""" Test functions for linalg module

"""
import itertools
import os
import sys
import textwrap
import threading
import traceback
import warnings

import pytest

import numpy as np
from numpy import (
    array,
    asarray,
    atleast_2d,
    cdouble,
    csingle,
    dot,
    double,
    identity,
    inf,
    linalg,
    matmul,
    multiply,
    single,
)
from numpy._core import swapaxes
from numpy.exceptions import AxisError
from numpy.linalg import LinAlgError, matrix_power, matrix_rank, multi_dot, norm
from numpy.linalg._linalg import _multi_dot_matrix_chain_order
from numpy.testing import (
    HAS_LAPACK64,
    IS_WASM,
    NOGIL_BUILD,
    assert_,
    assert_allclose,
    assert_almost_equal,
    assert_array_equal,
    assert_equal,
    assert_raises,
    assert_raises_regex,
)
from numpy.testing._private.utils import run_subprocess

try:
    import numpy.linalg.lapack_lite
except ImportError:
    # May be broken when numpy was built without BLAS/LAPACK present
    # If so, ensure we don't break the whole test suite - the `lapack_lite`
    # submodule should be removed, it's only used in two tests in this file.
    pass


def consistent_subclass(out, in_):
    # For ndarray subclass input, our output should have the same subclass
    # (non-ndarray input gets converted to ndarray).
    return type(out) is (type(in_) if isinstance(in_, np.ndarray)
                         else np.ndarray)


old_assert_almost_equal = assert_almost_equal


def assert_almost_equal(a, b, single_decimal=6, double_decimal=12, **kw):
    if asarray(a).dtype.type in (single, csingle):
        decimal = single_decimal
    else:
        decimal = double_decimal
    old_assert_almost_equal(a, b, decimal=decimal, **kw)


def get_real_dtype(dtype):
    return {single: single, double: double,
            csingle: single, cdouble: double}[dtype]


def get_complex_dtype(dtype):
    return {single: csingle, double: cdouble,
            csingle: csingle, cdouble: cdouble}[dtype]


def get_rtol(dtype):
    # Choose a safe rtol
    if dtype in (single, csingle):
        return 1e-5
    else:
        return 1e-11


# used to categorize tests
all_tags = {
  'square', 'nonsquare', 'hermitian',  # mutually exclusive
  'generalized', 'size-0', 'strided'  # optional additions
}


class LinalgCase:
    def __init__(self, name, a, b, tags=set()):
        """
        A bundle of arguments to be passed to a test case, with an identifying
        name, the operands a and b, and a set of tags to filter the tests
        """
        assert_(isinstance(name, str))
        self.name = name
        self.a = a
        self.b = b
        self.tags = frozenset(tags)  # prevent shared tags

    def check(self, do):
        """
        Run the function `do` on this test case, expanding arguments
        """
        do(self.a, self.b, tags=self.tags)

    def __repr__(self):
        return f'<LinalgCase: {self.name}>'


def apply_tag(tag, cases):
    """
    Add the given tag (a string) to each of the cases (a list of LinalgCase
    objects)
    """
    assert tag in all_tags, "Invalid tag"
    for case in cases:
        case.tags = case.tags | {tag}
    return cases


#
# Base test cases
#

np.random.seed(1234)

CASES = []

# square test cases
CASES += apply_tag('square', [
    LinalgCase("single",
               array([[1., 2.], [3., 4.]], dtype=single),
               array([2., 1.], dtype=single)),
    LinalgCase("double",
               array([[1., 2.], [3., 4.]], dtype=double),
               array([2., 1.], dtype=double)),
    LinalgCase("double_2",
               array([[1., 2.], [3., 4.]], dtype=double),
               array([[2., 1., 4.], [3., 4., 6.]], dtype=double)),
    LinalgCase("csingle",
               array([[1. + 2j, 2 + 3j], [3 + 4j, 4 + 5j]], dtype=csingle),
               array([2. + 1j, 1. + 2j], dtype=csingle)),
    LinalgCase("cdouble",
               array([[1. + 2j, 2 + 3j], [3 + 4j, 4 + 5j]], dtype=cdouble),
               array([2. + 1j, 1. + 2j], dtype=cdouble)),
    LinalgCase("cdouble_2",
               array([[1. + 2j, 2 + 3j], [3 + 4j, 4 + 5j]], dtype=cdouble),
               array(
                   [[2. + 1j, 1. + 2j, 1 + 3j], [1 - 2j, 1 - 3j, 1 - 6j]],
                   dtype=cdouble,
               )),
    LinalgCase("0x0",
               np.empty((0, 0), dtype=double),
               np.empty((0,), dtype=double),
               tags={'size-0'}),
    LinalgCase("8x8",
               np.random.rand(8, 8),
               np.random.rand(8)),
    LinalgCase("1x1",
               np.random.rand(1, 1),
               np.random.rand(1)),
    LinalgCase("nonarray",
               [[1, 2], [3, 4]],
               [2, 1]),
])

# non-square test-cases
CASES += apply_tag('nonsquare', [
    LinalgCase("single_nsq_1",
               array([[1., 2., 3.], [3., 4., 6.]], dtype=single),
               array([2., 1.], dtype=single)),
    LinalgCase("single_nsq_2",
               array([[1., 2.], [3., 4.], [5., 6.]], dtype=single),
               array([2., 1., 3.], dtype=single)),
    LinalgCase("double_nsq_1",
               array([[1., 2., 3.], [3., 4., 6.]], dtype=double),
               array([2., 1.], dtype=double)),
    LinalgCase("double_nsq_2",
               array([[1., 2.], [3., 4.], [5., 6.]], dtype=double),
               array([2., 1., 3.], dtype=double)),
    LinalgCase("csingle_nsq_1",
               array(
                   [[1. + 1j, 2. + 2j, 3. - 3j], [3. - 5j, 4. + 9j, 6. + 2j]],
                   dtype=csingle,
               ),
               array([2. + 1j, 1. + 2j], dtype=csingle)),
    LinalgCase("csingle_nsq_2",
               array(
                    [[1. + 1j, 2. + 2j], [3. - 3j, 4. - 9j], [5. - 4j, 6. + 8j]],
                    dtype=csingle,
                ),
               array([2. + 1j, 1. + 2j, 3. - 3j], dtype=csingle)),
    LinalgCase("cdouble_nsq_1",
               array(
                    [[1. + 1j, 2. + 2j, 3. - 3j], [3. - 5j, 4. + 9j, 6. + 2j]],
                    dtype=cdouble,
                ),
               array([2. + 1j, 1. + 2j], dtype=cdouble)),
    LinalgCase("cdouble_nsq_2",
               array(
                    [[1. + 1j, 2. + 2j], [3. - 3j, 4. - 9j], [5. - 4j, 6. + 8j]],
                    dtype=cdouble,
                ),
               array([2. + 1j, 1. + 2j, 3. - 3j], dtype=cdouble)),
    LinalgCase("cdouble_nsq_1_2",
               array(
                    [[1. + 1j, 2. + 2j, 3. - 3j], [3. - 5j, 4. + 9j, 6. + 2j]],
                    dtype=cdouble,
                ),
               array([[2. + 1j, 1. + 2j], [1 - 1j, 2 - 2j]], dtype=cdouble)),
    LinalgCase("cdouble_nsq_2_2",
               array(
                    [[1. + 1j, 2. + 2j], [3. - 3j, 4. - 9j], [5. - 4j, 6. + 8j]],
                    dtype=cdouble,
                ),
               array(
                    [[2. + 1j, 1. + 2j], [1 - 1j, 2 - 2j], [1 - 1j, 2 - 2j]],
                    dtype=cdouble,
                )),
    LinalgCase("8x11",
               np.random.rand(8, 11),
               np.random.rand(8)),
    LinalgCase("1x5",
               np.random.rand(1, 5),
               np.random.rand(1)),
    LinalgCase("5x1",
               np.random.rand(5, 1),
               np.random.rand(5)),
    LinalgCase("0x4",
               np.random.rand(0, 4),
               np.random.rand(0),
               tags={'size-0'}),
    LinalgCase("4x0",
               np.random.rand(4, 0),
               np.random.rand(4),
               tags={'size-0'}),
])

# hermitian test-cases
CASES += apply_tag('hermitian', [
    LinalgCase("hsingle",
               array([[1., 2.], [2., 1.]], dtype=single),
               None),
    LinalgCase("hdouble",
               array([[1., 2.], [2., 1.]], dtype=double),
               None),
    LinalgCase("hcsingle",
               array([[1., 2 + 3j], [2 - 3j, 1]], dtype=csingle),
               None),
    LinalgCase("hcdouble",
               array([[1., 2 + 3j], [2 - 3j, 1]], dtype=cdouble),
               None),
    LinalgCase("hempty",
               np.empty((0, 0), dtype=double),
               None,
               tags={'size-0'}),
    LinalgCase("hnonarray",
               [[1, 2], [2, 1]],
               None),
    LinalgCase("matrix_b_only",
               array([[1., 2.], [2., 1.]]),
               None),
    LinalgCase("hmatrix_1x1",
               np.random.rand(1, 1),
               None),
])


#
# Gufunc test cases
#
def _make_generalized_cases():
    new_cases = []

    for case in CASES:
        if not isinstance(case.a, np.ndarray):
            continue

        a = np.array([case.a, 2 * case.a, 3 * case.a])
        if case.b is None:
            b = None
        elif case.b.ndim == 1:
            b = case.b
        else:
            b = np.array([case.b, 7 * case.b, 6 * case.b])
        new_case = LinalgCase(case.name + "_tile3", a, b,
                              tags=case.tags | {'generalized'})
        new_cases.append(new_case)

        a = np.array([case.a] * 2 * 3).reshape((3, 2) + case.a.shape)
        if case.b is None:
            b = None
        elif case.b.ndim == 1:
            b = np.array([case.b] * 2 * 3 * a.shape[-1])\
                  .reshape((3, 2) + case.a.shape[-2:])
        else:
            b = np.array([case.b] * 2 * 3).reshape((3, 2) + case.b.shape)
        new_case = LinalgCase(case.name + "_tile213", a, b,
                              tags=case.tags | {'generalized'})
        new_cases.append(new_case)

    return new_cases


CASES += _make_generalized_cases()


#
# Generate stride combination variations of the above
#
def _stride_comb_iter(x):
    """
    Generate cartesian product of strides for all axes
    """

    if not isinstance(x, np.ndarray):
        yield x, "nop"
        return

    stride_set = [(1,)] * x.ndim
    stride_set[-1] = (1, 3, -4)
    if x.ndim > 1:
        stride_set[-2] = (1, 3, -4)
    if x.ndim > 2:
        stride_set[-3] = (1, -4)

    for repeats in itertools.product(*tuple(stride_set)):
        new_shape = [abs(a * b) for a, b in zip(x.shape, repeats)]
        slices = tuple(slice(None, None, repeat) for repeat in repeats)

        # new array with different strides, but same data
        xi = np.empty(new_shape, dtype=x.dtype)
        xi.view(np.uint32).fill(0xdeadbeef)
        xi = xi[slices]
        xi[...] = x
        xi = xi.view(x.__class__)
        assert_(np.all(xi == x))
        yield xi, "stride_" + "_".join(f"{j:+}" for j in repeats)

        # generate also zero strides if possible
        if x.ndim >= 1 and x.shape[-1] == 1:
            s = list(x.strides)
            s[-1] = 0
            xi = np.lib.stride_tricks.as_strided(x, strides=s)
            yield xi, "stride_xxx_0"
        if x.ndim >= 2 and x.shape[-2] == 1:
            s = list(x.strides)
            s[-2] = 0
            xi = np.lib.stride_tricks.as_strided(x, strides=s)
            yield xi, "stride_xxx_0_x"
        if x.ndim >= 2 and x.shape[:-2] == (1, 1):
            s = list(x.strides)
            s[-1] = 0
            s[-2] = 0
            xi = np.lib.stride_tricks.as_strided(x, strides=s)
            yield xi, "stride_xxx_0_0"


def _make_strided_cases():
    new_cases = []
    for case in CASES:
        for a, a_label in _stride_comb_iter(case.a):
            for b, b_label in _stride_comb_iter(case.b):
                new_case = LinalgCase(case.name + "_" + a_label + "_" + b_label, a, b,
                                      tags=case.tags | {'strided'})
                new_cases.append(new_case)
    return new_cases


CASES += _make_strided_cases()


#
# Test different routines against the above cases
#
class LinalgTestCase:
    TEST_CASES = CASES

    def check_cases(self, require=set(), exclude=set()):
        """
        Run func on each of the cases with all of the tags in require, and none
        of the tags in exclude
        """
        for case in self.TEST_CASES:
            # filter by require and exclude
            if case.tags & require != require:
                continue
            if case.tags & exclude:
                continue

            try:
                case.check(self.do)
            except Exception as e:
                msg = f'In test case: {case!r}\n\n'
                msg += traceback.format_exc()
                raise AssertionError(msg) from e


class LinalgSquareTestCase(LinalgTestCase):

    def test_sq_cases(self):
        self.check_cases(require={'square'},
                         exclude={'generalized', 'size-0'})

    def test_empty_sq_cases(self):
        self.check_cases(require={'square', 'size-0'},
                         exclude={'generalized'})


class LinalgNonsquareTestCase(LinalgTestCase):

    def test_nonsq_cases(self):
        self.check_cases(require={'nonsquare'},
                         exclude={'generalized', 'size-0'})

    def test_empty_nonsq_cases(self):
        self.check_cases(require={'nonsquare', 'size-0'},
                         exclude={'generalized'})


class HermitianTestCase(LinalgTestCase):

    def test_herm_cases(self):
        self.check_cases(require={'hermitian'},
                         exclude={'generalized', 'size-0'})

    def test_empty_herm_cases(self):
        self.check_cases(require={'hermitian', 'size-0'},
                         exclude={'generalized'})


class LinalgGeneralizedSquareTestCase(LinalgTestCase):

    @pytest.mark.slow
    def test_generalized_sq_cases(self):
        self.check_cases(require={'generalized', 'square'},
                         exclude={'size-0'})

    @pytest.mark.slow
    def test_generalized_empty_sq_cases(self):
        self.check_cases(require={'generalized', 'square', 'size-0'})


class LinalgGeneralizedNonsquareTestCase(LinalgTestCase):

    @pytest.mark.slow
    def test_generalized_nonsq_cases(self):
        self.check_cases(require={'generalized', 'nonsquare'},
                         exclude={'size-0'})

    @pytest.mark.slow
    def test_generalized_empty_nonsq_cases(self):
        self.check_cases(require={'generalized', 'nonsquare', 'size-0'})


class HermitianGeneralizedTestCase(LinalgTestCase):

    @pytest.mark.slow
    def test_generalized_herm_cases(self):
        self.check_cases(require={'generalized', 'hermitian'},
                         exclude={'size-0'})

    @pytest.mark.slow
    def test_generalized_empty_herm_cases(self):
        self.check_cases(require={'generalized', 'hermitian', 'size-0'},
                         exclude={'none'})


def identity_like_generalized(a):
    a = asarray(a)
    if a.ndim >= 3:
        r = np.empty(a.shape, dtype=a.dtype)
        r[...] = identity(a.shape[-2])
        return r
    else:
        return identity(a.shape[0])


class SolveCases(LinalgSquareTestCase, LinalgGeneralizedSquareTestCase):
    # kept apart from TestSolve for use for testing with matrices.
    def do(self, a, b, tags):
        x = linalg.solve(a, b)
        if np.array(b).ndim == 1:
            # When a is (..., M, M) and b is (M,), it is the same as when b is
            # (M, 1), except the result has shape (..., M)
            adotx = matmul(a, x[..., None])[..., 0]
            assert_almost_equal(np.broadcast_to(b, adotx.shape), adotx)
        else:
            adotx = matmul(a, x)
            assert_almost_equal(b, adotx)
        assert_(consistent_subclass(x, b))


class TestSolve(SolveCases):
    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        assert_equal(linalg.solve(x, x).dtype, dtype)

    def test_1_d(self):
        class ArraySubclass(np.ndarray):
            pass
        a = np.arange(8).reshape(2, 2, 2)
        b = np.arange(2).view(ArraySubclass)
        result = linalg.solve(a, b)
        assert result.shape == (2, 2)

        # If b is anything other than 1-D it should be treated as a stack of
        # matrices
        b = np.arange(4).reshape(2, 2).view(ArraySubclass)
        result = linalg.solve(a, b)
        assert result.shape == (2, 2, 2)

        b = np.arange(2).reshape(1, 2).view(ArraySubclass)
        assert_raises(ValueError, linalg.solve, a, b)

    def test_0_size(self):
        class ArraySubclass(np.ndarray):
            pass
        # Test system of 0x0 matrices
        a = np.arange(8).reshape(2, 2, 2)
        b = np.arange(6).reshape(1, 2, 3).view(ArraySubclass)

        expected = linalg.solve(a, b)[:, 0:0, :]
        result = linalg.solve(a[:, 0:0, 0:0], b[:, 0:0, :])
        assert_array_equal(result, expected)
        assert_(isinstance(result, ArraySubclass))

        # Test errors for non-square and only b's dimension being 0
        assert_raises(linalg.LinAlgError, linalg.solve, a[:, 0:0, 0:1], b)
        assert_raises(ValueError, linalg.solve, a, b[:, 0:0, :])

        # Test broadcasting error
        b = np.arange(6).reshape(1, 3, 2)  # broadcasting error
        assert_raises(ValueError, linalg.solve, a, b)
        assert_raises(ValueError, linalg.solve, a[0:0], b[0:0])

        # Test zero "single equations" with 0x0 matrices.
        b = np.arange(2).view(ArraySubclass)
        expected = linalg.solve(a, b)[:, 0:0]
        result = linalg.solve(a[:, 0:0, 0:0], b[0:0])
        assert_array_equal(result, expected)
        assert_(isinstance(result, ArraySubclass))

        b = np.arange(3).reshape(1, 3)
        assert_raises(ValueError, linalg.solve, a, b)
        assert_raises(ValueError, linalg.solve, a[0:0], b[0:0])
        assert_raises(ValueError, linalg.solve, a[:, 0:0, 0:0], b)

    def test_0_size_k(self):
        # test zero multiple equation (K=0) case.
        class ArraySubclass(np.ndarray):
            pass
        a = np.arange(4).reshape(1, 2, 2)
        b = np.arange(6).reshape(3, 2, 1).view(ArraySubclass)

        expected = linalg.solve(a, b)[:, :, 0:0]
        result = linalg.solve(a, b[:, :, 0:0])
        assert_array_equal(result, expected)
        assert_(isinstance(result, ArraySubclass))

        # test both zero.
        expected = linalg.solve(a, b)[:, 0:0, 0:0]
        result = linalg.solve(a[:, 0:0, 0:0], b[:, 0:0, 0:0])
        assert_array_equal(result, expected)
        assert_(isinstance(result, ArraySubclass))


class InvCases(LinalgSquareTestCase, LinalgGeneralizedSquareTestCase):

    def do(self, a, b, tags):
        a_inv = linalg.inv(a)
        assert_almost_equal(matmul(a, a_inv),
                            identity_like_generalized(a))
        assert_(consistent_subclass(a_inv, a))


class TestInv(InvCases):
    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        assert_equal(linalg.inv(x).dtype, dtype)

    def test_0_size(self):
        # Check that all kinds of 0-sized arrays work
        class ArraySubclass(np.ndarray):
            pass
        a = np.zeros((0, 1, 1), dtype=np.int_).view(ArraySubclass)
        res = linalg.inv(a)
        assert_(res.dtype.type is np.float64)
        assert_equal(a.shape, res.shape)
        assert_(isinstance(res, ArraySubclass))

        a = np.zeros((0, 0), dtype=np.complex64).view(ArraySubclass)
        res = linalg.inv(a)
        assert_(res.dtype.type is np.complex64)
        assert_equal(a.shape, res.shape)
        assert_(isinstance(res, ArraySubclass))


class EigvalsCases(LinalgSquareTestCase, LinalgGeneralizedSquareTestCase):

    def do(self, a, b, tags):
        ev = linalg.eigvals(a)
        evalues, evectors = linalg.eig(a)
        assert_almost_equal(ev, evalues)


class TestEigvals(EigvalsCases):
    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))
        x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
        assert_equal(linalg.eigvals(x).dtype, get_complex_dtype(dtype))

    def test_0_size(self):
        # Check that all kinds of 0-sized arrays work
        class ArraySubclass(np.ndarray):
            pass
        a = np.zeros((0, 1, 1), dtype=np.int_).view(ArraySubclass)
        res = linalg.eigvals(a)
        assert_(res.dtype.type is np.complex128)
        assert_equal((0, 1), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(res, np.ndarray))

        a = np.zeros((0, 0), dtype=np.complex64).view(ArraySubclass)
        res = linalg.eigvals(a)
        assert_(res.dtype.type is np.complex64)
        assert_equal((0,), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(res, np.ndarray))


class EigCases(LinalgSquareTestCase, LinalgGeneralizedSquareTestCase):

    def do(self, a, b, tags):
        res = linalg.eig(a)
        eigenvalues, eigenvectors = res.eigenvalues, res.eigenvectors
        assert_allclose(matmul(a, eigenvectors),
            np.asarray(eigenvectors) * np.asarray(eigenvalues)[..., None, :],
            rtol=get_rtol(eigenvalues.dtype))
        assert_(consistent_subclass(eigenvectors, a))


class TestEig(EigCases):
    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        w, v = np.linalg.eig(x)
        assert_equal(w.dtype, get_complex_dtype(dtype))
        assert_equal(v.dtype, get_complex_dtype(dtype))

        x = np.array([[1, 0.5], [-1, 1]], dtype=dtype)
        w, v = np.linalg.eig(x)
        assert_equal(w.dtype, get_complex_dtype(dtype))
        assert_equal(v.dtype, get_complex_dtype(dtype))

    def test_0_size(self):
        # Check that all kinds of 0-sized arrays work
        class ArraySubclass(np.ndarray):
            pass
        a = np.zeros((0, 1, 1), dtype=np.int_).view(ArraySubclass)
        res, res_v = linalg.eig(a)
        assert_(res_v.dtype.type is np.complex128)
        assert_(res.dtype.type is np.complex128)
        assert_equal(a.shape, res_v.shape)
        assert_equal((0, 1), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(a, np.ndarray))

        a = np.zeros((0, 0), dtype=np.complex64).view(ArraySubclass)
        res, res_v = linalg.eig(a)
        assert_(res_v.dtype.type is np.complex64)
        assert_(res.dtype.type is np.complex64)
        assert_equal(a.shape, res_v.shape)
        assert_equal((0,), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(a, np.ndarray))


class SVDBaseTests:
    hermitian = False

    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        res = linalg.svd(x)
        U, S, Vh = res.U, res.S, res.Vh
        assert_equal(U.dtype, dtype)
        assert_equal(S.dtype, get_real_dtype(dtype))
        assert_equal(Vh.dtype, dtype)
        s = linalg.svd(x, compute_uv=False, hermitian=self.hermitian)
        assert_equal(s.dtype, get_real_dtype(dtype))


class SVDCases(LinalgSquareTestCase, LinalgGeneralizedSquareTestCase):

    def do(self, a, b, tags):
        u, s, vt = linalg.svd(a, False)
        assert_allclose(a, matmul(np.asarray(u) * np.asarray(s)[..., None, :],
                                           np.asarray(vt)),
                        rtol=get_rtol(u.dtype))
        assert_(consistent_subclass(u, a))
        assert_(consistent_subclass(vt, a))


class TestSVD(SVDCases, SVDBaseTests):
    def test_empty_identity(self):
        """ Empty input should put an identity matrix in u or vh """
        x = np.empty((4, 0))
        u, s, vh = linalg.svd(x, compute_uv=True, hermitian=self.hermitian)
        assert_equal(u.shape, (4, 4))
        assert_equal(vh.shape, (0, 0))
        assert_equal(u, np.eye(4))

        x = np.empty((0, 4))
        u, s, vh = linalg.svd(x, compute_uv=True, hermitian=self.hermitian)
        assert_equal(u.shape, (0, 0))
        assert_equal(vh.shape, (4, 4))
        assert_equal(vh, np.eye(4))

    def test_svdvals(self):
        x = np.array([[1, 0.5], [0.5, 1]])
        s_from_svd = linalg.svd(x, compute_uv=False, hermitian=self.hermitian)
        s_from_svdvals = linalg.svdvals(x)
        assert_almost_equal(s_from_svd, s_from_svdvals)


class SVDHermitianCases(HermitianTestCase, HermitianGeneralizedTestCase):

    def do(self, a, b, tags):
        u, s, vt = linalg.svd(a, False, hermitian=True)
        assert_allclose(a, matmul(np.asarray(u) * np.asarray(s)[..., None, :],
                                           np.asarray(vt)),
                        rtol=get_rtol(u.dtype))

        def hermitian(mat):
            axes = list(range(mat.ndim))
            axes[-1], axes[-2] = axes[-2], axes[-1]
            return np.conj(np.transpose(mat, axes=axes))

        expected = np.broadcast_to(np.eye(u.shape[-1]), u.shape)
        assert_almost_equal(np.matmul(u, hermitian(u)), expected)
        expected = np.broadcast_to(np.eye(vt.shape[-1]), vt.shape)
        assert_almost_equal(np.matmul(vt, hermitian(vt)), expected)
        assert_equal(np.sort(s)[..., ::-1], s)
        assert_(consistent_subclass(u, a))
        assert_(consistent_subclass(vt, a))


class TestSVDHermitian(SVDHermitianCases, SVDBaseTests):
    hermitian = True

    def test_singular(self):
        x = np.array([[1, 0], [0, 0]])
        u, _, vh = linalg.svd(x, hermitian=self.hermitian)
        assert_allclose(u @ u.T.conj(), np.eye(2), rtol=1e-14)
        assert_allclose(vh @ vh.T.conj(), np.eye(2), rtol=1e-14)


class CondCases(LinalgSquareTestCase, LinalgGeneralizedSquareTestCase):
    # cond(x, p) for p in (None, 2, -2)

    def do(self, a, b, tags):
        c = asarray(a)  # a might be a matrix
        if 'size-0' in tags:
            assert_raises(LinAlgError, linalg.cond, c)
            return

        # +-2 norms
        s = linalg.svd(c, compute_uv=False)
        assert_almost_equal(
            linalg.cond(a), s[..., 0] / s[..., -1],
            single_decimal=5, double_decimal=11)
        assert_almost_equal(
            linalg.cond(a, 2), s[..., 0] / s[..., -1],
            single_decimal=5, double_decimal=11)
        assert_almost_equal(
            linalg.cond(a, -2), s[..., -1] / s[..., 0],
            single_decimal=5, double_decimal=11)

        # Other norms
        cinv = np.linalg.inv(c)
        assert_almost_equal(
            linalg.cond(a, 1),
            abs(c).sum(-2).max(-1) * abs(cinv).sum(-2).max(-1),
            single_decimal=5, double_decimal=11)
        assert_almost_equal(
            linalg.cond(a, -1),
            abs(c).sum(-2).min(-1) * abs(cinv).sum(-2).min(-1),
            single_decimal=5, double_decimal=11)
        assert_almost_equal(
            linalg.cond(a, np.inf),
            abs(c).sum(-1).max(-1) * abs(cinv).sum(-1).max(-1),
            single_decimal=5, double_decimal=11)
        assert_almost_equal(
            linalg.cond(a, -np.inf),
            abs(c).sum(-1).min(-1) * abs(cinv).sum(-1).min(-1),
            single_decimal=5, double_decimal=11)
        assert_almost_equal(
            linalg.cond(a, 'fro'),
            np.sqrt((abs(c)**2).sum(-1).sum(-1)
                    * (abs(cinv)**2).sum(-1).sum(-1)),
            single_decimal=5, double_decimal=11)


class TestCond(CondCases):
    @pytest.mark.parametrize('is_complex', [False, True])
    def test_basic_nonsvd(self, is_complex):
        # Smoketest the non-svd norms
        A = array([[1., 0, 1], [0, -2., 0], [0, 0, 3.]])
        if is_complex:
            # Since A is linearly scaled, the condition number should not change
            A = A * (1 + 1j)
        assert_almost_equal(linalg.cond(A, inf), 4)
        assert_almost_equal(linalg.cond(A, -inf), 2 / 3)
        assert_almost_equal(linalg.cond(A, 1), 4)
        assert_almost_equal(linalg.cond(A, -1), 0.5)
        assert_almost_equal(linalg.cond(A, 'fro'), np.sqrt(265 / 12))

    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    @pytest.mark.parametrize('norm_ord', [1, -1, 2, -2, 'fro', np.inf, -np.inf])
    def test_cond_dtypes(self, dtype, norm_ord):
        # Check that the condition number is computed in the same dtype
        # as the input matrix
        A = array([[1., 0, 1], [0, -2., 0], [0, 0, 3.]], dtype=dtype)
        out_type = get_real_dtype(dtype)
        assert_equal(linalg.cond(A, p=norm_ord).dtype, out_type)

    def test_singular(self):
        # Singular matrices have infinite condition number for
        # positive norms, and negative norms shouldn't raise
        # exceptions
        As = [np.zeros((2, 2)), np.ones((2, 2))]
        p_pos = [None, 1, 2, 'fro']
        p_neg = [-1, -2]
        for A, p in itertools.product(As, p_pos):
            # Inversion may not hit exact infinity, so just check the
            # number is large
            assert_(linalg.cond(A, p) > 1e15)
        for A, p in itertools.product(As, p_neg):
            linalg.cond(A, p)

    @pytest.mark.xfail(True, run=False,
                       reason="Platform/LAPACK-dependent failure, "
                              "see gh-18914")
    def test_nan(self):
        # nans should be passed through, not converted to infs
        ps = [None, 1, -1, 2, -2, 'fro']
        p_pos = [None, 1, 2, 'fro']

        A = np.ones((2, 2))
        A[0, 1] = np.nan
        for p in ps:
            c = linalg.cond(A, p)
            assert_(isinstance(c, np.float64))
            assert_(np.isnan(c))

        A = np.ones((3, 2, 2))
        A[1, 0, 1] = np.nan
        for p in ps:
            c = linalg.cond(A, p)
            assert_(np.isnan(c[1]))
            if p in p_pos:
                assert_(c[0] > 1e15)
                assert_(c[2] > 1e15)
            else:
                assert_(not np.isnan(c[0]))
                assert_(not np.isnan(c[2]))

    def test_stacked_singular(self):
        # Check behavior when only some of the stacked matrices are
        # singular
        np.random.seed(1234)
        A = np.random.rand(2, 2, 2, 2)
        A[0, 0] = 0
        A[1, 1] = 0

        for p in (None, 1, 2, 'fro', -1, -2):
            c = linalg.cond(A, p)
            assert_equal(c[0, 0], np.inf)
            assert_equal(c[1, 1], np.inf)
            assert_(np.isfinite(c[0, 1]))
            assert_(np.isfinite(c[1, 0]))


class PinvCases(LinalgSquareTestCase,
                LinalgNonsquareTestCase,
                LinalgGeneralizedSquareTestCase,
                LinalgGeneralizedNonsquareTestCase):

    def do(self, a, b, tags):
        a_ginv = linalg.pinv(a)
        # `a @ a_ginv == I` does not hold if a is singular
        dot = matmul
        result = dot(dot(a, a_ginv), a)
        assert_almost_equal(result, a, single_decimal=5, double_decimal=11)
        assert_(consistent_subclass(a_ginv, a))


class TestPinv(PinvCases):
    pass


class PinvHermitianCases(HermitianTestCase, HermitianGeneralizedTestCase):

    def do(self, a, b, tags):
        a_ginv = linalg.pinv(a, hermitian=True)
        # `a @ a_ginv == I` does not hold if a is singular
        dot = matmul
        result = dot(dot(a, a_ginv), a)
        assert_almost_equal(result, a, single_decimal=5, double_decimal=11)
        assert_(consistent_subclass(a_ginv, a))


class TestPinvHermitian(PinvHermitianCases):
    pass


def test_pinv_rtol_arg():
    a = np.array([[1, 2, 3], [4, 1, 1], [2, 3, 1]])

    assert_almost_equal(
        np.linalg.pinv(a, rcond=0.5),
        np.linalg.pinv(a, rtol=0.5),
    )

    with pytest.raises(
        ValueError, match=r"`rtol` and `rcond` can't be both set."
    ):
        np.linalg.pinv(a, rcond=0.5, rtol=0.5)


class DetCases(LinalgSquareTestCase, LinalgGeneralizedSquareTestCase):

    def do(self, a, b, tags):
        d = linalg.det(a)
        res = linalg.slogdet(a)
        s, ld = res.sign, res.logabsdet
        if asarray(a).dtype.type in (single, double):
            ad = asarray(a).astype(double)
        else:
            ad = asarray(a).astype(cdouble)
        ev = linalg.eigvals(ad)
        assert_almost_equal(d, multiply.reduce(ev, axis=-1))
        assert_almost_equal(s * np.exp(ld), multiply.reduce(ev, axis=-1))

        s = np.atleast_1d(s)
        ld = np.atleast_1d(ld)
        m = (s != 0)
        assert_almost_equal(np.abs(s[m]), 1)
        assert_equal(ld[~m], -inf)


class TestDet(DetCases):
    def test_zero(self):
        assert_equal(linalg.det([[0.0]]), 0.0)
        assert_equal(type(linalg.det([[0.0]])), double)
        assert_equal(linalg.det([[0.0j]]), 0.0)
        assert_equal(type(linalg.det([[0.0j]])), cdouble)

        assert_equal(linalg.slogdet([[0.0]]), (0.0, -inf))
        assert_equal(type(linalg.slogdet([[0.0]])[0]), double)
        assert_equal(type(linalg.slogdet([[0.0]])[1]), double)
        assert_equal(linalg.slogdet([[0.0j]]), (0.0j, -inf))
        assert_equal(type(linalg.slogdet([[0.0j]])[0]), cdouble)
        assert_equal(type(linalg.slogdet([[0.0j]])[1]), double)

    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        assert_equal(np.linalg.det(x).dtype, dtype)
        ph, s = np.linalg.slogdet(x)
        assert_equal(s.dtype, get_real_dtype(dtype))
        assert_equal(ph.dtype, dtype)

    def test_0_size(self):
        a = np.zeros((0, 0), dtype=np.complex64)
        res = linalg.det(a)
        assert_equal(res, 1.)
        assert_(res.dtype.type is np.complex64)
        res = linalg.slogdet(a)
        assert_equal(res, (1, 0))
        assert_(res[0].dtype.type is np.complex64)
        assert_(res[1].dtype.type is np.float32)

        a = np.zeros((0, 0), dtype=np.float64)
        res = linalg.det(a)
        assert_equal(res, 1.)
        assert_(res.dtype.type is np.float64)
        res = linalg.slogdet(a)
        assert_equal(res, (1, 0))
        assert_(res[0].dtype.type is np.float64)
        assert_(res[1].dtype.type is np.float64)


class LstsqCases(LinalgSquareTestCase, LinalgNonsquareTestCase):

    def do(self, a, b, tags):
        arr = np.asarray(a)
        m, n = arr.shape
        u, s, vt = linalg.svd(a, False)
        x, residuals, rank, sv = linalg.lstsq(a, b, rcond=-1)
        if m == 0:
            assert_((x == 0).all())
        if m <= n:
            assert_almost_equal(b, dot(a, x))
            assert_equal(rank, m)
        else:
            assert_equal(rank, n)
        assert_almost_equal(sv, sv.__array_wrap__(s))
        if rank == n and m > n:
            expect_resids = (
                np.asarray(abs(np.dot(a, x) - b)) ** 2).sum(axis=0)
            expect_resids = np.asarray(expect_resids)
            if np.asarray(b).ndim == 1:
                expect_resids = expect_resids.reshape((1,))
            assert_equal(residuals.shape, expect_resids.shape)
        else:
            expect_resids = np.array([]).view(type(x))
        assert_almost_equal(residuals, expect_resids)
        assert_(np.issubdtype(residuals.dtype, np.floating))
        assert_(consistent_subclass(x, b))
        assert_(consistent_subclass(residuals, b))


class TestLstsq(LstsqCases):
    def test_rcond(self):
        a = np.array([[0., 1.,  0.,  1.,  2.,  0.],
                      [0., 2.,  0.,  0.,  1.,  0.],
                      [1., 0.,  1.,  0.,  0.,  4.],
                      [0., 0.,  0.,  2.,  3.,  0.]]).T

        b = np.array([1, 0, 0, 0, 0, 0])

        x, residuals, rank, s = linalg.lstsq(a, b, rcond=-1)
        assert_(rank == 4)
        x, residuals, rank, s = linalg.lstsq(a, b)
        assert_(rank == 3)
        x, residuals, rank, s = linalg.lstsq(a, b, rcond=None)
        assert_(rank == 3)

    @pytest.mark.parametrize(["m", "n", "n_rhs"], [
        (4, 2, 2),
        (0, 4, 1),
        (0, 4, 2),
        (4, 0, 1),
        (4, 0, 2),
        (4, 2, 0),
        (0, 0, 0)
    ])
    def test_empty_a_b(self, m, n, n_rhs):
        a = np.arange(m * n).reshape(m, n)
        b = np.ones((m, n_rhs))
        x, residuals, rank, s = linalg.lstsq(a, b, rcond=None)
        if m == 0:
            assert_((x == 0).all())
        assert_equal(x.shape, (n, n_rhs))
        assert_equal(residuals.shape, ((n_rhs,) if m > n else (0,)))
        if m > n and n_rhs > 0:
            # residuals are exactly the squared norms of b's columns
            r = b - np.dot(a, x)
            assert_almost_equal(residuals, (r * r).sum(axis=-2))
        assert_equal(rank, min(m, n))
        assert_equal(s.shape, (min(m, n),))

    def test_incompatible_dims(self):
        # use modified version of docstring example
        x = np.array([0, 1, 2, 3])
        y = np.array([-1, 0.2, 0.9, 2.1, 3.3])
        A = np.vstack([x, np.ones(len(x))]).T
        with assert_raises_regex(LinAlgError, "Incompatible dimensions"):
            linalg.lstsq(A, y, rcond=None)


@pytest.mark.parametrize('dt', [np.dtype(c) for c in '?bBhHiIqQefdgFDGO'])
class TestMatrixPower:

    rshft_0 = np.eye(4)
    rshft_1 = rshft_0[[3, 0, 1, 2]]
    rshft_2 = rshft_0[[2, 3, 0, 1]]
    rshft_3 = rshft_0[[1, 2, 3, 0]]
    rshft_all = [rshft_0, rshft_1, rshft_2, rshft_3]
    noninv = array([[1, 0], [0, 0]])
    stacked = np.block([[[rshft_0]]] * 2)
    # FIXME the 'e' dtype might work in future
    dtnoinv = [object, np.dtype('e'), np.dtype('g'), np.dtype('G')]

    def test_large_power(self, dt):
        rshft = self.rshft_1.astype(dt)
        assert_equal(
            matrix_power(rshft, 2**100 + 2**10 + 2**5 + 0), self.rshft_0)
        assert_equal(
            matrix_power(rshft, 2**100 + 2**10 + 2**5 + 1), self.rshft_1)
        assert_equal(
            matrix_power(rshft, 2**100 + 2**10 + 2**5 + 2), self.rshft_2)
        assert_equal(
            matrix_power(rshft, 2**100 + 2**10 + 2**5 + 3), self.rshft_3)

    def test_power_is_zero(self, dt):
        def tz(M):
            mz = matrix_power(M, 0)
            assert_equal(mz, identity_like_generalized(M))
            assert_equal(mz.dtype, M.dtype)

        for mat in self.rshft_all:
            tz(mat.astype(dt))
            if np.dtype(dt).type is not np.object_:
                tz(self.stacked.astype(dt))

    def test_power_is_one(self, dt):
        def tz(mat):
            mz = matrix_power(mat, 1)
            assert_equal(mz, mat)
            assert_equal(mz.dtype, mat.dtype)

        for mat in self.rshft_all:
            tz(mat.astype(dt))
            if np.dtype(dt).type is not np.object_:
                tz(self.stacked.astype(dt))

    def test_power_is_two(self, dt):
        def tz(mat):
            mz = matrix_power(mat, 2)
            mmul = matmul if mat.dtype != object else dot
            assert_equal(mz, mmul(mat, mat))
            assert_equal(mz.dtype, mat.dtype)

        for mat in self.rshft_all:
            tz(mat.astype(dt))
            if np.dtype(dt).type is not np.object_:
                tz(self.stacked.astype(dt))

    def test_power_is_minus_one(self, dt):
        def tz(mat):
            invmat = matrix_power(mat, -1)
            mmul = matmul if mat.dtype != object else dot
            assert_almost_equal(
                mmul(invmat, mat), identity_like_generalized(mat))

        for mat in self.rshft_all:
            if dt not in self.dtnoinv:
                tz(mat.astype(dt))

    def test_exceptions_bad_power(self, dt):
        mat = self.rshft_0.astype(dt)
        assert_raises(TypeError, matrix_power, mat, 1.5)
        assert_raises(TypeError, matrix_power, mat, [1])

    def test_exceptions_non_square(self, dt):
        assert_raises(LinAlgError, matrix_power, np.array([1], dt), 1)
        assert_raises(LinAlgError, matrix_power, np.array([[1], [2]], dt), 1)
        assert_raises(LinAlgError, matrix_power, np.ones((4, 3, 2), dt), 1)

    @pytest.mark.skipif(IS_WASM, reason="fp errors don't work in wasm")
    def test_exceptions_not_invertible(self, dt):
        if dt in self.dtnoinv:
            return
        mat = self.noninv.astype(dt)
        assert_raises(LinAlgError, matrix_power, mat, -1)


class TestEigvalshCases(HermitianTestCase, HermitianGeneralizedTestCase):

    def do(self, a, b, tags):
        # note that eigenvalue arrays returned by eig must be sorted since
        # their order isn't guaranteed.
        ev = linalg.eigvalsh(a, 'L')
        evalues, evectors = linalg.eig(a)
        evalues.sort(axis=-1)
        assert_allclose(ev, evalues, rtol=get_rtol(ev.dtype))

        ev2 = linalg.eigvalsh(a, 'U')
        assert_allclose(ev2, evalues, rtol=get_rtol(ev.dtype))


class TestEigvalsh:
    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        w = np.linalg.eigvalsh(x)
        assert_equal(w.dtype, get_real_dtype(dtype))

    def test_invalid(self):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
        assert_raises(ValueError, np.linalg.eigvalsh, x, UPLO="lrong")
        assert_raises(ValueError, np.linalg.eigvalsh, x, "lower")
        assert_raises(ValueError, np.linalg.eigvalsh, x, "upper")

    def test_UPLO(self):
        Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
        Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
        tgt = np.array([-1, 1], dtype=np.double)
        rtol = get_rtol(np.double)

        # Check default is 'L'
        w = np.linalg.eigvalsh(Klo)
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'L'
        w = np.linalg.eigvalsh(Klo, UPLO='L')
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'l'
        w = np.linalg.eigvalsh(Klo, UPLO='l')
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'U'
        w = np.linalg.eigvalsh(Kup, UPLO='U')
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'u'
        w = np.linalg.eigvalsh(Kup, UPLO='u')
        assert_allclose(w, tgt, rtol=rtol)

    def test_0_size(self):
        # Check that all kinds of 0-sized arrays work
        class ArraySubclass(np.ndarray):
            pass
        a = np.zeros((0, 1, 1), dtype=np.int_).view(ArraySubclass)
        res = linalg.eigvalsh(a)
        assert_(res.dtype.type is np.float64)
        assert_equal((0, 1), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(res, np.ndarray))

        a = np.zeros((0, 0), dtype=np.complex64).view(ArraySubclass)
        res = linalg.eigvalsh(a)
        assert_(res.dtype.type is np.float32)
        assert_equal((0,), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(res, np.ndarray))


class TestEighCases(HermitianTestCase, HermitianGeneralizedTestCase):

    def do(self, a, b, tags):
        # note that eigenvalue arrays returned by eig must be sorted since
        # their order isn't guaranteed.
        res = linalg.eigh(a)
        ev, evc = res.eigenvalues, res.eigenvectors
        evalues, evectors = linalg.eig(a)
        evalues.sort(axis=-1)
        assert_almost_equal(ev, evalues)

        assert_allclose(matmul(a, evc),
                        np.asarray(ev)[..., None, :] * np.asarray(evc),
                        rtol=get_rtol(ev.dtype))

        ev2, evc2 = linalg.eigh(a, 'U')
        assert_almost_equal(ev2, evalues)

        assert_allclose(matmul(a, evc2),
                        np.asarray(ev2)[..., None, :] * np.asarray(evc2),
                        rtol=get_rtol(ev.dtype), err_msg=repr(a))


class TestEigh:
    @pytest.mark.parametrize('dtype', [single, double, csingle, cdouble])
    def test_types(self, dtype):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
        w, v = np.linalg.eigh(x)
        assert_equal(w.dtype, get_real_dtype(dtype))
        assert_equal(v.dtype, dtype)

    def test_invalid(self):
        x = np.array([[1, 0.5], [0.5, 1]], dtype=np.float32)
        assert_raises(ValueError, np.linalg.eigh, x, UPLO="lrong")
        assert_raises(ValueError, np.linalg.eigh, x, "lower")
        assert_raises(ValueError, np.linalg.eigh, x, "upper")

    def test_UPLO(self):
        Klo = np.array([[0, 0], [1, 0]], dtype=np.double)
        Kup = np.array([[0, 1], [0, 0]], dtype=np.double)
        tgt = np.array([-1, 1], dtype=np.double)
        rtol = get_rtol(np.double)

        # Check default is 'L'
        w, v = np.linalg.eigh(Klo)
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'L'
        w, v = np.linalg.eigh(Klo, UPLO='L')
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'l'
        w, v = np.linalg.eigh(Klo, UPLO='l')
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'U'
        w, v = np.linalg.eigh(Kup, UPLO='U')
        assert_allclose(w, tgt, rtol=rtol)
        # Check 'u'
        w, v = np.linalg.eigh(Kup, UPLO='u')
        assert_allclose(w, tgt, rtol=rtol)

    def test_0_size(self):
        # Check that all kinds of 0-sized arrays work
        class ArraySubclass(np.ndarray):
            pass
        a = np.zeros((0, 1, 1), dtype=np.int_).view(ArraySubclass)
        res, res_v = linalg.eigh(a)
        assert_(res_v.dtype.type is np.float64)
        assert_(res.dtype.type is np.float64)
        assert_equal(a.shape, res_v.shape)
        assert_equal((0, 1), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(a, np.ndarray))

        a = np.zeros((0, 0), dtype=np.complex64).view(ArraySubclass)
        res, res_v = linalg.eigh(a)
        assert_(res_v.dtype.type is np.complex64)
        assert_(res.dtype.type is np.float32)
        assert_equal(a.shape, res_v.shape)
        assert_equal((0,), res.shape)
        # This is just for documentation, it might make sense to change:
        assert_(isinstance(a, np.ndarray))


class _TestNormBase:
    dt = None
    dec = None

    @staticmethod
    def check_dtype(x, res):
        if issubclass(x.dtype.type, np.inexact):
            assert_equal(res.dtype, x.real.dtype)
        else:
            # For integer input, don't have to test float precision of output.
            assert_(issubclass(res.dtype.type, np.floating))


class _TestNormGeneral(_TestNormBase):

    def test_empty(self):
        assert_equal(norm([]), 0.0)
        assert_equal(norm(array([], dtype=self.dt)), 0.0)
        assert_equal(norm(atleast_2d(array([], dtype=self.dt))), 0.0)

    def test_vector_return_type(self):
        a = np.array([1, 0, 1])

        exact_types = np.typecodes['AllInteger']
        inexact_types = np.typecodes['AllFloat']

        all_types = exact_types + inexact_types

        for each_type in all_types:
            at = a.astype(each_type)

            an = norm(at, -np.inf)
            self.check_dtype(at, an)
            assert_almost_equal(an, 0.0)

            with warnings.catch_warnings():
                warnings.filterwarnings(
                    'ignore', "divide by zero encountered", RuntimeWarning)
                an = norm(at, -1)
                self.check_dtype(at, an)
                assert_almost_equal(an, 0.0)

            an = norm(at, 0)
            self.check_dtype(at, an)
            assert_almost_equal(an, 2)

            an = norm(at, 1)
            self.check_dtype(at, an)
            assert_almost_equal(an, 2.0)

            an = norm(at, 2)
            self.check_dtype(at, an)
            assert_almost_equal(an, an.dtype.type(2.0)**an.dtype.type(1.0 / 2.0))

            an = norm(at, 4)
            self.check_dtype(at, an)
            assert_almost_equal(an, an.dtype.type(2.0)**an.dtype.type(1.0 / 4.0))

            an = norm(at, np.inf)
            self.check_dtype(at, an)
            assert_almost_equal(an, 1.0)

    def test_vector(self):
        a = [1, 2, 3, 4]
        b = [-1, -2, -3, -4]
        c = [-1, 2, -3, 4]

        def _test(v):
            np.testing.assert_almost_equal(norm(v), 30 ** 0.5,
                                           decimal=self.dec)
            np.testing.assert_almost_equal(norm(v, inf), 4.0,
                                           decimal=self.dec)
            np.testing.assert_almost_equal(norm(v, -inf), 1.0,
                                           decimal=self.dec)
            np.testing.assert_almost_equal(norm(v, 1), 10.0,
                                           decimal=self.dec)
            np.testing.assert_almost_equal(norm(v, -1), 12.0 / 25,
                                           decimal=self.dec)
            np.testing.assert_almost_equal(norm(v, 2), 30 ** 0.5,
                                           decimal=self.dec)
            np.testing.assert_almost_equal(norm(v, -2), ((205. / 144) ** -0.5),
                                           decimal=self.dec)
            np.testing.assert_almost_equal(norm(v, 0), 4,
                                           decimal=self.dec)

        for v in (a, b, c,):
            _test(v)

        for v in (array(a, dtype=self.dt), array(b, dtype=self.dt),
                  array(c, dtype=self.dt)):
            _test(v)

    def test_axis(self):
        # Vector norms.
        # Compare the use of `axis` with computing the norm of each row
        # or column separately.
        A = array([[1, 2, 3], [4, 5, 6]], dtype=self.dt)
        for order in [None, -1, 0, 1, 2, 3, np.inf, -np.inf]:
            expected0 = [norm(A[:, k], ord=order) for k in range(A.shape[1])]
            assert_almost_equal(norm(A, ord=order, axis=0), expected0)
            expected1 = [norm(A[k, :], ord=order) for k in range(A.shape[0])]
            assert_almost_equal(norm(A, ord=order, axis=1), expected1)

        # Matrix norms.
        B = np.arange(1, 25, dtype=self.dt).reshape(2, 3, 4)
        nd = B.ndim
        for order in [None, -2, 2, -1, 1, np.inf, -np.inf, 'fro']:
            for axis in itertools.combinations(range(-nd, nd), 2):
                row_axis, col_axis = axis
                if row_axis < 0:
                    row_axis += nd
                if col_axis < 0:
                    col_axis += nd
                if row_axis == col_axis:
                    assert_raises(ValueError, norm, B, ord=order, axis=axis)
                else:
                    n = norm(B, ord=order, axis=axis)

                    # The logic using k_index only works for nd = 3.
                    # This has to be changed if nd is increased.
                    k_index = nd - (row_axis + col_axis)
                    if row_axis < col_axis:
                        expected = [norm(B[:].take(k, axis=k_index), ord=order)
                                    for k in range(B.shape[k_index])]
                    else:
                        expected = [norm(B[:].take(k, axis=k_index).T, ord=order)
                                    for k in range(B.shape[k_index])]
                    assert_almost_equal(n, expected)

    def test_keepdims(self):
        A = np.arange(1, 25, dtype=self.dt).reshape(2, 3, 4)

        allclose_err = 'order {0}, axis = {1}'
        shape_err = 'Shape mismatch found {0}, expected {1}, order={2}, axis={3}'

        # check the order=None, axis=None case
        expected = norm(A, ord=None, axis=None)
        found = norm(A, ord=None, axis=None, keepdims=True)
        assert_allclose(np.squeeze(found), expected,
                        err_msg=allclose_err.format(None, None))
        expected_shape = (1, 1, 1)
        assert_(found.shape == expected_shape,
                shape_err.format(found.shape, expected_shape, None, None))

        # Vector norms.
        for order in [None, -1, 0, 1, 2, 3, np.inf, -np.inf]:
            for k in range(A.ndim):
                expected = norm(A, ord=order, axis=k)
                found = norm(A, ord=order, axis=k, keepdims=True)
                assert_allclose(np.squeeze(found), expected,
                                err_msg=allclose_err.format(order, k))
                expected_shape = list(A.shape)
                expected_shape[k] = 1
                expected_shape = tuple(expected_shape)
                assert_(found.shape == expected_shape,
                        shape_err.format(found.shape, expected_shape, order, k))

        # Matrix norms.
        for order in [None, -2, 2, -1, 1, np.inf, -np.inf, 'fro', 'nuc']:
            for k in itertools.permutations(range(A.ndim), 2):
                expected = norm(A, ord=order, axis=k)
                found = norm(A, ord=order, axis=k, keepdims=True)
                assert_allclose(np.squeeze(found), expected,
                                err_msg=allclose_err.format(order, k))
                expected_shape = list(A.shape)
                expected_shape[k[0]] = 1
                expected_shape[k[1]] = 1
                expected_shape = tuple(expected_shape)
                assert_(found.shape == expected_shape,
                        shape_err.format(found.shape, expected_shape, order, k))


class _TestNorm2D(_TestNormBase):
    # Define the part for 2d arrays separately, so we can subclass this
    # and run the tests using np.matrix in matrixlib.tests.test_matrix_linalg.
    array = np.array

    def test_matrix_empty(self):
        assert_equal(norm(self.array([[]], dtype=self.dt)), 0.0)

    def test_matrix_return_type(self):
        a = self.array([[1, 0, 1], [0, 1, 1]])

        exact_types = np.typecodes['AllInteger']

        # float32, complex64, float64, complex128 types are the only types
        # allowed by `linalg`, which performs the matrix operations used
        # within `norm`.
        inexact_types = 'fdFD'

        all_types = exact_types + inexact_types

        for each_type in all_types:
            at = a.astype(each_type)

            an = norm(at, -np.inf)
            self.check_dtype(at, an)
            assert_almost_equal(an, 2.0)

            with warnings.catch_warnings():
                warnings.filterwarnings(
                    'ignore', "divide by zero encountered", RuntimeWarning)
                an = norm(at, -1)
                self.check_dtype(at, an)
                assert_almost_equal(an, 1.0)

            an = norm(at, 1)
            self.check_dtype(at, an)
            assert_almost_equal(an, 2.0)

            an = norm(at, 2)
            self.check_dtype(at, an)
            assert_almost_equal(an, 3.0**(1.0 / 2.0))

            an = norm(at, -2)
            self.check_dtype(at, an)
            assert_almost_equal(an, 1.0)

            an = norm(at, np.inf)
            self.check_dtype(at, an)
            assert_almost_equal(an, 2.0)

            an = norm(at, 'fro')
            self.check_dtype(at, an)
            assert_almost_equal(an, 2.0)

            an = norm(at, 'nuc')
            self.check_dtype(at, an)
            # Lower bar needed to support low precision floats.
            # They end up being off by 1 in the 7th place.
            np.testing.assert_almost_equal(an, 2.7320508075688772, decimal=6)

    def test_matrix_2x2(self):
        A = self.array([[1, 3], [5, 7]], dtype=self.dt)
        assert_almost_equal(norm(A), 84 ** 0.5)
        assert_almost_equal(norm(A, 'fro'), 84 ** 0.5)
        assert_almost_equal(norm(A, 'nuc'), 10.0)
        assert_almost_equal(norm(A, inf), 12.0)
        assert_almost_equal(norm(A, -inf), 4.0)
        assert_almost_equal(norm(A, 1), 10.0)
        assert_almost_equal(norm(A, -1), 6.0)
        assert_almost_equal(norm(A, 2), 9.1231056256176615)
        assert_almost_equal(norm(A, -2), 0.87689437438234041)

        assert_raises(ValueError, norm, A, 'nofro')
        assert_raises(ValueError, norm, A, -3)
        assert_raises(ValueError, norm, A, 0)

    def test_matrix_3x3(self):
        # This test has been added because the 2x2 example
        # happened to have equal nuclear norm and induced 1-norm.
        # The 1/10 scaling factor accommodates the absolute tolerance
        # used in assert_almost_equal.
        A = (1 / 10) * \
            self.array([[1, 2, 3], [6, 0, 5], [3, 2, 1]], dtype=self.dt)
        assert_almost_equal(norm(A), (1 / 10) * 89 ** 0.5)
        assert_almost_equal(norm(A, 'fro'), (1 / 10) * 89 ** 0.5)
        assert_almost_equal(norm(A, 'nuc'), 1.3366836911774836)
        assert_almost_equal(norm(A, inf), 1.1)
        assert_almost_equal(norm(A, -inf), 0.6)
        assert_almost_equal(norm(A, 1), 1.0)
        assert_almost_equal(norm(A, -1), 0.4)
        assert_almost_equal(norm(A, 2), 0.88722940323461277)
        assert_almost_equal(norm(A, -2), 0.19456584790481812)

    def test_bad_args(self):
        # Check that bad arguments raise the appropriate exceptions.

        A = self.array([[1, 2, 3], [4, 5, 6]], dtype=self.dt)
        B = np.arange(1, 25, dtype=self.dt).reshape(2, 3, 4)

        # Using `axis=<integer>` or passing in a 1-D array implies vector
        # norms are being computed, so also using `ord='fro'`
        # or `ord='nuc'` or any other string raises a ValueError.
        assert_raises(ValueError, norm, A, 'fro', 0)
        assert_raises(ValueError, norm, A, 'nuc', 0)
        assert_raises(ValueError, norm, [3, 4], 'fro', None)
        assert_raises(ValueError, norm, [3, 4], 'nuc', None)
        assert_raises(ValueError, norm, [3, 4], 'test', None)

        # Similarly, norm should raise an exception when ord is any finite
        # number other than 1, 2, -1 or -2 when computing matrix norms.
        for order in [0, 3]:
            assert_raises(ValueError, norm, A, order, None)
            assert_raises(ValueError, norm, A, order, (0, 1))
            assert_raises(ValueError, norm, B, order, (1, 2))

        # Invalid axis
        assert_raises(AxisError, norm, B, None, 3)
        assert_raises(AxisError, norm, B, None, (2, 3))
        assert_raises(ValueError, norm, B, None, (0, 1, 2))


class _TestNorm(_TestNorm2D, _TestNormGeneral):
    pass


class TestNorm_NonSystematic:

    def test_longdouble_norm(self):
        # Non-regression test: p-norm of longdouble would previously raise
        # UnboundLocalError.
        x = np.arange(10, dtype=np.longdouble)
        old_assert_almost_equal(norm(x, ord=3), 12.65, decimal=2)

    def test_intmin(self):
        # Non-regression test: p-norm of signed integer would previously do
        # float cast and abs in the wrong order.
        x = np.array([-2 ** 31], dtype=np.int32)
        old_assert_almost_equal(norm(x, ord=3), 2 ** 31, decimal=5)

    def test_complex_high_ord(self):
        # gh-4156
        d = np.empty((2,), dtype=np.clongdouble)
        d[0] = 6 + 7j
        d[1] = -6 + 7j
        res = 11.615898132184
        old_assert_almost_equal(np.linalg.norm(d, ord=3), res, decimal=10)
        d = d.astype(np.complex128)
        old_assert_almost_equal(np.linalg.norm(d, ord=3), res, decimal=9)
        d = d.astype(np.complex64)
        old_assert_almost_equal(np.linalg.norm(d, ord=3), res, decimal=5)


# Separate definitions so we can use them for matrix tests.
class _TestNormDoubleBase(_TestNormBase):
    dt = np.double
    dec = 12


class _TestNormSingleBase(_TestNormBase):
    dt = np.float32
    dec = 6


class _TestNormInt64Base(_TestNormBase):
    dt = np.int64
    dec = 12


class TestNormDouble(_TestNorm, _TestNormDoubleBase):
    pass


class TestNormSingle(_TestNorm, _TestNormSingleBase):
    pass


class TestNormInt64(_TestNorm, _TestNormInt64Base):
    pass


class TestMatrixRank:

    def test_matrix_rank(self):
        # Full rank matrix
        assert_equal(4, matrix_rank(np.eye(4)))
        # rank deficient matrix
        I = np.eye(4)
        I[-1, -1] = 0.
        assert_equal(matrix_rank(I), 3)
        # All zeros - zero rank
        assert_equal(matrix_rank(np.zeros((4, 4))), 0)
        # 1 dimension - rank 1 unless all 0
        assert_equal(matrix_rank([1, 0, 0, 0]), 1)
        assert_equal(matrix_rank(np.zeros((4,))), 0)
        # accepts array-like
        assert_equal(matrix_rank([1]), 1)
        # greater than 2 dimensions treated as stacked matrices
        ms = np.array([I, np.eye(4), np.zeros((4, 4))])
        assert_equal(matrix_rank(ms), np.array([3, 4, 0]))
        # works on scalar
        assert_equal(matrix_rank(1), 1)

        with assert_raises_regex(
            ValueError, "`tol` and `rtol` can\'t be both set."
        ):
            matrix_rank(I, tol=0.01, rtol=0.01)

    def test_symmetric_rank(self):
        assert_equal(4, matrix_rank(np.eye(4), hermitian=True))
        assert_equal(1, matrix_rank(np.ones((4, 4)), hermitian=True))
        assert_equal(0, matrix_rank(np.zeros((4, 4)), hermitian=True))
        # rank deficient matrix
        I = np.eye(4)
        I[-1, -1] = 0.
        assert_equal(3, matrix_rank(I, hermitian=True))
        # manually supplied tolerance
        I[-1, -1] = 1e-8
        assert_equal(4, matrix_rank(I, hermitian=True, tol=0.99e-8))
        assert_equal(3, matrix_rank(I, hermitian=True, tol=1.01e-8))


def test_reduced_rank():
    # Test matrices with reduced rank
    rng = np.random.RandomState(20120714)
    for i in range(100):
        # Make a rank deficient matrix
        X = rng.normal(size=(40, 10))
        X[:, 0] = X[:, 1] + X[:, 2]
        # Assert that matrix_rank detected deficiency
        assert_equal(matrix_rank(X), 9)
        X[:, 3] = X[:, 4] + X[:, 5]
        assert_equal(matrix_rank(X), 8)


class TestQR:
    # Define the array class here, so run this on matrices elsewhere.
    array = np.array

    def check_qr(self, a):
        # This test expects the argument `a` to be an ndarray or
        # a subclass of an ndarray of inexact type.
        a_type = type(a)
        a_dtype = a.dtype
        m, n = a.shape
        k = min(m, n)

        # mode == 'complete'
        res = linalg.qr(a, mode='complete')
        Q, R = res.Q, res.R
        assert_(Q.dtype == a_dtype)
        assert_(R.dtype == a_dtype)
        assert_(isinstance(Q, a_type))
        assert_(isinstance(R, a_type))
        assert_(Q.shape == (m, m))
        assert_(R.shape == (m, n))
        assert_almost_equal(dot(Q, R), a)
        assert_almost_equal(dot(Q.T.conj(), Q), np.eye(m))
        assert_almost_equal(np.triu(R), R)

        # mode == 'reduced'
        q1, r1 = linalg.qr(a, mode='reduced')
        assert_(q1.dtype == a_dtype)
        assert_(r1.dtype == a_dtype)
        assert_(isinstance(q1, a_type))
        assert_(isinstance(r1, a_type))
        assert_(q1.shape == (m, k))
        assert_(r1.shape == (k, n))
        assert_almost_equal(dot(q1, r1), a)
        assert_almost_equal(dot(q1.T.conj(), q1), np.eye(k))
        assert_almost_equal(np.triu(r1), r1)

        # mode == 'r'
        r2 = linalg.qr(a, mode='r')
        assert_(r2.dtype == a_dtype)
        assert_(isinstance(r2, a_type))
        assert_almost_equal(r2, r1)

    @pytest.mark.parametrize(["m", "n"], [
        (3, 0),
        (0, 3),
        (0, 0)
    ])
    def test_qr_empty(self, m, n):
        k = min(m, n)
        a = np.empty((m, n))

        self.check_qr(a)

        h, tau = np.linalg.qr(a, mode='raw')
        assert_equal(h.dtype, np.double)
        assert_equal(tau.dtype, np.double)
        assert_equal(h.shape, (n, m))
        assert_equal(tau.shape, (k,))

    def test_mode_raw(self):
        # The factorization is not unique and varies between libraries,
        # so it is not possible to check against known values. Functional
        # testing is a possibility, but awaits the exposure of more
        # of the functions in lapack_lite. Consequently, this test is
        # very limited in scope. Note that the results are in FORTRAN
        # order, hence the h arrays are transposed.
        a = self.array([[1, 2], [3, 4], [5, 6]], dtype=np.double)

        # Test double
        h, tau = linalg.qr(a, mode='raw')
        assert_(h.dtype == np.double)
        assert_(tau.dtype == np.double)
        assert_(h.shape == (2, 3))
        assert_(tau.shape == (2,))

        h, tau = linalg.qr(a.T, mode='raw')
        assert_(h.dtype == np.double)
        assert_(tau.dtype == np.double)
        assert_(h.shape == (3, 2))
        assert_(tau.shape == (2,))

    def test_mode_all_but_economic(self):
        a = self.array([[1, 2], [3, 4]])
        b = self.array([[1, 2], [3, 4], [5, 6]])
        for dt in "fd":
            m1 = a.astype(dt)
            m2 = b.astype(dt)
            self.check_qr(m1)
            self.check_qr(m2)
            self.check_qr(m2.T)

        for dt in "fd":
            m1 = 1 + 1j * a.astype(dt)
            m2 = 1 + 1j * b.astype(dt)
            self.check_qr(m1)
            self.check_qr(m2)
            self.check_qr(m2.T)

    def check_qr_stacked(self, a):
        # This test expects the argument `a` to be an ndarray or
        # a subclass of an ndarray of inexact type.
        a_type = type(a)
        a_dtype = a.dtype
        m, n = a.shape[-2:]
        k = min(m, n)

        # mode == 'complete'
        q, r = linalg.qr(a, mode='complete')
        assert_(q.dtype == a_dtype)
        assert_(r.dtype == a_dtype)
        assert_(isinstance(q, a_type))
        assert_(isinstance(r, a_type))
        assert_(q.shape[-2:] == (m, m))
        assert_(r.shape[-2:] == (m, n))
        assert_almost_equal(matmul(q, r), a)
        I_mat = np.identity(q.shape[-1])
        stack_I_mat = np.broadcast_to(I_mat,
                        q.shape[:-2] + (q.shape[-1],) * 2)
        assert_almost_equal(matmul(swapaxes(q, -1, -2).conj(), q), stack_I_mat)
        assert_almost_equal(np.triu(r[..., :, :]), r)

        # mode == 'reduced'
        q1, r1 = linalg.qr(a, mode='reduced')
        assert_(q1.dtype == a_dtype)
        assert_(r1.dtype == a_dtype)
        assert_(isinstance(q1, a_type))
        assert_(isinstance(r1, a_type))
        assert_(q1.shape[-2:] == (m, k))
        assert_(r1.shape[-2:] == (k, n))
        assert_almost_equal(matmul(q1, r1), a)
        I_mat = np.identity(q1.shape[-1])
        stack_I_mat = np.broadcast_to(I_mat,
                        q1.shape[:-2] + (q1.shape[-1],) * 2)
        assert_almost_equal(matmul(swapaxes(q1, -1, -2).conj(), q1),
                            stack_I_mat)
        assert_almost_equal(np.triu(r1[..., :, :]), r1)

        # mode == 'r'
        r2 = linalg.qr(a, mode='r')
        assert_(r2.dtype == a_dtype)
        assert_(isinstance(r2, a_type))
        assert_almost_equal(r2, r1)

    @pytest.mark.parametrize("size", [
        (3, 4), (4, 3), (4, 4),
        (3, 0), (0, 3)])
    @pytest.mark.parametrize("outer_size", [
        (2, 2), (2,), (2, 3, 4)])
    @pytest.mark.parametrize("dt", [
        np.single, np.double,
        np.csingle, np.cdouble])
    def test_stacked_inputs(self, outer_size, size, dt):

        rng = np.random.default_rng(123)
        A = rng.normal(size=outer_size + size).astype(dt)
        B = rng.normal(size=outer_size + size).astype(dt)
        self.check_qr_stacked(A)
        self.check_qr_stacked(A + 1.j * B)


class TestCholesky:

    @pytest.mark.parametrize(
        'shape', [(1, 1), (2, 2), (3, 3), (50, 50), (3, 10, 10)]
    )
    @pytest.mark.parametrize(
        'dtype', (np.float32, np.float64, np.complex64, np.complex128)
    )
    @pytest.mark.parametrize(
        'upper', [False, True])
    def test_basic_property(self, shape, dtype, upper):
        np.random.seed(1)
        a = np.random.randn(*shape)
        if np.issubdtype(dtype, np.complexfloating):
            a = a + 1j * np.random.randn(*shape)

        t = list(range(len(shape)))
        t[-2:] = -1, -2

        a = np.matmul(a.transpose(t).conj(), a)
        a = np.asarray(a, dtype=dtype)

        c = np.linalg.cholesky(a, upper=upper)

        # Check A = L L^H or A = U^H U
        if upper:
            b = np.matmul(c.transpose(t).conj(), c)
        else:
            b = np.matmul(c, c.transpose(t).conj())

        atol = 500 * a.shape[0] * np.finfo(dtype).eps
        assert_allclose(b, a, atol=atol, err_msg=f'{shape} {dtype}\n{a}\n{c}')

        # Check diag(L or U) is real and positive
        d = np.diagonal(c, axis1=-2, axis2=-1)
        assert_(np.all(np.isreal(d)))
        assert_(np.all(d >= 0))

    def test_0_size(self):
        class ArraySubclass(np.ndarray):
            pass
        a = np.zeros((0, 1, 1), dtype=np.int_).view(ArraySubclass)
        res = linalg.cholesky(a)
        assert_equal(a.shape, res.shape)
        assert_(res.dtype.type is np.float64)
        # for documentation purpose:
        assert_(isinstance(res, np.ndarray))

        a = np.zeros((1, 0, 0), dtype=np.complex64).view(ArraySubclass)
        res = linalg.cholesky(a)
        assert_equal(a.shape, res.shape)
        assert_(res.dtype.type is np.complex64)
        assert_(isinstance(res, np.ndarray))

    def test_upper_lower_arg(self):
        # Explicit test of upper argument that also checks the default.
        a = np.array([[1 + 0j, 0 - 2j], [0 + 2j, 5 + 0j]])

        assert_equal(linalg.cholesky(a), linalg.cholesky(a, upper=False))

        assert_equal(
            linalg.cholesky(a, upper=True),
            linalg.cholesky(a).T.conj()
        )


class TestOuter:
    arr1 = np.arange(3)
    arr2 = np.arange(3)
    expected = np.array(
        [[0, 0, 0],
         [0, 1, 2],
         [0, 2, 4]]
    )

    assert_array_equal(np.linalg.outer(arr1, arr2), expected)

    with assert_raises_regex(
        ValueError, "Input arrays must be one-dimensional"
    ):
        np.linalg.outer(arr1[:, np.newaxis], arr2)


def test_byteorder_check():
    # Byte order check should pass for native order
    if sys.byteorder == 'little':
        native = '<'
    else:
        native = '>'

    for dtt in (np.float32, np.float64):
        arr = np.eye(4, dtype=dtt)
        n_arr = arr.view(arr.dtype.newbyteorder(native))
        sw_arr = arr.view(arr.dtype.newbyteorder("S")).byteswap()
        assert_equal(arr.dtype.byteorder, '=')
        for routine in (linalg.inv, linalg.det, linalg.pinv):
            # Normal call
            res = routine(arr)
            # Native but not '='
            assert_array_equal(res, routine(n_arr))
            # Swapped
            assert_array_equal(res, routine(sw_arr))


@pytest.mark.skipif(IS_WASM, reason="fp errors don't work in wasm")
def test_generalized_raise_multiloop():
    # It should raise an error even if the error doesn't occur in the
    # last iteration of the ufunc inner loop

    invertible = np.array([[1, 2], [3, 4]])
    non_invertible = np.array([[1, 1], [1, 1]])

    x = np.zeros([4, 4, 2, 2])[1::2]
    x[...] = invertible
    x[0, 0] = non_invertible

    assert_raises(np.linalg.LinAlgError, np.linalg.inv, x)


@pytest.mark.filterwarnings(
    r"ignore:.*fork\(\) may lead to deadlocks.*:DeprecationWarning"
)
@pytest.mark.skipif(
    threading.active_count() > 1,
    reason="skipping test that uses fork because there are multiple threads")
@pytest.mark.skipif(
    NOGIL_BUILD,
    reason="Cannot safely use fork in tests on the free-threaded build")
def test_xerbla_override():
    # Check that our xerbla has been successfully linked in. If it is not,
    # the default xerbla routine is called, which prints a message to stdout
    # and may, or may not, abort the process depending on the LAPACK package.

    XERBLA_OK = 255

    try:
        pid = os.fork()
    except (OSError, AttributeError):
        # fork failed, or not running on POSIX
        pytest.skip("Not POSIX or fork failed.")

    if pid == 0:
        # child; close i/o file handles
        os.close(1)
        os.close(0)
        # Avoid producing core files.
        import resource
        resource.setrlimit(resource.RLIMIT_CORE, (0, 0))
        # These calls may abort.
        try:
            np.linalg.lapack_lite.xerbla()
        except ValueError:
            pass
        except Exception:
            os._exit(os.EX_CONFIG)

        try:
            a = np.array([[1.]])
            np.linalg.lapack_lite.dorgqr(
                1, 1, 1, a,
                0,  # <- invalid value
                a, a, 0, 0)
        except ValueError as e:
            if "DORGQR parameter number 5" in str(e):
                # success, reuse error code to mark success as
                # FORTRAN STOP returns as success.
                os._exit(XERBLA_OK)

        # Did not abort, but our xerbla was not linked in.
        os._exit(os.EX_CONFIG)
    else:
        # parent
        pid, status = os.wait()
        if os.WEXITSTATUS(status) != XERBLA_OK:
            pytest.skip('Numpy xerbla not linked in.')


@pytest.mark.skipif(IS_WASM, reason="Cannot start subprocess")
@pytest.mark.slow
def test_sdot_bug_8577():
    # Regression test that loading certain other libraries does not
    # result to wrong results in float32 linear algebra.
    #
    # There's a bug gh-8577 on OSX that can trigger this, and perhaps
    # there are also other situations in which it occurs.
    #
    # Do the check in a separate process.

    bad_libs = ['PyQt5.QtWidgets', 'IPython']

    template = textwrap.dedent("""
    import sys
    {before}
    try:
        import {bad_lib}
    except ImportError:
        sys.exit(0)
    {after}
    x = np.ones(2, dtype=np.float32)
    sys.exit(0 if np.allclose(x.dot(x), 2.0) else 1)
    """)

    for bad_lib in bad_libs:
        code = template.format(before="import numpy as np", after="",
                               bad_lib=bad_lib)
        run_subprocess([sys.executable, "-c", code])

        # Swapped import order
        code = template.format(after="import numpy as np", before="",
                               bad_lib=bad_lib)
        run_subprocess([sys.executable, "-c", code])


class TestMultiDot:

    def test_basic_function_with_three_arguments(self):
        # multi_dot with three arguments uses a fast hand coded algorithm to
        # determine the optimal order. Therefore test it separately.
        A = np.random.random((6, 2))
        B = np.random.random((2, 6))
        C = np.random.random((6, 2))

        assert_almost_equal(multi_dot([A, B, C]), A.dot(B).dot(C))
        assert_almost_equal(multi_dot([A, B, C]), np.dot(A, np.dot(B, C)))

    def test_basic_function_with_two_arguments(self):
        # separate code path with two arguments
        A = np.random.random((6, 2))
        B = np.random.random((2, 6))

        assert_almost_equal(multi_dot([A, B]), A.dot(B))
        assert_almost_equal(multi_dot([A, B]), np.dot(A, B))

    def test_basic_function_with_dynamic_programming_optimization(self):
        # multi_dot with four or more arguments uses the dynamic programming
        # optimization and therefore deserve a separate
        A = np.random.random((6, 2))
        B = np.random.random((2, 6))
        C = np.random.random((6, 2))
        D = np.random.random((2, 1))
        assert_almost_equal(multi_dot([A, B, C, D]), A.dot(B).dot(C).dot(D))

    def test_vector_as_first_argument(self):
        # The first argument can be 1-D
        A1d = np.random.random(2)  # 1-D
        B = np.random.random((2, 6))
        C = np.random.random((6, 2))
        D = np.random.random((2, 2))

        # the result should be 1-D
        assert_equal(multi_dot([A1d, B, C, D]).shape, (2,))

    def test_vector_as_last_argument(self):
        # The last argument can be 1-D
        A = np.random.random((6, 2))
        B = np.random.random((2, 6))
        C = np.random.random((6, 2))
        D1d = np.random.random(2)  # 1-D

        # the result should be 1-D
        assert_equal(multi_dot([A, B, C, D1d]).shape, (6,))

    def test_vector_as_first_and_last_argument(self):
        # The first and last arguments can be 1-D
        A1d = np.random.random(2)  # 1-D
        B = np.random.random((2, 6))
        C = np.random.random((6, 2))
        D1d = np.random.random(2)  # 1-D

        # the result should be a scalar
        assert_equal(multi_dot([A1d, B, C, D1d]).shape, ())

    def test_three_arguments_and_out(self):
        # multi_dot with three arguments uses a fast hand coded algorithm to
        # determine the optimal order. Therefore test it separately.
        A = np.random.random((6, 2))
        B = np.random.random((2, 6))
        C = np.random.random((6, 2))

        out = np.zeros((6, 2))
        ret = multi_dot([A, B, C], out=out)
        assert out is ret
        assert_almost_equal(out, A.dot(B).dot(C))
        assert_almost_equal(out, np.dot(A, np.dot(B, C)))

    def test_two_arguments_and_out(self):
        # separate code path with two arguments
        A = np.random.random((6, 2))
        B = np.random.random((2, 6))
        out = np.zeros((6, 6))
        ret = multi_dot([A, B], out=out)
        assert out is ret
        assert_almost_equal(out, A.dot(B))
        assert_almost_equal(out, np.dot(A, B))

    def test_dynamic_programming_optimization_and_out(self):
        # multi_dot with four or more arguments uses the dynamic programming
        # optimization and therefore deserve a separate test
        A = np.random.random((6, 2))
        B = np.random.random((2, 6))
        C = np.random.random((6, 2))
        D = np.random.random((2, 1))
        out = np.zeros((6, 1))
        ret = multi_dot([A, B, C, D], out=out)
        assert out is ret
        assert_almost_equal(out, A.dot(B).dot(C).dot(D))

    def test_dynamic_programming_logic(self):
        # Test for the dynamic programming part
        # This test is directly taken from Cormen page 376.
        arrays = [np.random.random((30, 35)),
                  np.random.random((35, 15)),
                  np.random.random((15, 5)),
                  np.random.random((5, 10)),
                  np.random.random((10, 20)),
                  np.random.random((20, 25))]
        m_expected = np.array([[0., 15750., 7875., 9375., 11875., 15125.],
                               [0.,     0., 2625., 4375.,  7125., 10500.],
                               [0.,     0.,    0.,  750.,  2500.,  5375.],
                               [0.,     0.,    0.,    0.,  1000.,  3500.],
                               [0.,     0.,    0.,    0.,     0.,  5000.],
                               [0.,     0.,    0.,    0.,     0.,     0.]])
        s_expected = np.array([[0,  1,  1,  3,  3,  3],
                               [0,  0,  2,  3,  3,  3],
                               [0,  0,  0,  3,  3,  3],
                               [0,  0,  0,  0,  4,  5],
                               [0,  0,  0,  0,  0,  5],
                               [0,  0,  0,  0,  0,  0]], dtype=int)
        s_expected -= 1  # Cormen uses 1-based index, python does not.

        s, m = _multi_dot_matrix_chain_order(arrays, return_costs=True)

        # Only the upper triangular part (without the diagonal) is interesting.
        assert_almost_equal(np.triu(s[:-1, 1:]),
                            np.triu(s_expected[:-1, 1:]))
        assert_almost_equal(np.triu(m), np.triu(m_expected))

    def test_too_few_input_arrays(self):
        assert_raises(ValueError, multi_dot, [])
        assert_raises(ValueError, multi_dot, [np.random.random((3, 3))])


class TestTensorinv:

    @pytest.mark.parametrize("arr, ind", [
        (np.ones((4, 6, 8, 2)), 2),
        (np.ones((3, 3, 2)), 1),
        ])
    def test_non_square_handling(self, arr, ind):
        with assert_raises(LinAlgError):
            linalg.tensorinv(arr, ind=ind)

    @pytest.mark.parametrize("shape, ind", [
        # examples from docstring
        ((4, 6, 8, 3), 2),
        ((24, 8, 3), 1),
        ])
    def test_tensorinv_shape(self, shape, ind):
        a = np.eye(24).reshape(shape)
        ainv = linalg.tensorinv(a=a, ind=ind)
        expected = a.shape[ind:] + a.shape[:ind]
        actual = ainv.shape
        assert_equal(actual, expected)

    @pytest.mark.parametrize("ind", [
        0, -2,
        ])
    def test_tensorinv_ind_limit(self, ind):
        a = np.eye(24).reshape((4, 6, 8, 3))
        with assert_raises(ValueError):
            linalg.tensorinv(a=a, ind=ind)

    def test_tensorinv_result(self):
        # mimic a docstring example
        a = np.eye(24).reshape((24, 8, 3))
        ainv = linalg.tensorinv(a, ind=1)
        b = np.ones(24)
        assert_allclose(np.tensordot(ainv, b, 1), np.linalg.tensorsolve(a, b))


class TestTensorsolve:

    @pytest.mark.parametrize("a, axes", [
        (np.ones((4, 6, 8, 2)), None),
        (np.ones((3, 3, 2)), (0, 2)),
        ])
    def test_non_square_handling(self, a, axes):
        with assert_raises(LinAlgError):
            b = np.ones(a.shape[:2])
            linalg.tensorsolve(a, b, axes=axes)

    @pytest.mark.parametrize("shape",
        [(2, 3, 6), (3, 4, 4, 3), (0, 3, 3, 0)],
    )
    def test_tensorsolve_result(self, shape):
        a = np.random.randn(*shape)
        b = np.ones(a.shape[:2])
        x = np.linalg.tensorsolve(a, b)
        assert_allclose(np.tensordot(a, x, axes=len(x.shape)), b)


def test_unsupported_commontype():
    # linalg gracefully handles unsupported type
    arr = np.array([[1, -2], [2, 5]], dtype='float16')
    with assert_raises_regex(TypeError, "unsupported in linalg"):
        linalg.cholesky(arr)


#@pytest.mark.slow
#@pytest.mark.xfail(not HAS_LAPACK64, run=False,
#                   reason="Numpy not compiled with 64-bit BLAS/LAPACK")
#@requires_memory(free_bytes=16e9)
@pytest.mark.skip(reason="Bad memory reports lead to OOM in ci testing")
def test_blas64_dot():
    n = 2**32
    a = np.zeros([1, n], dtype=np.float32)
    b = np.ones([1, 1], dtype=np.float32)
    a[0, -1] = 1
    c = np.dot(b, a)
    assert_equal(c[0, -1], 1)


@pytest.mark.xfail(not HAS_LAPACK64,
                   reason="Numpy not compiled with 64-bit BLAS/LAPACK")
def test_blas64_geqrf_lwork_smoketest():
    # Smoke test LAPACK geqrf lwork call with 64-bit integers
    dtype = np.float64
    lapack_routine = np.linalg.lapack_lite.dgeqrf

    m = 2**32 + 1
    n = 2**32 + 1
    lda = m

    # Dummy arrays, not referenced by the lapack routine, so don't
    # need to be of the right size
    a = np.zeros([1, 1], dtype=dtype)
    work = np.zeros([1], dtype=dtype)
    tau = np.zeros([1], dtype=dtype)

    # Size query
    results = lapack_routine(m, n, a, lda, tau, work, -1, 0)
    assert_equal(results['info'], 0)
    assert_equal(results['m'], m)
    assert_equal(results['n'], m)

    # Should result to an integer of a reasonable size
    lwork = int(work.item())
    assert_(2**32 < lwork < 2**42)


def test_diagonal():
    # Here we only test if selected axes are compatible
    # with Array API (last two). Core implementation
    # of `diagonal` is tested in `test_multiarray.py`.
    x = np.arange(60).reshape((3, 4, 5))
    actual = np.linalg.diagonal(x)
    expected = np.array(
        [
            [0,  6, 12, 18],
            [20, 26, 32, 38],
            [40, 46, 52, 58],
        ]
    )
    assert_equal(actual, expected)


def test_trace():
    # Here we only test if selected axes are compatible
    # with Array API (last two). Core implementation
    # of `trace` is tested in `test_multiarray.py`.
    x = np.arange(60).reshape((3, 4, 5))
    actual = np.linalg.trace(x)
    expected = np.array([36, 116, 196])

    assert_equal(actual, expected)


def test_cross():
    x = np.arange(9).reshape((3, 3))
    actual = np.linalg.cross(x, x + 1)
    expected = np.array([
        [-1, 2, -1],
        [-1, 2, -1],
        [-1, 2, -1],
    ])

    assert_equal(actual, expected)

    # We test that lists are converted to arrays.
    u = [1, 2, 3]
    v = [4, 5, 6]
    actual = np.linalg.cross(u, v)
    expected = array([-3,  6, -3])

    assert_equal(actual, expected)

    with assert_raises_regex(
        ValueError,
        r"input arrays must be \(arrays of\) 3-dimensional vectors"
    ):
        x_2dim = x[:, 1:]
        np.linalg.cross(x_2dim, x_2dim)


def test_tensordot():
    # np.linalg.tensordot is just an alias for np.tensordot
    x = np.arange(6).reshape((2, 3))

    assert np.linalg.tensordot(x, x) == 55
    assert np.linalg.tensordot(x, x, axes=[(0, 1), (0, 1)]) == 55


def test_matmul():
    # np.linalg.matmul and np.matmul only differs in the number
    # of arguments in the signature
    x = np.arange(6).reshape((2, 3))
    actual = np.linalg.matmul(x, x.T)
    expected = np.array([[5, 14], [14, 50]])

    assert_equal(actual, expected)


def test_matrix_transpose():
    x = np.arange(6).reshape((2, 3))
    actual = np.linalg.matrix_transpose(x)
    expected = x.T

    assert_equal(actual, expected)

    with assert_raises_regex(
        ValueError, "array must be at least 2-dimensional"
    ):
        np.linalg.matrix_transpose(x[:, 0])


def test_matrix_norm():
    x = np.arange(9).reshape((3, 3))
    actual = np.linalg.matrix_norm(x)

    assert_almost_equal(actual, np.float64(14.2828), double_decimal=3)

    actual = np.linalg.matrix_norm(x, keepdims=True)

    assert_almost_equal(actual, np.array([[14.2828]]), double_decimal=3)


def test_matrix_norm_empty():
    for shape in [(0, 2), (2, 0), (0, 0)]:
        for dtype in [np.float64, np.float32, np.int32]:
            x = np.zeros(shape, dtype)
            assert_equal(np.linalg.matrix_norm(x, ord="fro"), 0)
            assert_equal(np.linalg.matrix_norm(x, ord="nuc"), 0)
            assert_equal(np.linalg.matrix_norm(x, ord=1), 0)
            assert_equal(np.linalg.matrix_norm(x, ord=2), 0)
            assert_equal(np.linalg.matrix_norm(x, ord=np.inf), 0)

def test_vector_norm():
    x = np.arange(9).reshape((3, 3))
    actual = np.linalg.vector_norm(x)

    assert_almost_equal(actual, np.float64(14.2828), double_decimal=3)

    actual = np.linalg.vector_norm(x, axis=0)

    assert_almost_equal(
        actual, np.array([6.7082, 8.124, 9.6436]), double_decimal=3
    )

    actual = np.linalg.vector_norm(x, keepdims=True)
    expected = np.full((1, 1), 14.2828, dtype='float64')
    assert_equal(actual.shape, expected.shape)
    assert_almost_equal(actual, expected, double_decimal=3)


def test_vector_norm_empty():
    for dtype in [np.float64, np.float32, np.int32]:
        x = np.zeros(0, dtype)
        assert_equal(np.linalg.vector_norm(x, ord=1), 0)
        assert_equal(np.linalg.vector_norm(x, ord=2), 0)
        assert_equal(np.linalg.vector_norm(x, ord=np.inf), 0)

def test_empty_matrix_rank():
    assert_equal(matrix_rank(np.zeros((0, 0))), 0)
    assert_equal(matrix_rank(np.zeros((0, 5))), 0)
    assert_equal(matrix_rank(np.zeros((5, 0))), 0)

    result = matrix_rank(np.zeros((0, 5, 5)))
    assert_equal(result.shape, (0,))
    assert_equal(result.dtype, np.intp)

    result = matrix_rank(np.zeros((3, 0, 5)))
    assert_equal(result, np.array([0, 0, 0]))

    result = matrix_rank(np.zeros((2, 5, 0)))
    assert_equal(result, np.array([0, 0]))

    result = matrix_rank(np.zeros((2, 3, 0, 4)))
    assert_equal(result.shape, (2, 3))
    assert_equal(result, np.zeros((2, 3), dtype=np.intp))
