venv/lib/python3.11/site-packages/scipy/linalg/tests/test_decomp.py

import itertools
import platform
import sys

import numpy as np
from numpy.testing import (assert_equal, assert_almost_equal,
                           assert_array_almost_equal, assert_array_equal,
                           assert_, assert_allclose)

import pytest
from pytest import raises as assert_raises

from scipy.linalg import (eig, eigvals, lu, svd, svdvals, cholesky, qr,
                          schur, rsf2csf, lu_solve, lu_factor, solve, diagsvd,
                          hessenberg, rq, eig_banded, eigvals_banded, eigh,
                          eigvalsh, qr_multiply, qz, orth, ordqz,
                          subspace_angles, hadamard, eigvalsh_tridiagonal,
                          eigh_tridiagonal, null_space, cdf2rdf, LinAlgError)

from scipy.linalg.lapack import (dgbtrf, dgbtrs, zgbtrf, zgbtrs, dsbev,
                                 dsbevd, dsbevx, zhbevd, zhbevx)

from scipy.linalg._misc import norm
from scipy.linalg._decomp_qz import _select_function
from scipy.stats import ortho_group

from numpy import (array, diag, full, linalg, argsort, zeros, arange,
                   float32, complex64, ravel, sqrt, iscomplex, shape, sort,
                   sign, asarray, isfinite, ndarray, eye,)

from scipy.linalg._testutils import assert_no_overwrite
from scipy.sparse._sputils import matrix

from scipy._lib._testutils import check_free_memory
from scipy.linalg.blas import HAS_ILP64
try:
    from scipy.__config__ import CONFIG
except ImportError:
    CONFIG = None

IS_WASM = (sys.platform == "emscripten" or platform.machine() in ["wasm32", "wasm64"])


def _random_hermitian_matrix(n, posdef=False, dtype=float):
    "Generate random sym/hermitian array of the given size n"
    if dtype in COMPLEX_DTYPES:
        A = np.random.rand(n, n) + np.random.rand(n, n)*1.0j
        A = (A + A.conj().T)/2
    else:
        A = np.random.rand(n, n)
        A = (A + A.T)/2

    if posdef:
        A += sqrt(2*n)*np.eye(n)

    return A.astype(dtype)


REAL_DTYPES = [np.float32, np.float64]
COMPLEX_DTYPES = [np.complex64, np.complex128]
DTYPES = REAL_DTYPES + COMPLEX_DTYPES


# XXX: This function should not be defined here, but somewhere in
#      scipy.linalg namespace
def symrand(dim_or_eigv, rng):
    """Return a random symmetric (Hermitian) matrix.

    If 'dim_or_eigv' is an integer N, return a NxN matrix, with eigenvalues
        uniformly distributed on (-1,1).

    If 'dim_or_eigv' is  1-D real array 'a', return a matrix whose
                      eigenvalues are 'a'.
    """
    if isinstance(dim_or_eigv, int):
        dim = dim_or_eigv
        d = rng.random(dim)*2 - 1
    elif (isinstance(dim_or_eigv, ndarray) and
          len(dim_or_eigv.shape) == 1):
        dim = dim_or_eigv.shape[0]
        d = dim_or_eigv
    else:
        raise TypeError("input type not supported.")

    v = ortho_group.rvs(dim)
    h = v.T.conj() @ diag(d) @ v
    # to avoid roundoff errors, symmetrize the matrix (again)
    h = 0.5*(h.T+h)
    return h


class TestEigVals:

    def test_simple(self):
        a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
        w = eigvals(a)
        exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
        assert_array_almost_equal(w, exact_w)

    def test_simple_tr(self):
        a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6]], 'd').T
        a = a.copy()
        a = a.T
        w = eigvals(a)
        exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
        assert_array_almost_equal(w, exact_w)

    def test_simple_complex(self):
        a = [[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]]
        w = eigvals(a)
        exact_w = [(9+1j+sqrt(92+6j))/2,
                   0,
                   (9+1j-sqrt(92+6j))/2]
        assert_array_almost_equal(w, exact_w)

    def test_finite(self):
        a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
        w = eigvals(a, check_finite=False)
        exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
        assert_array_almost_equal(w, exact_w)

    @pytest.mark.parametrize('dt', [int, float, float32, complex, complex64])
    def test_empty(self, dt):
        a = np.empty((0, 0), dtype=dt)
        w = eigvals(a)
        assert w.shape == (0,)
        assert w.dtype == eigvals(np.eye(2, dtype=dt)).dtype

        w = eigvals(a, homogeneous_eigvals=True)
        assert w.shape == (2, 0)
        assert w.dtype == eigvals(np.eye(2, dtype=dt)).dtype


class TestEig:

    def test_simple(self):
        a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6]])
        w, v = eig(a)
        exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
        v0 = array([1, 1, (1+sqrt(93)/3)/2])
        v1 = array([3., 0, -1])
        v2 = array([1, 1, (1-sqrt(93)/3)/2])
        v0 = v0 / norm(v0)
        v1 = v1 / norm(v1)
        v2 = v2 / norm(v2)
        assert_array_almost_equal(w, exact_w)
        assert_array_almost_equal(v0, v[:, 0]*sign(v[0, 0]))
        assert_array_almost_equal(v1, v[:, 1]*sign(v[0, 1]))
        assert_array_almost_equal(v2, v[:, 2]*sign(v[0, 2]))
        for i in range(3):
            assert_array_almost_equal(a @ v[:, i], w[i]*v[:, i])
        w, v = eig(a, left=1, right=0)
        for i in range(3):
            assert_array_almost_equal(a.T @ v[:, i], w[i]*v[:, i])

    def test_simple_complex_eig(self):
        a = array([[1, 2], [-2, 1]])
        w, vl, vr = eig(a, left=1, right=1)
        assert_array_almost_equal(w, array([1+2j, 1-2j]))
        for i in range(2):
            assert_array_almost_equal(a @ vr[:, i], w[i]*vr[:, i])
        for i in range(2):
            assert_array_almost_equal(a.conj().T @ vl[:, i],
                                      w[i].conj()*vl[:, i])

    def test_simple_complex(self):
        a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]])
        w, vl, vr = eig(a, left=1, right=1)
        for i in range(3):
            assert_array_almost_equal(a @ vr[:, i], w[i]*vr[:, i])
        for i in range(3):
            assert_array_almost_equal(a.conj().T @ vl[:, i],
                                      w[i].conj()*vl[:, i])

    def test_gh_3054(self):
        a = [[1]]
        b = [[0]]
        w, vr = eig(a, b, homogeneous_eigvals=True)
        assert_allclose(w[1, 0], 0)
        assert_(w[0, 0] != 0)
        assert_allclose(vr, 1)

        w, vr = eig(a, b)
        assert_equal(w, np.inf)
        assert_allclose(vr, 1)

    def _check_gen_eig(self, A, B, atol_homog=1e-13, rtol_homog=1e-13,
                                   atol=1e-13, rtol=1e-13):
        if B is not None:
            A, B = asarray(A), asarray(B)
            B0 = B
        else:
            A = asarray(A)
            B0 = B
            B = np.eye(*A.shape)
        msg = f"\n{A!r}\n{B!r}"

        # Eigenvalues in homogeneous coordinates
        w, vr = eig(A, B0, homogeneous_eigvals=True)
        wt = eigvals(A, B0, homogeneous_eigvals=True)
        val1 = A @ vr * w[1, :]
        val2 = B @ vr * w[0, :]
        for i in range(val1.shape[1]):
            assert_allclose(val1[:, i], val2[:, i],
                            rtol=rtol_homog, atol=atol_homog, err_msg=msg)

        if B0 is None:
            assert_allclose(w[1, :], 1)
            assert_allclose(wt[1, :], 1)

        perm = np.lexsort(w)
        permt = np.lexsort(wt)
        assert_allclose(w[:, perm], wt[:, permt], atol=1e-7, rtol=1e-7,
                        err_msg=msg)

        length = np.empty(len(vr))

        for i in range(len(vr)):
            length[i] = norm(vr[:, i])

        assert_allclose(length, np.ones(length.size), err_msg=msg,
                        atol=1e-7, rtol=1e-7)

        # Convert homogeneous coordinates
        beta_nonzero = (w[1, :] != 0)
        wh = w[0, beta_nonzero] / w[1, beta_nonzero]

        # Eigenvalues in standard coordinates
        w, vr = eig(A, B0)
        wt = eigvals(A, B0)
        val1 = A @ vr
        val2 = B @ vr * w
        res = val1 - val2
        for i in range(res.shape[1]):
            if np.all(isfinite(res[:, i])):
                assert_allclose(res[:, i], 0,
                                rtol=rtol, atol=atol, err_msg=msg)

        # try to consistently order eigenvalues, including complex conjugate pairs
        w_fin = w[isfinite(w)]
        wt_fin = wt[isfinite(wt)]

        # prune noise in the real parts
        w_fin = -1j * np.real_if_close(1j*w_fin, tol=1e-10)
        wt_fin = -1j * np.real_if_close(1j*wt_fin, tol=1e-10)

        perm = argsort(abs(w_fin) + w_fin.imag)
        permt = argsort(abs(wt_fin) + wt_fin.imag)

        assert_allclose(w_fin[perm], wt_fin[permt],
                        atol=1e-7, rtol=1e-7, err_msg=msg)

        length = np.empty(len(vr))
        for i in range(len(vr)):
            length[i] = norm(vr[:, i])
        assert_allclose(length, np.ones(length.size), err_msg=msg)

        # Compare homogeneous and nonhomogeneous versions
        assert_allclose(sort(wh), sort(w[np.isfinite(w)]))

    def test_singular(self):
        # Example taken from
        # https://web.archive.org/web/20040903121217/http://www.cs.umu.se/research/nla/singular_pairs/guptri/matlab.html
        A = array([[22, 34, 31, 31, 17],
                   [45, 45, 42, 19, 29],
                   [39, 47, 49, 26, 34],
                   [27, 31, 26, 21, 15],
                   [38, 44, 44, 24, 30]])
        B = array([[13, 26, 25, 17, 24],
                   [31, 46, 40, 26, 37],
                   [26, 40, 19, 25, 25],
                   [16, 25, 27, 14, 23],
                   [24, 35, 18, 21, 22]])

        with np.errstate(all='ignore'):
            self._check_gen_eig(A, B, atol_homog=5e-13, atol=5e-13)

    def test_falker(self):
        # Test matrices giving some Nan generalized eigenvalues.
        M = diag(array([1, 0, 3]))
        K = array(([2, -1, -1], [-1, 2, -1], [-1, -1, 2]))
        D = array(([1, -1, 0], [-1, 1, 0], [0, 0, 0]))
        Z = zeros((3, 3))
        I3 = eye(3)
        A = np.block([[I3, Z], [Z, -K]])
        B = np.block([[Z, I3], [M, D]])

        with np.errstate(all='ignore'):
            self._check_gen_eig(A, B)

    def test_bad_geneig(self):
        # Ticket #709 (strange return values from DGGEV)

        def matrices(omega):
            c1 = -9 + omega**2
            c2 = 2*omega
            A = [[1, 0, 0, 0],
                 [0, 1, 0, 0],
                 [0, 0, c1, 0],
                 [0, 0, 0, c1]]
            B = [[0, 0, 1, 0],
                 [0, 0, 0, 1],
                 [1, 0, 0, -c2],
                 [0, 1, c2, 0]]
            return A, B

        # With a buggy LAPACK, this can fail for different omega on different
        # machines -- so we need to test several values
        with np.errstate(all='ignore'):
            for k in range(100):
                A, B = matrices(omega=k*5./100)
                self._check_gen_eig(A, B)

    def test_make_eigvals(self):
        # Step through all paths in _make_eigvals
        # Real eigenvalues
        rng = np.random.RandomState(1234)
        A = symrand(3, rng)
        self._check_gen_eig(A, None)
        B = symrand(3, rng)
        self._check_gen_eig(A, B)
        # Complex eigenvalues
        A = rng.random((3, 3)) + 1j*rng.random((3, 3))
        self._check_gen_eig(A, None)
        B = rng.random((3, 3)) + 1j*rng.random((3, 3))
        self._check_gen_eig(A, B)

    def test_check_finite(self):
        a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
        w, v = eig(a, check_finite=False)
        exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
        v0 = array([1, 1, (1+sqrt(93)/3)/2])
        v1 = array([3., 0, -1])
        v2 = array([1, 1, (1-sqrt(93)/3)/2])
        v0 = v0 / norm(v0)
        v1 = v1 / norm(v1)
        v2 = v2 / norm(v2)
        assert_array_almost_equal(w, exact_w)
        assert_array_almost_equal(v0, v[:, 0]*sign(v[0, 0]))
        assert_array_almost_equal(v1, v[:, 1]*sign(v[0, 1]))
        assert_array_almost_equal(v2, v[:, 2]*sign(v[0, 2]))
        for i in range(3):
            assert_array_almost_equal(a @ v[:, i], w[i]*v[:, i])

    def test_not_square_error(self):
        """Check that passing a non-square array raises a ValueError."""
        A = np.arange(6).reshape(3, 2)
        assert_raises(ValueError, eig, A)

    def test_shape_mismatch(self):
        """Check that passing arrays of with different shapes
        raises a ValueError."""
        A = eye(2)
        B = np.arange(9.0).reshape(3, 3)
        assert_raises(ValueError, eig, A, B)
        assert_raises(ValueError, eig, B, A)

    def test_gh_11577(self):
        # https://github.com/scipy/scipy/issues/11577
        # `A - lambda B` should have 4 and 8 among the eigenvalues, and this
        # was apparently broken on some platforms
        A = np.array([[12.0, 28.0, 76.0, 220.0],
                      [16.0, 32.0, 80.0, 224.0],
                      [24.0, 40.0, 88.0, 232.0],
                      [40.0, 56.0, 104.0, 248.0]], dtype='float64')
        B = np.array([[2.0, 4.0, 10.0, 28.0],
                      [3.0, 5.0, 11.0, 29.0],
                      [5.0, 7.0, 13.0, 31.0],
                      [9.0, 11.0, 17.0, 35.0]], dtype='float64')

        D, V = eig(A, B)

        # The problem is ill-conditioned, and two other eigenvalues
        # depend on ATLAS/OpenBLAS version, compiler version etc
        # see gh-11577 for discussion
        #
        # NB: it is tempting to use `assert_allclose(D[:2], [4, 8])` instead but
        # the ordering of eigenvalues also comes out different on different
        # systems depending on who knows what.
        with np.testing.suppress_warnings() as sup:
            # isclose chokes on inf/nan values
            sup.filter(RuntimeWarning, "invalid value encountered in multiply")
            assert np.isclose(D, 4.0, atol=1e-14).any()
            assert np.isclose(D, 8.0, atol=1e-14).any()

    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
    def test_empty(self, dt):
        a = np.empty((0, 0), dtype=dt)
        w, vr = eig(a)

        w_n, vr_n = eig(np.eye(2, dtype=dt))

        assert w.shape == (0,)
        assert w.dtype == w_n.dtype  #eigvals(np.eye(2, dtype=dt)).dtype

        assert_allclose(vr, np.empty((0, 0)))
        assert vr.shape == (0, 0)
        assert vr.dtype == vr_n.dtype

        w, vr = eig(a, homogeneous_eigvals=True)
        assert w.shape == (2, 0)
        assert w.dtype == w_n.dtype

        assert vr.shape == (0, 0)
        assert vr.dtype == vr_n.dtype



class TestEigBanded:
    def setup_method(self):
        self.create_bandmat()

    def create_bandmat(self):
        """Create the full matrix `self.fullmat` and
           the corresponding band matrix `self.bandmat`."""
        N = 10
        self.KL = 2   # number of subdiagonals (below the diagonal)
        self.KU = 2   # number of superdiagonals (above the diagonal)

        # symmetric band matrix
        self.sym_mat = (diag(full(N, 1.0))
                        + diag(full(N-1, -1.0), -1) + diag(full(N-1, -1.0), 1)
                        + diag(full(N-2, -2.0), -2) + diag(full(N-2, -2.0), 2))

        # hermitian band matrix
        self.herm_mat = (diag(full(N, -1.0))
                         + 1j*diag(full(N-1, 1.0), -1)
                         - 1j*diag(full(N-1, 1.0), 1)
                         + diag(full(N-2, -2.0), -2)
                         + diag(full(N-2, -2.0), 2))

        # general real band matrix
        self.real_mat = (diag(full(N, 1.0))
                         + diag(full(N-1, -1.0), -1) + diag(full(N-1, -3.0), 1)
                         + diag(full(N-2, 2.0), -2) + diag(full(N-2, -2.0), 2))

        # general complex band matrix
        self.comp_mat = (1j*diag(full(N, 1.0))
                         + diag(full(N-1, -1.0), -1)
                         + 1j*diag(full(N-1, -3.0), 1)
                         + diag(full(N-2, 2.0), -2)
                         + diag(full(N-2, -2.0), 2))

        # Eigenvalues and -vectors from linalg.eig
        ew, ev = linalg.eig(self.sym_mat)
        ew = ew.real
        args = argsort(ew)
        self.w_sym_lin = ew[args]
        self.evec_sym_lin = ev[:, args]

        ew, ev = linalg.eig(self.herm_mat)
        ew = ew.real
        args = argsort(ew)
        self.w_herm_lin = ew[args]
        self.evec_herm_lin = ev[:, args]

        # Extract upper bands from symmetric and hermitian band matrices
        # (for use in dsbevd, dsbevx, zhbevd, zhbevx
        #  and their single precision versions)
        LDAB = self.KU + 1
        self.bandmat_sym = zeros((LDAB, N), dtype=float)
        self.bandmat_herm = zeros((LDAB, N), dtype=complex)
        for i in range(LDAB):
            self.bandmat_sym[LDAB-i-1, i:N] = diag(self.sym_mat, i)
            self.bandmat_herm[LDAB-i-1, i:N] = diag(self.herm_mat, i)

        # Extract bands from general real and complex band matrix
        # (for use in dgbtrf, dgbtrs and their single precision versions)
        LDAB = 2*self.KL + self.KU + 1
        self.bandmat_real = zeros((LDAB, N), dtype=float)
        self.bandmat_real[2*self.KL, :] = diag(self.real_mat)  # diagonal
        for i in range(self.KL):
            # superdiagonals
            self.bandmat_real[2*self.KL-1-i, i+1:N] = diag(self.real_mat, i+1)
            # subdiagonals
            self.bandmat_real[2*self.KL+1+i, 0:N-1-i] = diag(self.real_mat,
                                                             -i-1)

        self.bandmat_comp = zeros((LDAB, N), dtype=complex)
        self.bandmat_comp[2*self.KL, :] = diag(self.comp_mat)  # diagonal
        for i in range(self.KL):
            # superdiagonals
            self.bandmat_comp[2*self.KL-1-i, i+1:N] = diag(self.comp_mat, i+1)
            # subdiagonals
            self.bandmat_comp[2*self.KL+1+i, 0:N-1-i] = diag(self.comp_mat,
                                                             -i-1)

        # absolute value for linear equation system A*x = b
        self.b = 1.0*arange(N)
        self.bc = self.b * (1 + 1j)

    #####################################################################

    def test_dsbev(self):
        """Compare dsbev eigenvalues and eigenvectors with
           the result of linalg.eig."""
        w, evec, info = dsbev(self.bandmat_sym, compute_v=1)
        evec_ = evec[:, argsort(w)]
        assert_array_almost_equal(sort(w), self.w_sym_lin)
        assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin))

    def test_dsbevd(self):
        """Compare dsbevd eigenvalues and eigenvectors with
           the result of linalg.eig."""
        w, evec, info = dsbevd(self.bandmat_sym, compute_v=1)
        evec_ = evec[:, argsort(w)]
        assert_array_almost_equal(sort(w), self.w_sym_lin)
        assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin))

    def test_dsbevx(self):
        """Compare dsbevx eigenvalues and eigenvectors
           with the result of linalg.eig."""
        N, N = shape(self.sym_mat)
        # Achtung: Argumente 0.0,0.0,range?
        w, evec, num, ifail, info = dsbevx(self.bandmat_sym, 0.0, 0.0, 1, N,
                                           compute_v=1, range=2)
        evec_ = evec[:, argsort(w)]
        assert_array_almost_equal(sort(w), self.w_sym_lin)
        assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin))

    def test_zhbevd(self):
        """Compare zhbevd eigenvalues and eigenvectors
           with the result of linalg.eig."""
        w, evec, info = zhbevd(self.bandmat_herm, compute_v=1)
        evec_ = evec[:, argsort(w)]
        assert_array_almost_equal(sort(w), self.w_herm_lin)
        assert_array_almost_equal(abs(evec_), abs(self.evec_herm_lin))

    def test_zhbevx(self):
        """Compare zhbevx eigenvalues and eigenvectors
           with the result of linalg.eig."""
        N, N = shape(self.herm_mat)
        # Achtung: Argumente 0.0,0.0,range?
        w, evec, num, ifail, info = zhbevx(self.bandmat_herm, 0.0, 0.0, 1, N,
                                           compute_v=1, range=2)
        evec_ = evec[:, argsort(w)]
        assert_array_almost_equal(sort(w), self.w_herm_lin)
        assert_array_almost_equal(abs(evec_), abs(self.evec_herm_lin))

    def test_eigvals_banded(self):
        """Compare eigenvalues of eigvals_banded with those of linalg.eig."""
        w_sym = eigvals_banded(self.bandmat_sym)
        w_sym = w_sym.real
        assert_array_almost_equal(sort(w_sym), self.w_sym_lin)

        w_herm = eigvals_banded(self.bandmat_herm)
        w_herm = w_herm.real
        assert_array_almost_equal(sort(w_herm), self.w_herm_lin)

        # extracting eigenvalues with respect to an index range
        ind1 = 2
        ind2 = np.longlong(6)
        w_sym_ind = eigvals_banded(self.bandmat_sym,
                                   select='i', select_range=(ind1, ind2))
        assert_array_almost_equal(sort(w_sym_ind),
                                  self.w_sym_lin[ind1:ind2+1])
        w_herm_ind = eigvals_banded(self.bandmat_herm,
                                    select='i', select_range=(ind1, ind2))
        assert_array_almost_equal(sort(w_herm_ind),
                                  self.w_herm_lin[ind1:ind2+1])

        # extracting eigenvalues with respect to a value range
        v_lower = self.w_sym_lin[ind1] - 1.0e-5
        v_upper = self.w_sym_lin[ind2] + 1.0e-5
        w_sym_val = eigvals_banded(self.bandmat_sym,
                                   select='v', select_range=(v_lower, v_upper))
        assert_array_almost_equal(sort(w_sym_val),
                                  self.w_sym_lin[ind1:ind2+1])

        v_lower = self.w_herm_lin[ind1] - 1.0e-5
        v_upper = self.w_herm_lin[ind2] + 1.0e-5
        w_herm_val = eigvals_banded(self.bandmat_herm,
                                    select='v',
                                    select_range=(v_lower, v_upper))
        assert_array_almost_equal(sort(w_herm_val),
                                  self.w_herm_lin[ind1:ind2+1])

        w_sym = eigvals_banded(self.bandmat_sym, check_finite=False)
        w_sym = w_sym.real
        assert_array_almost_equal(sort(w_sym), self.w_sym_lin)

    def test_eig_banded(self):
        """Compare eigenvalues and eigenvectors of eig_banded
           with those of linalg.eig. """
        w_sym, evec_sym = eig_banded(self.bandmat_sym)
        evec_sym_ = evec_sym[:, argsort(w_sym.real)]
        assert_array_almost_equal(sort(w_sym), self.w_sym_lin)
        assert_array_almost_equal(abs(evec_sym_), abs(self.evec_sym_lin))

        w_herm, evec_herm = eig_banded(self.bandmat_herm)
        evec_herm_ = evec_herm[:, argsort(w_herm.real)]
        assert_array_almost_equal(sort(w_herm), self.w_herm_lin)
        assert_array_almost_equal(abs(evec_herm_), abs(self.evec_herm_lin))

        # extracting eigenvalues with respect to an index range
        ind1 = 2
        ind2 = 6
        w_sym_ind, evec_sym_ind = eig_banded(self.bandmat_sym,
                                             select='i',
                                             select_range=(ind1, ind2))
        assert_array_almost_equal(sort(w_sym_ind),
                                  self.w_sym_lin[ind1:ind2+1])
        assert_array_almost_equal(abs(evec_sym_ind),
                                  abs(self.evec_sym_lin[:, ind1:ind2+1]))

        w_herm_ind, evec_herm_ind = eig_banded(self.bandmat_herm,
                                               select='i',
                                               select_range=(ind1, ind2))
        assert_array_almost_equal(sort(w_herm_ind),
                                  self.w_herm_lin[ind1:ind2+1])
        assert_array_almost_equal(abs(evec_herm_ind),
                                  abs(self.evec_herm_lin[:, ind1:ind2+1]))

        # extracting eigenvalues with respect to a value range
        v_lower = self.w_sym_lin[ind1] - 1.0e-5
        v_upper = self.w_sym_lin[ind2] + 1.0e-5
        w_sym_val, evec_sym_val = eig_banded(self.bandmat_sym,
                                             select='v',
                                             select_range=(v_lower, v_upper))
        assert_array_almost_equal(sort(w_sym_val),
                                  self.w_sym_lin[ind1:ind2+1])
        assert_array_almost_equal(abs(evec_sym_val),
                                  abs(self.evec_sym_lin[:, ind1:ind2+1]))

        v_lower = self.w_herm_lin[ind1] - 1.0e-5
        v_upper = self.w_herm_lin[ind2] + 1.0e-5
        w_herm_val, evec_herm_val = eig_banded(self.bandmat_herm,
                                               select='v',
                                               select_range=(v_lower, v_upper))
        assert_array_almost_equal(sort(w_herm_val),
                                  self.w_herm_lin[ind1:ind2+1])
        assert_array_almost_equal(abs(evec_herm_val),
                                  abs(self.evec_herm_lin[:, ind1:ind2+1]))

        w_sym, evec_sym = eig_banded(self.bandmat_sym, check_finite=False)
        evec_sym_ = evec_sym[:, argsort(w_sym.real)]
        assert_array_almost_equal(sort(w_sym), self.w_sym_lin)
        assert_array_almost_equal(abs(evec_sym_), abs(self.evec_sym_lin))

    def test_dgbtrf(self):
        """Compare dgbtrf  LU factorisation with the LU factorisation result
           of linalg.lu."""
        M, N = shape(self.real_mat)
        lu_symm_band, ipiv, info = dgbtrf(self.bandmat_real, self.KL, self.KU)

        # extract matrix u from lu_symm_band
        u = diag(lu_symm_band[2*self.KL, :])
        for i in range(self.KL + self.KU):
            u += diag(lu_symm_band[2*self.KL-1-i, i+1:N], i+1)

        p_lin, l_lin, u_lin = lu(self.real_mat, permute_l=0)
        assert_array_almost_equal(u, u_lin)

    def test_zgbtrf(self):
        """Compare zgbtrf  LU factorisation with the LU factorisation result
           of linalg.lu."""
        M, N = shape(self.comp_mat)
        lu_symm_band, ipiv, info = zgbtrf(self.bandmat_comp, self.KL, self.KU)

        # extract matrix u from lu_symm_band
        u = diag(lu_symm_band[2*self.KL, :])
        for i in range(self.KL + self.KU):
            u += diag(lu_symm_band[2*self.KL-1-i, i+1:N], i+1)

        p_lin, l_lin, u_lin = lu(self.comp_mat, permute_l=0)
        assert_array_almost_equal(u, u_lin)

    def test_dgbtrs(self):
        """Compare dgbtrs  solutions for linear equation system  A*x = b
           with solutions of linalg.solve."""

        lu_symm_band, ipiv, info = dgbtrf(self.bandmat_real, self.KL, self.KU)
        y, info = dgbtrs(lu_symm_band, self.KL, self.KU, self.b, ipiv)

        y_lin = linalg.solve(self.real_mat, self.b)
        assert_array_almost_equal(y, y_lin)

    def test_zgbtrs(self):
        """Compare zgbtrs  solutions for linear equation system  A*x = b
           with solutions of linalg.solve."""

        lu_symm_band, ipiv, info = zgbtrf(self.bandmat_comp, self.KL, self.KU)
        y, info = zgbtrs(lu_symm_band, self.KL, self.KU, self.bc, ipiv)

        y_lin = linalg.solve(self.comp_mat, self.bc)
        assert_array_almost_equal(y, y_lin)

    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
    def test_empty(self, dt):
        a_band = np.empty((0, 0), dtype=dt)
        w, v = eig_banded(a_band)

        w_n, v_n = eig_banded(np.array([[0, 0], [1, 1]], dtype=dt))

        assert w.shape == (0,)
        assert w.dtype == w_n.dtype

        assert v.shape == (0, 0)
        assert v.dtype == v_n.dtype

        w = eig_banded(a_band, eigvals_only=True)
        assert w.shape == (0,)
        assert w.dtype == w_n.dtype

class TestEigTridiagonal:
    def setup_method(self):
        self.create_trimat()

    def create_trimat(self):
        """Create the full matrix `self.fullmat`, `self.d`, and `self.e`."""
        N = 10

        # symmetric band matrix
        self.d = full(N, 1.0)
        self.e = full(N-1, -1.0)
        self.full_mat = (diag(self.d) + diag(self.e, -1) + diag(self.e, 1))

        ew, ev = linalg.eig(self.full_mat)
        ew = ew.real
        args = argsort(ew)
        self.w = ew[args]
        self.evec = ev[:, args]

    def test_degenerate(self):
        """Test error conditions."""
        # Wrong sizes
        assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e[:-1])
        # Must be real
        assert_raises(TypeError, eigvalsh_tridiagonal, self.d, self.e * 1j)
        # Bad driver
        assert_raises(TypeError, eigvalsh_tridiagonal, self.d, self.e,
                      lapack_driver=1.)
        assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e,
                      lapack_driver='foo')
        # Bad bounds
        assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e,
                      select='i', select_range=(0, -1))

    def test_eigvalsh_tridiagonal(self):
        """Compare eigenvalues of eigvalsh_tridiagonal with those of eig."""
        # can't use ?STERF with subselection
        for driver in ('sterf', 'stev', 'stebz', 'stemr', 'auto'):
            w = eigvalsh_tridiagonal(self.d, self.e, lapack_driver=driver)
            assert_array_almost_equal(sort(w), self.w)

        for driver in ('sterf', 'stev'):
            assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e,
                          lapack_driver=driver, select='i',
                          select_range=(0, 1))
        for driver in ('stebz', 'stemr', 'auto'):
            # extracting eigenvalues with respect to the full index range
            w_ind = eigvalsh_tridiagonal(
                self.d, self.e, select='i', select_range=(0, len(self.d)-1),
                lapack_driver=driver)
            assert_array_almost_equal(sort(w_ind), self.w)

            # extracting eigenvalues with respect to an index range
            ind1 = 2
            ind2 = 6
            w_ind = eigvalsh_tridiagonal(
                self.d, self.e, select='i', select_range=(ind1, ind2),
                lapack_driver=driver)
            assert_array_almost_equal(sort(w_ind), self.w[ind1:ind2+1])

            # extracting eigenvalues with respect to a value range
            v_lower = self.w[ind1] - 1.0e-5
            v_upper = self.w[ind2] + 1.0e-5
            w_val = eigvalsh_tridiagonal(
                self.d, self.e, select='v', select_range=(v_lower, v_upper),
                lapack_driver=driver)
            assert_array_almost_equal(sort(w_val), self.w[ind1:ind2+1])

    def test_eigh_tridiagonal(self):
        """Compare eigenvalues and eigenvectors of eigh_tridiagonal
           with those of eig. """
        # can't use ?STERF when eigenvectors are requested
        assert_raises(ValueError, eigh_tridiagonal, self.d, self.e,
                      lapack_driver='sterf')
        for driver in ('stebz', 'stev', 'stemr', 'auto'):
            w, evec = eigh_tridiagonal(self.d, self.e, lapack_driver=driver)
            evec_ = evec[:, argsort(w)]
            assert_array_almost_equal(sort(w), self.w)
            assert_array_almost_equal(abs(evec_), abs(self.evec))

        assert_raises(ValueError, eigh_tridiagonal, self.d, self.e,
                      lapack_driver='stev', select='i', select_range=(0, 1))
        for driver in ('stebz', 'stemr', 'auto'):
            # extracting eigenvalues with respect to an index range
            ind1 = 0
            ind2 = len(self.d)-1
            w, evec = eigh_tridiagonal(
                self.d, self.e, select='i', select_range=(ind1, ind2),
                lapack_driver=driver)
            assert_array_almost_equal(sort(w), self.w)
            assert_array_almost_equal(abs(evec), abs(self.evec))
            ind1 = 2
            ind2 = 6
            w, evec = eigh_tridiagonal(
                self.d, self.e, select='i', select_range=(ind1, ind2),
                lapack_driver=driver)
            assert_array_almost_equal(sort(w), self.w[ind1:ind2+1])
            assert_array_almost_equal(abs(evec),
                                      abs(self.evec[:, ind1:ind2+1]))

            # extracting eigenvalues with respect to a value range
            v_lower = self.w[ind1] - 1.0e-5
            v_upper = self.w[ind2] + 1.0e-5
            w, evec = eigh_tridiagonal(
                self.d, self.e, select='v', select_range=(v_lower, v_upper),
                lapack_driver=driver)
            assert_array_almost_equal(sort(w), self.w[ind1:ind2+1])
            assert_array_almost_equal(abs(evec),
                                      abs(self.evec[:, ind1:ind2+1]))

    def test_eigh_tridiagonal_1x1(self):
        """See gh-20075"""
        a = np.array([-2.0])
        b = np.array([])
        x = eigh_tridiagonal(a, b, eigvals_only=True)
        assert x.ndim == 1
        assert_allclose(x, a)
        x, V = eigh_tridiagonal(a, b, select="i", select_range=(0, 0))
        assert x.ndim == 1
        assert V.ndim == 2
        assert_allclose(x, a)
        assert_allclose(V, array([[1.]]))

        x, V = eigh_tridiagonal(a, b, select="v", select_range=(-2, 0))
        assert x.size == 0
        assert x.shape == (0,)
        assert V.shape == (1, 0)


class TestEigh:
    def setup_class(self):
        np.random.seed(1234)

    def test_wrong_inputs(self):
        # Nonsquare a
        assert_raises(ValueError, eigh, np.ones([1, 2]))
        # Nonsquare b
        assert_raises(ValueError, eigh, np.ones([2, 2]), np.ones([2, 1]))
        # Incompatible a, b sizes
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([2, 2]))
        # Wrong type parameter for generalized problem
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      type=4)
        # Both value and index subsets requested
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      subset_by_value=[1, 2], subset_by_index=[2, 4])
        # Invalid upper index spec
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      subset_by_index=[0, 4])
        # Invalid lower index
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      subset_by_index=[-2, 2])
        # Invalid index spec #2
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      subset_by_index=[2, 0])
        # Invalid value spec
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      subset_by_value=[2, 0])
        # Invalid driver name
        assert_raises(ValueError, eigh, np.ones([2, 2]), driver='wrong')
        # Generalized driver selection without b
        assert_raises(ValueError, eigh, np.ones([3, 3]), None, driver='gvx')
        # Standard driver with b
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      driver='evr')
        # Subset request from invalid driver
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      driver='gvd', subset_by_index=[1, 2])
        assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
                      driver='gvd', subset_by_index=[1, 2])

    def test_nonpositive_b(self):
        assert_raises(LinAlgError, eigh, np.ones([3, 3]), np.ones([3, 3]))

    # index based subsets are done in the legacy test_eigh()
    def test_value_subsets(self):
        for ind, dt in enumerate(DTYPES):

            a = _random_hermitian_matrix(20, dtype=dt)
            w, v = eigh(a, subset_by_value=[-2, 2])
            assert_equal(v.shape[1], len(w))
            assert all((w > -2) & (w < 2))

            b = _random_hermitian_matrix(20, posdef=True, dtype=dt)
            w, v = eigh(a, b, subset_by_value=[-2, 2])
            assert_equal(v.shape[1], len(w))
            assert all((w > -2) & (w < 2))

    def test_eigh_integer(self):
        a = array([[1, 2], [2, 7]])
        b = array([[3, 1], [1, 5]])
        w, z = eigh(a)
        w, z = eigh(a, b)

    def test_eigh_of_sparse(self):
        # This tests the rejection of inputs that eigh cannot currently handle.
        import scipy.sparse
        a = scipy.sparse.identity(2).tocsc()
        b = np.atleast_2d(a)
        assert_raises(ValueError, eigh, a)
        assert_raises(ValueError, eigh, b)

    @pytest.mark.parametrize('dtype_', DTYPES)
    @pytest.mark.parametrize('driver', ("ev", "evd", "evr", "evx"))
    def test_various_drivers_standard(self, driver, dtype_):
        a = _random_hermitian_matrix(n=20, dtype=dtype_)
        w, v = eigh(a, driver=driver)
        assert_allclose(a @ v - (v * w), 0.,
                        atol=1000*np.finfo(dtype_).eps,
                        rtol=0.)

    @pytest.mark.parametrize('driver', ("ev", "evd", "evr", "evx"))
    def test_1x1_lwork(self, driver):
        w, v = eigh([[1]], driver=driver)
        assert_allclose(w, array([1.]), atol=1e-15)
        assert_allclose(v, array([[1.]]), atol=1e-15)

        # complex case now
        w, v = eigh([[1j]], driver=driver)
        assert_allclose(w, array([0]), atol=1e-15)
        assert_allclose(v, array([[1.]]), atol=1e-15)

    @pytest.mark.parametrize('type', (1, 2, 3))
    @pytest.mark.parametrize('driver', ("gv", "gvd", "gvx"))
    def test_various_drivers_generalized(self, driver, type):
        atol = np.spacing(5000.)
        a = _random_hermitian_matrix(20)
        b = _random_hermitian_matrix(20, posdef=True)
        w, v = eigh(a=a, b=b, driver=driver, type=type)
        if type == 1:
            assert_allclose(a @ v - w*(b @ v), 0., atol=atol, rtol=0.)
        elif type == 2:
            assert_allclose(a @ b @ v - v * w, 0., atol=atol, rtol=0.)
        else:
            assert_allclose(b @ a @ v - v * w, 0., atol=atol, rtol=0.)

    def test_eigvalsh_new_args(self):
        a = _random_hermitian_matrix(5)
        w = eigvalsh(a, subset_by_index=[1, 2])
        assert_equal(len(w), 2)

        w2 = eigvalsh(a, subset_by_index=[1, 2])
        assert_equal(len(w2), 2)
        assert_allclose(w, w2)

        b = np.diag([1, 1.2, 1.3, 1.5, 2])
        w3 = eigvalsh(b, subset_by_value=[1, 1.4])
        assert_equal(len(w3), 2)
        assert_allclose(w3, np.array([1.2, 1.3]))

    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
    def test_empty(self, dt):
        a = np.empty((0, 0), dtype=dt)
        w, v = eigh(a)

        w_n, v_n = eigh(np.eye(2, dtype=dt))

        assert w.shape == (0,)
        assert w.dtype == w_n.dtype

        assert v.shape == (0, 0)
        assert v.dtype == v_n.dtype

        w = eigh(a, eigvals_only=True)
        assert_allclose(w, np.empty((0,)))

        assert w.shape == (0,)
        assert w.dtype == w_n.dtype

class TestSVD_GESDD:
    lapack_driver = 'gesdd'

    def test_degenerate(self):
        assert_raises(TypeError, svd, [[1.]], lapack_driver=1.)
        assert_raises(ValueError, svd, [[1.]], lapack_driver='foo')

    def test_simple(self):
        a = [[1, 2, 3], [1, 20, 3], [2, 5, 6]]
        for full_matrices in (True, False):
            u, s, vh = svd(a, full_matrices=full_matrices,
                           lapack_driver=self.lapack_driver)
            assert_array_almost_equal(u.T @ u, eye(3))
            assert_array_almost_equal(vh.T @ vh, eye(3))
            sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
            for i in range(len(s)):
                sigma[i, i] = s[i]
            assert_array_almost_equal(u @ sigma @ vh, a)

    def test_simple_singular(self):
        a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
        for full_matrices in (True, False):
            u, s, vh = svd(a, full_matrices=full_matrices,
                           lapack_driver=self.lapack_driver)
            assert_array_almost_equal(u.T @ u, eye(3))
            assert_array_almost_equal(vh.T @ vh, eye(3))
            sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
            for i in range(len(s)):
                sigma[i, i] = s[i]
            assert_array_almost_equal(u @ sigma @ vh, a)

    def test_simple_underdet(self):
        a = [[1, 2, 3], [4, 5, 6]]
        for full_matrices in (True, False):
            u, s, vh = svd(a, full_matrices=full_matrices,
                           lapack_driver=self.lapack_driver)
            assert_array_almost_equal(u.T @ u, eye(u.shape[0]))
            sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
            for i in range(len(s)):
                sigma[i, i] = s[i]
            assert_array_almost_equal(u @ sigma @ vh, a)

    def test_simple_overdet(self):
        a = [[1, 2], [4, 5], [3, 4]]
        for full_matrices in (True, False):
            u, s, vh = svd(a, full_matrices=full_matrices,
                           lapack_driver=self.lapack_driver)
            assert_array_almost_equal(u.T @ u, eye(u.shape[1]))
            assert_array_almost_equal(vh.T @ vh, eye(2))
            sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char)
            for i in range(len(s)):
                sigma[i, i] = s[i]
            assert_array_almost_equal(u @ sigma @ vh, a)

    def test_random(self):
        rng = np.random.RandomState(1234)
        n = 20
        m = 15
        for i in range(3):
            for a in [rng.random([n, m]), rng.random([m, n])]:
                for full_matrices in (True, False):
                    u, s, vh = svd(a, full_matrices=full_matrices,
                                   lapack_driver=self.lapack_driver)
                    assert_array_almost_equal(u.T @ u, eye(u.shape[1]))
                    assert_array_almost_equal(vh @ vh.T, eye(vh.shape[0]))
                    sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char)
                    for i in range(len(s)):
                        sigma[i, i] = s[i]
                    assert_array_almost_equal(u @ sigma @ vh, a)

    def test_simple_complex(self):
        a = [[1, 2, 3], [1, 2j, 3], [2, 5, 6]]
        for full_matrices in (True, False):
            u, s, vh = svd(a, full_matrices=full_matrices,
                           lapack_driver=self.lapack_driver)
            assert_array_almost_equal(u.conj().T @ u, eye(u.shape[1]))
            assert_array_almost_equal(vh.conj().T @ vh, eye(vh.shape[0]))
            sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
            for i in range(len(s)):
                sigma[i, i] = s[i]
            assert_array_almost_equal(u @ sigma @ vh, a)

    def test_random_complex(self):
        rng = np.random.RandomState(1234)
        n = 20
        m = 15
        for i in range(3):
            for full_matrices in (True, False):
                for a in [rng.random([n, m]), rng.random([m, n])]:
                    a = a + 1j*rng.random(list(a.shape))
                    u, s, vh = svd(a, full_matrices=full_matrices,
                                   lapack_driver=self.lapack_driver)
                    assert_array_almost_equal(u.conj().T @ u,
                                              eye(u.shape[1]))
                    # This fails when [m,n]
                    # assert_array_almost_equal(vh.conj().T @ vh,
                    #                        eye(len(vh),dtype=vh.dtype.char))
                    sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char)
                    for i in range(len(s)):
                        sigma[i, i] = s[i]
                    assert_array_almost_equal(u @ sigma @ vh, a)

    def test_crash_1580(self):
        rng = np.random.RandomState(1234)
        sizes = [(13, 23), (30, 50), (60, 100)]
        for sz in sizes:
            for dt in [np.float32, np.float64, np.complex64, np.complex128]:
                a = rng.rand(*sz).astype(dt)
                # should not crash
                svd(a, lapack_driver=self.lapack_driver)

    def test_check_finite(self):
        a = [[1, 2, 3], [1, 20, 3], [2, 5, 6]]
        u, s, vh = svd(a, check_finite=False, lapack_driver=self.lapack_driver)
        assert_array_almost_equal(u.T @ u, eye(3))
        assert_array_almost_equal(vh.T @ vh, eye(3))
        sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
        for i in range(len(s)):
            sigma[i, i] = s[i]
        assert_array_almost_equal(u @ sigma @ vh, a)

    def test_gh_5039(self):
        # This is a smoke test for https://github.com/scipy/scipy/issues/5039
        #
        # The following is reported to raise "ValueError: On entry to DGESDD
        # parameter number 12 had an illegal value".
        # `interp1d([1,2,3,4], [1,2,3,4], kind='cubic')`
        # This is reported to only show up on LAPACK 3.0.3.
        #
        # The matrix below is taken from the call to
        # `B = _fitpack._bsplmat(order, xk)` in interpolate._find_smoothest
        b = np.array(
            [[0.16666667, 0.66666667, 0.16666667, 0., 0., 0.],
             [0., 0.16666667, 0.66666667, 0.16666667, 0., 0.],
             [0., 0., 0.16666667, 0.66666667, 0.16666667, 0.],
             [0., 0., 0., 0.16666667, 0.66666667, 0.16666667]])
        svd(b, lapack_driver=self.lapack_driver)

    @pytest.mark.skipif(not HAS_ILP64, reason="64-bit LAPACK required")
    @pytest.mark.slow
    def test_large_matrix(self):
        check_free_memory(free_mb=17000)
        A = np.zeros([1, 2**31], dtype=np.float32)
        A[0, -1] = 1
        u, s, vh = svd(A, full_matrices=False)
        assert_allclose(s[0], 1.0)
        assert_allclose(u[0, 0] * vh[0, -1], 1.0)

    @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):
        a = np.zeros((m, n), dtype=dtype)
        k = min(m, n)
        dchar = a.dtype.char
        real_dchar = dchar.lower() if dchar in 'FD' else dchar

        u, s, v = svd(a)
        assert_equal(u.shape, (m, m))
        assert_equal(u.dtype, dtype)
        assert_equal(s.shape, (k,))
        assert_equal(s.dtype, np.dtype(real_dchar))
        assert_equal(v.shape, (n, n))
        assert_equal(v.dtype, dtype)

        u, s, v = svd(a, full_matrices=False)
        assert_equal(u.shape, (m, k))
        assert_equal(u.dtype, dtype)
        assert_equal(s.shape, (k,))
        assert_equal(s.dtype, np.dtype(real_dchar))
        assert_equal(v.shape, (k, n))
        assert_equal(v.dtype, dtype)

        s = svd(a, compute_uv=False)
        assert_equal(s.shape, (k,))
        assert_equal(s.dtype, np.dtype(real_dchar))

    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
    @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)])
    def test_empty(self, dt, m, n):
        a0 = np.eye(3, dtype=dt)
        u0, s0, v0 = svd(a0)

        a = np.empty((m, n), dtype=dt)
        u, s, v = svd(a)
        assert_allclose(u, np.identity(m))
        assert_allclose(s, np.empty((0,)))
        assert_allclose(v, np.identity(n))

        assert u.dtype == u0.dtype
        assert v.dtype == v0.dtype
        assert s.dtype == s0.dtype

        u, s, v = svd(a, full_matrices=False)
        assert_allclose(u, np.empty((m, 0)))
        assert_allclose(s, np.empty((0,)))
        assert_allclose(v, np.empty((0, n)))

        assert u.dtype == u0.dtype
        assert v.dtype == v0.dtype
        assert s.dtype == s0.dtype

        s = svd(a, compute_uv=False)
        assert_allclose(s, np.empty((0,)))

        assert s.dtype == s0.dtype

class TestSVD_GESVD(TestSVD_GESDD):
    lapack_driver = 'gesvd'


# Allocating an array of such a size leads to _ArrayMemoryError(s)
# since the maximum memory that can be in 32-bit (WASM) is 4GB
@pytest.mark.skipif(IS_WASM, reason="out of memory in WASM")
@pytest.mark.fail_slow(10)
def test_svd_gesdd_nofegfault():
    # svd(a) with {U,VT}.size > INT_MAX does not segfault
    # cf https://github.com/scipy/scipy/issues/14001
    df=np.ones((4799, 53130), dtype=np.float64)
    with assert_raises(ValueError):
        svd(df)


class TestSVDVals:

    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
    def test_empty(self, dt):
        for a in [[]], np.empty((2, 0)), np.ones((0, 3)):
            a = np.array(a, dtype=dt)
            s = svdvals(a)
            assert_equal(s, np.empty(0))

            s0 = svdvals(np.eye(2, dtype=dt))
            assert s.dtype == s0.dtype

    def test_simple(self):
        a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
        s = svdvals(a)
        assert_(len(s) == 3)
        assert_(s[0] >= s[1] >= s[2])

    def test_simple_underdet(self):
        a = [[1, 2, 3], [4, 5, 6]]
        s = svdvals(a)
        assert_(len(s) == 2)
        assert_(s[0] >= s[1])

    def test_simple_overdet(self):
        a = [[1, 2], [4, 5], [3, 4]]
        s = svdvals(a)
        assert_(len(s) == 2)
        assert_(s[0] >= s[1])

    def test_simple_complex(self):
        a = [[1, 2, 3], [1, 20, 3j], [2, 5, 6]]
        s = svdvals(a)
        assert_(len(s) == 3)
        assert_(s[0] >= s[1] >= s[2])

    def test_simple_underdet_complex(self):
        a = [[1, 2, 3], [4, 5j, 6]]
        s = svdvals(a)
        assert_(len(s) == 2)
        assert_(s[0] >= s[1])

    def test_simple_overdet_complex(self):
        a = [[1, 2], [4, 5], [3j, 4]]
        s = svdvals(a)
        assert_(len(s) == 2)
        assert_(s[0] >= s[1])

    def test_check_finite(self):
        a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
        s = svdvals(a, check_finite=False)
        assert_(len(s) == 3)
        assert_(s[0] >= s[1] >= s[2])

    @pytest.mark.slow
    def test_crash_2609(self):
        np.random.seed(1234)
        a = np.random.rand(1500, 2800)
        # Shouldn't crash:
        svdvals(a)


class TestDiagSVD:

    def test_simple(self):
        assert_array_almost_equal(diagsvd([1, 0, 0], 3, 3),
                                  [[1, 0, 0], [0, 0, 0], [0, 0, 0]])


class TestQR:
    def test_simple(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        q, r = qr(a)
        assert_array_almost_equal(q.T @ q, eye(3))
        assert_array_almost_equal(q @ r, a)

    def test_simple_left(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        q, r = qr(a)
        c = [1, 2, 3]
        qc, r2 = qr_multiply(a, c, "left")
        assert_array_almost_equal(q @ c, qc)
        assert_array_almost_equal(r, r2)
        qc, r2 = qr_multiply(a, eye(3), "left")
        assert_array_almost_equal(q, qc)

    def test_simple_right(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        q, r = qr(a)
        c = [1, 2, 3]
        qc, r2 = qr_multiply(a, c)
        assert_array_almost_equal(c @ q, qc)
        assert_array_almost_equal(r, r2)
        qc, r = qr_multiply(a, eye(3))
        assert_array_almost_equal(q, qc)

    def test_simple_pivoting(self):
        a = np.asarray([[8, 2, 3], [2, 9, 3], [5, 3, 6]])
        q, r, p = qr(a, pivoting=True)
        d = abs(diag(r))
        assert_(np.all(d[1:] <= d[:-1]))
        assert_array_almost_equal(q.T @ q, eye(3))
        assert_array_almost_equal(q @ r, a[:, p])
        q2, r2 = qr(a[:, p])
        assert_array_almost_equal(q, q2)
        assert_array_almost_equal(r, r2)

    def test_simple_left_pivoting(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        q, r, jpvt = qr(a, pivoting=True)
        c = [1, 2, 3]
        qc, r, jpvt = qr_multiply(a, c, "left", True)
        assert_array_almost_equal(q @ c, qc)

    def test_simple_right_pivoting(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        q, r, jpvt = qr(a, pivoting=True)
        c = [1, 2, 3]
        qc, r, jpvt = qr_multiply(a, c, pivoting=True)
        assert_array_almost_equal(c @ q, qc)

    def test_simple_trap(self):
        a = [[8, 2, 3], [2, 9, 3]]
        q, r = qr(a)
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a)

    def test_simple_trap_pivoting(self):
        a = np.asarray([[8, 2, 3], [2, 9, 3]])
        q, r, p = qr(a, pivoting=True)
        d = abs(diag(r))
        assert_(np.all(d[1:] <= d[:-1]))
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a[:, p])
        q2, r2 = qr(a[:, p])
        assert_array_almost_equal(q, q2)
        assert_array_almost_equal(r, r2)

    def test_simple_tall(self):
        # full version
        a = [[8, 2], [2, 9], [5, 3]]
        q, r = qr(a)
        assert_array_almost_equal(q.T @ q, eye(3))
        assert_array_almost_equal(q @ r, a)

    def test_simple_tall_pivoting(self):
        # full version pivoting
        a = np.asarray([[8, 2], [2, 9], [5, 3]])
        q, r, p = qr(a, pivoting=True)
        d = abs(diag(r))
        assert_(np.all(d[1:] <= d[:-1]))
        assert_array_almost_equal(q.T @ q, eye(3))
        assert_array_almost_equal(q @ r, a[:, p])
        q2, r2 = qr(a[:, p])
        assert_array_almost_equal(q, q2)
        assert_array_almost_equal(r, r2)

    def test_simple_tall_e(self):
        # economy version
        a = [[8, 2], [2, 9], [5, 3]]
        q, r = qr(a, mode='economic')
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a)
        assert_equal(q.shape, (3, 2))
        assert_equal(r.shape, (2, 2))

    def test_simple_tall_e_pivoting(self):
        # economy version pivoting
        a = np.asarray([[8, 2], [2, 9], [5, 3]])
        q, r, p = qr(a, pivoting=True, mode='economic')
        d = abs(diag(r))
        assert_(np.all(d[1:] <= d[:-1]))
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a[:, p])
        q2, r2 = qr(a[:, p], mode='economic')
        assert_array_almost_equal(q, q2)
        assert_array_almost_equal(r, r2)

    def test_simple_tall_left(self):
        a = [[8, 2], [2, 9], [5, 3]]
        q, r = qr(a, mode="economic")
        c = [1, 2]
        qc, r2 = qr_multiply(a, c, "left")
        assert_array_almost_equal(q @ c, qc)
        assert_array_almost_equal(r, r2)
        c = array([1, 2, 0])
        qc, r2 = qr_multiply(a, c, "left", overwrite_c=True)
        assert_array_almost_equal(q @ c[:2], qc)
        qc, r = qr_multiply(a, eye(2), "left")
        assert_array_almost_equal(qc, q)

    def test_simple_tall_left_pivoting(self):
        a = [[8, 2], [2, 9], [5, 3]]
        q, r, jpvt = qr(a, mode="economic", pivoting=True)
        c = [1, 2]
        qc, r, kpvt = qr_multiply(a, c, "left", True)
        assert_array_equal(jpvt, kpvt)
        assert_array_almost_equal(q @ c, qc)
        qc, r, jpvt = qr_multiply(a, eye(2), "left", True)
        assert_array_almost_equal(qc, q)

    def test_simple_tall_right(self):
        a = [[8, 2], [2, 9], [5, 3]]
        q, r = qr(a, mode="economic")
        c = [1, 2, 3]
        cq, r2 = qr_multiply(a, c)
        assert_array_almost_equal(c @ q, cq)
        assert_array_almost_equal(r, r2)
        cq, r = qr_multiply(a, eye(3))
        assert_array_almost_equal(cq, q)

    def test_simple_tall_right_pivoting(self):
        a = [[8, 2], [2, 9], [5, 3]]
        q, r, jpvt = qr(a, pivoting=True, mode="economic")
        c = [1, 2, 3]
        cq, r, jpvt = qr_multiply(a, c, pivoting=True)
        assert_array_almost_equal(c @ q, cq)
        cq, r, jpvt = qr_multiply(a, eye(3), pivoting=True)
        assert_array_almost_equal(cq, q)

    def test_simple_fat(self):
        # full version
        a = [[8, 2, 5], [2, 9, 3]]
        q, r = qr(a)
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a)
        assert_equal(q.shape, (2, 2))
        assert_equal(r.shape, (2, 3))

    def test_simple_fat_pivoting(self):
        # full version pivoting
        a = np.asarray([[8, 2, 5], [2, 9, 3]])
        q, r, p = qr(a, pivoting=True)
        d = abs(diag(r))
        assert_(np.all(d[1:] <= d[:-1]))
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a[:, p])
        assert_equal(q.shape, (2, 2))
        assert_equal(r.shape, (2, 3))
        q2, r2 = qr(a[:, p])
        assert_array_almost_equal(q, q2)
        assert_array_almost_equal(r, r2)

    def test_simple_fat_e(self):
        # economy version
        a = [[8, 2, 3], [2, 9, 5]]
        q, r = qr(a, mode='economic')
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a)
        assert_equal(q.shape, (2, 2))
        assert_equal(r.shape, (2, 3))

    def test_simple_fat_e_pivoting(self):
        # economy version pivoting
        a = np.asarray([[8, 2, 3], [2, 9, 5]])
        q, r, p = qr(a, pivoting=True, mode='economic')
        d = abs(diag(r))
        assert_(np.all(d[1:] <= d[:-1]))
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(q @ r, a[:, p])
        assert_equal(q.shape, (2, 2))
        assert_equal(r.shape, (2, 3))
        q2, r2 = qr(a[:, p], mode='economic')
        assert_array_almost_equal(q, q2)
        assert_array_almost_equal(r, r2)

    def test_simple_fat_left(self):
        a = [[8, 2, 3], [2, 9, 5]]
        q, r = qr(a, mode="economic")
        c = [1, 2]
        qc, r2 = qr_multiply(a, c, "left")
        assert_array_almost_equal(q @ c, qc)
        assert_array_almost_equal(r, r2)
        qc, r = qr_multiply(a, eye(2), "left")
        assert_array_almost_equal(qc, q)

    def test_simple_fat_left_pivoting(self):
        a = [[8, 2, 3], [2, 9, 5]]
        q, r, jpvt = qr(a, mode="economic", pivoting=True)
        c = [1, 2]
        qc, r, jpvt = qr_multiply(a, c, "left", True)
        assert_array_almost_equal(q @ c, qc)
        qc, r, jpvt = qr_multiply(a, eye(2), "left", True)
        assert_array_almost_equal(qc, q)

    def test_simple_fat_right(self):
        a = [[8, 2, 3], [2, 9, 5]]
        q, r = qr(a, mode="economic")
        c = [1, 2]
        cq, r2 = qr_multiply(a, c)
        assert_array_almost_equal(c @ q, cq)
        assert_array_almost_equal(r, r2)
        cq, r = qr_multiply(a, eye(2))
        assert_array_almost_equal(cq, q)

    def test_simple_fat_right_pivoting(self):
        a = [[8, 2, 3], [2, 9, 5]]
        q, r, jpvt = qr(a, pivoting=True, mode="economic")
        c = [1, 2]
        cq, r, jpvt = qr_multiply(a, c, pivoting=True)
        assert_array_almost_equal(c @ q, cq)
        cq, r, jpvt = qr_multiply(a, eye(2), pivoting=True)
        assert_array_almost_equal(cq, q)

    def test_simple_complex(self):
        a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
        q, r = qr(a)
        assert_array_almost_equal(q.conj().T @ q, eye(3))
        assert_array_almost_equal(q @ r, a)

    def test_simple_complex_left(self):
        a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
        q, r = qr(a)
        c = [1, 2, 3+4j]
        qc, r = qr_multiply(a, c, "left")
        assert_array_almost_equal(q @ c, qc)
        qc, r = qr_multiply(a, eye(3), "left")
        assert_array_almost_equal(q, qc)

    def test_simple_complex_right(self):
        a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
        q, r = qr(a)
        c = [1, 2, 3+4j]
        qc, r = qr_multiply(a, c)
        assert_array_almost_equal(c @ q, qc)
        qc, r = qr_multiply(a, eye(3))
        assert_array_almost_equal(q, qc)

    def test_simple_tall_complex_left(self):
        a = [[8, 2+3j], [2, 9], [5+7j, 3]]
        q, r = qr(a, mode="economic")
        c = [1, 2+2j]
        qc, r2 = qr_multiply(a, c, "left")
        assert_array_almost_equal(q @ c, qc)
        assert_array_almost_equal(r, r2)
        c = array([1, 2, 0])
        qc, r2 = qr_multiply(a, c, "left", overwrite_c=True)
        assert_array_almost_equal(q @ c[:2], qc)
        qc, r = qr_multiply(a, eye(2), "left")
        assert_array_almost_equal(qc, q)

    def test_simple_complex_left_conjugate(self):
        a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
        q, r = qr(a)
        c = [1, 2, 3+4j]
        qc, r = qr_multiply(a, c, "left", conjugate=True)
        assert_array_almost_equal(q.conj() @ c, qc)

    def test_simple_complex_tall_left_conjugate(self):
        a = [[3, 3+4j], [5, 2+2j], [3, 2]]
        q, r = qr(a, mode='economic')
        c = [1, 3+4j]
        qc, r = qr_multiply(a, c, "left", conjugate=True)
        assert_array_almost_equal(q.conj() @ c, qc)

    def test_simple_complex_right_conjugate(self):
        a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
        q, r = qr(a)
        c = np.array([1, 2, 3+4j])
        qc, r = qr_multiply(a, c, conjugate=True)
        assert_array_almost_equal(c @ q.conj(), qc)

    def test_simple_complex_pivoting(self):
        a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]])
        q, r, p = qr(a, pivoting=True)
        d = abs(diag(r))
        assert_(np.all(d[1:] <= d[:-1]))
        assert_array_almost_equal(q.conj().T @ q, eye(3))
        assert_array_almost_equal(q @ r, a[:, p])
        q2, r2 = qr(a[:, p])
        assert_array_almost_equal(q, q2)
        assert_array_almost_equal(r, r2)

    def test_simple_complex_left_pivoting(self):
        a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]])
        q, r, jpvt = qr(a, pivoting=True)
        c = [1, 2, 3+4j]
        qc, r, jpvt = qr_multiply(a, c, "left", True)
        assert_array_almost_equal(q @ c, qc)

    def test_simple_complex_right_pivoting(self):
        a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]])
        q, r, jpvt = qr(a, pivoting=True)
        c = [1, 2, 3+4j]
        qc, r, jpvt = qr_multiply(a, c, pivoting=True)
        assert_array_almost_equal(c @ q, qc)

    def test_random(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n])
            q, r = qr(a)
            assert_array_almost_equal(q.T @ q, eye(n))
            assert_array_almost_equal(q @ r, a)

    def test_random_left(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n])
            q, r = qr(a)
            c = rng.random([n])
            qc, r = qr_multiply(a, c, "left")
            assert_array_almost_equal(q @ c, qc)
            qc, r = qr_multiply(a, eye(n), "left")
            assert_array_almost_equal(q, qc)

    def test_random_right(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n])
            q, r = qr(a)
            c = rng.random([n])
            cq, r = qr_multiply(a, c)
            assert_array_almost_equal(c @ q, cq)
            cq, r = qr_multiply(a, eye(n))
            assert_array_almost_equal(q, cq)

    def test_random_pivoting(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n])
            q, r, p = qr(a, pivoting=True)
            d = abs(diag(r))
            assert_(np.all(d[1:] <= d[:-1]))
            assert_array_almost_equal(q.T @ q, eye(n))
            assert_array_almost_equal(q @ r, a[:, p])
            q2, r2 = qr(a[:, p])
            assert_array_almost_equal(q, q2)
            assert_array_almost_equal(r, r2)

    def test_random_tall(self):
        rng = np.random.RandomState(1234)
        # full version
        m = 200
        n = 100
        for k in range(2):
            a = rng.random([m, n])
            q, r = qr(a)
            assert_array_almost_equal(q.T @ q, eye(m))
            assert_array_almost_equal(q @ r, a)

    def test_random_tall_left(self):
        rng = np.random.RandomState(1234)
        # full version
        m = 200
        n = 100
        for k in range(2):
            a = rng.random([m, n])
            q, r = qr(a, mode="economic")
            c = rng.random([n])
            qc, r = qr_multiply(a, c, "left")
            assert_array_almost_equal(q @ c, qc)
            qc, r = qr_multiply(a, eye(n), "left")
            assert_array_almost_equal(qc, q)

    def test_random_tall_right(self):
        rng = np.random.RandomState(1234)
        # full version
        m = 200
        n = 100
        for k in range(2):
            a = rng.random([m, n])
            q, r = qr(a, mode="economic")
            c = rng.random([m])
            cq, r = qr_multiply(a, c)
            assert_array_almost_equal(c @ q, cq)
            cq, r = qr_multiply(a, eye(m))
            assert_array_almost_equal(cq, q)

    def test_random_tall_pivoting(self):
        rng = np.random.RandomState(1234)
        # full version pivoting
        m = 200
        n = 100
        for k in range(2):
            a = rng.random([m, n])
            q, r, p = qr(a, pivoting=True)
            d = abs(diag(r))
            assert_(np.all(d[1:] <= d[:-1]))
            assert_array_almost_equal(q.T @ q, eye(m))
            assert_array_almost_equal(q @ r, a[:, p])
            q2, r2 = qr(a[:, p])
            assert_array_almost_equal(q, q2)
            assert_array_almost_equal(r, r2)

    def test_random_tall_e(self):
        rng = np.random.RandomState(1234)
        # economy version
        m = 200
        n = 100
        for k in range(2):
            a = rng.random([m, n])
            q, r = qr(a, mode='economic')
            assert_array_almost_equal(q.T @ q, eye(n))
            assert_array_almost_equal(q @ r, a)
            assert_equal(q.shape, (m, n))
            assert_equal(r.shape, (n, n))

    def test_random_tall_e_pivoting(self):
        rng = np.random.RandomState(1234)
        # economy version pivoting
        m = 200
        n = 100
        for k in range(2):
            a = rng.random([m, n])
            q, r, p = qr(a, pivoting=True, mode='economic')
            d = abs(diag(r))
            assert_(np.all(d[1:] <= d[:-1]))
            assert_array_almost_equal(q.T @ q, eye(n))
            assert_array_almost_equal(q @ r, a[:, p])
            assert_equal(q.shape, (m, n))
            assert_equal(r.shape, (n, n))
            q2, r2 = qr(a[:, p], mode='economic')
            assert_array_almost_equal(q, q2)
            assert_array_almost_equal(r, r2)

    def test_random_trap(self):
        rng = np.random.RandomState(1234)
        m = 100
        n = 200
        for k in range(2):
            a = rng.random([m, n])
            q, r = qr(a)
            assert_array_almost_equal(q.T @ q, eye(m))
            assert_array_almost_equal(q @ r, a)

    def test_random_trap_pivoting(self):
        rng = np.random.RandomState(1234)
        m = 100
        n = 200
        for k in range(2):
            a = rng.random([m, n])
            q, r, p = qr(a, pivoting=True)
            d = abs(diag(r))
            assert_(np.all(d[1:] <= d[:-1]))
            assert_array_almost_equal(q.T @ q, eye(m))
            assert_array_almost_equal(q @ r, a[:, p])
            q2, r2 = qr(a[:, p])
            assert_array_almost_equal(q, q2)
            assert_array_almost_equal(r, r2)

    def test_random_complex(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n]) + 1j*rng.random([n, n])
            q, r = qr(a)
            assert_array_almost_equal(q.conj().T @ q, eye(n))
            assert_array_almost_equal(q @ r, a)

    def test_random_complex_left(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n]) + 1j*rng.random([n, n])
            q, r = qr(a)
            c = rng.random([n]) + 1j*rng.random([n])
            qc, r = qr_multiply(a, c, "left")
            assert_array_almost_equal(q @ c, qc)
            qc, r = qr_multiply(a, eye(n), "left")
            assert_array_almost_equal(q, qc)

    def test_random_complex_right(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n]) + 1j*rng.random([n, n])
            q, r = qr(a)
            c = rng.random([n]) + 1j*rng.random([n])
            cq, r = qr_multiply(a, c)
            assert_array_almost_equal(c @ q, cq)
            cq, r = qr_multiply(a, eye(n))
            assert_array_almost_equal(q, cq)

    def test_random_complex_pivoting(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n]) + 1j*rng.random([n, n])
            q, r, p = qr(a, pivoting=True)
            d = abs(diag(r))
            assert_(np.all(d[1:] <= d[:-1]))
            assert_array_almost_equal(q.conj().T @ q, eye(n))
            assert_array_almost_equal(q @ r, a[:, p])
            q2, r2 = qr(a[:, p])
            assert_array_almost_equal(q, q2)
            assert_array_almost_equal(r, r2)

    def test_check_finite(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        q, r = qr(a, check_finite=False)
        assert_array_almost_equal(q.T @ q, eye(3))
        assert_array_almost_equal(q @ r, a)

    def test_lwork(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        # Get comparison values
        q, r = qr(a, lwork=None)

        # Test against minimum valid lwork
        q2, r2 = qr(a, lwork=3)
        assert_array_almost_equal(q2, q)
        assert_array_almost_equal(r2, r)

        # Test against larger lwork
        q3, r3 = qr(a, lwork=10)
        assert_array_almost_equal(q3, q)
        assert_array_almost_equal(r3, r)

        # Test against explicit lwork=-1
        q4, r4 = qr(a, lwork=-1)
        assert_array_almost_equal(q4, q)
        assert_array_almost_equal(r4, r)

        # Test against invalid lwork
        assert_raises(Exception, qr, (a,), {'lwork': 0})
        assert_raises(Exception, qr, (a,), {'lwork': 2})

    @pytest.mark.parametrize("m", [0, 1, 2])
    @pytest.mark.parametrize("n", [0, 1, 2])
    @pytest.mark.parametrize("pivoting", [False, True])
    @pytest.mark.parametrize('dtype', DTYPES)
    def test_shape_dtype(self, m, n, pivoting, dtype):
        k = min(m, n)

        a = np.zeros((m, n), dtype=dtype)
        q, r, *other = qr(a, pivoting=pivoting)
        assert_equal(q.shape, (m, m))
        assert_equal(q.dtype, dtype)
        assert_equal(r.shape, (m, n))
        assert_equal(r.dtype, dtype)
        assert len(other) == (1 if pivoting else 0)
        if pivoting:
            p, = other
            assert_equal(p.shape, (n,))
            assert_equal(p.dtype, np.int32)

        r, *other = qr(a, mode='r', pivoting=pivoting)
        assert_equal(r.shape, (m, n))
        assert_equal(r.dtype, dtype)
        assert len(other) == (1 if pivoting else 0)
        if pivoting:
            p, = other
            assert_equal(p.shape, (n,))
            assert_equal(p.dtype, np.int32)

        q, r, *other = qr(a, mode='economic', pivoting=pivoting)
        assert_equal(q.shape, (m, k))
        assert_equal(q.dtype, dtype)
        assert_equal(r.shape, (k, n))
        assert_equal(r.dtype, dtype)
        assert len(other) == (1 if pivoting else 0)
        if pivoting:
            p, = other
            assert_equal(p.shape, (n,))
            assert_equal(p.dtype, np.int32)

        (raw, tau), r, *other = qr(a, mode='raw', pivoting=pivoting)
        assert_equal(raw.shape, (m, n))
        assert_equal(raw.dtype, dtype)
        assert_equal(tau.shape, (k,))
        assert_equal(tau.dtype, dtype)
        assert_equal(r.shape, (k, n))
        assert_equal(r.dtype, dtype)
        assert len(other) == (1 if pivoting else 0)
        if pivoting:
            p, = other
            assert_equal(p.shape, (n,))
            assert_equal(p.dtype, np.int32)

    @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)])
    def test_empty(self, m, n):
        k = min(m, n)

        a = np.empty((m, n))
        q, r = qr(a)
        assert_allclose(q, np.identity(m))
        assert_allclose(r, np.empty((m, n)))

        q, r, p = qr(a, pivoting=True)
        assert_allclose(q, np.identity(m))
        assert_allclose(r, np.empty((m, n)))
        assert_allclose(p, np.arange(n))

        r, = qr(a, mode='r')
        assert_allclose(r, np.empty((m, n)))

        q, r = qr(a, mode='economic')
        assert_allclose(q, np.empty((m, k)))
        assert_allclose(r, np.empty((k, n)))

        (raw, tau), r = qr(a, mode='raw')
        assert_allclose(raw, np.empty((m, n)))
        assert_allclose(tau, np.empty((k,)))
        assert_allclose(r, np.empty((k, n)))

    def test_multiply_empty(self):
        a = np.empty((0, 0))
        c = np.empty((0, 0))
        cq, r = qr_multiply(a, c)
        assert_allclose(cq, np.empty((0, 0)))

        a = np.empty((0, 2))
        c = np.empty((2, 0))
        cq, r = qr_multiply(a, c)
        assert_allclose(cq, np.empty((2, 0)))

        a = np.empty((2, 0))
        c = np.empty((0, 2))
        cq, r = qr_multiply(a, c)
        assert_allclose(cq, np.empty((0, 2)))


class TestRQ:
    def test_simple(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        r, q = rq(a)
        assert_array_almost_equal(q @ q.T, eye(3))
        assert_array_almost_equal(r @ q, a)

    def test_r(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        r, q = rq(a)
        r2 = rq(a, mode='r')
        assert_array_almost_equal(r, r2)

    def test_random(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n])
            r, q = rq(a)
            assert_array_almost_equal(q @ q.T, eye(n))
            assert_array_almost_equal(r @ q, a)

    def test_simple_trap(self):
        a = [[8, 2, 3], [2, 9, 3]]
        r, q = rq(a)
        assert_array_almost_equal(q.T @ q, eye(3))
        assert_array_almost_equal(r @ q, a)

    def test_simple_tall(self):
        a = [[8, 2], [2, 9], [5, 3]]
        r, q = rq(a)
        assert_array_almost_equal(q.T @ q, eye(2))
        assert_array_almost_equal(r @ q, a)

    def test_simple_fat(self):
        a = [[8, 2, 5], [2, 9, 3]]
        r, q = rq(a)
        assert_array_almost_equal(q @ q.T, eye(3))
        assert_array_almost_equal(r @ q, a)

    def test_simple_complex(self):
        a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
        r, q = rq(a)
        assert_array_almost_equal(q @ q.conj().T, eye(3))
        assert_array_almost_equal(r @ q, a)

    def test_random_tall(self):
        rng = np.random.RandomState(1234)
        m = 200
        n = 100
        for k in range(2):
            a = rng.random([m, n])
            r, q = rq(a)
            assert_array_almost_equal(q @ q.T, eye(n))
            assert_array_almost_equal(r @ q, a)

    def test_random_trap(self):
        rng = np.random.RandomState(1234)
        m = 100
        n = 200
        for k in range(2):
            a = rng.random([m, n])
            r, q = rq(a)
            assert_array_almost_equal(q @ q.T, eye(n))
            assert_array_almost_equal(r @ q, a)

    def test_random_trap_economic(self):
        rng = np.random.RandomState(1234)
        m = 100
        n = 200
        for k in range(2):
            a = rng.random([m, n])
            r, q = rq(a, mode='economic')
            assert_array_almost_equal(q @ q.T, eye(m))
            assert_array_almost_equal(r @ q, a)
            assert_equal(q.shape, (m, n))
            assert_equal(r.shape, (m, m))

    def test_random_complex(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n]) + 1j*rng.random([n, n])
            r, q = rq(a)
            assert_array_almost_equal(q @ q.conj().T, eye(n))
            assert_array_almost_equal(r @ q, a)

    def test_random_complex_economic(self):
        rng = np.random.RandomState(1234)
        m = 100
        n = 200
        for k in range(2):
            a = rng.random([m, n]) + 1j*rng.random([m, n])
            r, q = rq(a, mode='economic')
            assert_array_almost_equal(q @ q.conj().T, eye(m))
            assert_array_almost_equal(r @ q, a)
            assert_equal(q.shape, (m, n))
            assert_equal(r.shape, (m, m))

    def test_check_finite(self):
        a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
        r, q = rq(a, check_finite=False)
        assert_array_almost_equal(q @ q.T, eye(3))
        assert_array_almost_equal(r @ q, a)

    @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.zeros((m, n), dtype=dtype)
        r, q = rq(a)
        assert_equal(q.shape, (n, n))
        assert_equal(r.shape, (m, n))
        assert_equal(r.dtype, dtype)
        assert_equal(q.dtype, dtype)

        r = rq(a, mode='r')
        assert_equal(r.shape, (m, n))
        assert_equal(r.dtype, dtype)

        r, q = rq(a, mode='economic')
        assert_equal(r.shape, (m, k))
        assert_equal(r.dtype, dtype)
        assert_equal(q.shape, (k, n))
        assert_equal(q.dtype, dtype)

    @pytest.mark.parametrize(("m", "n"), [(0, 0), (0, 2), (2, 0)])
    def test_empty(self, m, n):
        k = min(m, n)

        a = np.empty((m, n))
        r, q = rq(a)
        assert_allclose(r, np.empty((m, n)))
        assert_allclose(q, np.identity(n))

        r = rq(a, mode='r')
        assert_allclose(r, np.empty((m, n)))

        r, q = rq(a, mode='economic')
        assert_allclose(r, np.empty((m, k)))
        assert_allclose(q, np.empty((k, n)))


class TestSchur:

    def check_schur(self, a, t, u, rtol, atol):
        # Check that the Schur decomposition is correct.
        assert_allclose(u @ t @ u.conj().T, a, rtol=rtol, atol=atol,
                        err_msg="Schur decomposition does not match 'a'")
        # The expected value of u @ u.H - I is all zeros, so test
        # with absolute tolerance only.
        assert_allclose(u @ u.conj().T - np.eye(len(u)), 0, rtol=0, atol=atol,
                        err_msg="u is not unitary")

    def test_simple(self):
        a = [[8, 12, 3], [2, 9, 3], [10, 3, 6]]
        t, z = schur(a)
        self.check_schur(a, t, z, rtol=1e-14, atol=5e-15)
        tc, zc = schur(a, 'complex')
        assert_(np.any(ravel(iscomplex(zc))) and np.any(ravel(iscomplex(tc))))
        self.check_schur(a, tc, zc, rtol=1e-14, atol=5e-15)
        tc2, zc2 = rsf2csf(tc, zc)
        self.check_schur(a, tc2, zc2, rtol=1e-14, atol=5e-15)

    @pytest.mark.parametrize(
        'sort, expected_diag',
        [('lhp', [-np.sqrt(2), -0.5, np.sqrt(2), 0.5]),
         ('rhp', [np.sqrt(2), 0.5, -np.sqrt(2), -0.5]),
         ('iuc', [-0.5, 0.5, np.sqrt(2), -np.sqrt(2)]),
         ('ouc', [np.sqrt(2), -np.sqrt(2), -0.5, 0.5]),
         (lambda x: x >= 0.0, [np.sqrt(2), 0.5, -np.sqrt(2), -0.5])]
    )
    def test_sort(self, sort, expected_diag):
        # The exact eigenvalues of this matrix are
        #   -sqrt(2), sqrt(2), -1/2, 1/2.
        a = [[4., 3., 1., -1.],
             [-4.5, -3.5, -1., 1.],
             [9., 6., -4., 4.5],
             [6., 4., -3., 3.5]]
        t, u, sdim = schur(a, sort=sort)
        self.check_schur(a, t, u, rtol=1e-14, atol=5e-15)
        assert_allclose(np.diag(t), expected_diag, rtol=1e-12)
        assert_equal(2, sdim)

    def test_sort_errors(self):
        a = [[4., 3., 1., -1.],
             [-4.5, -3.5, -1., 1.],
             [9., 6., -4., 4.5],
             [6., 4., -3., 3.5]]
        assert_raises(ValueError, schur, a, sort='unsupported')
        assert_raises(ValueError, schur, a, sort=1)

    def test_check_finite(self):
        a = [[8, 12, 3], [2, 9, 3], [10, 3, 6]]
        t, z = schur(a, check_finite=False)
        assert_array_almost_equal(z @ t @ z.conj().T, a)

    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
    def test_empty(self, dt):
        a = np.empty((0, 0), dtype=dt)
        t, z = schur(a)
        t0, z0 = schur(np.eye(2, dtype=dt))
        assert_allclose(t, np.empty((0, 0)))
        assert_allclose(z, np.empty((0, 0)))
        assert t.dtype == t0.dtype
        assert z.dtype == z0.dtype

        t, z, sdim = schur(a, sort='lhp')
        assert_allclose(t, np.empty((0, 0)))
        assert_allclose(z, np.empty((0, 0)))
        assert_equal(sdim, 0)
        assert t.dtype == t0.dtype
        assert z.dtype == z0.dtype

    @pytest.mark.parametrize('sort', ['iuc', 'ouc'])
    @pytest.mark.parametrize('output', ['real', 'complex'])
    @pytest.mark.parametrize('dtype', [np.float32, np.float64,
                                       np.complex64, np.complex128])
    def test_gh_13137_sort_str(self, sort, output, dtype):
        # gh-13137 reported that sort values 'iuc' and 'ouc' were not
        # correct because the callables assumed that the eigenvalues would
        # always be expressed as a single complex number.
        # In fact, when `output='real'` and the dtype is real, the
        # eigenvalues are passed as separate real and imaginary components
        # (yet no error is raised if the callable accepts only one argument).
        #
        # This tests these sort values by counting the number of eigenvalues
        # `schur` reports as being inside/outside the unit circle.

        # Real matrix with eigenvalues 0.1 +- 2j
        A = np.asarray([[0.1, -2], [2, 0.1]])

        # Previously, this would fail for `output='real'` with real dtypes
        sdim = schur(A.astype(dtype), sort=sort, output=output)[-1]
        assert sdim == 0 if sort == 'iuc' else sdim == 2

    @pytest.mark.parametrize('output', ['real', 'complex'])
    @pytest.mark.parametrize('dtype', [np.float32, np.float64,
                                       np.complex64, np.complex128])
    def test_gh_13137_sort_custom(self, output, dtype):
        # This simply tests our understanding of how eigenvalues are
        # passed to a sort callable. If `output='real'` and the dtype is real,
        # real and imaginary parts are passed as separate real arguments;
        # otherwise, they are passed a single complex argument.
        # Also, if `output='real'` and the dtype is real, when either
        # eigenvalue in a complex conjugate pair satisfies the sort condition,
        # `sdim` is incremented by TWO.

        # Real matrix with eigenvalues 0.1 +- 2j
        A = np.asarray([[0.1, -2], [2, 0.1]])

        all_real = output=='real' and dtype in {np.float32, np.float64}

        def sort(x, y=None):
            if all_real:
                assert not np.iscomplexobj(x)
                assert y is not None and np.isreal(y)
                z = x + y*1j
            else:
                assert np.iscomplexobj(x)
                assert y is None
                z = x
            return z.imag > 1e-15

        # Only one complex eigenvalue satisfies the condition, but when
        # `all_real` applies, both eigenvalues in the complex conjugate pair
        # are counted.
        sdim = schur(A.astype(dtype), sort=sort, output=output)[-1]
        assert sdim == 2 if all_real else sdim == 1


class TestHessenberg:

    def test_simple(self):
        a = [[-149, -50, -154],
             [537, 180, 546],
             [-27, -9, -25]]
        h1 = [[-149.0000, 42.2037, -156.3165],
              [-537.6783, 152.5511, -554.9272],
              [0, 0.0728, 2.4489]]
        h, q = hessenberg(a, calc_q=1)
        assert_array_almost_equal(q.T @ a @ q, h)
        assert_array_almost_equal(h, h1, decimal=4)

    def test_simple_complex(self):
        a = [[-149, -50, -154],
             [537, 180j, 546],
             [-27j, -9, -25]]
        h, q = hessenberg(a, calc_q=1)
        assert_array_almost_equal(q.conj().T @ a @ q, h)

    def test_simple2(self):
        a = [[1, 2, 3, 4, 5, 6, 7],
             [0, 2, 3, 4, 6, 7, 2],
             [0, 2, 2, 3, 0, 3, 2],
             [0, 0, 2, 8, 0, 0, 2],
             [0, 3, 1, 2, 0, 1, 2],
             [0, 1, 2, 3, 0, 1, 0],
             [0, 0, 0, 0, 0, 1, 2]]
        h, q = hessenberg(a, calc_q=1)
        assert_array_almost_equal(q.T @ a @ q, h)

    def test_simple3(self):
        a = np.eye(3)
        a[-1, 0] = 2
        h, q = hessenberg(a, calc_q=1)
        assert_array_almost_equal(q.T @ a @ q, h)

    def test_random(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n])
            h, q = hessenberg(a, calc_q=1)
            assert_array_almost_equal(q.T @ a @ q, h)

    def test_random_complex(self):
        rng = np.random.RandomState(1234)
        n = 20
        for k in range(2):
            a = rng.random([n, n]) + 1j*rng.random([n, n])
            h, q = hessenberg(a, calc_q=1)
            assert_array_almost_equal(q.conj().T @ a @ q, h)

    def test_check_finite(self):
        a = [[-149, -50, -154],
             [537, 180, 546],
             [-27, -9, -25]]
        h1 = [[-149.0000, 42.2037, -156.3165],
              [-537.6783, 152.5511, -554.9272],
              [0, 0.0728, 2.4489]]
        h, q = hessenberg(a, calc_q=1, check_finite=False)
        assert_array_almost_equal(q.T @ a @ q, h)
        assert_array_almost_equal(h, h1, decimal=4)

    def test_2x2(self):
        a = [[2, 1], [7, 12]]

        h, q = hessenberg(a, calc_q=1)
        assert_array_almost_equal(q, np.eye(2))
        assert_array_almost_equal(h, a)

        b = [[2-7j, 1+2j], [7+3j, 12-2j]]
        h2, q2 = hessenberg(b, calc_q=1)
        assert_array_almost_equal(q2, np.eye(2))
        assert_array_almost_equal(h2, b)

    @pytest.mark.parametrize('dt', [int, float, float32, complex, complex64])
    def test_empty(self, dt):
        a = np.empty((0, 0), dtype=dt)
        h = hessenberg(a)
        assert h.shape == (0, 0)
        assert h.dtype == hessenberg(np.eye(3, dtype=dt)).dtype

        h, q = hessenberg(a, calc_q=True)
        h3, q3 = hessenberg(a, calc_q=True)
        assert h.shape == (0, 0)
        assert h.dtype == h3.dtype

        assert q.shape == (0, 0)
        assert q.dtype == q3.dtype


blas_provider = blas_version = None
if CONFIG is not None:
    blas_provider = CONFIG['Build Dependencies']['blas']['name']
    blas_version = CONFIG['Build Dependencies']['blas']['version']


class TestQZ:
    def test_qz_single(self):
        rng = np.random.RandomState(12345)
        n = 5
        A = rng.random([n, n]).astype(float32)
        B = rng.random([n, n]).astype(float32)
        AA, BB, Q, Z = qz(A, B)
        assert_array_almost_equal(Q @ AA @ Z.T, A, decimal=5)
        assert_array_almost_equal(Q @ BB @ Z.T, B, decimal=5)
        assert_array_almost_equal(Q @ Q.T, eye(n), decimal=5)
        assert_array_almost_equal(Z @ Z.T, eye(n), decimal=5)
        assert_(np.all(diag(BB) >= 0))

    def test_qz_double(self):
        rng = np.random.RandomState(12345)
        n = 5
        A = rng.random([n, n])
        B = rng.random([n, n])
        AA, BB, Q, Z = qz(A, B)
        assert_array_almost_equal(Q @ AA @ Z.T, A)
        assert_array_almost_equal(Q @ BB @ Z.T, B)
        assert_array_almost_equal(Q @ Q.T, eye(n))
        assert_array_almost_equal(Z @ Z.T, eye(n))
        assert_(np.all(diag(BB) >= 0))

    def test_qz_complex(self):
        rng = np.random.RandomState(12345)
        n = 5
        A = rng.random([n, n]) + 1j*rng.random([n, n])
        B = rng.random([n, n]) + 1j*rng.random([n, n])
        AA, BB, Q, Z = qz(A, B)
        assert_array_almost_equal(Q @ AA @ Z.conj().T, A)
        assert_array_almost_equal(Q @ BB @ Z.conj().T, B)
        assert_array_almost_equal(Q @ Q.conj().T, eye(n))
        assert_array_almost_equal(Z @ Z.conj().T, eye(n))
        assert_(np.all(diag(BB) >= 0))
        assert_(np.all(diag(BB).imag == 0))

    def test_qz_complex64(self):
        rng = np.random.RandomState(12345)
        n = 5
        A = (rng.random([n, n]) + 1j*rng.random([n, n])).astype(complex64)
        B = (rng.random([n, n]) + 1j*rng.random([n, n])).astype(complex64)
        AA, BB, Q, Z = qz(A, B)
        assert_array_almost_equal(Q @ AA @ Z.conj().T, A, decimal=5)
        assert_array_almost_equal(Q @ BB @ Z.conj().T, B, decimal=5)
        assert_array_almost_equal(Q @ Q.conj().T, eye(n), decimal=5)
        assert_array_almost_equal(Z @ Z.conj().T, eye(n), decimal=5)
        assert_(np.all(diag(BB) >= 0))
        assert_(np.all(diag(BB).imag == 0))

    def test_qz_double_complex(self):
        rng = np.random.RandomState(12345)
        n = 5
        A = rng.random([n, n])
        B = rng.random([n, n])
        AA, BB, Q, Z = qz(A, B, output='complex')
        aa = Q @ AA @ Z.conj().T
        assert_array_almost_equal(aa.real, A)
        assert_array_almost_equal(aa.imag, 0)
        bb = Q @ BB @ Z.conj().T
        assert_array_almost_equal(bb.real, B)
        assert_array_almost_equal(bb.imag, 0)
        assert_array_almost_equal(Q @ Q.conj().T, eye(n))
        assert_array_almost_equal(Z @ Z.conj().T, eye(n))
        assert_(np.all(diag(BB) >= 0))

    def test_qz_double_sort(self):
        # from https://www.nag.com/lapack-ex/node119.html
        # NOTE: These matrices may be ill-conditioned and lead to a
        # seg fault on certain python versions when compiled with
        # sse2 or sse3 older ATLAS/LAPACK binaries for windows
        # A =   np.array([[3.9,  12.5, -34.5,  -0.5],
        #                [ 4.3,  21.5, -47.5,   7.5],
        #                [ 4.3,  21.5, -43.5,   3.5],
        #                [ 4.4,  26.0, -46.0,   6.0 ]])

        # B = np.array([[ 1.0,   2.0,  -3.0,   1.0],
        #              [1.0,   3.0,  -5.0,   4.0],
        #              [1.0,   3.0,  -4.0,   3.0],
        #              [1.0,   3.0,  -4.0,   4.0]])
        A = np.array([[3.9, 12.5, -34.5, 2.5],
                      [4.3, 21.5, -47.5, 7.5],
                      [4.3, 1.5, -43.5, 3.5],
                      [4.4, 6.0, -46.0, 6.0]])

        B = np.array([[1.0, 1.0, -3.0, 1.0],
                      [1.0, 3.0, -5.0, 4.4],
                      [1.0, 2.0, -4.0, 1.0],
                      [1.2, 3.0, -4.0, 4.0]])

        assert_raises(ValueError, qz, A, B, sort=lambda ar, ai, beta: ai == 0)
        if False:
            AA, BB, Q, Z, sdim = qz(A, B, sort=lambda ar, ai, beta: ai == 0)
            # assert_(sdim == 2)
            assert_(sdim == 4)
            assert_array_almost_equal(Q @ AA @ Z.T, A)
            assert_array_almost_equal(Q @ BB @ Z.T, B)

            # test absolute values bc the sign is ambiguous and
            # might be platform dependent
            assert_array_almost_equal(np.abs(AA), np.abs(np.array(
                            [[35.7864, -80.9061, -12.0629, -9.498],
                             [0., 2.7638, -2.3505, 7.3256],
                             [0., 0., 0.6258, -0.0398],
                             [0., 0., 0., -12.8217]])), 4)
            assert_array_almost_equal(np.abs(BB), np.abs(np.array(
                            [[4.5324, -8.7878, 3.2357, -3.5526],
                             [0., 1.4314, -2.1894, 0.9709],
                             [0., 0., 1.3126, -0.3468],
                             [0., 0., 0., 0.559]])), 4)
            assert_array_almost_equal(np.abs(Q), np.abs(np.array(
                            [[-0.4193, -0.605, -0.1894, -0.6498],
                             [-0.5495, 0.6987, 0.2654, -0.3734],
                             [-0.4973, -0.3682, 0.6194, 0.4832],
                             [-0.5243, 0.1008, -0.7142, 0.4526]])), 4)
            assert_array_almost_equal(np.abs(Z), np.abs(np.array(
                            [[-0.9471, -0.2971, -0.1217, 0.0055],
                             [-0.0367, 0.1209, 0.0358, 0.9913],
                             [0.3171, -0.9041, -0.2547, 0.1312],
                             [0.0346, 0.2824, -0.9587, 0.0014]])), 4)

        # test absolute values bc the sign is ambiguous and might be platform
        # dependent
        # assert_array_almost_equal(abs(AA), abs(np.array([
        #                [3.8009, -69.4505, 50.3135, -43.2884],
        #                [0.0000, 9.2033, -0.2001, 5.9881],
        #                [0.0000, 0.0000, 1.4279, 4.4453],
        #                [0.0000, 0.0000, 0.9019, -1.1962]])), 4)
        # assert_array_almost_equal(abs(BB), abs(np.array([
        #                [1.9005, -10.2285, 0.8658, -5.2134],
        #                [0.0000,   2.3008, 0.7915,  0.4262],
        #                [0.0000,   0.0000, 0.8101,  0.0000],
        #                [0.0000,   0.0000, 0.0000, -0.2823]])), 4)
        # assert_array_almost_equal(abs(Q), abs(np.array([
        #                [0.4642,  0.7886,  0.2915, -0.2786],
        #                [0.5002, -0.5986,  0.5638, -0.2713],
        #                [0.5002,  0.0154, -0.0107,  0.8657],
        #                [0.5331, -0.1395, -0.7727, -0.3151]])), 4)
        # assert_array_almost_equal(dot(Q,Q.T), eye(4))
        # assert_array_almost_equal(abs(Z), abs(np.array([
        #                [0.9961, -0.0014,  0.0887, -0.0026],
        #                [0.0057, -0.0404, -0.0938, -0.9948],
        #                [0.0626,  0.7194, -0.6908,  0.0363],
        #                [0.0626, -0.6934, -0.7114,  0.0956]])), 4)
        # assert_array_almost_equal(dot(Z,Z.T), eye(4))

    # def test_qz_complex_sort(self):
    #    cA = np.array([
    #   [-21.10+22.50*1j, 53.50+-50.50*1j, -34.50+127.50*1j, 7.50+  0.50*1j],
    #   [-0.46+ -7.78*1j, -3.50+-37.50*1j, -15.50+ 58.50*1j,-10.50+ -1.50*1j],
    #   [ 4.30+ -5.50*1j, 39.70+-17.10*1j, -68.50+ 12.50*1j, -7.50+ -3.50*1j],
    #   [ 5.50+  4.40*1j, 14.40+ 43.30*1j, -32.50+-46.00*1j,-19.00+-32.50*1j]])

    #    cB =  np.array([
    #   [1.00+ -5.00*1j, 1.60+  1.20*1j,-3.00+  0.00*1j, 0.00+ -1.00*1j],
    #   [0.80+ -0.60*1j, 3.00+ -5.00*1j,-4.00+  3.00*1j,-2.40+ -3.20*1j],
    #   [1.00+  0.00*1j, 2.40+  1.80*1j,-4.00+ -5.00*1j, 0.00+ -3.00*1j],
    #   [0.00+  1.00*1j,-1.80+  2.40*1j, 0.00+ -4.00*1j, 4.00+ -5.00*1j]])

    #    AAS,BBS,QS,ZS,sdim = qz(cA,cB,sort='lhp')

    #    eigenvalues = diag(AAS)/diag(BBS)
    #    assert_(np.all(np.real(eigenvalues[:sdim] < 0)))
    #    assert_(np.all(np.real(eigenvalues[sdim:] > 0)))

    def test_check_finite(self):
        rng = np.random.RandomState(12345)
        n = 5
        A = rng.random([n, n])
        B = rng.random([n, n])
        AA, BB, Q, Z = qz(A, B, check_finite=False)
        assert_array_almost_equal(Q @ AA @ Z.T, A)
        assert_array_almost_equal(Q @ BB @ Z.T, B)
        assert_array_almost_equal(Q @ Q.T, eye(n))
        assert_array_almost_equal(Z @ Z.T, eye(n))
        assert_(np.all(diag(BB) >= 0))


class TestOrdQZ:
    @classmethod
    def setup_class(cls):
        # https://www.nag.com/lapack-ex/node119.html
        A1 = np.array([[-21.10 - 22.50j, 53.5 - 50.5j, -34.5 + 127.5j,
                        7.5 + 0.5j],
                       [-0.46 - 7.78j, -3.5 - 37.5j, -15.5 + 58.5j,
                        -10.5 - 1.5j],
                       [4.30 - 5.50j, 39.7 - 17.1j, -68.5 + 12.5j,
                        -7.5 - 3.5j],
                       [5.50 + 4.40j, 14.4 + 43.3j, -32.5 - 46.0j,
                        -19.0 - 32.5j]])

        B1 = np.array([[1.0 - 5.0j, 1.6 + 1.2j, -3 + 0j, 0.0 - 1.0j],
                       [0.8 - 0.6j, .0 - 5.0j, -4 + 3j, -2.4 - 3.2j],
                       [1.0 + 0.0j, 2.4 + 1.8j, -4 - 5j, 0.0 - 3.0j],
                       [0.0 + 1.0j, -1.8 + 2.4j, 0 - 4j, 4.0 - 5.0j]])

        # https://www.nag.com/numeric/fl/nagdoc_fl23/xhtml/F08/f08yuf.xml
        A2 = np.array([[3.9, 12.5, -34.5, -0.5],
                       [4.3, 21.5, -47.5, 7.5],
                       [4.3, 21.5, -43.5, 3.5],
                       [4.4, 26.0, -46.0, 6.0]])

        B2 = np.array([[1, 2, -3, 1],
                       [1, 3, -5, 4],
                       [1, 3, -4, 3],
                       [1, 3, -4, 4]])

        # example with the eigenvalues
        # -0.33891648, 1.61217396+0.74013521j, 1.61217396-0.74013521j,
        # 0.61244091
        # thus featuring:
        #  * one complex conjugate eigenvalue pair,
        #  * one eigenvalue in the lhp
        #  * 2 eigenvalues in the unit circle
        #  * 2 non-real eigenvalues
        A3 = np.array([[5., 1., 3., 3.],
                       [4., 4., 2., 7.],
                       [7., 4., 1., 3.],
                       [0., 4., 8., 7.]])
        B3 = np.array([[8., 10., 6., 10.],
                       [7., 7., 2., 9.],
                       [9., 1., 6., 6.],
                       [5., 1., 4., 7.]])

        # example with infinite eigenvalues
        A4 = np.eye(2)
        B4 = np.diag([0, 1])

        # example with (alpha, beta) = (0, 0)
        A5 = np.diag([1, 0])

        cls.A = [A1, A2, A3, A4, A5]
        cls.B = [B1, B2, B3, B4, A5]

    def qz_decomp(self, sort):
        with np.errstate(all='raise'):
            ret = [ordqz(Ai, Bi, sort=sort) for Ai, Bi in zip(self.A, self.B)]
        return tuple(ret)

    def check(self, A, B, sort, AA, BB, alpha, beta, Q, Z):
        Id = np.eye(*A.shape)
        # make sure Q and Z are orthogonal
        assert_array_almost_equal(Q @ Q.T.conj(), Id)
        assert_array_almost_equal(Z @ Z.T.conj(), Id)
        # check factorization
        assert_array_almost_equal(Q @ AA, A @ Z)
        assert_array_almost_equal(Q @ BB, B @ Z)
        # check shape of AA and BB
        assert_array_equal(np.tril(AA, -2), np.zeros(AA.shape))
        assert_array_equal(np.tril(BB, -1), np.zeros(BB.shape))
        # check eigenvalues
        for i in range(A.shape[0]):
            # does the current diagonal element belong to a 2-by-2 block
            # that was already checked?
            if i > 0 and A[i, i - 1] != 0:
                continue
            # take care of 2-by-2 blocks
            if i < AA.shape[0] - 1 and AA[i + 1, i] != 0:
                evals, _ = eig(AA[i:i + 2, i:i + 2], BB[i:i + 2, i:i + 2])
                # make sure the pair of complex conjugate eigenvalues
                # is ordered consistently (positive imaginary part first)
                if evals[0].imag < 0:
                    evals = evals[[1, 0]]
                tmp = alpha[i:i + 2]/beta[i:i + 2]
                if tmp[0].imag < 0:
                    tmp = tmp[[1, 0]]
                assert_array_almost_equal(evals, tmp)
            else:
                if alpha[i] == 0 and beta[i] == 0:
                    assert_equal(AA[i, i], 0)
                    assert_equal(BB[i, i], 0)
                elif beta[i] == 0:
                    assert_equal(BB[i, i], 0)
                else:
                    assert_almost_equal(AA[i, i]/BB[i, i], alpha[i]/beta[i])
        sortfun = _select_function(sort)
        lastsort = True
        for i in range(A.shape[0]):
            cursort = sortfun(np.array([alpha[i]]), np.array([beta[i]]))
            # once the sorting criterion was not matched all subsequent
            # eigenvalues also shouldn't match
            if not lastsort:
                assert not cursort
            lastsort = cursort

    def check_all(self, sort):
        ret = self.qz_decomp(sort)

        for reti, Ai, Bi in zip(ret, self.A, self.B):
            self.check(Ai, Bi, sort, *reti)

    def test_lhp(self):
        self.check_all('lhp')

    def test_rhp(self):
        self.check_all('rhp')

    def test_iuc(self):
        self.check_all('iuc')

    def test_ouc(self):
        self.check_all('ouc')

    def test_ref(self):
        # real eigenvalues first (top-left corner)
        def sort(x, y):
            out = np.empty_like(x, dtype=bool)
            nonzero = (y != 0)
            out[~nonzero] = False
            out[nonzero] = (x[nonzero]/y[nonzero]).imag == 0
            return out

        self.check_all(sort)

    def test_cef(self):
        # complex eigenvalues first (top-left corner)
        def sort(x, y):
            out = np.empty_like(x, dtype=bool)
            nonzero = (y != 0)
            out[~nonzero] = False
            out[nonzero] = (x[nonzero]/y[nonzero]).imag != 0
            return out

        self.check_all(sort)

    def test_diff_input_types(self):
        ret = ordqz(self.A[1], self.B[2], sort='lhp')
        self.check(self.A[1], self.B[2], 'lhp', *ret)

        ret = ordqz(self.B[2], self.A[1], sort='lhp')
        self.check(self.B[2], self.A[1], 'lhp', *ret)

    def test_sort_explicit(self):
        # Test order of the eigenvalues in the 2 x 2 case where we can
        # explicitly compute the solution
        A1 = np.eye(2)
        B1 = np.diag([-2, 0.5])
        expected1 = [('lhp', [-0.5, 2]),
                     ('rhp', [2, -0.5]),
                     ('iuc', [-0.5, 2]),
                     ('ouc', [2, -0.5])]
        A2 = np.eye(2)
        B2 = np.diag([-2 + 1j, 0.5 + 0.5j])
        expected2 = [('lhp', [1/(-2 + 1j), 1/(0.5 + 0.5j)]),
                     ('rhp', [1/(0.5 + 0.5j), 1/(-2 + 1j)]),
                     ('iuc', [1/(-2 + 1j), 1/(0.5 + 0.5j)]),
                     ('ouc', [1/(0.5 + 0.5j), 1/(-2 + 1j)])]
        # 'lhp' is ambiguous so don't test it
        A3 = np.eye(2)
        B3 = np.diag([2, 0])
        expected3 = [('rhp', [0.5, np.inf]),
                     ('iuc', [0.5, np.inf]),
                     ('ouc', [np.inf, 0.5])]
        # 'rhp' is ambiguous so don't test it
        A4 = np.eye(2)
        B4 = np.diag([-2, 0])
        expected4 = [('lhp', [-0.5, np.inf]),
                     ('iuc', [-0.5, np.inf]),
                     ('ouc', [np.inf, -0.5])]
        A5 = np.diag([0, 1])
        B5 = np.diag([0, 0.5])
        # 'lhp' and 'iuc' are ambiguous so don't test them
        expected5 = [('rhp', [2, np.nan]),
                     ('ouc', [2, np.nan])]

        A = [A1, A2, A3, A4, A5]
        B = [B1, B2, B3, B4, B5]
        expected = [expected1, expected2, expected3, expected4, expected5]
        for Ai, Bi, expectedi in zip(A, B, expected):
            for sortstr, expected_eigvals in expectedi:
                _, _, alpha, beta, _, _ = ordqz(Ai, Bi, sort=sortstr)
                azero = (alpha == 0)
                bzero = (beta == 0)
                x = np.empty_like(alpha)
                x[azero & bzero] = np.nan
                x[~azero & bzero] = np.inf
                x[~bzero] = alpha[~bzero]/beta[~bzero]
                assert_allclose(expected_eigvals, x)


class TestOrdQZWorkspaceSize:
    @pytest.mark.fail_slow(5)
    def test_decompose(self):
        rng = np.random.RandomState(12345)
        N = 202
        # raises error if lwork parameter to dtrsen is too small
        for ddtype in [np.float32, np.float64]:
            A = rng.random((N, N)).astype(ddtype)
            B = rng.random((N, N)).astype(ddtype)
            # sort = lambda ar, ai, b: ar**2 + ai**2 < b**2
            _ = ordqz(A, B, sort=lambda alpha, beta: alpha < beta,
                      output='real')

        for ddtype in [np.complex128, np.complex64]:
            A = rng.random((N, N)).astype(ddtype)
            B = rng.random((N, N)).astype(ddtype)
            _ = ordqz(A, B, sort=lambda alpha, beta: alpha < beta,
                      output='complex')

    @pytest.mark.slow
    def test_decompose_ouc(self):
        rng = np.random.RandomState(12345)
        N = 202
        # segfaults if lwork parameter to dtrsen is too small
        for ddtype in [np.float32, np.float64, np.complex128, np.complex64]:
            A = rng.random((N, N)).astype(ddtype)
            B = rng.random((N, N)).astype(ddtype)
            S, T, alpha, beta, U, V = ordqz(A, B, sort='ouc')


class TestDatacopied:

    def test_datacopied(self):
        from scipy.linalg._decomp import _datacopied

        M = matrix([[0, 1], [2, 3]])
        A = asarray(M)
        L = M.tolist()
        M2 = M.copy()

        class Fake1:
            def __array__(self, dtype=None, copy=None):
                return A

        class Fake2:
            __array_interface__ = A.__array_interface__

        F1 = Fake1()
        F2 = Fake2()

        for item, status in [(M, False), (A, False), (L, True),
                             (M2, False), (F1, False), (F2, False)]:
            arr = asarray(item)
            assert_equal(_datacopied(arr, item), status,
                         err_msg=repr(item))


def test_aligned_mem_float():
    """Check linalg works with non-aligned memory (float32)"""
    # Allocate 402 bytes of memory (allocated on boundary)
    a = arange(402, dtype=np.uint8)

    # Create an array with boundary offset 4
    z = np.frombuffer(a.data, offset=2, count=100, dtype=float32)
    z.shape = 10, 10

    eig(z, overwrite_a=True)
    eig(z.T, overwrite_a=True)


@pytest.mark.skipif(platform.machine() == 'ppc64le',
                    reason="crashes on ppc64le")
def test_aligned_mem():
    """Check linalg works with non-aligned memory (float64)"""
    # Allocate 804 bytes of memory (allocated on boundary)
    a = arange(804, dtype=np.uint8)

    # Create an array with boundary offset 4
    z = np.frombuffer(a.data, offset=4, count=100, dtype=float)
    z.shape = 10, 10

    eig(z, overwrite_a=True)
    eig(z.T, overwrite_a=True)


def test_aligned_mem_complex():
    """Check that complex objects don't need to be completely aligned"""
    # Allocate 1608 bytes of memory (allocated on boundary)
    a = zeros(1608, dtype=np.uint8)

    # Create an array with boundary offset 8
    z = np.frombuffer(a.data, offset=8, count=100, dtype=complex)
    z.shape = 10, 10

    eig(z, overwrite_a=True)
    # This does not need special handling
    eig(z.T, overwrite_a=True)


def check_lapack_misaligned(func, args, kwargs):
    args = list(args)
    for i in range(len(args)):
        a = args[:]
        if isinstance(a[i], np.ndarray):
            # Try misaligning a[i]
            aa = np.zeros(a[i].size*a[i].dtype.itemsize+8, dtype=np.uint8)
            aa = np.frombuffer(aa.data, offset=4, count=a[i].size,
                               dtype=a[i].dtype)
            aa.shape = a[i].shape
            aa[...] = a[i]
            a[i] = aa
            func(*a, **kwargs)
            if len(a[i].shape) > 1:
                a[i] = a[i].T
                func(*a, **kwargs)


@pytest.mark.xfail(run=False,
                   reason="Ticket #1152, triggers a segfault in rare cases.")
def test_lapack_misaligned():
    M = np.eye(10, dtype=float)
    R = np.arange(100)
    R.shape = 10, 10
    S = np.arange(20000, dtype=np.uint8)
    S = np.frombuffer(S.data, offset=4, count=100, dtype=float)
    S.shape = 10, 10
    b = np.ones(10)
    LU, piv = lu_factor(S)
    for (func, args, kwargs) in [
            (eig, (S,), dict(overwrite_a=True)),  # crash
            (eigvals, (S,), dict(overwrite_a=True)),  # no crash
            (lu, (S,), dict(overwrite_a=True)),  # no crash
            (lu_factor, (S,), dict(overwrite_a=True)),  # no crash
            (lu_solve, ((LU, piv), b), dict(overwrite_b=True)),
            (solve, (S, b), dict(overwrite_a=True, overwrite_b=True)),
            (svd, (M,), dict(overwrite_a=True)),  # no crash
            (svd, (R,), dict(overwrite_a=True)),  # no crash
            (svd, (S,), dict(overwrite_a=True)),  # crash
            (svdvals, (S,), dict()),  # no crash
            (svdvals, (S,), dict(overwrite_a=True)),  # crash
            (cholesky, (M,), dict(overwrite_a=True)),  # no crash
            (qr, (S,), dict(overwrite_a=True)),  # crash
            (rq, (S,), dict(overwrite_a=True)),  # crash
            (hessenberg, (S,), dict(overwrite_a=True)),  # crash
            (schur, (S,), dict(overwrite_a=True)),  # crash
            ]:
        check_lapack_misaligned(func, args, kwargs)
# not properly tested
# cholesky, rsf2csf, lu_solve, solve, eig_banded, eigvals_banded, eigh, diagsvd


class TestOverwrite:
    def test_eig(self):
        assert_no_overwrite(eig, [(3, 3)])
        assert_no_overwrite(eig, [(3, 3), (3, 3)])

    def test_eigh(self):
        assert_no_overwrite(eigh, [(3, 3)])
        assert_no_overwrite(eigh, [(3, 3), (3, 3)])

    def test_eig_banded(self):
        assert_no_overwrite(eig_banded, [(3, 2)])

    def test_eigvals(self):
        assert_no_overwrite(eigvals, [(3, 3)])

    def test_eigvalsh(self):
        assert_no_overwrite(eigvalsh, [(3, 3)])

    def test_eigvals_banded(self):
        assert_no_overwrite(eigvals_banded, [(3, 2)])

    def test_hessenberg(self):
        assert_no_overwrite(hessenberg, [(3, 3)])

    def test_lu_factor(self):
        assert_no_overwrite(lu_factor, [(3, 3)])

    def test_lu_solve(self):
        x = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 8]])
        xlu = lu_factor(x)
        assert_no_overwrite(lambda b: lu_solve(xlu, b), [(3,)])

    def test_lu(self):
        assert_no_overwrite(lu, [(3, 3)])

    def test_qr(self):
        assert_no_overwrite(qr, [(3, 3)])

    def test_rq(self):
        assert_no_overwrite(rq, [(3, 3)])

    def test_schur(self):
        assert_no_overwrite(schur, [(3, 3)])

    def test_schur_complex(self):
        assert_no_overwrite(lambda a: schur(a, 'complex'), [(3, 3)],
                            dtypes=[np.float32, np.float64])

    def test_svd(self):
        assert_no_overwrite(svd, [(3, 3)])
        assert_no_overwrite(lambda a: svd(a, lapack_driver='gesvd'), [(3, 3)])

    def test_svdvals(self):
        assert_no_overwrite(svdvals, [(3, 3)])


def _check_orth(n, dtype, skip_big=False):
    X = np.ones((n, 2), dtype=float).astype(dtype)

    eps = np.finfo(dtype).eps
    tol = 1000 * eps

    Y = orth(X)
    assert_equal(Y.shape, (n, 1))
    assert_allclose(Y, Y.mean(), atol=tol)

    Y = orth(X.T)
    assert_equal(Y.shape, (2, 1))
    assert_allclose(Y, Y.mean(), atol=tol)

    if n > 5 and not skip_big:
        rng = np.random.RandomState(1)
        X = rng.rand(n, 5) @ rng.rand(5, n)
        X = X + 1e-4 * rng.rand(n, 1) @ rng.rand(1, n)
        X = X.astype(dtype)

        Y = orth(X, rcond=1e-3)
        assert_equal(Y.shape, (n, 5))

        Y = orth(X, rcond=1e-6)
        assert_equal(Y.shape, (n, 5 + 1))


@pytest.mark.slow
@pytest.mark.skipif(np.dtype(np.intp).itemsize < 8,
                    reason="test only on 64-bit, else too slow")
def test_orth_memory_efficiency():
    # Pick n so that 16*n bytes is reasonable but 8*n*n bytes is unreasonable.
    # Keep in mind that @pytest.mark.slow tests are likely to be running
    # under configurations that support 4Gb+ memory for tests related to
    # 32 bit overflow.
    n = 10*1000*1000
    try:
        _check_orth(n, np.float64, skip_big=True)
    except MemoryError as e:
        raise AssertionError(
            'memory error perhaps caused by orth regression'
        ) from e


def test_orth():
    dtypes = [np.float32, np.float64, np.complex64, np.complex128]
    sizes = [1, 2, 3, 10, 100]
    for dt, n in itertools.product(dtypes, sizes):
        _check_orth(n, dt)

@pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
def test_orth_empty(dt):
    a = np.empty((0, 0), dtype=dt)
    a0 = np.eye(2, dtype=dt)

    oa = orth(a)
    assert oa.dtype == orth(a0).dtype
    assert oa.shape == (0, 0)


class TestNullSpace:
    def test_null_space(self):
        rng = np.random.RandomState(1)

        dtypes = [np.float32, np.float64, np.complex64, np.complex128]
        sizes = [1, 2, 3, 10, 100]

        for dt, n in itertools.product(dtypes, sizes):
            X = np.ones((2, n), dtype=dt)

            eps = np.finfo(dt).eps
            tol = 1000 * eps

            Y = null_space(X)
            assert_equal(Y.shape, (n, n-1))
            assert_allclose(X @ Y, 0, atol=tol)

            Y = null_space(X.T)
            assert_equal(Y.shape, (2, 1))
            assert_allclose(X.T @ Y, 0, atol=tol)

            X = rng.randn(1 + n//2, n)
            Y = null_space(X)
            assert_equal(Y.shape, (n, n - 1 - n//2))
            assert_allclose(X @ Y, 0, atol=tol)

            if n > 5:
                rng = np.random.RandomState(1)
                X = rng.rand(n, 5) @ rng.rand(5, n)
                X = X + 1e-4 * rng.rand(n, 1) @ rng.rand(1, n)
                X = X.astype(dt)

                Y = null_space(X, rcond=1e-3)
                assert_equal(Y.shape, (n, n - 5))

                Y = null_space(X, rcond=1e-6)
                assert_equal(Y.shape, (n, n - 6))

    @pytest.mark.parametrize('dt', [int, float, np.float32, complex, np.complex64])
    def test_null_space_empty(self, dt):
        a = np.empty((0, 0), dtype=dt)
        a0 = np.eye(2, dtype=dt)
        nsa = null_space(a)

        assert nsa.shape == (0, 0)
        assert nsa.dtype == null_space(a0).dtype

    @pytest.mark.parametrize("overwrite_a", [True, False])
    @pytest.mark.parametrize("check_finite", [True, False])
    @pytest.mark.parametrize("lapack_driver", ["gesdd", "gesvd"])
    def test_null_space_options(self, overwrite_a, check_finite, lapack_driver):
        rng = np.random.default_rng(42887289350573064398746)
        n = 10
        X = rng.standard_normal((1 + n//2, n))
        Y = null_space(X.copy(), overwrite_a=overwrite_a, check_finite=check_finite,
                       lapack_driver=lapack_driver)
        assert_allclose(X @ Y, 0, atol=np.finfo(X.dtype).eps*100)


def test_subspace_angles():
    H = hadamard(8, float)
    A = H[:, :3]
    B = H[:, 3:]
    assert_allclose(subspace_angles(A, B), [np.pi / 2.] * 3, atol=1e-14)
    assert_allclose(subspace_angles(B, A), [np.pi / 2.] * 3, atol=1e-14)
    for x in (A, B):
        assert_allclose(subspace_angles(x, x), np.zeros(x.shape[1]),
                        atol=1e-14)
    # From MATLAB function "subspace", which effectively only returns the
    # last value that we calculate
    x = np.array(
        [[0.537667139546100, 0.318765239858981, 3.578396939725760, 0.725404224946106],  # noqa: E501
         [1.833885014595086, -1.307688296305273, 2.769437029884877, -0.063054873189656],  # noqa: E501
         [-2.258846861003648, -0.433592022305684, -1.349886940156521, 0.714742903826096],  # noqa: E501
         [0.862173320368121, 0.342624466538650, 3.034923466331855, -0.204966058299775]])  # noqa: E501
    expected = 1.481454682101605
    assert_allclose(subspace_angles(x[:, :2], x[:, 2:])[0], expected,
                    rtol=1e-12)
    assert_allclose(subspace_angles(x[:, 2:], x[:, :2])[0], expected,
                    rtol=1e-12)
    expected = 0.746361174247302
    assert_allclose(subspace_angles(x[:, :2], x[:, [2]]), expected, rtol=1e-12)
    assert_allclose(subspace_angles(x[:, [2]], x[:, :2]), expected, rtol=1e-12)
    expected = 0.487163718534313
    assert_allclose(subspace_angles(x[:, :3], x[:, [3]]), expected, rtol=1e-12)
    assert_allclose(subspace_angles(x[:, [3]], x[:, :3]), expected, rtol=1e-12)
    expected = 0.328950515907756
    assert_allclose(subspace_angles(x[:, :2], x[:, 1:]), [expected, 0],
                    atol=1e-12)
    # Degenerate conditions
    assert_raises(ValueError, subspace_angles, x[0], x)
    assert_raises(ValueError, subspace_angles, x, x[0])
    assert_raises(ValueError, subspace_angles, x[:-1], x)

    # Test branch if mask.any is True:
    A = np.array([[1, 0, 0],
                  [0, 1, 0],
                  [0, 0, 1],
                  [0, 0, 0],
                  [0, 0, 0]])
    B = np.array([[1, 0, 0],
                  [0, 1, 0],
                  [0, 0, 0],
                  [0, 0, 0],
                  [0, 0, 1]])
    expected = np.array([np.pi/2, 0, 0])
    assert_allclose(subspace_angles(A, B), expected, rtol=1e-12)

    # Complex
    # second column in "b" does not affect result, just there so that
    # b can have more cols than a, and vice-versa (both conditional code paths)
    a = [[1 + 1j], [0]]
    b = [[1 - 1j, 0], [0, 1]]
    assert_allclose(subspace_angles(a, b), 0., atol=1e-14)
    assert_allclose(subspace_angles(b, a), 0., atol=1e-14)

    # Empty
    a = np.empty((0, 0))
    b = np.empty((0, 0))
    assert_allclose(subspace_angles(a, b), np.empty((0,)))
    a = np.empty((2, 0))
    b = np.empty((2, 0))
    assert_allclose(subspace_angles(a, b), np.empty((0,)))
    a = np.empty((0, 2))
    b = np.empty((0, 3))
    assert_allclose(subspace_angles(a, b), np.empty((0,)))


class TestCDF2RDF:

    def matmul(self, a, b):
        return np.einsum('...ij,...jk->...ik', a, b)

    def assert_eig_valid(self, w, v, x):
        assert_array_almost_equal(
            self.matmul(v, w),
            self.matmul(x, v)
        )

    def test_single_array0x0real(self):
        # eig doesn't support 0x0 in old versions of numpy
        X = np.empty((0, 0))
        w, v = np.empty(0), np.empty((0, 0))
        wr, vr = cdf2rdf(w, v)
        self.assert_eig_valid(wr, vr, X)

    def test_single_array2x2_real(self):
        X = np.array([[1, 2], [3, -1]])
        w, v = np.linalg.eig(X)
        wr, vr = cdf2rdf(w, v)
        self.assert_eig_valid(wr, vr, X)

    def test_single_array2x2_complex(self):
        X = np.array([[1, 2], [-2, 1]])
        w, v = np.linalg.eig(X)
        wr, vr = cdf2rdf(w, v)
        self.assert_eig_valid(wr, vr, X)

    def test_single_array3x3_real(self):
        X = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6]])
        w, v = np.linalg.eig(X)
        wr, vr = cdf2rdf(w, v)
        self.assert_eig_valid(wr, vr, X)

    def test_single_array3x3_complex(self):
        X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]])
        w, v = np.linalg.eig(X)
        wr, vr = cdf2rdf(w, v)
        self.assert_eig_valid(wr, vr, X)

    def test_random_1d_stacked_arrays(self):
        # cannot test M == 0 due to bug in old numpy
        for M in range(1, 7):
            np.random.seed(999999999)
            X = np.random.rand(100, M, M)
            w, v = np.linalg.eig(X)
            wr, vr = cdf2rdf(w, v)
            self.assert_eig_valid(wr, vr, X)

    def test_random_2d_stacked_arrays(self):
        # cannot test M == 0 due to bug in old numpy
        for M in range(1, 7):
            X = np.random.rand(10, 10, M, M)
            w, v = np.linalg.eig(X)
            wr, vr = cdf2rdf(w, v)
            self.assert_eig_valid(wr, vr, X)

    def test_low_dimensionality_error(self):
        w, v = np.empty(()), np.array((2,))
        assert_raises(ValueError, cdf2rdf, w, v)

    def test_not_square_error(self):
        # Check that passing a non-square array raises a ValueError.
        w, v = np.arange(3), np.arange(6).reshape(3, 2)
        assert_raises(ValueError, cdf2rdf, w, v)

    def test_swapped_v_w_error(self):
        # Check that exchanging places of w and v raises ValueError.
        X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]])
        w, v = np.linalg.eig(X)
        assert_raises(ValueError, cdf2rdf, v, w)

    def test_non_associated_error(self):
        # Check that passing non-associated eigenvectors raises a ValueError.
        w, v = np.arange(3), np.arange(16).reshape(4, 4)
        assert_raises(ValueError, cdf2rdf, w, v)

    def test_not_conjugate_pairs(self):
        # Check that passing non-conjugate pairs raises a ValueError.
        X = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]])
        w, v = np.linalg.eig(X)
        assert_raises(ValueError, cdf2rdf, w, v)

        # different arrays in the stack, so not conjugate
        X = np.array([
            [[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]],
            [[1, 2, 3], [1, 2, 3], [2, 5, 6-1j]],
        ])
        w, v = np.linalg.eig(X)
        assert_raises(ValueError, cdf2rdf, w, v)