comparison DEPENDENCIES/mingw32/Python27/Lib/site-packages/numpy/linalg/tests/test_linalg.py @ 87:2a2c65a20a8b

Add Python libs and headers

+""" Test functions for linalg module
+
+"""
+from __future__ import division, absolute_import, print_function
+
+import os
+import sys
+import itertools
+import traceback
+
+import numpy as np
+from numpy import array, single, double, csingle, cdouble, dot, identity
+from numpy import multiply, atleast_2d, inf, asarray, matrix
+from numpy import linalg
+from numpy.linalg import matrix_power, norm, matrix_rank
+from numpy.testing import (
+    assert_, assert_equal, assert_raises, assert_array_equal,
+    assert_almost_equal, assert_allclose, run_module_suite,
+    dec
+)
+
+
+def ifthen(a, b):
+    return not a or b
+
+
+def imply(a, b):
+    return not a or b
+
+
+old_assert_almost_equal = assert_almost_equal
+
+
+def assert_almost_equal(a, b, **kw):
+    if asarray(a).dtype.type in (single, csingle):
+        decimal = 6
+    else:
+        decimal = 12
+    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
+
+class LinalgCase(object):
+    def __init__(self, name, a, b, exception_cls=None):
+        assert isinstance(name, str)
+        self.name = name
+        self.a = a
+        self.b = b
+        self.exception_cls = exception_cls
+
+    def check(self, do):
+        if self.exception_cls is None:
+            do(self.a, self.b)
+        else:
+            assert_raises(self.exception_cls, do, self.a, self.b)
+
+    def __repr__(self):
+        return "<LinalgCase: %s>" % (self.name,)
+
+
+#
+# Base test cases
+#
+
+np.random.seed(1234)
+
+SQUARE_CASES = [
+    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("empty",
+               atleast_2d(array([], dtype = double)),
+               atleast_2d(array([], dtype = double)),
+               linalg.LinAlgError),
+    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]),
+    LinalgCase("matrix_b_only",
+               array([[1., 2.], [3., 4.]]),
+               matrix([2., 1.]).T),
+    LinalgCase("matrix_a_and_b",
+               matrix([[1., 2.], [3., 4.]]),
+               matrix([2., 1.]).T),
+]
+
+NONSQUARE_CASES = [
+    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(11)),
+    LinalgCase("1x5",
+               np.random.rand(1, 5),
+               np.random.rand(5)),
+    LinalgCase("5x1",
+               np.random.rand(5, 1),
+               np.random.rand(1)),
+]
+
+HERMITIAN_CASES = [
+    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",
+               atleast_2d(array([], dtype = double)),
+               None,
+               linalg.LinAlgError),
+    LinalgCase("hnonarray",
+               [[1, 2], [2, 1]],
+               None),
+    LinalgCase("matrix_b_only",
+               array([[1., 2.], [2., 1.]]),
+               None),
+    LinalgCase("hmatrix_a_and_b",
+               matrix([[1., 2.], [2., 1.]]),
+               None),
+    LinalgCase("hmatrix_1x1",
+               np.random.rand(1, 1),
+               None),
+]
+
+
+#
+# Gufunc test cases
+#
+
+GENERALIZED_SQUARE_CASES = []
+GENERALIZED_NONSQUARE_CASES = []
+GENERALIZED_HERMITIAN_CASES = []
+
+for tgt, src in ((GENERALIZED_SQUARE_CASES, SQUARE_CASES),
+                 (GENERALIZED_NONSQUARE_CASES, NONSQUARE_CASES),
+                 (GENERALIZED_HERMITIAN_CASES, HERMITIAN_CASES)):
+    for case in src:
+        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,
+                              case.exception_cls)
+        tgt.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,
+                              case.exception_cls)
+        tgt.append(new_case)
+
+#
+# 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"
+
+for src in (SQUARE_CASES,
+            NONSQUARE_CASES,
+            HERMITIAN_CASES,
+            GENERALIZED_SQUARE_CASES,
+            GENERALIZED_NONSQUARE_CASES,
+            GENERALIZED_HERMITIAN_CASES):
+
+    new_cases = []
+    for case in src:
+        for a, a_tag in _stride_comb_iter(case.a):
+            for b, b_tag in _stride_comb_iter(case.b):
+                new_case = LinalgCase(case.name + "_" + a_tag + "_" + b_tag, a, b,
+                                      exception_cls=case.exception_cls)
+                new_cases.append(new_case)
+    src.extend(new_cases)
+
+
+#
+# Test different routines against the above cases
+#
+
+def _check_cases(func, cases):
+    for case in cases:
+        try:
+            case.check(func)
+        except Exception:
+            msg = "In test case: %r\n\n" % case
+            msg += traceback.format_exc()
+            raise AssertionError(msg)
+
+class LinalgTestCase(object):
+    def test_sq_cases(self):
+        _check_cases(self.do, SQUARE_CASES)
+
+
+class LinalgNonsquareTestCase(object):
+    def test_sq_cases(self):
+        _check_cases(self.do, NONSQUARE_CASES)
+
+
+class LinalgGeneralizedTestCase(object):
+    @dec.slow
+    def test_generalized_sq_cases(self):
+        _check_cases(self.do, GENERALIZED_SQUARE_CASES)
+
+
+class LinalgGeneralizedNonsquareTestCase(object):
+    @dec.slow
+    def test_generalized_nonsq_cases(self):
+        _check_cases(self.do, GENERALIZED_NONSQUARE_CASES)
+
+
+class HermitianTestCase(object):
+    def test_herm_cases(self):
+        _check_cases(self.do, HERMITIAN_CASES)
+
+
+class HermitianGeneralizedTestCase(object):
+    @dec.slow
+    def test_generalized_herm_cases(self):
+        _check_cases(self.do, GENERALIZED_HERMITIAN_CASES)
+
+
+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)
+        for c in itertools.product(*map(range, a.shape[:-2])):
+            r[c] = identity(a.shape[-2])
+        return r
+    else:
+        return identity(a.shape[0])
+
+
+class TestSolve(LinalgTestCase, LinalgGeneralizedTestCase):
+    def do(self, a, b):
+        x = linalg.solve(a, b)
+        assert_almost_equal(b, dot_generalized(a, x))
+        assert_(imply(isinstance(b, matrix), isinstance(x, matrix)))
+
+    def test_types(self):
+        def check(dtype):
+            x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
+            assert_equal(linalg.solve(x, x).dtype, dtype)
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, 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 TestInv(LinalgTestCase, LinalgGeneralizedTestCase):
+    def do(self, a, b):
+        a_inv = linalg.inv(a)
+        assert_almost_equal(dot_generalized(a, a_inv),
+                            identity_like_generalized(a))
+        assert_(imply(isinstance(a, matrix), isinstance(a_inv, matrix)))
+
+    def test_types(self):
+        def check(dtype):
+            x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
+            assert_equal(linalg.inv(x).dtype, dtype)
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, 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(a, 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)
+
+
+class TestEigvals(LinalgTestCase, LinalgGeneralizedTestCase):
+    def do(self, a, b):
+        ev = linalg.eigvals(a)
+        evalues, evectors = linalg.eig(a)
+        assert_almost_equal(ev, evalues)
+
+    def test_types(self):
+        def check(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))
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, dtype
+
+
+class TestEig(LinalgTestCase, LinalgGeneralizedTestCase):
+    def do(self, a, b):
+        evalues, evectors = linalg.eig(a)
+        assert_allclose(dot_generalized(a, evectors),
+                        np.asarray(evectors) * np.asarray(evalues)[...,None,:],
+                        rtol=get_rtol(evalues.dtype))
+        assert_(imply(isinstance(a, matrix), isinstance(evectors, matrix)))
+
+    def test_types(self):
+        def check(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))
+
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, dtype
+
+
+class TestSVD(LinalgTestCase, LinalgGeneralizedTestCase):
+    def do(self, a, b):
+        u, s, vt = linalg.svd(a, 0)
+        assert_allclose(a, dot_generalized(np.asarray(u) * np.asarray(s)[...,None,:],
+                                           np.asarray(vt)),
+                        rtol=get_rtol(u.dtype))
+        assert_(imply(isinstance(a, matrix), isinstance(u, matrix)))
+        assert_(imply(isinstance(a, matrix), isinstance(vt, matrix)))
+
+    def test_types(self):
+        def check(dtype):
+            x = np.array([[1, 0.5], [0.5, 1]], dtype=dtype)
+            u, s, vh = linalg.svd(x)
+            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)
+            assert_equal(s.dtype, get_real_dtype(dtype))
+
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, dtype
+
+
+class TestCondSVD(LinalgTestCase, LinalgGeneralizedTestCase):
+    def do(self, a, b):
+        c = asarray(a) # a might be a matrix
+        s = linalg.svd(c, compute_uv=False)
+        old_assert_almost_equal(s[0]/s[-1], linalg.cond(a), decimal=5)
+
+
+class TestCond2(LinalgTestCase):
+    def do(self, a, b):
+        c = asarray(a) # a might be a matrix
+        s = linalg.svd(c, compute_uv=False)
+        old_assert_almost_equal(s[0]/s[-1], linalg.cond(a, 2), decimal=5)
+
+
+class TestCondInf(object):
+    def test(self):
+        A = array([[1., 0, 0], [0, -2., 0], [0, 0, 3.]])
+        assert_almost_equal(linalg.cond(A, inf), 3.)
+
+
+class TestPinv(LinalgTestCase):
+    def do(self, a, b):
+        a_ginv = linalg.pinv(a)
+        assert_almost_equal(dot(a, a_ginv), identity(asarray(a).shape[0]))
+        assert_(imply(isinstance(a, matrix), isinstance(a_ginv, matrix)))
+
+
+class TestDet(LinalgTestCase, LinalgGeneralizedTestCase):
+    def do(self, a, b):
+        d = linalg.det(a)
+        (s, ld) = linalg.slogdet(a)
+        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)
+
+    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)
+
+    def test_types(self):
+        def check(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)
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, dtype
+
+
+class TestLstsq(LinalgTestCase, LinalgNonsquareTestCase):
+    def do(self, a, b):
+        arr = np.asarray(a)
+        m, n = arr.shape
+        u, s, vt = linalg.svd(a, 0)
+        x, residuals, rank, sv = linalg.lstsq(a, b)
+        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 len(np.asarray(b).shape) == 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_(imply(isinstance(b, matrix), isinstance(x, matrix)))
+        assert_(imply(isinstance(b, matrix), isinstance(residuals, matrix)))
+
+
+class TestMatrixPower(object):
+    R90 = array([[0, 1], [-1, 0]])
+    Arb22 = array([[4, -7], [-2, 10]])
+    noninv = array([[1, 0], [0, 0]])
+    arbfloat = array([[0.1, 3.2], [1.2, 0.7]])
+
+    large = identity(10)
+    t = large[1,:].copy()
+    large[1,:] = large[0,:]
+    large[0,:] = t
+
+    def test_large_power(self):
+        assert_equal(matrix_power(self.R90, 2**100+2**10+2**5+1), self.R90)
+
+    def test_large_power_trailing_zero(self):
+        assert_equal(matrix_power(self.R90, 2**100+2**10+2**5), identity(2))
+
+    def testip_zero(self):
+        def tz(M):
+            mz = matrix_power(M, 0)
+            assert_equal(mz, identity(M.shape[0]))
+            assert_equal(mz.dtype, M.dtype)
+        for M in [self.Arb22, self.arbfloat, self.large]:
+            yield tz, M
+
+    def testip_one(self):
+        def tz(M):
+            mz = matrix_power(M, 1)
+            assert_equal(mz, M)
+            assert_equal(mz.dtype, M.dtype)
+        for M in [self.Arb22, self.arbfloat, self.large]:
+            yield tz, M
+
+    def testip_two(self):
+        def tz(M):
+            mz = matrix_power(M, 2)
+            assert_equal(mz, dot(M, M))
+            assert_equal(mz.dtype, M.dtype)
+        for M in [self.Arb22, self.arbfloat, self.large]:
+            yield tz, M
+
+    def testip_invert(self):
+        def tz(M):
+            mz = matrix_power(M, -1)
+            assert_almost_equal(identity(M.shape[0]), dot(mz, M))
+        for M in [self.R90, self.Arb22, self.arbfloat, self.large]:
+            yield tz, M
+
+    def test_invert_noninvertible(self):
+        import numpy.linalg
+        assert_raises(numpy.linalg.linalg.LinAlgError,
+                      lambda: matrix_power(self.noninv, -1))
+
+
+class TestBoolPower(object):
+    def test_square(self):
+        A = array([[True, False], [True, True]])
+        assert_equal(matrix_power(A, 2), A)
+
+
+class TestEigvalsh(HermitianTestCase, HermitianGeneralizedTestCase):
+    def do(self, a, b):
+        # note that eigenvalue arrays must be sorted since
+        # their order isn't guaranteed.
+        ev = linalg.eigvalsh(a, 'L')
+        evalues, evectors = linalg.eig(a)
+        ev.sort(axis=-1)
+        evalues.sort(axis=-1)
+        assert_allclose(ev, evalues,
+                        rtol=get_rtol(ev.dtype))
+
+        ev2 = linalg.eigvalsh(a, 'U')
+        ev2.sort(axis=-1)
+        assert_allclose(ev2, evalues,
+                        rtol=get_rtol(ev.dtype))
+
+    def test_types(self):
+        def check(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))
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, 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(np.sort(w), tgt, rtol=rtol)
+        # Check 'L'
+        w = np.linalg.eigvalsh(Klo, UPLO='L')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+        # Check 'l'
+        w = np.linalg.eigvalsh(Klo, UPLO='l')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+        # Check 'U'
+        w = np.linalg.eigvalsh(Kup, UPLO='U')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+        # Check 'u'
+        w = np.linalg.eigvalsh(Kup, UPLO='u')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+
+
+class TestEigh(HermitianTestCase, HermitianGeneralizedTestCase):
+    def do(self, a, b):
+        # note that eigenvalue arrays must be sorted since
+        # their order isn't guaranteed.
+        ev, evc = linalg.eigh(a)
+        evalues, evectors = linalg.eig(a)
+        ev.sort(axis=-1)
+        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')
+        ev2.sort(axis=-1)
+        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))
+
+    def test_types(self):
+        def check(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)
+        for dtype in [single, double, csingle, cdouble]:
+            yield check, 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(np.sort(w), tgt, rtol=rtol)
+        # Check 'L'
+        w, v = np.linalg.eigh(Klo, UPLO='L')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+        # Check 'l'
+        w, v = np.linalg.eigh(Klo, UPLO='l')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+        # Check 'U'
+        w, v = np.linalg.eigh(Kup, UPLO='U')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+        # Check 'u'
+        w, v = np.linalg.eigh(Kup, UPLO='u')
+        assert_allclose(np.sort(w), tgt, rtol=rtol)
+
+
+class _TestNorm(object):
+
+    dt = None
+    dec = None
+
+    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(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_matrix(self):
+        A = matrix([[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, 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_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)
+
+        for order in [None, -2, 2, -1, 1, np.Inf, -np.Inf, 'fro']:
+            assert_almost_equal(norm(A, ord=order), norm(A, ord=order,
+                                                         axis=(0, 1)))
+
+            n = norm(B, ord=order, axis=(1, 2))
+            expected = [norm(B[k], ord=order) for k in range(B.shape[0])]
+            assert_almost_equal(n, expected)
+
+            n = norm(B, ord=order, axis=(2, 1))
+            expected = [norm(B[k].T, ord=order) for k in range(B.shape[0])]
+            assert_almost_equal(n, expected)
+
+            n = norm(B, ord=order, axis=(0, 2))
+            expected = [norm(B[:, k,:], ord=order) for k in range(B.shape[1])]
+            assert_almost_equal(n, expected)
+
+            n = norm(B, ord=order, axis=(0, 1))
+            expected = [norm(B[:,:, k], ord=order) for k in range(B.shape[2])]
+            assert_almost_equal(n, expected)
+
+    def test_bad_args(self):
+        # Check that bad arguments raise the appropriate exceptions.
+
+        A = 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'` raises a
+        # ValueError.
+        assert_raises(ValueError, norm, A, 'fro', 0)
+        assert_raises(ValueError, norm, [3, 4], 'fro', 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(ValueError, norm, B, None, 3)
+        assert_raises(ValueError, norm, B, None, (2, 3))
+        assert_raises(ValueError, norm, B, None, (0, 1, 2))
+
+    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)
+
+
+class TestNormDouble(_TestNorm):
+    dt = np.double
+    dec = 12
+
+
+class TestNormSingle(_TestNorm):
+    dt = np.float32
+    dec = 6
+
+
+class TestNormInt64(_TestNorm):
+    dt = np.int64
+    dec = 12
+
+
+class TestMatrixRank(object):
+    def test_matrix_rank(self):
+        # Full rank matrix
+        yield assert_equal, 4, matrix_rank(np.eye(4))
+        # rank deficient matrix
+        I=np.eye(4); I[-1, -1] = 0.
+        yield assert_equal, matrix_rank(I), 3
+        # All zeros - zero rank
+        yield assert_equal, matrix_rank(np.zeros((4, 4))), 0
+        # 1 dimension - rank 1 unless all 0
+        yield assert_equal, matrix_rank([1, 0, 0, 0]), 1
+        yield assert_equal, matrix_rank(np.zeros((4,))), 0
+        # accepts array-like
+        yield assert_equal, matrix_rank([1]), 1
+        # greater than 2 dimensions raises error
+        yield assert_raises, TypeError, matrix_rank, np.zeros((2, 2, 2))
+        # works on scalar
+        yield assert_equal, matrix_rank(1), 1
+
+
+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(object):
+
+    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'
+        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 == (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)
+
+    def test_qr_empty(self):
+        a = np.zeros((0, 2))
+        assert_raises(linalg.LinAlgError, linalg.qr, a)
+
+    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 = array([[1, 2], [3, 4], [5, 6]], dtype=np.double)
+        b = a.astype(np.single)
+
+        # 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 = array([[1, 2], [3, 4]])
+        b = 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)
+            self.check_qr(matrix(m1))
+        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)
+            self.check_qr(matrix(m1))
+
+
+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))
+
+
+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.
+    from nose import SkipTest
+
+    try:
+        pid = os.fork()
+    except (OSError, AttributeError):
+        # fork failed, or not running on POSIX
+        raise SkipTest("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:
+            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
+                os._exit(os.EX_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) != os.EX_OK or os.WIFSIGNALED(status):
+            raise SkipTest('Numpy xerbla not linked in.')
+
+
+if __name__ == "__main__":
+    run_module_suite()