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diff --git a/.venv/lib/python3.12/site-packages/numpy/linalg/tests/test_linalg.py b/.venv/lib/python3.12/site-packages/numpy/linalg/tests/test_linalg.py new file mode 100644 index 00000000..5dabdfdf --- /dev/null +++ b/.venv/lib/python3.12/site-packages/numpy/linalg/tests/test_linalg.py @@ -0,0 +1,2198 @@ +""" Test functions for linalg module + +""" +import os +import sys +import itertools +import traceback +import textwrap +import subprocess +import pytest + +import numpy as np +from numpy import array, single, double, csingle, cdouble, dot, identity, matmul +from numpy.core import swapaxes +from numpy import multiply, atleast_2d, inf, asarray +from numpy import linalg +from numpy.linalg import matrix_power, norm, matrix_rank, multi_dot, LinAlgError +from numpy.linalg.linalg import _multi_dot_matrix_chain_order +from numpy.testing import ( + assert_, assert_equal, assert_raises, assert_array_equal, + assert_almost_equal, assert_allclose, suppress_warnings, + assert_raises_regex, HAS_LAPACK64, IS_WASM + ) +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 + 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 + 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(["%+d" % 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 dot_generalized(a, b): + a = asarray(a) + if a.ndim >= 3: + if a.ndim == b.ndim: + # matrix x matrix + new_shape = a.shape[:-1] + b.shape[-1:] + elif a.ndim == b.ndim + 1: + # matrix x vector + new_shape = a.shape[:-1] + else: + raise ValueError("Not implemented...") + r = np.empty(new_shape, dtype=np.common_type(a, b)) + for c in itertools.product(*map(range, a.shape[:-2])): + r[c] = dot(a[c], b[c]) + return r + else: + return dot(a, b) + + +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) + assert_almost_equal(b, dot_generalized(a, x)) + 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_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).reshape(1, 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(dot_generalized(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, 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.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.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(dot_generalized(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, dtype) + assert_equal(v.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.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.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, dot_generalized(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)) + + +class SVDHermitianCases(HermitianTestCase, HermitianGeneralizedTestCase): + + def do(self, a, b, tags): + u, s, vt = linalg.svd(a, False, hermitian=True) + assert_allclose(a, dot_generalized(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)) + + assert_almost_equal(np.matmul(u, hermitian(u)), np.broadcast_to(np.eye(u.shape[-1]), u.shape)) + assert_almost_equal(np.matmul(vt, hermitian(vt)), np.broadcast_to(np.eye(vt.shape[-1]), vt.shape)) + assert_equal(np.sort(s)[..., ::-1], s) + assert_(consistent_subclass(u, a)) + assert_(consistent_subclass(vt, a)) + + +class TestSVDHermitian(SVDHermitianCases, SVDBaseTests): + hermitian = True + + +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): + def test_basic_nonsvd(self): + # Smoketest the non-svd norms + A = array([[1., 0, 1], [0, -2., 0], [0, 0, 3.]]) + 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)) + + 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.float_)) + 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 = dot_generalized + assert_almost_equal(dot(dot(a, a_ginv), a), 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 = dot_generalized + assert_almost_equal(dot(dot(a, a_ginv), a), a, single_decimal=5, double_decimal=11) + assert_(consistent_subclass(a_ginv, a)) + + +class TestPinvHermitian(PinvHermitianCases): + pass + + +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.shape = (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_future_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]) + with suppress_warnings() as sup: + w = sup.record(FutureWarning, "`rcond` parameter will change") + x, residuals, rank, s = linalg.lstsq(a, b) + assert_(rank == 4) + x, residuals, rank, s = linalg.lstsq(a, b, rcond=-1) + assert_(rank == 4) + x, residuals, rank, s = linalg.lstsq(a, b, rcond=None) + assert_(rank == 3) + # Warning should be raised exactly once (first command) + assert_(len(w) == 1) + + @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 dt != 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 dt != 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 dt != 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(dot_generalized(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(dot_generalized(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 suppress_warnings() as sup: + sup.filter(RuntimeWarning, "divide by zero encountered") + 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 suppress_warnings() as sup: + sup.filter(RuntimeWarning, "divide by zero encountered") + 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(np.AxisError, norm, B, None, 3) + assert_raises(np.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) + + 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): + + A = np.random.normal(size=outer_size + size).astype(dt) + B = np.random.normal(size=outer_size + size).astype(dt) + self.check_qr_stacked(A) + self.check_qr_stacked(A + 1.j*B) + + +class TestCholesky: + # TODO: are there no other tests for cholesky? + + @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) + ) + def test_basic_property(self, shape, dtype): + # Check A = L L^H + 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) + + b = np.matmul(c, c.transpose(t).conj()) + with np._no_nep50_warning(): + atol = 500 * a.shape[0] * np.finfo(dtype).eps + assert_allclose(b, a, atol=atol, err_msg=f'{shape} {dtype}\n{a}\n{c}') + + 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_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.newbyteorder(native) + sw_arr = arr.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) + + +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) + subprocess.check_call([sys.executable, "-c", code]) + + # Swapped import order + code = template.format(after="import numpy as np", before="", + bad_lib=bad_lib) + subprocess.check_call([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) + a.shape = 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) + a.shape = (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) + a.shape = (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) |