 11],
+           [12, 13, 14, 15]])
+    >>> labels = np.zeros_like(a)
+    >>> labels[:2,:2] = 1
+    >>> labels[2:, 1:3] = 2
+    >>> labels
+    array([[1, 1, 0, 0],
+           [1, 1, 0, 0],
+           [0, 2, 2, 0],
+           [0, 2, 2, 0]])
+    >>> from scipy import ndimage
+    >>> ndimage.maximum(a)
+    15
+    >>> ndimage.maximum(a, labels=labels, index=[1,2])
+    [5, 14]
+    >>> ndimage.maximum(a, labels=labels)
+    14
+
+    >>> b = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> labels, labels_nb = ndimage.label(b)
+    >>> labels
+    array([[1, 1, 0, 0],
+           [1, 1, 0, 2],
+           [0, 0, 0, 2],
+           [3, 3, 0, 0]], dtype=int32)
+    >>> ndimage.maximum(b, labels=labels, index=np.arange(1, labels_nb + 1))
+    [5, 7, 9]
+
+    """
+    return _select(input, labels, index, find_max=True)[0]
+
+
+def median(input, labels=None, index=None):
+    """
+    Calculate the median of the values of an array over labeled regions.
+
+    Parameters
+    ----------
+    input : array_like
+        Array_like of values. For each region specified by `labels`, the
+        median value of `input` over the region is computed.
+    labels : array_like, optional
+        An array_like of integers marking different regions over which the
+        median value of `input` is to be computed. `labels` must have the
+        same shape as `input`. If `labels` is not specified, the median
+        over the whole array is returned.
+    index : array_like, optional
+        A list of region labels that are taken into account for computing the
+        medians. If index is None, the median over all elements where `labels`
+        is non-zero is returned.
+
+    Returns
+    -------
+    median : float or list of floats
+        List of medians of `input` over the regions determined by `labels` and
+        whose index is in `index`. If `index` or `labels` are not specified, a
+        float is returned: the median value of `input` if `labels` is None,
+        and the median value of elements where `labels` is greater than zero
+        if `index` is None.
+
+    See Also
+    --------
+    label, minimum, maximum, extrema, sum, mean, variance, standard_deviation
+
+    Notes
+    -----
+    The function returns a Python list and not a NumPy array, use
+    `np.array` to convert the list to an array.
+
+    Examples
+    --------
+    >>> from scipy import ndimage
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 1],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> labels, labels_nb = ndimage.label(a)
+    >>> labels
+    array([[1, 1, 0, 2],
+           [1, 1, 0, 2],
+           [0, 0, 0, 2],
+           [3, 3, 0, 0]], dtype=int32)
+    >>> ndimage.median(a, labels=labels, index=np.arange(1, labels_nb + 1))
+    [2.5, 4.0, 6.0]
+    >>> ndimage.median(a)
+    1.0
+    >>> ndimage.median(a, labels=labels)
+    3.0
+
+    """
+    return _select(input, labels, index, find_median=True)[0]
+
+
+def minimum_position(input, labels=None, index=None):
+    """
+    Find the positions of the minimums of the values of an array at labels.
+
+    Parameters
+    ----------
+    input : array_like
+        Array_like of values.
+    labels : array_like, optional
+        An array of integers marking different regions over which the
+        position of the minimum value of `input` is to be computed.
+        `labels` must have the same shape as `input`. If `labels` is not
+        specified, the location of the first minimum over the whole
+        array is returned.
+
+        The `labels` argument only works when `index` is specified.
+    index : array_like, optional
+        A list of region labels that are taken into account for finding the
+        location of the minima. If `index` is None, the ``first`` minimum
+        over all elements where `labels` is non-zero is returned.
+
+        The `index` argument only works when `labels` is specified.
+
+    Returns
+    -------
+    output : list of tuples of ints
+        Tuple of ints or list of tuples of ints that specify the location
+        of minima of `input` over the regions determined by `labels` and
+        whose index is in `index`.
+
+        If `index` or `labels` are not specified, a tuple of ints is
+        returned specifying the location of the first minimal value of `input`.
+
+    See Also
+    --------
+    label, minimum, median, maximum_position, extrema, sum, mean, variance,
+    standard_deviation
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.array([[10, 20, 30],
+    ...               [40, 80, 100],
+    ...               [1, 100, 200]])
+    >>> b = np.array([[1, 2, 0, 1],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+
+    >>> from scipy import ndimage
+
+    >>> ndimage.minimum_position(a)
+    (2, 0)
+    >>> ndimage.minimum_position(b)
+    (0, 2)
+
+    Features to process can be specified using `labels` and `index`:
+
+    >>> label, pos = ndimage.label(a)
+    >>> ndimage.minimum_position(a, label, index=np.arange(1, pos+1))
+    [(2, 0)]
+
+    >>> label, pos = ndimage.label(b)
+    >>> ndimage.minimum_position(b, label, index=np.arange(1, pos+1))
+    [(0, 0), (0, 3), (3, 1)]
+
+    """
+    dims = np.array(np.asarray(input).shape)
+    # see np.unravel_index to understand this line.
+    dim_prod = np.cumprod([1] + list(dims[:0:-1]))[::-1]
+
+    result = _select(input, labels, index, find_min_positions=True)[0]
+
+    if np.isscalar(result):
+        return tuple((result // dim_prod) % dims)
+
+    return [tuple(v) for v in (result.reshape(-1, 1) // dim_prod) % dims]
+
+
+def maximum_position(input, labels=None, index=None):
+    """
+    Find the positions of the maximums of the values of an array at labels.
+
+    For each region specified by `labels`, the position of the maximum
+    value of `input` within the region is returned.
+
+    Parameters
+    ----------
+    input : array_like
+        Array_like of values.
+    labels : array_like, optional
+        An array of integers marking different regions over which the
+        position of the maximum value of `input` is to be computed.
+        `labels` must have the same shape as `input`. If `labels` is not
+        specified, the location of the first maximum over the whole
+        array is returned.
+
+        The `labels` argument only works when `index` is specified.
+    index : array_like, optional
+        A list of region labels that are taken into account for finding the
+        location of the maxima. If `index` is None, the first maximum
+        over all elements where `labels` is non-zero is returned.
+
+        The `index` argument only works when `labels` is specified.
+
+    Returns
+    -------
+    output : list of tuples of ints
+        List of tuples of ints that specify the location of maxima of
+        `input` over the regions determined by `labels` and whose index
+        is in `index`.
+
+        If `index` or `labels` are not specified, a tuple of ints is
+        returned specifying the location of the ``first`` maximal value
+        of `input`.
+
+    See Also
+    --------
+    label, minimum, median, maximum_position, extrema, sum, mean, variance,
+    standard_deviation
+
+    Examples
+    --------
+    >>> from scipy import ndimage
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> ndimage.maximum_position(a)
+    (3, 0)
+
+    Features to process can be specified using `labels` and `index`:
+
+    >>> lbl = np.array([[0, 1, 2, 3],
+    ...                 [0, 1, 2, 3],
+    ...                 [0, 1, 2, 3],
+    ...                 [0, 1, 2, 3]])
+    >>> ndimage.maximum_position(a, lbl, 1)
+    (1, 1)
+
+    If no index is given, non-zero `labels` are processed:
+
+    >>> ndimage.maximum_position(a, lbl)
+    (2, 3)
+
+    If there are no maxima, the position of the first element is returned:
+
+    >>> ndimage.maximum_position(a, lbl, 2)
+    (0, 2)
+
+    """
+    dims = np.array(np.asarray(input).shape)
+    # see np.unravel_index to understand this line.
+    dim_prod = np.cumprod([1] + list(dims[:0:-1]))[::-1]
+
+    result = _select(input, labels, index, find_max_positions=True)[0]
+
+    if np.isscalar(result):
+        return tuple((result // dim_prod) % dims)
+
+    return [tuple(v) for v in (result.reshape(-1, 1) // dim_prod) % dims]
+
+
+def extrema(input, labels=None, index=None):
+    """
+    Calculate the minimums and maximums of the values of an array
+    at labels, along with their positions.
+
+    Parameters
+    ----------
+    input : ndarray
+        N-D image data to process.
+    labels : ndarray, optional
+        Labels of features in input.
+        If not None, must be same shape as `input`.
+    index : int or sequence of ints, optional
+        Labels to include in output.  If None (default), all values where
+        non-zero `labels` are used.
+
+    Returns
+    -------
+    minimums, maximums : int or ndarray
+        Values of minimums and maximums in each feature.
+    min_positions, max_positions : tuple or list of tuples
+        Each tuple gives the N-D coordinates of the corresponding minimum
+        or maximum.
+
+    See Also
+    --------
+    maximum, minimum, maximum_position, minimum_position, center_of_mass
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.array([[1, 2, 0, 0],
+    ...               [5, 3, 0, 4],
+    ...               [0, 0, 0, 7],
+    ...               [9, 3, 0, 0]])
+    >>> from scipy import ndimage
+    >>> ndimage.extrema(a)
+    (0, 9, (0, 2), (3, 0))
+
+    Features to process can be specified using `labels` and `index`:
+
+    >>> lbl, nlbl = ndimage.label(a)
+    >>> ndimage.extrema(a, lbl, index=np.arange(1, nlbl+1))
+    (array([1, 4, 3]),
+     array([5, 7, 9]),
+     [(0, 0), (1, 3), (3, 1)],
+     [(1, 0), (2, 3), (3, 0)])
+
+    If no index is given, non-zero `labels` are processed:
+
+    >>> ndimage.extrema(a, lbl)
+    (1, 9, (0, 0), (3, 0))
+
+    """
+    dims = np.array(np.asarray(input).shape)
+    # see np.unravel_index to understand this line.
+    dim_prod = np.cumprod([1] + list(dims[:0:-1]))[::-1]
+
+    minimums, min_positions, maximums, max_positions = _select(input, labels,
+                                                               index,
+                                                               find_min=True,
+                                                               find_max=True,
+                                                               find_min_positions=True,
+                                                               find_max_positions=True)
+
+    if np.isscalar(minimums):
+        return (minimums, maximums, tuple((min_positions // dim_prod) % dims),
+                tuple((max_positions // dim_prod) % dims))
+
+    min_positions = [
+        tuple(v) for v in (min_positions.reshape(-1, 1) // dim_prod) % dims
+    ]
+    max_positions = [
+        tuple(v) for v in (max_positions.reshape(-1, 1) // dim_prod) % dims
+    ]
+
+    return minimums, maximums, min_positions, max_positions
+
+
+def center_of_mass(input, labels=None, index=None):
+    """
+    Calculate the center of mass of the values of an array at labels.
+
+    Parameters
+    ----------
+    input : ndarray
+        Data from which to calculate center-of-mass. The masses can either
+        be positive or negative.
+    labels : ndarray, optional
+        Labels for objects in `input`, as generated by `ndimage.label`.
+        Only used with `index`. Dimensions must be the same as `input`.
+    index : int or sequence of ints, optional
+        Labels for which to calculate centers-of-mass. If not specified,
+        the combined center of mass of all labels greater than zero
+        will be calculated. Only used with `labels`.
+
+    Returns
+    -------
+    center_of_mass : tuple, or list of tuples
+        Coordinates of centers-of-mass.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.array(([0,0,0,0],
+    ...               [0,1,1,0],
+    ...               [0,1,1,0],
+    ...               [0,1,1,0]))
+    >>> from scipy import ndimage
+    >>> ndimage.center_of_mass(a)
+    (2.0, 1.5)
+
+    Calculation of multiple objects in an image
+
+    >>> b = np.array(([0,1,1,0],
+    ...               [0,1,0,0],
+    ...               [0,0,0,0],
+    ...               [0,0,1,1],
+    ...               [0,0,1,1]))
+    >>> lbl = ndimage.label(b)[0]
+    >>> ndimage.center_of_mass(b, lbl, [1,2])
+    [(0.33333333333333331, 1.3333333333333333), (3.5, 2.5)]
+
+    Negative masses are also accepted, which can occur for example when
+    bias is removed from measured data due to random noise.
+
+    >>> c = np.array(([-1,0,0,0],
+    ...               [0,-1,-1,0],
+    ...               [0,1,-1,0],
+    ...               [0,1,1,0]))
+    >>> ndimage.center_of_mass(c)
+    (-4.0, 1.0)
+
+    If there are division by zero issues, the function does not raise an
+    error but rather issues a RuntimeWarning before returning inf and/or NaN.
+
+    >>> d = np.array([-1, 1])
+    >>> ndimage.center_of_mass(d)
+    (inf,)
+    """
+    input = np.asarray(input)
+    normalizer = sum_labels(input, labels, index)
+    grids = np.ogrid[[slice(0, i) for i in input.shape]]
+
+    results = [sum_labels(input * grids[dir].astype(float), labels, index) / normalizer
+               for dir in range(input.ndim)]
+
+    if np.isscalar(results[0]):
+        return tuple(results)
+
+    return [tuple(v) for v in np.array(results).T]
+
+
+def histogram(input, min, max, bins, labels=None, index=None):
+    """
+    Calculate the histogram of the values of an array, optionally at labels.
+
+    Histogram calculates the frequency of values in an array within bins
+    determined by `min`, `max`, and `bins`. The `labels` and `index`
+    keywords can limit the scope of the histogram to specified sub-regions
+    within the array.
+
+    Parameters
+    ----------
+    input : array_like
+        Data for which to calculate histogram.
+    min, max : int
+        Minimum and maximum values of range of histogram bins.
+    bins : int
+        Number of bins.
+    labels : array_like, optional
+        Labels for objects in `input`.
+        If not None, must be same shape as `input`.
+    index : int or sequence of ints, optional
+        Label or labels for which to calculate histogram. If None, all values
+        where label is greater than zero are used
+
+    Returns
+    -------
+    hist : ndarray
+        Histogram counts.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> a = np.array([[ 0.    ,  0.2146,  0.5962,  0.    ],
+    ...               [ 0.    ,  0.7778,  0.    ,  0.    ],
+    ...               [ 0.    ,  0.    ,  0.    ,  0.    ],
+    ...               [ 0.    ,  0.    ,  0.7181,  0.2787],
+    ...               [ 0.    ,  0.    ,  0.6573,  0.3094]])
+    >>> from scipy import ndimage
+    >>> ndimage.histogram(a, 0, 1, 10)
+    array([13,  0,  2,  1,  0,  1,  1,  2,  0,  0])
+
+    With labels and no indices, non-zero elements are counted:
+
+    >>> lbl, nlbl = ndimage.label(a)
+    >>> ndimage.histogram(a, 0, 1, 10, lbl)
+    array([0, 0, 2, 1, 0, 1, 1, 2, 0, 0])
+
+    Indices can be used to count only certain objects:
+
+    >>> ndimage.histogram(a, 0, 1, 10, lbl, 2)
+    array([0, 0, 1, 1, 0, 0, 1, 1, 0, 0])
+
+    """
+    _bins = np.linspace(min, max, bins + 1)
+
+    def _hist(vals):
+        return np.histogram(vals, _bins)[0]
+
+    return labeled_comprehension(input, labels, index, _hist, object, None,
+                                 pass_positions=False)
+
+
+def watershed_ift(input, markers, structure=None, output=None):
+    """
+    Apply watershed from markers using image foresting transform algorithm.
+
+    Parameters
+    ----------
+    input : array_like
+        Input.
+    markers : array_like
+        Markers are points within each watershed that form the beginning
+        of the process. Negative markers are considered background markers
+        which are processed after the other markers.
+    structure : structure element, optional
+        A structuring element defining the connectivity of the object can be
+        provided. If None, an element is generated with a squared
+        connectivity equal to one.
+    output : ndarray, optional
+        An output array can optionally be provided. The same shape as input.
+
+    Returns
+    -------
+    watershed_ift : ndarray
+        Output.  Same shape as `input`.
+
+    References
+    ----------
+    .. [1] A.X. Falcao, J. Stolfi and R. de Alencar Lotufo, "The image
+           foresting transform: theory, algorithms, and applications",
+           Pattern Analysis and Machine Intelligence, vol. 26, pp. 19-29, 2004.
+
+    """
+    input = np.asarray(input)
+    if input.dtype.type not in [np.uint8, np.uint16]:
+        raise TypeError('only 8 and 16 unsigned inputs are supported')
+
+    if structure is None:
+        structure = _morphology.generate_binary_structure(input.ndim, 1)
+    structure = np.asarray(structure, dtype=bool)
+    if structure.ndim != input.ndim:
+        raise RuntimeError('structure and input must have equal rank')
+    for ii in structure.shape:
+        if ii != 3:
+            raise RuntimeError('structure dimensions must be equal to 3')
+
+    if not structure.flags.contiguous:
+        structure = structure.copy()
+    markers = np.asarray(markers)
+    if input.shape != markers.shape:
+        raise RuntimeError('input and markers must have equal shape')
+
+    integral_types = [np.int8,
+                      np.int16,
+                      np.int32,
+                      np.int64,
+                      np.intc,
+                      np.intp]
+
+    if markers.dtype.type not in integral_types:
+        raise RuntimeError('marker should be of integer type')
+
+    if isinstance(output, np.ndarray):
+        if output.dtype.type not in integral_types:
+            raise RuntimeError('output should be of integer type')
+    else:
+        output = markers.dtype
+
+    output = _ni_support._get_output(output, input)
+    _nd_image.watershed_ift(input, markers, structure, output)
+    return output

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.venv/lib/python3.12/site-packages/scipy/ndimage/_ndimage_api.py

@@ -0,0 +1,16 @@
+"""This is the 'bare' ndimage API.
+
+This --- private! --- module only collects implementations of public ndimage API
+for _support_alternative_backends.
+The latter --- also private! --- module adds delegation to CuPy etc and
+re-exports decorated names to __init__.py
+"""
+
+from ._filters import *    # noqa: F403
+from ._fourier import *   # noqa: F403
+from ._interpolation import *   # noqa: F403
+from ._measurements import *   # noqa: F403
+from ._morphology import *   # noqa: F403
+
+# '@' due to pytest bug, scipy/scipy#22236
+__all__ = [s for s in dir() if not s.startswith(('_', '@'))]

.venv/lib/python3.12/site-packages/scipy/ndimage/_ni_docstrings.py

@@ -0,0 +1,220 @@
+"""Docstring components common to several ndimage functions."""
+from typing import Final
+
+from scipy._lib import doccer
+
+__all__ = ['docfiller']
+
+
+_input_doc = (
+"""input : array_like
+    The input array.""")
+_axis_doc = (
+"""axis : int, optional
+    The axis of `input` along which to calculate. Default is -1.""")
+_output_doc = (
+"""output : array or dtype, optional
+    The array in which to place the output, or the dtype of the
+    returned array. By default an array of the same dtype as input
+    will be created.""")
+_nan_doc = (
+"""The behavior of this function with NaN elements is undefined. To control
+behavior in the presence of NaNs, consider using `vectorized_filter`.""")
+_size_foot_doc = (
+"""size : scalar or tuple, optional
+    See footprint, below. Ignored if footprint is given.
+footprint : array, optional
+    Either `size` or `footprint` must be defined. `size` gives
+    the shape that is taken from the input array, at every element
+    position, to define the input to the filter function.
+    `footprint` is a boolean array that specifies (implicitly) a
+    shape, but also which of the elements within this shape will get
+    passed to the filter function. Thus ``size=(n,m)`` is equivalent
+    to ``footprint=np.ones((n,m))``.  We adjust `size` to the number
+    of dimensions of the input array, so that, if the input array is
+    shape (10,10,10), and `size` is 2, then the actual size used is
+    (2,2,2). When `footprint` is given, `size` is ignored.""")
+_mode_reflect_doc = (
+"""mode : {'reflect', 'constant', 'nearest', 'mirror', 'wrap'}, optional
+    The `mode` parameter determines how the input array is extended
+    beyond its boundaries. Default is 'reflect'. Behavior for each valid
+    value is as follows:
+
+    'reflect' (`d c b a | a b c d | d c b a`)
+        The input is extended by reflecting about the edge of the last
+        pixel. This mode is also sometimes referred to as half-sample
+        symmetric.
+
+    'constant' (`k k k k | a b c d | k k k k`)
+        The input is extended by filling all values beyond the edge with
+        the same constant value, defined by the `cval` parameter.
+
+    'nearest' (`a a a a | a b c d | d d d d`)
+        The input is extended by replicating the last pixel.
+
+    'mirror' (`d c b | a b c d | c b a`)
+        The input is extended by reflecting about the center of the last
+        pixel. This mode is also sometimes referred to as whole-sample
+        symmetric.
+
+    'wrap' (`a b c d | a b c d | a b c d`)
+        The input is extended by wrapping around to the opposite edge.
+
+    For consistency with the interpolation functions, the following mode
+    names can also be used:
+
+    'grid-mirror'
+        This is a synonym for 'reflect'.
+
+    'grid-constant'
+        This is a synonym for 'constant'.
+
+    'grid-wrap'
+        This is a synonym for 'wrap'.""")
+
+_mode_interp_constant_doc = (
+"""mode : {'reflect', 'grid-mirror', 'constant', 'grid-constant', 'nearest', \
+'mirror', 'grid-wrap', 'wrap'}, optional
+    The `mode` parameter determines how the input array is extended
+    beyond its boundaries. Default is 'constant'. Behavior for each valid
+    value is as follows (see additional plots and details on
+    :ref:`boundary modes <ndimage-interpolation-modes>`):
+
+    'reflect' (`d c b a | a b c d | d c b a`)
+        The input is extended by reflecting about the edge of the last
+        pixel. This mode is also sometimes referred to as half-sample
+        symmetric.
+
+    'grid-mirror'
+        This is a synonym for 'reflect'.
+
+    'constant' (`k k k k | a b c d | k k k k`)
+        The input is extended by filling all values beyond the edge with
+        the same constant value, defined by the `cval` parameter. No
+        interpolation is performed beyond the edges of the input.
+
+    'grid-constant' (`k k k k | a b c d | k k k k`)
+        The input is extended by filling all values beyond the edge with
+        the same constant value, defined by the `cval` parameter. Interpolation
+        occurs for samples outside the input's extent  as well.
+
+    'nearest' (`a a a a | a b c d | d d d d`)
+        The input is extended by replicating the last pixel.
+
+    'mirror' (`d c b | a b c d | c b a`)
+        The input is extended by reflecting about the center of the last
+        pixel. This mode is also sometimes referred to as whole-sample
+        symmetric.
+
+    'grid-wrap' (`a b c d | a b c d | a b c d`)
+        The input is extended by wrapping around to the opposite edge.
+
+    'wrap' (`b c d b c | a b c d | b c a b c`)
+        The input is extended by wrapping around to the opposite edge, but in a
+        way such that the last point and initial point exactly overlap. In this
+        case it is not well defined which sample will be chosen at the point of
+        overlap.
+
+
+        For the example illustrated above, the edge values ``a`` and ``d`` overlap,
+        while the inner pattern ``b c`` repeats. The example demonstrates one
+        possible output, while ``b c a b c | a b c d | b c d b c`` is another,
+        since either ``a`` or ``d`` may occur at each point of overlap.""")
+_mode_interp_mirror_doc = (
+    _mode_interp_constant_doc.replace("Default is 'constant'",
+                                      "Default is 'mirror'")
+)
+assert _mode_interp_mirror_doc != _mode_interp_constant_doc, \
+    'Default not replaced'
+
+_mode_multiple_doc = (
+"""mode : str or sequence, optional
+    The `mode` parameter determines how the input array is extended
+    when the filter overlaps a border. By passing a sequence of modes
+    with length equal to the number of dimensions of the input array,
+    different modes can be specified along each axis. Default value is
+    'reflect'. The valid values and their behavior is as follows:
+
+    'reflect' (`d c b a | a b c d | d c b a`)
+        The input is extended by reflecting about the edge of the last
+        pixel. This mode is also sometimes referred to as half-sample
+        symmetric.
+
+    'constant' (`k k k k | a b c d | k k k k`)
+        The input is extended by filling all values beyond the edge with
+        the same constant value, defined by the `cval` parameter.
+
+    'nearest' (`a a a a | a b c d | d d d d`)
+        The input is extended by replicating the last pixel.
+
+    'mirror' (`d c b | a b c d | c b a`)
+        The input is extended by reflecting about the center of the last
+        pixel. This mode is also sometimes referred to as whole-sample
+        symmetric.
+
+    'wrap' (`a b c d | a b c d | a b c d`)
+        The input is extended by wrapping around to the opposite edge.
+
+    For consistency with the interpolation functions, the following mode
+    names can also be used:
+
+    'grid-constant'
+        This is a synonym for 'constant'.
+
+    'grid-mirror'
+        This is a synonym for 'reflect'.
+
+    'grid-wrap'
+        This is a synonym for 'wrap'.""")
+_cval_doc = (
+"""cval : scalar, optional
+    Value to fill past edges of input if `mode` is 'constant'. Default
+    is 0.0.""")
+_origin_doc = (
+"""origin : int, optional
+    Controls the placement of the filter on the input array's pixels.
+    A value of 0 (the default) centers the filter over the pixel, with
+    positive values shifting the filter to the left, and negative ones
+    to the right.""")
+_origin_multiple_doc = (
+"""origin : int or sequence, optional
+    Controls the placement of the filter on the input array's pixels.
+    A value of 0 (the default) centers the filter over the pixel, with
+    positive values shifting the filter to the left, and negative ones
+    to the right. By passing a sequence of origins with length equal to
+    the number of dimensions of the input array, different shifts can
+    be specified along each axis.""")
+_extra_arguments_doc = (
+"""extra_arguments : sequence, optional
+    Sequence of extra positional arguments to pass to passed function.""")
+_extra_keywords_doc = (
+"""extra_keywords : dict, optional
+    dict of extra keyword arguments to pass to passed function.""")
+_prefilter_doc = (
+"""prefilter : bool, optional
+    Determines if the input array is prefiltered with `spline_filter`
+    before interpolation. The default is True, which will create a
+    temporary `float64` array of filtered values if ``order > 1``. If
+    setting this to False, the output will be slightly blurred if
+    ``order > 1``, unless the input is prefiltered, i.e. it is the result
+    of calling `spline_filter` on the original input.""")
+
+docdict = {
+    'input': _input_doc,
+    'axis': _axis_doc,
+    'output': _output_doc,
+    'size_foot': _size_foot_doc,
+    'mode_interp_constant': _mode_interp_constant_doc,
+    'mode_interp_mirror': _mode_interp_mirror_doc,
+    'mode_reflect': _mode_reflect_doc,
+    'mode_multiple': _mode_multiple_doc,
+    'cval': _cval_doc,
+    'origin': _origin_doc,
+    'origin_multiple': _origin_multiple_doc,
+    'extra_arguments': _extra_arguments_doc,
+    'extra_keywords': _extra_keywords_doc,
+    'prefilter': _prefilter_doc,
+    'nan': _nan_doc,
+    }
+
+docfiller: Final = doccer.filldoc(docdict)

.venv/lib/python3.12/site-packages/scipy/ndimage/_ni_label.cpython-312-x86_64-linux-gnu.so

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:639e113aee21ad4d53663fa9023f66aaff72c9706ed093aaf2f18c8cde8e4908
+size 228496

.venv/lib/python3.12/site-packages/scipy/ndimage/_ni_support.py

@@ -0,0 +1,139 @@
+# Copyright (C) 2003-2005 Peter J. Verveer
+#
+# Redistribution and use in source and binary forms, with or without
+# modification, are permitted provided that the following conditions
+# are met:
+#
+# 1. Redistributions of source code must retain the above copyright
+#    notice, this list of conditions and the following disclaimer.
+#
+# 2. Redistributions in binary form must reproduce the above
+#    copyright notice, this list of conditions and the following
+#    disclaimer in the documentation and/or other materials provided
+#    with the distribution.
+#
+# 3. The name of the author may not be used to endorse or promote
+#    products derived from this software without specific prior
+#    written permission.
+#
+# THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS
+# OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
+# WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+# ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
+# DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+# DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
+# GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY,
+# WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
+# NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
+# SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+
+from collections.abc import Iterable
+import operator
+import warnings
+import numpy as np
+
+
+def _extend_mode_to_code(mode, is_filter=False):
+    """Convert an extension mode to the corresponding integer code.
+    """
+    if mode == 'nearest':
+        return 0
+    elif mode == 'wrap':
+        return 1
+    elif mode in ['reflect', 'grid-mirror']:
+        return 2
+    elif mode == 'mirror':
+        return 3
+    elif mode == 'constant':
+        return 4
+    elif mode == 'grid-wrap' and is_filter:
+        return 1
+    elif mode == 'grid-wrap':
+        return 5
+    elif mode == 'grid-constant' and is_filter:
+        return 4
+    elif mode == 'grid-constant':
+        return 6
+    else:
+        raise RuntimeError('boundary mode not supported')
+
+
+def _normalize_sequence(input, rank):
+    """If input is a scalar, create a sequence of length equal to the
+    rank by duplicating the input. If input is a sequence,
+    check if its length is equal to the length of array.
+    """
+    is_str = isinstance(input, str)
+    if not is_str and np.iterable(input):
+        normalized = list(input)
+        if len(normalized) != rank:
+            err = "sequence argument must have length equal to input rank"
+            raise RuntimeError(err)
+    else:
+        normalized = [input] * rank
+    return normalized
+
+
+def _get_output(output, input, shape=None, complex_output=False):
+    if shape is None:
+        shape = input.shape
+    if output is None:
+        if not complex_output:
+            output = np.zeros(shape, dtype=input.dtype.name)
+        else:
+            complex_type = np.promote_types(input.dtype, np.complex64)
+            output = np.zeros(shape, dtype=complex_type)
+    elif isinstance(output, type | np.dtype):
+        # Classes (like `np.float32`) and dtypes are interpreted as dtype
+        if complex_output and np.dtype(output).kind != 'c':
+            warnings.warn("promoting specified output dtype to complex", stacklevel=3)
+            output = np.promote_types(output, np.complex64)
+        output = np.zeros(shape, dtype=output)
+    elif isinstance(output, str):
+        output = np.dtype(output)
+        if complex_output and output.kind != 'c':
+            raise RuntimeError("output must have complex dtype")
+        elif not issubclass(output.type, np.number):
+            raise RuntimeError("output must have numeric dtype")
+        output = np.zeros(shape, dtype=output)
+    else:
+        # output was supplied as an array
+        output = np.asarray(output)
+        if output.shape != shape:
+            raise RuntimeError("output shape not correct")
+        elif complex_output and output.dtype.kind != 'c':
+            raise RuntimeError("output must have complex dtype")
+    return output
+
+
+def _check_axes(axes, ndim):
+    if axes is None:
+        return tuple(range(ndim))
+    elif np.isscalar(axes):
+        axes = (operator.index(axes),)
+    elif isinstance(axes, Iterable):
+        for ax in axes:
+            axes = tuple(operator.index(ax) for ax in axes)
+            if ax < -ndim or ax > ndim - 1:
+                raise ValueError(f"specified axis: {ax} is out of range")
+        axes = tuple(ax % ndim if ax < 0 else ax for ax in axes)
+    else:
+        message = "axes must be an integer, iterable of integers, or None"
+        raise ValueError(message)
+    if len(tuple(set(axes))) != len(axes):
+        raise ValueError("axes must be unique")
+    return axes
+
+def _skip_if_dtype(arg):
+    """'array or dtype' polymorphism.
+
+    Return None for np.int8, dtype('float32') or 'f' etc
+           arg for np.empty(3) etc
+    """
+    if isinstance(arg, str):
+        return None
+    if type(arg) is type:
+        return None if issubclass(arg, np.generic) else arg
+    else:
+        return None if isinstance(arg, np.dtype) else arg

.venv/lib/python3.12/site-packages/scipy/ndimage/_support_alternative_backends.py

@@ -0,0 +1,122 @@
+import functools
+from scipy._lib._array_api import (
+    is_cupy, is_jax, scipy_namespace_for, SCIPY_ARRAY_API, xp_capabilities
+)
+
+import numpy as np
+from ._ndimage_api import *   # noqa: F403
+from . import _ndimage_api
+from . import _delegators
+__all__ = _ndimage_api.__all__
+
+
+MODULE_NAME = 'ndimage'
+
+
+def _maybe_convert_arg(arg, xp):
+    """Convert arrays/scalars hiding in the sequence `arg`."""
+    if isinstance(arg, np.ndarray | np.generic):
+        return xp.asarray(arg)
+    elif isinstance(arg, list | tuple):
+        return type(arg)(_maybe_convert_arg(x, xp) for x in arg)
+    else:
+        return arg
+
+
+# Some cupyx.scipy.ndimage functions don't exist or are incompatible with
+# their SciPy counterparts
+CUPY_BLOCKLIST = [
+    'distance_transform_bf',
+    'distance_transform_cdt',
+    'find_objects',
+    'geometric_transform',
+    'vectorized_filter',
+]
+
+
+def delegate_xp(delegator, module_name):
+    def inner(func):
+        @functools.wraps(func)
+        def wrapper(*args, **kwds):
+            xp = delegator(*args, **kwds)
+
+            # try delegating to a cupyx/jax namesake
+            if is_cupy(xp) and func.__name__ not in CUPY_BLOCKLIST:
+                # https://github.com/cupy/cupy/issues/8336
+                import importlib
+                cupyx_module = importlib.import_module(f"cupyx.scipy.{module_name}")
+                cupyx_func = getattr(cupyx_module, func.__name__)
+                return cupyx_func(*args, **kwds)
+            elif is_jax(xp) and func.__name__ == "map_coordinates":
+                spx = scipy_namespace_for(xp)
+                jax_module = getattr(spx, module_name)
+                jax_func = getattr(jax_module, func.__name__)
+                return jax_func(*args, **kwds)
+            else:
+                # the original function (does all np.asarray internally)
+                # XXX: output arrays
+                result = func(*args, **kwds)
+
+                if isinstance(result, np.ndarray | np.generic):
+                    # XXX: np.int32->np.array_0D
+                    return xp.asarray(result)
+                elif isinstance(result, int):
+                    return result
+                elif isinstance(result, dict):
+                    # value_indices:
+                    # result is {np.int64(1): (array(0), array(1))} etc
+                    return {
+                        k.item(): tuple(xp.asarray(vv) for vv in v)
+                        for k,v in result.items()
+                    }
+                elif result is None:
+                    # inplace operations
+                    return result
+                else:
+                    # lists/tuples
+                    return _maybe_convert_arg(result, xp)
+        return wrapper
+    return inner
+
+default_capabilities = xp_capabilities(
+    cpu_only=True, exceptions=["cupy"], allow_dask_compute=True, jax_jit=False
+)
+
+capabilities_dict = {
+    "geometric_transform": xp_capabilities(
+        cpu_only=True, allow_dask_compute=True, jax_jit=False
+    ),
+    "find_objects": xp_capabilities(
+        cpu_only=True, allow_dask_compute=True, jax_jit=False
+    ),
+    "distance_transform_bf": xp_capabilities(
+        cpu_only=True, allow_dask_compute=True, jax_jit=False
+    ),
+    "distance_transform_cdt": xp_capabilities(
+        cpu_only=True, allow_dask_compute=True, jax_jit=False
+    ),
+    "vectorized_filter": xp_capabilities(
+        cpu_only=True, allow_dask_compute=True, jax_jit=False
+    ),
+    "generate_binary_structure": xp_capabilities(out_of_scope=True),
+    "map_coordinates": xp_capabilities(
+        cpu_only=True, exceptions=["cupy", "jax.numpy"],
+        allow_dask_compute=True, jax_jit=True
+    ),
+    "labeled_comprehension": xp_capabilities(np_only=True),
+}
+
+# ### decorate ###
+for func_name in _ndimage_api.__all__:
+    bare_func = getattr(_ndimage_api, func_name)
+    delegator = getattr(_delegators, func_name + "_signature")
+
+    capabilities = capabilities_dict.get(func_name, default_capabilities)
+
+    # pyrefly:ignore[not-callable]
+    f = capabilities(
+        delegate_xp(delegator, MODULE_NAME)(bare_func)
+        if SCIPY_ARRAY_API else bare_func
+    )
+    # add the decorated function to the namespace, to be imported in __init__.py
+    vars()[func_name] = f

.venv/lib/python3.12/site-packages/scipy/ndimage/filters.py

@@ -0,0 +1,27 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.ndimage` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+
+__all__ = [  # noqa: F822
+    'correlate1d', 'convolve1d', 'gaussian_filter1d',
+    'gaussian_filter', 'prewitt', 'sobel', 'generic_laplace',
+    'laplace', 'gaussian_laplace', 'generic_gradient_magnitude',
+    'gaussian_gradient_magnitude', 'correlate', 'convolve',
+    'uniform_filter1d', 'uniform_filter', 'minimum_filter1d',
+    'maximum_filter1d', 'minimum_filter', 'maximum_filter',
+    'rank_filter', 'median_filter', 'percentile_filter',
+    'generic_filter1d', 'generic_filter'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package='ndimage', module='filters',
+                                   private_modules=['_filters'], all=__all__,
+                                   attribute=name)

.venv/lib/python3.12/site-packages/scipy/ndimage/fourier.py

@@ -0,0 +1,21 @@
+# This file is not meant for public use and will be removed in SciPy v2.0.0.
+# Use the `scipy.ndimage` namespace for importing the functions
+# included below.
+
+from scipy._lib.deprecation import _sub_module_deprecation
+
+
+__all__ = [  # noqa: F822
+    'fourier_gaussian', 'fourier_uniform',
+    'fourier_ellipsoid', 'fourier_shift'
+]
+
+
+def __dir__():
+    return __all__
+
+
+def __getattr__(name):
+    return _sub_module_deprecation(sub_package='ndimage', module='fourier',
+                                   private_modules=['_fourier'], all=__all__,
+                                   attribute=name)