3283 lines
122 KiB
Python
3283 lines
122 KiB
Python
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#
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# Created by: Pearu Peterson, September 2002
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#
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import sys
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from functools import reduce
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from numpy.testing import (assert_equal, assert_array_almost_equal, assert_,
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assert_allclose, assert_almost_equal,
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assert_array_equal)
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import pytest
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from pytest import raises as assert_raises
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import numpy as np
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from numpy import (eye, ones, zeros, zeros_like, triu, tril, tril_indices,
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triu_indices)
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from numpy.random import rand, randint, seed
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from scipy.linalg import (_flapack as flapack, lapack, inv, svd, cholesky,
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solve, ldl, norm, block_diag, qr, eigh)
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from scipy.linalg.lapack import _compute_lwork
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from scipy.stats import ortho_group, unitary_group
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import scipy.sparse as sps
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try:
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from scipy.linalg import _clapack as clapack
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except ImportError:
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clapack = None
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from scipy.linalg.lapack import get_lapack_funcs
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from scipy.linalg.blas import get_blas_funcs
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REAL_DTYPES = [np.float32, np.float64]
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COMPLEX_DTYPES = [np.complex64, np.complex128]
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DTYPES = REAL_DTYPES + COMPLEX_DTYPES
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def generate_random_dtype_array(shape, dtype):
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# generates a random matrix of desired data type of shape
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if dtype in COMPLEX_DTYPES:
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return (np.random.rand(*shape)
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+ np.random.rand(*shape)*1.0j).astype(dtype)
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return np.random.rand(*shape).astype(dtype)
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def test_lapack_documented():
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"""Test that all entries are in the doc."""
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if lapack.__doc__ is None: # just in case there is a python -OO
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pytest.skip('lapack.__doc__ is None')
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names = set(lapack.__doc__.split())
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ignore_list = set([
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'absolute_import', 'clapack', 'division', 'find_best_lapack_type',
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'flapack', 'print_function', 'HAS_ILP64',
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])
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missing = list()
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for name in dir(lapack):
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if (not name.startswith('_') and name not in ignore_list and
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name not in names):
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missing.append(name)
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assert missing == [], 'Name(s) missing from lapack.__doc__ or ignore_list'
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class TestFlapackSimple:
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def test_gebal(self):
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a = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
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a1 = [[1, 0, 0, 3e-4],
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[4, 0, 0, 2e-3],
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[7, 1, 0, 0],
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[0, 1, 0, 0]]
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for p in 'sdzc':
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f = getattr(flapack, p+'gebal', None)
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if f is None:
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continue
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ba, lo, hi, pivscale, info = f(a)
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assert_(not info, repr(info))
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assert_array_almost_equal(ba, a)
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assert_equal((lo, hi), (0, len(a[0])-1))
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assert_array_almost_equal(pivscale, np.ones(len(a)))
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ba, lo, hi, pivscale, info = f(a1, permute=1, scale=1)
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assert_(not info, repr(info))
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# print(a1)
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# print(ba, lo, hi, pivscale)
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def test_gehrd(self):
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a = [[-149, -50, -154],
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[537, 180, 546],
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[-27, -9, -25]]
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for p in 'd':
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f = getattr(flapack, p+'gehrd', None)
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if f is None:
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continue
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ht, tau, info = f(a)
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assert_(not info, repr(info))
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def test_trsyl(self):
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a = np.array([[1, 2], [0, 4]])
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b = np.array([[5, 6], [0, 8]])
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c = np.array([[9, 10], [11, 12]])
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trans = 'T'
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# Test single and double implementations, including most
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# of the options
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for dtype in 'fdFD':
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a1, b1, c1 = a.astype(dtype), b.astype(dtype), c.astype(dtype)
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trsyl, = get_lapack_funcs(('trsyl',), (a1,))
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if dtype.isupper(): # is complex dtype
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a1[0] += 1j
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trans = 'C'
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x, scale, info = trsyl(a1, b1, c1)
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assert_array_almost_equal(np.dot(a1, x) + np.dot(x, b1),
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scale * c1)
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x, scale, info = trsyl(a1, b1, c1, trana=trans, tranb=trans)
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assert_array_almost_equal(
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np.dot(a1.conjugate().T, x) + np.dot(x, b1.conjugate().T),
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scale * c1, decimal=4)
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x, scale, info = trsyl(a1, b1, c1, isgn=-1)
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assert_array_almost_equal(np.dot(a1, x) - np.dot(x, b1),
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scale * c1, decimal=4)
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def test_lange(self):
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a = np.array([
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[-149, -50, -154],
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[537, 180, 546],
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[-27, -9, -25]])
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for dtype in 'fdFD':
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for norm_str in 'Mm1OoIiFfEe':
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a1 = a.astype(dtype)
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if dtype.isupper():
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# is complex dtype
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a1[0, 0] += 1j
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lange, = get_lapack_funcs(('lange',), (a1,))
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value = lange(norm_str, a1)
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if norm_str in 'FfEe':
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if dtype in 'Ff':
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decimal = 3
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else:
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decimal = 7
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ref = np.sqrt(np.sum(np.square(np.abs(a1))))
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assert_almost_equal(value, ref, decimal)
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else:
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if norm_str in 'Mm':
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ref = np.max(np.abs(a1))
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elif norm_str in '1Oo':
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ref = np.max(np.sum(np.abs(a1), axis=0))
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elif norm_str in 'Ii':
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ref = np.max(np.sum(np.abs(a1), axis=1))
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assert_equal(value, ref)
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class TestLapack:
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def test_flapack(self):
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if hasattr(flapack, 'empty_module'):
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# flapack module is empty
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pass
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def test_clapack(self):
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if hasattr(clapack, 'empty_module'):
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# clapack module is empty
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pass
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class TestLeastSquaresSolvers:
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def test_gels(self):
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seed(1234)
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# Test fat/tall matrix argument handling - gh-issue #8329
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for ind, dtype in enumerate(DTYPES):
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m = 10
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n = 20
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nrhs = 1
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a1 = rand(m, n).astype(dtype)
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b1 = rand(n).astype(dtype)
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gls, glslw = get_lapack_funcs(('gels', 'gels_lwork'), dtype=dtype)
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# Request of sizes
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lwork = _compute_lwork(glslw, m, n, nrhs)
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_, _, info = gls(a1, b1, lwork=lwork)
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assert_(info >= 0)
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_, _, info = gls(a1, b1, trans='TTCC'[ind], lwork=lwork)
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assert_(info >= 0)
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for dtype in REAL_DTYPES:
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a1 = np.array([[1.0, 2.0],
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[4.0, 5.0],
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[7.0, 8.0]], dtype=dtype)
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b1 = np.array([16.0, 17.0, 20.0], dtype=dtype)
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gels, gels_lwork, geqrf = get_lapack_funcs(
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('gels', 'gels_lwork', 'geqrf'), (a1, b1))
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m, n = a1.shape
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if len(b1.shape) == 2:
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nrhs = b1.shape[1]
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else:
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nrhs = 1
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# Request of sizes
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lwork = _compute_lwork(gels_lwork, m, n, nrhs)
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lqr, x, info = gels(a1, b1, lwork=lwork)
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assert_allclose(x[:-1], np.array([-14.333333333333323,
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14.999999999999991],
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dtype=dtype),
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rtol=25*np.finfo(dtype).eps)
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lqr_truth, _, _, _ = geqrf(a1)
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assert_array_equal(lqr, lqr_truth)
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for dtype in COMPLEX_DTYPES:
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a1 = np.array([[1.0+4.0j, 2.0],
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[4.0+0.5j, 5.0-3.0j],
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[7.0-2.0j, 8.0+0.7j]], dtype=dtype)
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b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype)
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gels, gels_lwork, geqrf = get_lapack_funcs(
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('gels', 'gels_lwork', 'geqrf'), (a1, b1))
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m, n = a1.shape
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if len(b1.shape) == 2:
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nrhs = b1.shape[1]
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else:
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nrhs = 1
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# Request of sizes
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lwork = _compute_lwork(gels_lwork, m, n, nrhs)
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lqr, x, info = gels(a1, b1, lwork=lwork)
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assert_allclose(x[:-1],
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np.array([1.161753632288328-1.901075709391912j,
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1.735882340522193+1.521240901196909j],
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dtype=dtype), rtol=25*np.finfo(dtype).eps)
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lqr_truth, _, _, _ = geqrf(a1)
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assert_array_equal(lqr, lqr_truth)
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def test_gelsd(self):
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for dtype in REAL_DTYPES:
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a1 = np.array([[1.0, 2.0],
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[4.0, 5.0],
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[7.0, 8.0]], dtype=dtype)
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b1 = np.array([16.0, 17.0, 20.0], dtype=dtype)
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gelsd, gelsd_lwork = get_lapack_funcs(('gelsd', 'gelsd_lwork'),
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(a1, b1))
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m, n = a1.shape
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if len(b1.shape) == 2:
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nrhs = b1.shape[1]
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else:
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nrhs = 1
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# Request of sizes
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work, iwork, info = gelsd_lwork(m, n, nrhs, -1)
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lwork = int(np.real(work))
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iwork_size = iwork
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x, s, rank, info = gelsd(a1, b1, lwork, iwork_size,
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-1, False, False)
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assert_allclose(x[:-1], np.array([-14.333333333333323,
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14.999999999999991],
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dtype=dtype),
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rtol=25*np.finfo(dtype).eps)
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assert_allclose(s, np.array([12.596017180511966,
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0.583396253199685], dtype=dtype),
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rtol=25*np.finfo(dtype).eps)
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for dtype in COMPLEX_DTYPES:
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a1 = np.array([[1.0+4.0j, 2.0],
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[4.0+0.5j, 5.0-3.0j],
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[7.0-2.0j, 8.0+0.7j]], dtype=dtype)
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b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype)
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gelsd, gelsd_lwork = get_lapack_funcs(('gelsd', 'gelsd_lwork'),
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(a1, b1))
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m, n = a1.shape
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if len(b1.shape) == 2:
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nrhs = b1.shape[1]
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else:
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nrhs = 1
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# Request of sizes
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work, rwork, iwork, info = gelsd_lwork(m, n, nrhs, -1)
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lwork = int(np.real(work))
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rwork_size = int(rwork)
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iwork_size = iwork
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x, s, rank, info = gelsd(a1, b1, lwork, rwork_size, iwork_size,
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-1, False, False)
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assert_allclose(x[:-1],
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np.array([1.161753632288328-1.901075709391912j,
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1.735882340522193+1.521240901196909j],
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dtype=dtype), rtol=25*np.finfo(dtype).eps)
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assert_allclose(s,
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np.array([13.035514762572043, 4.337666985231382],
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dtype=dtype), rtol=25*np.finfo(dtype).eps)
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def test_gelss(self):
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for dtype in REAL_DTYPES:
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a1 = np.array([[1.0, 2.0],
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[4.0, 5.0],
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[7.0, 8.0]], dtype=dtype)
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b1 = np.array([16.0, 17.0, 20.0], dtype=dtype)
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gelss, gelss_lwork = get_lapack_funcs(('gelss', 'gelss_lwork'),
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(a1, b1))
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m, n = a1.shape
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if len(b1.shape) == 2:
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nrhs = b1.shape[1]
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else:
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nrhs = 1
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# Request of sizes
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work, info = gelss_lwork(m, n, nrhs, -1)
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lwork = int(np.real(work))
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v, x, s, rank, work, info = gelss(a1, b1, -1, lwork, False, False)
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assert_allclose(x[:-1], np.array([-14.333333333333323,
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14.999999999999991],
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dtype=dtype),
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rtol=25*np.finfo(dtype).eps)
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assert_allclose(s, np.array([12.596017180511966,
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0.583396253199685], dtype=dtype),
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rtol=25*np.finfo(dtype).eps)
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for dtype in COMPLEX_DTYPES:
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a1 = np.array([[1.0+4.0j, 2.0],
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[4.0+0.5j, 5.0-3.0j],
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[7.0-2.0j, 8.0+0.7j]], dtype=dtype)
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|
|
b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype)
|
||
|
|
gelss, gelss_lwork = get_lapack_funcs(('gelss', 'gelss_lwork'),
|
||
|
|
(a1, b1))
|
||
|
|
|
||
|
|
m, n = a1.shape
|
||
|
|
if len(b1.shape) == 2:
|
||
|
|
nrhs = b1.shape[1]
|
||
|
|
else:
|
||
|
|
nrhs = 1
|
||
|
|
|
||
|
|
# Request of sizes
|
||
|
|
work, info = gelss_lwork(m, n, nrhs, -1)
|
||
|
|
lwork = int(np.real(work))
|
||
|
|
|
||
|
|
v, x, s, rank, work, info = gelss(a1, b1, -1, lwork, False, False)
|
||
|
|
assert_allclose(x[:-1],
|
||
|
|
np.array([1.161753632288328-1.901075709391912j,
|
||
|
|
1.735882340522193+1.521240901196909j],
|
||
|
|
dtype=dtype),
|
||
|
|
rtol=25*np.finfo(dtype).eps)
|
||
|
|
assert_allclose(s, np.array([13.035514762572043,
|
||
|
|
4.337666985231382], dtype=dtype),
|
||
|
|
rtol=25*np.finfo(dtype).eps)
|
||
|
|
|
||
|
|
def test_gelsy(self):
|
||
|
|
|
||
|
|
for dtype in REAL_DTYPES:
|
||
|
|
a1 = np.array([[1.0, 2.0],
|
||
|
|
[4.0, 5.0],
|
||
|
|
[7.0, 8.0]], dtype=dtype)
|
||
|
|
b1 = np.array([16.0, 17.0, 20.0], dtype=dtype)
|
||
|
|
gelsy, gelsy_lwork = get_lapack_funcs(('gelsy', 'gelss_lwork'),
|
||
|
|
(a1, b1))
|
||
|
|
|
||
|
|
m, n = a1.shape
|
||
|
|
if len(b1.shape) == 2:
|
||
|
|
nrhs = b1.shape[1]
|
||
|
|
else:
|
||
|
|
nrhs = 1
|
||
|
|
|
||
|
|
# Request of sizes
|
||
|
|
work, info = gelsy_lwork(m, n, nrhs, 10*np.finfo(dtype).eps)
|
||
|
|
lwork = int(np.real(work))
|
||
|
|
|
||
|
|
jptv = np.zeros((a1.shape[1], 1), dtype=np.int32)
|
||
|
|
v, x, j, rank, info = gelsy(a1, b1, jptv, np.finfo(dtype).eps,
|
||
|
|
lwork, False, False)
|
||
|
|
assert_allclose(x[:-1], np.array([-14.333333333333323,
|
||
|
|
14.999999999999991],
|
||
|
|
dtype=dtype),
|
||
|
|
rtol=25*np.finfo(dtype).eps)
|
||
|
|
|
||
|
|
for dtype in COMPLEX_DTYPES:
|
||
|
|
a1 = np.array([[1.0+4.0j, 2.0],
|
||
|
|
[4.0+0.5j, 5.0-3.0j],
|
||
|
|
[7.0-2.0j, 8.0+0.7j]], dtype=dtype)
|
||
|
|
b1 = np.array([16.0, 17.0+2.0j, 20.0-4.0j], dtype=dtype)
|
||
|
|
gelsy, gelsy_lwork = get_lapack_funcs(('gelsy', 'gelss_lwork'),
|
||
|
|
(a1, b1))
|
||
|
|
|
||
|
|
m, n = a1.shape
|
||
|
|
if len(b1.shape) == 2:
|
||
|
|
nrhs = b1.shape[1]
|
||
|
|
else:
|
||
|
|
nrhs = 1
|
||
|
|
|
||
|
|
# Request of sizes
|
||
|
|
work, info = gelsy_lwork(m, n, nrhs, 10*np.finfo(dtype).eps)
|
||
|
|
lwork = int(np.real(work))
|
||
|
|
|
||
|
|
jptv = np.zeros((a1.shape[1], 1), dtype=np.int32)
|
||
|
|
v, x, j, rank, info = gelsy(a1, b1, jptv, np.finfo(dtype).eps,
|
||
|
|
lwork, False, False)
|
||
|
|
assert_allclose(x[:-1],
|
||
|
|
np.array([1.161753632288328-1.901075709391912j,
|
||
|
|
1.735882340522193+1.521240901196909j],
|
||
|
|
dtype=dtype),
|
||
|
|
rtol=25*np.finfo(dtype).eps)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
@pytest.mark.parametrize('shape', [(3, 4), (5, 2), (2**18, 2**18)])
|
||
|
|
def test_geqrf_lwork(dtype, shape):
|
||
|
|
geqrf_lwork = get_lapack_funcs(('geqrf_lwork'), dtype=dtype)
|
||
|
|
m, n = shape
|
||
|
|
lwork, info = geqrf_lwork(m=m, n=n)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
|
||
|
|
class TestRegression:
|
||
|
|
|
||
|
|
def test_ticket_1645(self):
|
||
|
|
# Check that RQ routines have correct lwork
|
||
|
|
for dtype in DTYPES:
|
||
|
|
a = np.zeros((300, 2), dtype=dtype)
|
||
|
|
|
||
|
|
gerqf, = get_lapack_funcs(['gerqf'], [a])
|
||
|
|
assert_raises(Exception, gerqf, a, lwork=2)
|
||
|
|
rq, tau, work, info = gerqf(a)
|
||
|
|
|
||
|
|
if dtype in REAL_DTYPES:
|
||
|
|
orgrq, = get_lapack_funcs(['orgrq'], [a])
|
||
|
|
assert_raises(Exception, orgrq, rq[-2:], tau, lwork=1)
|
||
|
|
orgrq(rq[-2:], tau, lwork=2)
|
||
|
|
elif dtype in COMPLEX_DTYPES:
|
||
|
|
ungrq, = get_lapack_funcs(['ungrq'], [a])
|
||
|
|
assert_raises(Exception, ungrq, rq[-2:], tau, lwork=1)
|
||
|
|
ungrq(rq[-2:], tau, lwork=2)
|
||
|
|
|
||
|
|
|
||
|
|
class TestDpotr:
|
||
|
|
def test_gh_2691(self):
|
||
|
|
# 'lower' argument of dportf/dpotri
|
||
|
|
for lower in [True, False]:
|
||
|
|
for clean in [True, False]:
|
||
|
|
np.random.seed(42)
|
||
|
|
x = np.random.normal(size=(3, 3))
|
||
|
|
a = x.dot(x.T)
|
||
|
|
|
||
|
|
dpotrf, dpotri = get_lapack_funcs(("potrf", "potri"), (a, ))
|
||
|
|
|
||
|
|
c, info = dpotrf(a, lower, clean=clean)
|
||
|
|
dpt = dpotri(c, lower)[0]
|
||
|
|
|
||
|
|
if lower:
|
||
|
|
assert_allclose(np.tril(dpt), np.tril(inv(a)))
|
||
|
|
else:
|
||
|
|
assert_allclose(np.triu(dpt), np.triu(inv(a)))
|
||
|
|
|
||
|
|
|
||
|
|
class TestDlasd4:
|
||
|
|
def test_sing_val_update(self):
|
||
|
|
|
||
|
|
sigmas = np.array([4., 3., 2., 0])
|
||
|
|
m_vec = np.array([3.12, 5.7, -4.8, -2.2])
|
||
|
|
|
||
|
|
M = np.hstack((np.vstack((np.diag(sigmas[0:-1]),
|
||
|
|
np.zeros((1, len(m_vec) - 1)))),
|
||
|
|
m_vec[:, np.newaxis]))
|
||
|
|
SM = svd(M, full_matrices=False, compute_uv=False, overwrite_a=False,
|
||
|
|
check_finite=False)
|
||
|
|
|
||
|
|
it_len = len(sigmas)
|
||
|
|
sgm = np.concatenate((sigmas[::-1], [sigmas[0] + it_len*norm(m_vec)]))
|
||
|
|
mvc = np.concatenate((m_vec[::-1], (0,)))
|
||
|
|
|
||
|
|
lasd4 = get_lapack_funcs('lasd4', (sigmas,))
|
||
|
|
|
||
|
|
roots = []
|
||
|
|
for i in range(0, it_len):
|
||
|
|
res = lasd4(i, sgm, mvc)
|
||
|
|
roots.append(res[1])
|
||
|
|
|
||
|
|
assert_((res[3] <= 0), "LAPACK root finding dlasd4 failed to find \
|
||
|
|
the singular value %i" % i)
|
||
|
|
roots = np.array(roots)[::-1]
|
||
|
|
|
||
|
|
assert_((not np.any(np.isnan(roots)), "There are NaN roots"))
|
||
|
|
assert_allclose(SM, roots, atol=100*np.finfo(np.float64).eps,
|
||
|
|
rtol=100*np.finfo(np.float64).eps)
|
||
|
|
|
||
|
|
|
||
|
|
class TestTbtrs:
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
def test_nag_example_f07vef_f07vsf(self, dtype):
|
||
|
|
"""Test real (f07vef) and complex (f07vsf) examples from NAG
|
||
|
|
|
||
|
|
Examples available from:
|
||
|
|
* https://www.nag.com/numeric/fl/nagdoc_latest/html/f07/f07vef.html
|
||
|
|
* https://www.nag.com/numeric/fl/nagdoc_latest/html/f07/f07vsf.html
|
||
|
|
|
||
|
|
"""
|
||
|
|
if dtype in REAL_DTYPES:
|
||
|
|
ab = np.array([[-4.16, 4.78, 6.32, 0.16],
|
||
|
|
[-2.25, 5.86, -4.82, 0]],
|
||
|
|
dtype=dtype)
|
||
|
|
b = np.array([[-16.64, -4.16],
|
||
|
|
[-13.78, -16.59],
|
||
|
|
[13.10, -4.94],
|
||
|
|
[-14.14, -9.96]],
|
||
|
|
dtype=dtype)
|
||
|
|
x_out = np.array([[4, 1],
|
||
|
|
[-1, -3],
|
||
|
|
[3, 2],
|
||
|
|
[2, -2]],
|
||
|
|
dtype=dtype)
|
||
|
|
elif dtype in COMPLEX_DTYPES:
|
||
|
|
ab = np.array([[-1.94+4.43j, 4.12-4.27j, 0.43-2.66j, 0.44+0.1j],
|
||
|
|
[-3.39+3.44j, -1.84+5.52j, 1.74 - 0.04j, 0],
|
||
|
|
[1.62+3.68j, -2.77-1.93j, 0, 0]],
|
||
|
|
dtype=dtype)
|
||
|
|
b = np.array([[-8.86 - 3.88j, -24.09 - 5.27j],
|
||
|
|
[-15.57 - 23.41j, -57.97 + 8.14j],
|
||
|
|
[-7.63 + 22.78j, 19.09 - 29.51j],
|
||
|
|
[-14.74 - 2.40j, 19.17 + 21.33j]],
|
||
|
|
dtype=dtype)
|
||
|
|
x_out = np.array([[2j, 1 + 5j],
|
||
|
|
[1 - 3j, -7 - 2j],
|
||
|
|
[-4.001887 - 4.988417j, 3.026830 + 4.003182j],
|
||
|
|
[1.996158 - 1.045105j, -6.103357 - 8.986653j]],
|
||
|
|
dtype=dtype)
|
||
|
|
else:
|
||
|
|
raise ValueError(f"Datatype {dtype} not understood.")
|
||
|
|
|
||
|
|
tbtrs = get_lapack_funcs(('tbtrs'), dtype=dtype)
|
||
|
|
x, info = tbtrs(ab=ab, b=b, uplo='L')
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_allclose(x, x_out, rtol=0, atol=1e-5)
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype,trans',
|
||
|
|
[(dtype, trans)
|
||
|
|
for dtype in DTYPES for trans in ['N', 'T', 'C']
|
||
|
|
if not (trans == 'C' and dtype in REAL_DTYPES)])
|
||
|
|
@pytest.mark.parametrize('uplo', ['U', 'L'])
|
||
|
|
@pytest.mark.parametrize('diag', ['N', 'U'])
|
||
|
|
def test_random_matrices(self, dtype, trans, uplo, diag):
|
||
|
|
seed(1724)
|
||
|
|
# n, nrhs, kd are used to specify A and b.
|
||
|
|
# A is of shape n x n with kd super/sub-diagonals
|
||
|
|
# b is of shape n x nrhs matrix
|
||
|
|
n, nrhs, kd = 4, 3, 2
|
||
|
|
tbtrs = get_lapack_funcs('tbtrs', dtype=dtype)
|
||
|
|
|
||
|
|
is_upper = (uplo == 'U')
|
||
|
|
ku = kd * is_upper
|
||
|
|
kl = kd - ku
|
||
|
|
|
||
|
|
# Construct the diagonal and kd super/sub diagonals of A with
|
||
|
|
# the corresponding offsets.
|
||
|
|
band_offsets = range(ku, -kl - 1, -1)
|
||
|
|
band_widths = [n - abs(x) for x in band_offsets]
|
||
|
|
bands = [generate_random_dtype_array((width,), dtype)
|
||
|
|
for width in band_widths]
|
||
|
|
|
||
|
|
if diag == 'U': # A must be unit triangular
|
||
|
|
bands[ku] = np.ones(n, dtype=dtype)
|
||
|
|
|
||
|
|
# Construct the diagonal banded matrix A from the bands and offsets.
|
||
|
|
a = sps.diags(bands, band_offsets, format='dia')
|
||
|
|
|
||
|
|
# Convert A into banded storage form
|
||
|
|
ab = np.zeros((kd + 1, n), dtype)
|
||
|
|
for row, k in enumerate(band_offsets):
|
||
|
|
ab[row, max(k, 0):min(n+k, n)] = a.diagonal(k)
|
||
|
|
|
||
|
|
# The RHS values.
|
||
|
|
b = generate_random_dtype_array((n, nrhs), dtype)
|
||
|
|
|
||
|
|
x, info = tbtrs(ab=ab, b=b, uplo=uplo, trans=trans, diag=diag)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
if trans == 'N':
|
||
|
|
assert_allclose(a @ x, b, rtol=5e-5)
|
||
|
|
elif trans == 'T':
|
||
|
|
assert_allclose(a.T @ x, b, rtol=5e-5)
|
||
|
|
elif trans == 'C':
|
||
|
|
assert_allclose(a.H @ x, b, rtol=5e-5)
|
||
|
|
else:
|
||
|
|
raise ValueError('Invalid trans argument')
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('uplo,trans,diag',
|
||
|
|
[['U', 'N', 'Invalid'],
|
||
|
|
['U', 'Invalid', 'N'],
|
||
|
|
['Invalid', 'N', 'N']])
|
||
|
|
def test_invalid_argument_raises_exception(self, uplo, trans, diag):
|
||
|
|
"""Test if invalid values of uplo, trans and diag raise exceptions"""
|
||
|
|
# Argument checks occur independently of used datatype.
|
||
|
|
# This mean we must not parameterize all available datatypes.
|
||
|
|
tbtrs = get_lapack_funcs('tbtrs', dtype=np.float64)
|
||
|
|
ab = rand(4, 2)
|
||
|
|
b = rand(2, 4)
|
||
|
|
assert_raises(Exception, tbtrs, ab, b, uplo, trans, diag)
|
||
|
|
|
||
|
|
def test_zero_element_in_diagonal(self):
|
||
|
|
"""Test if a matrix with a zero diagonal element is singular
|
||
|
|
|
||
|
|
If the i-th diagonal of A is zero, ?tbtrs should return `i` in `info`
|
||
|
|
indicating the provided matrix is singular.
|
||
|
|
|
||
|
|
Note that ?tbtrs requires the matrix A to be stored in banded form.
|
||
|
|
In this form the diagonal corresponds to the last row."""
|
||
|
|
ab = np.ones((3, 4), dtype=float)
|
||
|
|
b = np.ones(4, dtype=float)
|
||
|
|
tbtrs = get_lapack_funcs('tbtrs', dtype=float)
|
||
|
|
|
||
|
|
ab[-1, 3] = 0
|
||
|
|
_, info = tbtrs(ab=ab, b=b, uplo='U')
|
||
|
|
assert_equal(info, 4)
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('ldab,n,ldb,nrhs', [
|
||
|
|
(5, 5, 0, 5),
|
||
|
|
(5, 5, 3, 5)
|
||
|
|
])
|
||
|
|
def test_invalid_matrix_shapes(self, ldab, n, ldb, nrhs):
|
||
|
|
"""Test ?tbtrs fails correctly if shapes are invalid."""
|
||
|
|
ab = np.ones((ldab, n), dtype=float)
|
||
|
|
b = np.ones((ldb, nrhs), dtype=float)
|
||
|
|
tbtrs = get_lapack_funcs('tbtrs', dtype=float)
|
||
|
|
assert_raises(Exception, tbtrs, ab, b)
|
||
|
|
|
||
|
|
|
||
|
|
def test_lartg():
|
||
|
|
for dtype in 'fdFD':
|
||
|
|
lartg = get_lapack_funcs('lartg', dtype=dtype)
|
||
|
|
|
||
|
|
f = np.array(3, dtype)
|
||
|
|
g = np.array(4, dtype)
|
||
|
|
|
||
|
|
if np.iscomplexobj(g):
|
||
|
|
g *= 1j
|
||
|
|
|
||
|
|
cs, sn, r = lartg(f, g)
|
||
|
|
|
||
|
|
assert_allclose(cs, 3.0/5.0)
|
||
|
|
assert_allclose(r, 5.0)
|
||
|
|
|
||
|
|
if np.iscomplexobj(g):
|
||
|
|
assert_allclose(sn, -4.0j/5.0)
|
||
|
|
assert_(type(r) == complex)
|
||
|
|
assert_(type(cs) == float)
|
||
|
|
else:
|
||
|
|
assert_allclose(sn, 4.0/5.0)
|
||
|
|
|
||
|
|
|
||
|
|
def test_rot():
|
||
|
|
# srot, drot from blas and crot and zrot from lapack.
|
||
|
|
|
||
|
|
for dtype in 'fdFD':
|
||
|
|
c = 0.6
|
||
|
|
s = 0.8
|
||
|
|
|
||
|
|
u = np.full(4, 3, dtype)
|
||
|
|
v = np.full(4, 4, dtype)
|
||
|
|
atol = 10**-(np.finfo(dtype).precision-1)
|
||
|
|
|
||
|
|
if dtype in 'fd':
|
||
|
|
rot = get_blas_funcs('rot', dtype=dtype)
|
||
|
|
f = 4
|
||
|
|
else:
|
||
|
|
rot = get_lapack_funcs('rot', dtype=dtype)
|
||
|
|
s *= -1j
|
||
|
|
v *= 1j
|
||
|
|
f = 4j
|
||
|
|
|
||
|
|
assert_allclose(rot(u, v, c, s), [[5, 5, 5, 5],
|
||
|
|
[0, 0, 0, 0]], atol=atol)
|
||
|
|
assert_allclose(rot(u, v, c, s, n=2), [[5, 5, 3, 3],
|
||
|
|
[0, 0, f, f]], atol=atol)
|
||
|
|
assert_allclose(rot(u, v, c, s, offx=2, offy=2),
|
||
|
|
[[3, 3, 5, 5], [f, f, 0, 0]], atol=atol)
|
||
|
|
assert_allclose(rot(u, v, c, s, incx=2, offy=2, n=2),
|
||
|
|
[[5, 3, 5, 3], [f, f, 0, 0]], atol=atol)
|
||
|
|
assert_allclose(rot(u, v, c, s, offx=2, incy=2, n=2),
|
||
|
|
[[3, 3, 5, 5], [0, f, 0, f]], atol=atol)
|
||
|
|
assert_allclose(rot(u, v, c, s, offx=2, incx=2, offy=2, incy=2, n=1),
|
||
|
|
[[3, 3, 5, 3], [f, f, 0, f]], atol=atol)
|
||
|
|
assert_allclose(rot(u, v, c, s, incx=-2, incy=-2, n=2),
|
||
|
|
[[5, 3, 5, 3], [0, f, 0, f]], atol=atol)
|
||
|
|
|
||
|
|
a, b = rot(u, v, c, s, overwrite_x=1, overwrite_y=1)
|
||
|
|
assert_(a is u)
|
||
|
|
assert_(b is v)
|
||
|
|
assert_allclose(a, [5, 5, 5, 5], atol=atol)
|
||
|
|
assert_allclose(b, [0, 0, 0, 0], atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
def test_larfg_larf():
|
||
|
|
np.random.seed(1234)
|
||
|
|
a0 = np.random.random((4, 4))
|
||
|
|
a0 = a0.T.dot(a0)
|
||
|
|
|
||
|
|
a0j = np.random.random((4, 4)) + 1j*np.random.random((4, 4))
|
||
|
|
a0j = a0j.T.conj().dot(a0j)
|
||
|
|
|
||
|
|
# our test here will be to do one step of reducing a hermetian matrix to
|
||
|
|
# tridiagonal form using householder transforms.
|
||
|
|
|
||
|
|
for dtype in 'fdFD':
|
||
|
|
larfg, larf = get_lapack_funcs(['larfg', 'larf'], dtype=dtype)
|
||
|
|
|
||
|
|
if dtype in 'FD':
|
||
|
|
a = a0j.copy()
|
||
|
|
else:
|
||
|
|
a = a0.copy()
|
||
|
|
|
||
|
|
# generate a householder transform to clear a[2:,0]
|
||
|
|
alpha, x, tau = larfg(a.shape[0]-1, a[1, 0], a[2:, 0])
|
||
|
|
|
||
|
|
# create expected output
|
||
|
|
expected = np.zeros_like(a[:, 0])
|
||
|
|
expected[0] = a[0, 0]
|
||
|
|
expected[1] = alpha
|
||
|
|
|
||
|
|
# assemble householder vector
|
||
|
|
v = np.zeros_like(a[1:, 0])
|
||
|
|
v[0] = 1.0
|
||
|
|
v[1:] = x
|
||
|
|
|
||
|
|
# apply transform from the left
|
||
|
|
a[1:, :] = larf(v, tau.conjugate(), a[1:, :], np.zeros(a.shape[1]))
|
||
|
|
|
||
|
|
# apply transform from the right
|
||
|
|
a[:, 1:] = larf(v, tau, a[:, 1:], np.zeros(a.shape[0]), side='R')
|
||
|
|
|
||
|
|
assert_allclose(a[:, 0], expected, atol=1e-5)
|
||
|
|
assert_allclose(a[0, :], expected, atol=1e-5)
|
||
|
|
|
||
|
|
|
||
|
|
def test_sgesdd_lwork_bug_workaround():
|
||
|
|
# Test that SGESDD lwork is sufficiently large for LAPACK.
|
||
|
|
#
|
||
|
|
# This checks that _compute_lwork() correctly works around a bug in
|
||
|
|
# LAPACK versions older than 3.10.1.
|
||
|
|
|
||
|
|
sgesdd_lwork = get_lapack_funcs('gesdd_lwork', dtype=np.float32,
|
||
|
|
ilp64='preferred')
|
||
|
|
n = 9537
|
||
|
|
lwork = _compute_lwork(sgesdd_lwork, n, n,
|
||
|
|
compute_uv=True, full_matrices=True)
|
||
|
|
# If we called the Fortran function SGESDD directly with IWORK=-1, the
|
||
|
|
# LAPACK bug would result in lwork being 272929856, which was too small.
|
||
|
|
# (The result was returned in a single precision float, which does not
|
||
|
|
# have sufficient precision to represent the exact integer value that it
|
||
|
|
# computed internally.) The work-around implemented in _compute_lwork()
|
||
|
|
# will convert that to 272929888. If we are using LAPACK 3.10.1 or later
|
||
|
|
# (such as in OpenBLAS 0.3.21 or later), the work-around will return
|
||
|
|
# 272929920, because it does not know which version of LAPACK is being
|
||
|
|
# used, so it always applies the correction to whatever it is given. We
|
||
|
|
# will accept either 272929888 or 272929920.
|
||
|
|
# Note that the acceptable values are a LAPACK implementation detail.
|
||
|
|
# If a future version of LAPACK changes how SGESDD works, and therefore
|
||
|
|
# changes the required LWORK size, the acceptable values might have to
|
||
|
|
# be updated.
|
||
|
|
assert lwork == 272929888 or lwork == 272929920
|
||
|
|
|
||
|
|
|
||
|
|
class TestSytrd:
|
||
|
|
@pytest.mark.parametrize('dtype', REAL_DTYPES)
|
||
|
|
def test_sytrd_with_zero_dim_array(self, dtype):
|
||
|
|
# Assert that a 0x0 matrix raises an error
|
||
|
|
A = np.zeros((0, 0), dtype=dtype)
|
||
|
|
sytrd = get_lapack_funcs('sytrd', (A,))
|
||
|
|
assert_raises(ValueError, sytrd, A)
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', REAL_DTYPES)
|
||
|
|
@pytest.mark.parametrize('n', (1, 3))
|
||
|
|
def test_sytrd(self, dtype, n):
|
||
|
|
A = np.zeros((n, n), dtype=dtype)
|
||
|
|
|
||
|
|
sytrd, sytrd_lwork = \
|
||
|
|
get_lapack_funcs(('sytrd', 'sytrd_lwork'), (A,))
|
||
|
|
|
||
|
|
# some upper triangular array
|
||
|
|
A[np.triu_indices_from(A)] = \
|
||
|
|
np.arange(1, n*(n+1)//2+1, dtype=dtype)
|
||
|
|
|
||
|
|
# query lwork
|
||
|
|
lwork, info = sytrd_lwork(n)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
# check lower=1 behavior (shouldn't do much since the matrix is
|
||
|
|
# upper triangular)
|
||
|
|
data, d, e, tau, info = sytrd(A, lower=1, lwork=lwork)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
assert_allclose(data, A, atol=5*np.finfo(dtype).eps, rtol=1.0)
|
||
|
|
assert_allclose(d, np.diag(A))
|
||
|
|
assert_allclose(e, 0.0)
|
||
|
|
assert_allclose(tau, 0.0)
|
||
|
|
|
||
|
|
# and now for the proper test (lower=0 is the default)
|
||
|
|
data, d, e, tau, info = sytrd(A, lwork=lwork)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
# assert Q^T*A*Q = tridiag(e, d, e)
|
||
|
|
|
||
|
|
# build tridiagonal matrix
|
||
|
|
T = np.zeros_like(A, dtype=dtype)
|
||
|
|
k = np.arange(A.shape[0])
|
||
|
|
T[k, k] = d
|
||
|
|
k2 = np.arange(A.shape[0]-1)
|
||
|
|
T[k2+1, k2] = e
|
||
|
|
T[k2, k2+1] = e
|
||
|
|
|
||
|
|
# build Q
|
||
|
|
Q = np.eye(n, n, dtype=dtype)
|
||
|
|
for i in range(n-1):
|
||
|
|
v = np.zeros(n, dtype=dtype)
|
||
|
|
v[:i] = data[:i, i+1]
|
||
|
|
v[i] = 1.0
|
||
|
|
H = np.eye(n, n, dtype=dtype) - tau[i] * np.outer(v, v)
|
||
|
|
Q = np.dot(H, Q)
|
||
|
|
|
||
|
|
# Make matrix fully symmetric
|
||
|
|
i_lower = np.tril_indices(n, -1)
|
||
|
|
A[i_lower] = A.T[i_lower]
|
||
|
|
|
||
|
|
QTAQ = np.dot(Q.T, np.dot(A, Q))
|
||
|
|
|
||
|
|
# disable rtol here since some values in QTAQ and T are very close
|
||
|
|
# to 0.
|
||
|
|
assert_allclose(QTAQ, T, atol=5*np.finfo(dtype).eps, rtol=1.0)
|
||
|
|
|
||
|
|
|
||
|
|
class TestHetrd:
|
||
|
|
@pytest.mark.parametrize('complex_dtype', COMPLEX_DTYPES)
|
||
|
|
def test_hetrd_with_zero_dim_array(self, complex_dtype):
|
||
|
|
# Assert that a 0x0 matrix raises an error
|
||
|
|
A = np.zeros((0, 0), dtype=complex_dtype)
|
||
|
|
hetrd = get_lapack_funcs('hetrd', (A,))
|
||
|
|
assert_raises(ValueError, hetrd, A)
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('real_dtype,complex_dtype',
|
||
|
|
zip(REAL_DTYPES, COMPLEX_DTYPES))
|
||
|
|
@pytest.mark.parametrize('n', (1, 3))
|
||
|
|
def test_hetrd(self, n, real_dtype, complex_dtype):
|
||
|
|
A = np.zeros((n, n), dtype=complex_dtype)
|
||
|
|
hetrd, hetrd_lwork = \
|
||
|
|
get_lapack_funcs(('hetrd', 'hetrd_lwork'), (A,))
|
||
|
|
|
||
|
|
# some upper triangular array
|
||
|
|
A[np.triu_indices_from(A)] = (
|
||
|
|
np.arange(1, n*(n+1)//2+1, dtype=real_dtype)
|
||
|
|
+ 1j * np.arange(1, n*(n+1)//2+1, dtype=real_dtype)
|
||
|
|
)
|
||
|
|
np.fill_diagonal(A, np.real(np.diag(A)))
|
||
|
|
|
||
|
|
# test query lwork
|
||
|
|
for x in [0, 1]:
|
||
|
|
_, info = hetrd_lwork(n, lower=x)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
# lwork returns complex which segfaults hetrd call (gh-10388)
|
||
|
|
# use the safe and recommended option
|
||
|
|
lwork = _compute_lwork(hetrd_lwork, n)
|
||
|
|
|
||
|
|
# check lower=1 behavior (shouldn't do much since the matrix is
|
||
|
|
# upper triangular)
|
||
|
|
data, d, e, tau, info = hetrd(A, lower=1, lwork=lwork)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
assert_allclose(data, A, atol=5*np.finfo(real_dtype).eps, rtol=1.0)
|
||
|
|
|
||
|
|
assert_allclose(d, np.real(np.diag(A)))
|
||
|
|
assert_allclose(e, 0.0)
|
||
|
|
assert_allclose(tau, 0.0)
|
||
|
|
|
||
|
|
# and now for the proper test (lower=0 is the default)
|
||
|
|
data, d, e, tau, info = hetrd(A, lwork=lwork)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
# assert Q^T*A*Q = tridiag(e, d, e)
|
||
|
|
|
||
|
|
# build tridiagonal matrix
|
||
|
|
T = np.zeros_like(A, dtype=real_dtype)
|
||
|
|
k = np.arange(A.shape[0], dtype=int)
|
||
|
|
T[k, k] = d
|
||
|
|
k2 = np.arange(A.shape[0]-1, dtype=int)
|
||
|
|
T[k2+1, k2] = e
|
||
|
|
T[k2, k2+1] = e
|
||
|
|
|
||
|
|
# build Q
|
||
|
|
Q = np.eye(n, n, dtype=complex_dtype)
|
||
|
|
for i in range(n-1):
|
||
|
|
v = np.zeros(n, dtype=complex_dtype)
|
||
|
|
v[:i] = data[:i, i+1]
|
||
|
|
v[i] = 1.0
|
||
|
|
H = np.eye(n, n, dtype=complex_dtype) \
|
||
|
|
- tau[i] * np.outer(v, np.conj(v))
|
||
|
|
Q = np.dot(H, Q)
|
||
|
|
|
||
|
|
# Make matrix fully Hermitian
|
||
|
|
i_lower = np.tril_indices(n, -1)
|
||
|
|
A[i_lower] = np.conj(A.T[i_lower])
|
||
|
|
|
||
|
|
QHAQ = np.dot(np.conj(Q.T), np.dot(A, Q))
|
||
|
|
|
||
|
|
# disable rtol here since some values in QTAQ and T are very close
|
||
|
|
# to 0.
|
||
|
|
assert_allclose(
|
||
|
|
QHAQ, T, atol=10*np.finfo(real_dtype).eps, rtol=1.0
|
||
|
|
)
|
||
|
|
|
||
|
|
|
||
|
|
def test_gglse():
|
||
|
|
# Example data taken from NAG manual
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
# DTYPES = <s,d,c,z> gglse
|
||
|
|
func, func_lwork = get_lapack_funcs(('gglse', 'gglse_lwork'),
|
||
|
|
dtype=dtype)
|
||
|
|
lwork = _compute_lwork(func_lwork, m=6, n=4, p=2)
|
||
|
|
# For <s,d>gglse
|
||
|
|
if ind < 2:
|
||
|
|
a = np.array([[-0.57, -1.28, -0.39, 0.25],
|
||
|
|
[-1.93, 1.08, -0.31, -2.14],
|
||
|
|
[2.30, 0.24, 0.40, -0.35],
|
||
|
|
[-1.93, 0.64, -0.66, 0.08],
|
||
|
|
[0.15, 0.30, 0.15, -2.13],
|
||
|
|
[-0.02, 1.03, -1.43, 0.50]], dtype=dtype)
|
||
|
|
c = np.array([-1.50, -2.14, 1.23, -0.54, -1.68, 0.82], dtype=dtype)
|
||
|
|
d = np.array([0., 0.], dtype=dtype)
|
||
|
|
# For <s,d>gglse
|
||
|
|
else:
|
||
|
|
a = np.array([[0.96-0.81j, -0.03+0.96j, -0.91+2.06j, -0.05+0.41j],
|
||
|
|
[-0.98+1.98j, -1.20+0.19j, -0.66+0.42j, -0.81+0.56j],
|
||
|
|
[0.62-0.46j, 1.01+0.02j, 0.63-0.17j, -1.11+0.60j],
|
||
|
|
[0.37+0.38j, 0.19-0.54j, -0.98-0.36j, 0.22-0.20j],
|
||
|
|
[0.83+0.51j, 0.20+0.01j, -0.17-0.46j, 1.47+1.59j],
|
||
|
|
[1.08-0.28j, 0.20-0.12j, -0.07+1.23j, 0.26+0.26j]])
|
||
|
|
c = np.array([[-2.54+0.09j],
|
||
|
|
[1.65-2.26j],
|
||
|
|
[-2.11-3.96j],
|
||
|
|
[1.82+3.30j],
|
||
|
|
[-6.41+3.77j],
|
||
|
|
[2.07+0.66j]])
|
||
|
|
d = np.zeros(2, dtype=dtype)
|
||
|
|
|
||
|
|
b = np.array([[1., 0., -1., 0.], [0., 1., 0., -1.]], dtype=dtype)
|
||
|
|
|
||
|
|
_, _, _, result, _ = func(a, b, c, d, lwork=lwork)
|
||
|
|
if ind < 2:
|
||
|
|
expected = np.array([0.48904455,
|
||
|
|
0.99754786,
|
||
|
|
0.48904455,
|
||
|
|
0.99754786])
|
||
|
|
else:
|
||
|
|
expected = np.array([1.08742917-1.96205783j,
|
||
|
|
-0.74093902+3.72973919j,
|
||
|
|
1.08742917-1.96205759j,
|
||
|
|
-0.74093896+3.72973895j])
|
||
|
|
assert_array_almost_equal(result, expected, decimal=4)
|
||
|
|
|
||
|
|
|
||
|
|
def test_sycon_hecon():
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES+COMPLEX_DTYPES):
|
||
|
|
# DTYPES + COMPLEX DTYPES = <s,d,c,z> sycon + <c,z>hecon
|
||
|
|
n = 10
|
||
|
|
# For <s,d,c,z>sycon
|
||
|
|
if ind < 4:
|
||
|
|
func_lwork = get_lapack_funcs('sytrf_lwork', dtype=dtype)
|
||
|
|
funcon, functrf = get_lapack_funcs(('sycon', 'sytrf'), dtype=dtype)
|
||
|
|
A = (rand(n, n)).astype(dtype)
|
||
|
|
# For <c,z>hecon
|
||
|
|
else:
|
||
|
|
func_lwork = get_lapack_funcs('hetrf_lwork', dtype=dtype)
|
||
|
|
funcon, functrf = get_lapack_funcs(('hecon', 'hetrf'), dtype=dtype)
|
||
|
|
A = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
|
||
|
|
# Since sycon only refers to upper/lower part, conj() is safe here.
|
||
|
|
A = (A + A.conj().T)/2 + 2*np.eye(n, dtype=dtype)
|
||
|
|
|
||
|
|
anorm = norm(A, 1)
|
||
|
|
lwork = _compute_lwork(func_lwork, n)
|
||
|
|
ldu, ipiv, _ = functrf(A, lwork=lwork, lower=1)
|
||
|
|
rcond, _ = funcon(a=ldu, ipiv=ipiv, anorm=anorm, lower=1)
|
||
|
|
# The error is at most 1-fold
|
||
|
|
assert_(abs(1/rcond - np.linalg.cond(A, p=1))*rcond < 1)
|
||
|
|
|
||
|
|
|
||
|
|
def test_sygst():
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(REAL_DTYPES):
|
||
|
|
# DTYPES = <s,d> sygst
|
||
|
|
n = 10
|
||
|
|
|
||
|
|
potrf, sygst, syevd, sygvd = get_lapack_funcs(('potrf', 'sygst',
|
||
|
|
'syevd', 'sygvd'),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
A = rand(n, n).astype(dtype)
|
||
|
|
A = (A + A.T)/2
|
||
|
|
# B must be positive definite
|
||
|
|
B = rand(n, n).astype(dtype)
|
||
|
|
B = (B + B.T)/2 + 2 * np.eye(n, dtype=dtype)
|
||
|
|
|
||
|
|
# Perform eig (sygvd)
|
||
|
|
eig_gvd, _, info = sygvd(A, B)
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
# Convert to std problem potrf
|
||
|
|
b, info = potrf(B)
|
||
|
|
assert_(info == 0)
|
||
|
|
a, info = sygst(A, b)
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
eig, _, info = syevd(a)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_allclose(eig, eig_gvd, rtol=1e-4)
|
||
|
|
|
||
|
|
|
||
|
|
def test_hegst():
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(COMPLEX_DTYPES):
|
||
|
|
# DTYPES = <c,z> hegst
|
||
|
|
n = 10
|
||
|
|
|
||
|
|
potrf, hegst, heevd, hegvd = get_lapack_funcs(('potrf', 'hegst',
|
||
|
|
'heevd', 'hegvd'),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
A = rand(n, n).astype(dtype) + 1j * rand(n, n).astype(dtype)
|
||
|
|
A = (A + A.conj().T)/2
|
||
|
|
# B must be positive definite
|
||
|
|
B = rand(n, n).astype(dtype) + 1j * rand(n, n).astype(dtype)
|
||
|
|
B = (B + B.conj().T)/2 + 2 * np.eye(n, dtype=dtype)
|
||
|
|
|
||
|
|
# Perform eig (hegvd)
|
||
|
|
eig_gvd, _, info = hegvd(A, B)
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
# Convert to std problem potrf
|
||
|
|
b, info = potrf(B)
|
||
|
|
assert_(info == 0)
|
||
|
|
a, info = hegst(A, b)
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
eig, _, info = heevd(a)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_allclose(eig, eig_gvd, rtol=1e-4)
|
||
|
|
|
||
|
|
|
||
|
|
def test_tzrzf():
|
||
|
|
"""
|
||
|
|
This test performs an RZ decomposition in which an m x n upper trapezoidal
|
||
|
|
array M (m <= n) is factorized as M = [R 0] * Z where R is upper triangular
|
||
|
|
and Z is unitary.
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
m, n = 10, 15
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
tzrzf, tzrzf_lw = get_lapack_funcs(('tzrzf', 'tzrzf_lwork'),
|
||
|
|
dtype=dtype)
|
||
|
|
lwork = _compute_lwork(tzrzf_lw, m, n)
|
||
|
|
|
||
|
|
if ind < 2:
|
||
|
|
A = triu(rand(m, n).astype(dtype))
|
||
|
|
else:
|
||
|
|
A = triu((rand(m, n) + rand(m, n)*1j).astype(dtype))
|
||
|
|
|
||
|
|
# assert wrong shape arg, f2py returns generic error
|
||
|
|
assert_raises(Exception, tzrzf, A.T)
|
||
|
|
rz, tau, info = tzrzf(A, lwork=lwork)
|
||
|
|
# Check success
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
# Get Z manually for comparison
|
||
|
|
R = np.hstack((rz[:, :m], np.zeros((m, n-m), dtype=dtype)))
|
||
|
|
V = np.hstack((np.eye(m, dtype=dtype), rz[:, m:]))
|
||
|
|
Id = np.eye(n, dtype=dtype)
|
||
|
|
ref = [Id-tau[x]*V[[x], :].T.dot(V[[x], :].conj()) for x in range(m)]
|
||
|
|
Z = reduce(np.dot, ref)
|
||
|
|
assert_allclose(R.dot(Z) - A, zeros_like(A, dtype=dtype),
|
||
|
|
atol=10*np.spacing(dtype(1.0).real), rtol=0.)
|
||
|
|
|
||
|
|
|
||
|
|
def test_tfsm():
|
||
|
|
"""
|
||
|
|
Test for solving a linear system with the coefficient matrix is a
|
||
|
|
triangular array stored in Full Packed (RFP) format.
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
if ind > 1:
|
||
|
|
A = triu(rand(n, n) + rand(n, n)*1j + eye(n)).astype(dtype)
|
||
|
|
trans = 'C'
|
||
|
|
else:
|
||
|
|
A = triu(rand(n, n) + eye(n)).astype(dtype)
|
||
|
|
trans = 'T'
|
||
|
|
|
||
|
|
trttf, tfttr, tfsm = get_lapack_funcs(('trttf', 'tfttr', 'tfsm'),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
Afp, _ = trttf(A)
|
||
|
|
B = rand(n, 2).astype(dtype)
|
||
|
|
soln = tfsm(-1, Afp, B)
|
||
|
|
assert_array_almost_equal(soln, solve(-A, B),
|
||
|
|
decimal=4 if ind % 2 == 0 else 6)
|
||
|
|
|
||
|
|
soln = tfsm(-1, Afp, B, trans=trans)
|
||
|
|
assert_array_almost_equal(soln, solve(-A.conj().T, B),
|
||
|
|
decimal=4 if ind % 2 == 0 else 6)
|
||
|
|
|
||
|
|
# Make A, unit diagonal
|
||
|
|
A[np.arange(n), np.arange(n)] = dtype(1.)
|
||
|
|
soln = tfsm(-1, Afp, B, trans=trans, diag='U')
|
||
|
|
assert_array_almost_equal(soln, solve(-A.conj().T, B),
|
||
|
|
decimal=4 if ind % 2 == 0 else 6)
|
||
|
|
|
||
|
|
# Change side
|
||
|
|
B2 = rand(3, n).astype(dtype)
|
||
|
|
soln = tfsm(-1, Afp, B2, trans=trans, diag='U', side='R')
|
||
|
|
assert_array_almost_equal(soln, solve(-A, B2.T).conj().T,
|
||
|
|
decimal=4 if ind % 2 == 0 else 6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_ormrz_unmrz():
|
||
|
|
"""
|
||
|
|
This test performs a matrix multiplication with an arbitrary m x n matric C
|
||
|
|
and a unitary matrix Q without explicitly forming the array. The array data
|
||
|
|
is encoded in the rectangular part of A which is obtained from ?TZRZF. Q
|
||
|
|
size is inferred by m, n, side keywords.
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
qm, qn, cn = 10, 15, 15
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
tzrzf, tzrzf_lw = get_lapack_funcs(('tzrzf', 'tzrzf_lwork'),
|
||
|
|
dtype=dtype)
|
||
|
|
lwork_rz = _compute_lwork(tzrzf_lw, qm, qn)
|
||
|
|
|
||
|
|
if ind < 2:
|
||
|
|
A = triu(rand(qm, qn).astype(dtype))
|
||
|
|
C = rand(cn, cn).astype(dtype)
|
||
|
|
orun_mrz, orun_mrz_lw = get_lapack_funcs(('ormrz', 'ormrz_lwork'),
|
||
|
|
dtype=dtype)
|
||
|
|
else:
|
||
|
|
A = triu((rand(qm, qn) + rand(qm, qn)*1j).astype(dtype))
|
||
|
|
C = (rand(cn, cn) + rand(cn, cn)*1j).astype(dtype)
|
||
|
|
orun_mrz, orun_mrz_lw = get_lapack_funcs(('unmrz', 'unmrz_lwork'),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
lwork_mrz = _compute_lwork(orun_mrz_lw, cn, cn)
|
||
|
|
rz, tau, info = tzrzf(A, lwork=lwork_rz)
|
||
|
|
|
||
|
|
# Get Q manually for comparison
|
||
|
|
V = np.hstack((np.eye(qm, dtype=dtype), rz[:, qm:]))
|
||
|
|
Id = np.eye(qn, dtype=dtype)
|
||
|
|
ref = [Id-tau[x]*V[[x], :].T.dot(V[[x], :].conj()) for x in range(qm)]
|
||
|
|
Q = reduce(np.dot, ref)
|
||
|
|
|
||
|
|
# Now that we have Q, we can test whether lapack results agree with
|
||
|
|
# each case of CQ, CQ^H, QC, and QC^H
|
||
|
|
trans = 'T' if ind < 2 else 'C'
|
||
|
|
tol = 10*np.spacing(dtype(1.0).real)
|
||
|
|
|
||
|
|
cq, info = orun_mrz(rz, tau, C, lwork=lwork_mrz)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_allclose(cq - Q.dot(C), zeros_like(C), atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
cq, info = orun_mrz(rz, tau, C, trans=trans, lwork=lwork_mrz)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_allclose(cq - Q.conj().T.dot(C), zeros_like(C), atol=tol,
|
||
|
|
rtol=0.)
|
||
|
|
|
||
|
|
cq, info = orun_mrz(rz, tau, C, side='R', lwork=lwork_mrz)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_allclose(cq - C.dot(Q), zeros_like(C), atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
cq, info = orun_mrz(rz, tau, C, side='R', trans=trans, lwork=lwork_mrz)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_allclose(cq - C.dot(Q.conj().T), zeros_like(C), atol=tol,
|
||
|
|
rtol=0.)
|
||
|
|
|
||
|
|
|
||
|
|
def test_tfttr_trttf():
|
||
|
|
"""
|
||
|
|
Test conversion routines between the Rectengular Full Packed (RFP) format
|
||
|
|
and Standard Triangular Array (TR)
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
if ind > 1:
|
||
|
|
A_full = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
transr = 'C'
|
||
|
|
else:
|
||
|
|
A_full = (rand(n, n)).astype(dtype)
|
||
|
|
transr = 'T'
|
||
|
|
|
||
|
|
trttf, tfttr = get_lapack_funcs(('trttf', 'tfttr'), dtype=dtype)
|
||
|
|
A_tf_U, info = trttf(A_full)
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tf_L, info = trttf(A_full, uplo='L')
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tf_U_T, info = trttf(A_full, transr=transr, uplo='U')
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tf_L_T, info = trttf(A_full, transr=transr, uplo='L')
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
# Create the RFP array manually (n is even!)
|
||
|
|
A_tf_U_m = zeros((n+1, n//2), dtype=dtype)
|
||
|
|
A_tf_U_m[:-1, :] = triu(A_full)[:, n//2:]
|
||
|
|
A_tf_U_m[n//2+1:, :] += triu(A_full)[:n//2, :n//2].conj().T
|
||
|
|
|
||
|
|
A_tf_L_m = zeros((n+1, n//2), dtype=dtype)
|
||
|
|
A_tf_L_m[1:, :] = tril(A_full)[:, :n//2]
|
||
|
|
A_tf_L_m[:n//2, :] += tril(A_full)[n//2:, n//2:].conj().T
|
||
|
|
|
||
|
|
assert_array_almost_equal(A_tf_U, A_tf_U_m.reshape(-1, order='F'))
|
||
|
|
assert_array_almost_equal(A_tf_U_T,
|
||
|
|
A_tf_U_m.conj().T.reshape(-1, order='F'))
|
||
|
|
|
||
|
|
assert_array_almost_equal(A_tf_L, A_tf_L_m.reshape(-1, order='F'))
|
||
|
|
assert_array_almost_equal(A_tf_L_T,
|
||
|
|
A_tf_L_m.conj().T.reshape(-1, order='F'))
|
||
|
|
|
||
|
|
# Get the original array from RFP
|
||
|
|
A_tr_U, info = tfttr(n, A_tf_U)
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tr_L, info = tfttr(n, A_tf_L, uplo='L')
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tr_U_T, info = tfttr(n, A_tf_U_T, transr=transr, uplo='U')
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tr_L_T, info = tfttr(n, A_tf_L_T, transr=transr, uplo='L')
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
assert_array_almost_equal(A_tr_U, triu(A_full))
|
||
|
|
assert_array_almost_equal(A_tr_U_T, triu(A_full))
|
||
|
|
assert_array_almost_equal(A_tr_L, tril(A_full))
|
||
|
|
assert_array_almost_equal(A_tr_L_T, tril(A_full))
|
||
|
|
|
||
|
|
|
||
|
|
def test_tpttr_trttp():
|
||
|
|
"""
|
||
|
|
Test conversion routines between the Rectengular Full Packed (RFP) format
|
||
|
|
and Standard Triangular Array (TR)
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
if ind > 1:
|
||
|
|
A_full = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
else:
|
||
|
|
A_full = (rand(n, n)).astype(dtype)
|
||
|
|
|
||
|
|
trttp, tpttr = get_lapack_funcs(('trttp', 'tpttr'), dtype=dtype)
|
||
|
|
A_tp_U, info = trttp(A_full)
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tp_L, info = trttp(A_full, uplo='L')
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
# Create the TP array manually
|
||
|
|
inds = tril_indices(n)
|
||
|
|
A_tp_U_m = zeros(n*(n+1)//2, dtype=dtype)
|
||
|
|
A_tp_U_m[:] = (triu(A_full).T)[inds]
|
||
|
|
|
||
|
|
inds = triu_indices(n)
|
||
|
|
A_tp_L_m = zeros(n*(n+1)//2, dtype=dtype)
|
||
|
|
A_tp_L_m[:] = (tril(A_full).T)[inds]
|
||
|
|
|
||
|
|
assert_array_almost_equal(A_tp_U, A_tp_U_m)
|
||
|
|
assert_array_almost_equal(A_tp_L, A_tp_L_m)
|
||
|
|
|
||
|
|
# Get the original array from TP
|
||
|
|
A_tr_U, info = tpttr(n, A_tp_U)
|
||
|
|
assert_(info == 0)
|
||
|
|
A_tr_L, info = tpttr(n, A_tp_L, uplo='L')
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
assert_array_almost_equal(A_tr_U, triu(A_full))
|
||
|
|
assert_array_almost_equal(A_tr_L, tril(A_full))
|
||
|
|
|
||
|
|
|
||
|
|
def test_pftrf():
|
||
|
|
"""
|
||
|
|
Test Cholesky factorization of a positive definite Rectengular Full
|
||
|
|
Packed (RFP) format array
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
if ind > 1:
|
||
|
|
A = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
A = A + A.conj().T + n*eye(n)
|
||
|
|
else:
|
||
|
|
A = (rand(n, n)).astype(dtype)
|
||
|
|
A = A + A.T + n*eye(n)
|
||
|
|
|
||
|
|
pftrf, trttf, tfttr = get_lapack_funcs(('pftrf', 'trttf', 'tfttr'),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
# Get the original array from TP
|
||
|
|
Afp, info = trttf(A)
|
||
|
|
Achol_rfp, info = pftrf(n, Afp)
|
||
|
|
assert_(info == 0)
|
||
|
|
A_chol_r, _ = tfttr(n, Achol_rfp)
|
||
|
|
Achol = cholesky(A)
|
||
|
|
assert_array_almost_equal(A_chol_r, Achol)
|
||
|
|
|
||
|
|
|
||
|
|
def test_pftri():
|
||
|
|
"""
|
||
|
|
Test Cholesky factorization of a positive definite Rectengular Full
|
||
|
|
Packed (RFP) format array to find its inverse
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
if ind > 1:
|
||
|
|
A = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
A = A + A.conj().T + n*eye(n)
|
||
|
|
else:
|
||
|
|
A = (rand(n, n)).astype(dtype)
|
||
|
|
A = A + A.T + n*eye(n)
|
||
|
|
|
||
|
|
pftri, pftrf, trttf, tfttr = get_lapack_funcs(('pftri',
|
||
|
|
'pftrf',
|
||
|
|
'trttf',
|
||
|
|
'tfttr'),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
# Get the original array from TP
|
||
|
|
Afp, info = trttf(A)
|
||
|
|
A_chol_rfp, info = pftrf(n, Afp)
|
||
|
|
A_inv_rfp, info = pftri(n, A_chol_rfp)
|
||
|
|
assert_(info == 0)
|
||
|
|
A_inv_r, _ = tfttr(n, A_inv_rfp)
|
||
|
|
Ainv = inv(A)
|
||
|
|
assert_array_almost_equal(A_inv_r, triu(Ainv),
|
||
|
|
decimal=4 if ind % 2 == 0 else 6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_pftrs():
|
||
|
|
"""
|
||
|
|
Test Cholesky factorization of a positive definite Rectengular Full
|
||
|
|
Packed (RFP) format array and solve a linear system
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
if ind > 1:
|
||
|
|
A = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
A = A + A.conj().T + n*eye(n)
|
||
|
|
else:
|
||
|
|
A = (rand(n, n)).astype(dtype)
|
||
|
|
A = A + A.T + n*eye(n)
|
||
|
|
|
||
|
|
B = ones((n, 3), dtype=dtype)
|
||
|
|
Bf1 = ones((n+2, 3), dtype=dtype)
|
||
|
|
Bf2 = ones((n-2, 3), dtype=dtype)
|
||
|
|
pftrs, pftrf, trttf, tfttr = get_lapack_funcs(('pftrs',
|
||
|
|
'pftrf',
|
||
|
|
'trttf',
|
||
|
|
'tfttr'),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
# Get the original array from TP
|
||
|
|
Afp, info = trttf(A)
|
||
|
|
A_chol_rfp, info = pftrf(n, Afp)
|
||
|
|
# larger B arrays shouldn't segfault
|
||
|
|
soln, info = pftrs(n, A_chol_rfp, Bf1)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_raises(Exception, pftrs, n, A_chol_rfp, Bf2)
|
||
|
|
soln, info = pftrs(n, A_chol_rfp, B)
|
||
|
|
assert_(info == 0)
|
||
|
|
assert_array_almost_equal(solve(A, B), soln,
|
||
|
|
decimal=4 if ind % 2 == 0 else 6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_sfrk_hfrk():
|
||
|
|
"""
|
||
|
|
Test for performing a symmetric rank-k operation for matrix in RFP format.
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
if ind > 1:
|
||
|
|
A = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
A = A + A.conj().T + n*eye(n)
|
||
|
|
else:
|
||
|
|
A = (rand(n, n)).astype(dtype)
|
||
|
|
A = A + A.T + n*eye(n)
|
||
|
|
|
||
|
|
prefix = 's'if ind < 2 else 'h'
|
||
|
|
trttf, tfttr, shfrk = get_lapack_funcs(('trttf', 'tfttr', '{}frk'
|
||
|
|
''.format(prefix)),
|
||
|
|
dtype=dtype)
|
||
|
|
|
||
|
|
Afp, _ = trttf(A)
|
||
|
|
C = np.random.rand(n, 2).astype(dtype)
|
||
|
|
Afp_out = shfrk(n, 2, -1, C, 2, Afp)
|
||
|
|
A_out, _ = tfttr(n, Afp_out)
|
||
|
|
assert_array_almost_equal(A_out, triu(-C.dot(C.conj().T) + 2*A),
|
||
|
|
decimal=4 if ind % 2 == 0 else 6)
|
||
|
|
|
||
|
|
|
||
|
|
def test_syconv():
|
||
|
|
"""
|
||
|
|
Test for going back and forth between the returned format of he/sytrf to
|
||
|
|
L and D factors/permutations.
|
||
|
|
"""
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 10
|
||
|
|
|
||
|
|
if ind > 1:
|
||
|
|
A = (randint(-30, 30, (n, n)) +
|
||
|
|
randint(-30, 30, (n, n))*1j).astype(dtype)
|
||
|
|
|
||
|
|
A = A + A.conj().T
|
||
|
|
else:
|
||
|
|
A = randint(-30, 30, (n, n)).astype(dtype)
|
||
|
|
A = A + A.T + n*eye(n)
|
||
|
|
|
||
|
|
tol = 100*np.spacing(dtype(1.0).real)
|
||
|
|
syconv, trf, trf_lwork = get_lapack_funcs(('syconv', 'sytrf',
|
||
|
|
'sytrf_lwork'), dtype=dtype)
|
||
|
|
lw = _compute_lwork(trf_lwork, n, lower=1)
|
||
|
|
L, D, perm = ldl(A, lower=1, hermitian=False)
|
||
|
|
lw = _compute_lwork(trf_lwork, n, lower=1)
|
||
|
|
ldu, ipiv, info = trf(A, lower=1, lwork=lw)
|
||
|
|
a, e, info = syconv(ldu, ipiv, lower=1)
|
||
|
|
assert_allclose(tril(a, -1,), tril(L[perm, :], -1), atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
# Test also upper
|
||
|
|
U, D, perm = ldl(A, lower=0, hermitian=False)
|
||
|
|
ldu, ipiv, info = trf(A, lower=0)
|
||
|
|
a, e, info = syconv(ldu, ipiv, lower=0)
|
||
|
|
assert_allclose(triu(a, 1), triu(U[perm, :], 1), atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
|
||
|
|
class TestBlockedQR:
|
||
|
|
"""
|
||
|
|
Tests for the blocked QR factorization, namely through geqrt, gemqrt, tpqrt
|
||
|
|
and tpmqr.
|
||
|
|
"""
|
||
|
|
|
||
|
|
def test_geqrt_gemqrt(self):
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
|
||
|
|
if ind > 1:
|
||
|
|
A = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
else:
|
||
|
|
A = (rand(n, n)).astype(dtype)
|
||
|
|
|
||
|
|
tol = 100*np.spacing(dtype(1.0).real)
|
||
|
|
geqrt, gemqrt = get_lapack_funcs(('geqrt', 'gemqrt'), dtype=dtype)
|
||
|
|
|
||
|
|
a, t, info = geqrt(n, A)
|
||
|
|
assert info == 0
|
||
|
|
|
||
|
|
# Extract elementary reflectors from lower triangle, adding the
|
||
|
|
# main diagonal of ones.
|
||
|
|
v = np.tril(a, -1) + np.eye(n, dtype=dtype)
|
||
|
|
# Generate the block Householder transform I - VTV^H
|
||
|
|
Q = np.eye(n, dtype=dtype) - v @ t @ v.T.conj()
|
||
|
|
R = np.triu(a)
|
||
|
|
|
||
|
|
# Test columns of Q are orthogonal
|
||
|
|
assert_allclose(Q.T.conj() @ Q, np.eye(n, dtype=dtype), atol=tol,
|
||
|
|
rtol=0.)
|
||
|
|
assert_allclose(Q @ R, A, atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
if ind > 1:
|
||
|
|
C = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
transpose = 'C'
|
||
|
|
else:
|
||
|
|
C = (rand(n, n)).astype(dtype)
|
||
|
|
transpose = 'T'
|
||
|
|
|
||
|
|
for side in ('L', 'R'):
|
||
|
|
for trans in ('N', transpose):
|
||
|
|
c, info = gemqrt(a, t, C, side=side, trans=trans)
|
||
|
|
assert info == 0
|
||
|
|
|
||
|
|
if trans == transpose:
|
||
|
|
q = Q.T.conj()
|
||
|
|
else:
|
||
|
|
q = Q
|
||
|
|
|
||
|
|
if side == 'L':
|
||
|
|
qC = q @ C
|
||
|
|
else:
|
||
|
|
qC = C @ q
|
||
|
|
|
||
|
|
assert_allclose(c, qC, atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
# Test default arguments
|
||
|
|
if (side, trans) == ('L', 'N'):
|
||
|
|
c_default, info = gemqrt(a, t, C)
|
||
|
|
assert info == 0
|
||
|
|
assert_equal(c_default, c)
|
||
|
|
|
||
|
|
# Test invalid side/trans
|
||
|
|
assert_raises(Exception, gemqrt, a, t, C, side='A')
|
||
|
|
assert_raises(Exception, gemqrt, a, t, C, trans='A')
|
||
|
|
|
||
|
|
def test_tpqrt_tpmqrt(self):
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
n = 20
|
||
|
|
|
||
|
|
if ind > 1:
|
||
|
|
A = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
B = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
else:
|
||
|
|
A = (rand(n, n)).astype(dtype)
|
||
|
|
B = (rand(n, n)).astype(dtype)
|
||
|
|
|
||
|
|
tol = 100*np.spacing(dtype(1.0).real)
|
||
|
|
tpqrt, tpmqrt = get_lapack_funcs(('tpqrt', 'tpmqrt'), dtype=dtype)
|
||
|
|
|
||
|
|
# Test for the range of pentagonal B, from square to upper
|
||
|
|
# triangular
|
||
|
|
for l in (0, n // 2, n):
|
||
|
|
a, b, t, info = tpqrt(l, n, A, B)
|
||
|
|
assert info == 0
|
||
|
|
|
||
|
|
# Check that lower triangular part of A has not been modified
|
||
|
|
assert_equal(np.tril(a, -1), np.tril(A, -1))
|
||
|
|
# Check that elements not part of the pentagonal portion of B
|
||
|
|
# have not been modified.
|
||
|
|
assert_equal(np.tril(b, l - n - 1), np.tril(B, l - n - 1))
|
||
|
|
|
||
|
|
# Extract pentagonal portion of B
|
||
|
|
B_pent, b_pent = np.triu(B, l - n), np.triu(b, l - n)
|
||
|
|
|
||
|
|
# Generate elementary reflectors
|
||
|
|
v = np.concatenate((np.eye(n, dtype=dtype), b_pent))
|
||
|
|
# Generate the block Householder transform I - VTV^H
|
||
|
|
Q = np.eye(2 * n, dtype=dtype) - v @ t @ v.T.conj()
|
||
|
|
R = np.concatenate((np.triu(a), np.zeros_like(a)))
|
||
|
|
|
||
|
|
# Test columns of Q are orthogonal
|
||
|
|
assert_allclose(Q.T.conj() @ Q, np.eye(2 * n, dtype=dtype),
|
||
|
|
atol=tol, rtol=0.)
|
||
|
|
assert_allclose(Q @ R, np.concatenate((np.triu(A), B_pent)),
|
||
|
|
atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
if ind > 1:
|
||
|
|
C = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
D = (rand(n, n) + rand(n, n)*1j).astype(dtype)
|
||
|
|
transpose = 'C'
|
||
|
|
else:
|
||
|
|
C = (rand(n, n)).astype(dtype)
|
||
|
|
D = (rand(n, n)).astype(dtype)
|
||
|
|
transpose = 'T'
|
||
|
|
|
||
|
|
for side in ('L', 'R'):
|
||
|
|
for trans in ('N', transpose):
|
||
|
|
c, d, info = tpmqrt(l, b, t, C, D, side=side,
|
||
|
|
trans=trans)
|
||
|
|
assert info == 0
|
||
|
|
|
||
|
|
if trans == transpose:
|
||
|
|
q = Q.T.conj()
|
||
|
|
else:
|
||
|
|
q = Q
|
||
|
|
|
||
|
|
if side == 'L':
|
||
|
|
cd = np.concatenate((c, d), axis=0)
|
||
|
|
CD = np.concatenate((C, D), axis=0)
|
||
|
|
qCD = q @ CD
|
||
|
|
else:
|
||
|
|
cd = np.concatenate((c, d), axis=1)
|
||
|
|
CD = np.concatenate((C, D), axis=1)
|
||
|
|
qCD = CD @ q
|
||
|
|
|
||
|
|
assert_allclose(cd, qCD, atol=tol, rtol=0.)
|
||
|
|
|
||
|
|
if (side, trans) == ('L', 'N'):
|
||
|
|
c_default, d_default, info = tpmqrt(l, b, t, C, D)
|
||
|
|
assert info == 0
|
||
|
|
assert_equal(c_default, c)
|
||
|
|
assert_equal(d_default, d)
|
||
|
|
|
||
|
|
# Test invalid side/trans
|
||
|
|
assert_raises(Exception, tpmqrt, l, b, t, C, D, side='A')
|
||
|
|
assert_raises(Exception, tpmqrt, l, b, t, C, D, trans='A')
|
||
|
|
|
||
|
|
|
||
|
|
def test_pstrf():
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
# DTYPES = <s, d, c, z> pstrf
|
||
|
|
n = 10
|
||
|
|
r = 2
|
||
|
|
pstrf = get_lapack_funcs('pstrf', dtype=dtype)
|
||
|
|
|
||
|
|
# Create positive semidefinite A
|
||
|
|
if ind > 1:
|
||
|
|
A = rand(n, n-r).astype(dtype) + 1j * rand(n, n-r).astype(dtype)
|
||
|
|
A = A @ A.conj().T
|
||
|
|
else:
|
||
|
|
A = rand(n, n-r).astype(dtype)
|
||
|
|
A = A @ A.T
|
||
|
|
|
||
|
|
c, piv, r_c, info = pstrf(A)
|
||
|
|
U = triu(c)
|
||
|
|
U[r_c - n:, r_c - n:] = 0.
|
||
|
|
|
||
|
|
assert_equal(info, 1)
|
||
|
|
# python-dbg 3.5.2 runs cause trouble with the following assertion.
|
||
|
|
# assert_equal(r_c, n - r)
|
||
|
|
single_atol = 1000 * np.finfo(np.float32).eps
|
||
|
|
double_atol = 1000 * np.finfo(np.float64).eps
|
||
|
|
atol = single_atol if ind in [0, 2] else double_atol
|
||
|
|
assert_allclose(A[piv-1][:, piv-1], U.conj().T @ U, rtol=0., atol=atol)
|
||
|
|
|
||
|
|
c, piv, r_c, info = pstrf(A, lower=1)
|
||
|
|
L = tril(c)
|
||
|
|
L[r_c - n:, r_c - n:] = 0.
|
||
|
|
|
||
|
|
assert_equal(info, 1)
|
||
|
|
# assert_equal(r_c, n - r)
|
||
|
|
single_atol = 1000 * np.finfo(np.float32).eps
|
||
|
|
double_atol = 1000 * np.finfo(np.float64).eps
|
||
|
|
atol = single_atol if ind in [0, 2] else double_atol
|
||
|
|
assert_allclose(A[piv-1][:, piv-1], L @ L.conj().T, rtol=0., atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
def test_pstf2():
|
||
|
|
seed(1234)
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
# DTYPES = <s, d, c, z> pstf2
|
||
|
|
n = 10
|
||
|
|
r = 2
|
||
|
|
pstf2 = get_lapack_funcs('pstf2', dtype=dtype)
|
||
|
|
|
||
|
|
# Create positive semidefinite A
|
||
|
|
if ind > 1:
|
||
|
|
A = rand(n, n-r).astype(dtype) + 1j * rand(n, n-r).astype(dtype)
|
||
|
|
A = A @ A.conj().T
|
||
|
|
else:
|
||
|
|
A = rand(n, n-r).astype(dtype)
|
||
|
|
A = A @ A.T
|
||
|
|
|
||
|
|
c, piv, r_c, info = pstf2(A)
|
||
|
|
U = triu(c)
|
||
|
|
U[r_c - n:, r_c - n:] = 0.
|
||
|
|
|
||
|
|
assert_equal(info, 1)
|
||
|
|
# python-dbg 3.5.2 runs cause trouble with the commented assertions.
|
||
|
|
# assert_equal(r_c, n - r)
|
||
|
|
single_atol = 1000 * np.finfo(np.float32).eps
|
||
|
|
double_atol = 1000 * np.finfo(np.float64).eps
|
||
|
|
atol = single_atol if ind in [0, 2] else double_atol
|
||
|
|
assert_allclose(A[piv-1][:, piv-1], U.conj().T @ U, rtol=0., atol=atol)
|
||
|
|
|
||
|
|
c, piv, r_c, info = pstf2(A, lower=1)
|
||
|
|
L = tril(c)
|
||
|
|
L[r_c - n:, r_c - n:] = 0.
|
||
|
|
|
||
|
|
assert_equal(info, 1)
|
||
|
|
# assert_equal(r_c, n - r)
|
||
|
|
single_atol = 1000 * np.finfo(np.float32).eps
|
||
|
|
double_atol = 1000 * np.finfo(np.float64).eps
|
||
|
|
atol = single_atol if ind in [0, 2] else double_atol
|
||
|
|
assert_allclose(A[piv-1][:, piv-1], L @ L.conj().T, rtol=0., atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
def test_geequ():
|
||
|
|
desired_real = np.array([[0.6250, 1.0000, 0.0393, -0.4269],
|
||
|
|
[1.0000, -0.5619, -1.0000, -1.0000],
|
||
|
|
[0.5874, -1.0000, -0.0596, -0.5341],
|
||
|
|
[-1.0000, -0.5946, -0.0294, 0.9957]])
|
||
|
|
|
||
|
|
desired_cplx = np.array([[-0.2816+0.5359*1j,
|
||
|
|
0.0812+0.9188*1j,
|
||
|
|
-0.7439-0.2561*1j],
|
||
|
|
[-0.3562-0.2954*1j,
|
||
|
|
0.9566-0.0434*1j,
|
||
|
|
-0.0174+0.1555*1j],
|
||
|
|
[0.8607+0.1393*1j,
|
||
|
|
-0.2759+0.7241*1j,
|
||
|
|
-0.1642-0.1365*1j]])
|
||
|
|
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
if ind < 2:
|
||
|
|
# Use examples from the NAG documentation
|
||
|
|
A = np.array([[1.80e+10, 2.88e+10, 2.05e+00, -8.90e+09],
|
||
|
|
[5.25e+00, -2.95e+00, -9.50e-09, -3.80e+00],
|
||
|
|
[1.58e+00, -2.69e+00, -2.90e-10, -1.04e+00],
|
||
|
|
[-1.11e+00, -6.60e-01, -5.90e-11, 8.00e-01]])
|
||
|
|
A = A.astype(dtype)
|
||
|
|
else:
|
||
|
|
A = np.array([[-1.34e+00, 0.28e+10, -6.39e+00],
|
||
|
|
[-1.70e+00, 3.31e+10, -0.15e+00],
|
||
|
|
[2.41e-10, -0.56e+00, -0.83e-10]], dtype=dtype)
|
||
|
|
A += np.array([[2.55e+00, 3.17e+10, -2.20e+00],
|
||
|
|
[-1.41e+00, -0.15e+10, 1.34e+00],
|
||
|
|
[0.39e-10, 1.47e+00, -0.69e-10]])*1j
|
||
|
|
|
||
|
|
A = A.astype(dtype)
|
||
|
|
|
||
|
|
geequ = get_lapack_funcs('geequ', dtype=dtype)
|
||
|
|
r, c, rowcnd, colcnd, amax, info = geequ(A)
|
||
|
|
|
||
|
|
if ind < 2:
|
||
|
|
assert_allclose(desired_real.astype(dtype), r[:, None]*A*c,
|
||
|
|
rtol=0, atol=1e-4)
|
||
|
|
else:
|
||
|
|
assert_allclose(desired_cplx.astype(dtype), r[:, None]*A*c,
|
||
|
|
rtol=0, atol=1e-4)
|
||
|
|
|
||
|
|
|
||
|
|
def test_syequb():
|
||
|
|
desired_log2s = np.array([0, 0, 0, 0, 0, 0, -1, -1, -2, -3])
|
||
|
|
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
A = np.eye(10, dtype=dtype)
|
||
|
|
alpha = dtype(1. if ind < 2 else 1.j)
|
||
|
|
d = np.array([alpha * 2.**x for x in range(-5, 5)], dtype=dtype)
|
||
|
|
A += np.rot90(np.diag(d))
|
||
|
|
|
||
|
|
syequb = get_lapack_funcs('syequb', dtype=dtype)
|
||
|
|
s, scond, amax, info = syequb(A)
|
||
|
|
|
||
|
|
assert_equal(np.log2(s).astype(int), desired_log2s)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.skipif(True,
|
||
|
|
reason="Failing on some OpenBLAS version, see gh-12276")
|
||
|
|
def test_heequb():
|
||
|
|
# zheequb has a bug for versions =< LAPACK 3.9.0
|
||
|
|
# See Reference-LAPACK gh-61 and gh-408
|
||
|
|
# Hence the zheequb test is customized accordingly to avoid
|
||
|
|
# work scaling.
|
||
|
|
A = np.diag([2]*5 + [1002]*5) + np.diag(np.ones(9), k=1)*1j
|
||
|
|
s, scond, amax, info = lapack.zheequb(A)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_allclose(np.log2(s), [0., -1.]*2 + [0.] + [-4]*5)
|
||
|
|
|
||
|
|
A = np.diag(2**np.abs(np.arange(-5, 6)) + 0j)
|
||
|
|
A[5, 5] = 1024
|
||
|
|
A[5, 0] = 16j
|
||
|
|
s, scond, amax, info = lapack.cheequb(A.astype(np.complex64), lower=1)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_allclose(np.log2(s), [-2, -1, -1, 0, 0, -5, 0, -1, -1, -2, -2])
|
||
|
|
|
||
|
|
|
||
|
|
def test_getc2_gesc2():
|
||
|
|
np.random.seed(42)
|
||
|
|
n = 10
|
||
|
|
desired_real = np.random.rand(n)
|
||
|
|
desired_cplx = np.random.rand(n) + np.random.rand(n)*1j
|
||
|
|
|
||
|
|
for ind, dtype in enumerate(DTYPES):
|
||
|
|
if ind < 2:
|
||
|
|
A = np.random.rand(n, n)
|
||
|
|
A = A.astype(dtype)
|
||
|
|
b = A @ desired_real
|
||
|
|
b = b.astype(dtype)
|
||
|
|
else:
|
||
|
|
A = np.random.rand(n, n) + np.random.rand(n, n)*1j
|
||
|
|
A = A.astype(dtype)
|
||
|
|
b = A @ desired_cplx
|
||
|
|
b = b.astype(dtype)
|
||
|
|
|
||
|
|
getc2 = get_lapack_funcs('getc2', dtype=dtype)
|
||
|
|
gesc2 = get_lapack_funcs('gesc2', dtype=dtype)
|
||
|
|
lu, ipiv, jpiv, info = getc2(A, overwrite_a=0)
|
||
|
|
x, scale = gesc2(lu, b, ipiv, jpiv, overwrite_rhs=0)
|
||
|
|
|
||
|
|
if ind < 2:
|
||
|
|
assert_array_almost_equal(desired_real.astype(dtype),
|
||
|
|
x/scale, decimal=4)
|
||
|
|
else:
|
||
|
|
assert_array_almost_equal(desired_cplx.astype(dtype),
|
||
|
|
x/scale, decimal=4)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('size', [(6, 5), (5, 5)])
|
||
|
|
@pytest.mark.parametrize('dtype', REAL_DTYPES)
|
||
|
|
@pytest.mark.parametrize('joba', range(6)) # 'C', 'E', 'F', 'G', 'A', 'R'
|
||
|
|
@pytest.mark.parametrize('jobu', range(4)) # 'U', 'F', 'W', 'N'
|
||
|
|
@pytest.mark.parametrize('jobv', range(4)) # 'V', 'J', 'W', 'N'
|
||
|
|
@pytest.mark.parametrize('jobr', [0, 1])
|
||
|
|
@pytest.mark.parametrize('jobp', [0, 1])
|
||
|
|
def test_gejsv_general(size, dtype, joba, jobu, jobv, jobr, jobp, jobt=0):
|
||
|
|
"""Test the lapack routine ?gejsv.
|
||
|
|
|
||
|
|
This function tests that a singular value decomposition can be performed
|
||
|
|
on the random M-by-N matrix A. The test performs the SVD using ?gejsv
|
||
|
|
then performs the following checks:
|
||
|
|
|
||
|
|
* ?gejsv exist successfully (info == 0)
|
||
|
|
* The returned singular values are correct
|
||
|
|
* `A` can be reconstructed from `u`, `SIGMA`, `v`
|
||
|
|
* Ensure that u.T @ u is the identity matrix
|
||
|
|
* Ensure that v.T @ v is the identity matrix
|
||
|
|
* The reported matrix rank
|
||
|
|
* The reported number of singular values
|
||
|
|
* If denormalized floats are required
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
joba specifies several choices effecting the calculation's accuracy
|
||
|
|
Although all arguments are tested, the tests only check that the correct
|
||
|
|
solution is returned - NOT that the prescribed actions are performed
|
||
|
|
internally.
|
||
|
|
|
||
|
|
jobt is, as of v3.9.0, still experimental and removed to cut down number of
|
||
|
|
test cases. However keyword itself is tested externally.
|
||
|
|
"""
|
||
|
|
seed(42)
|
||
|
|
|
||
|
|
# Define some constants for later use:
|
||
|
|
m, n = size
|
||
|
|
atol = 100 * np.finfo(dtype).eps
|
||
|
|
A = generate_random_dtype_array(size, dtype)
|
||
|
|
gejsv = get_lapack_funcs('gejsv', dtype=dtype)
|
||
|
|
|
||
|
|
# Set up checks for invalid job? combinations
|
||
|
|
# if an invalid combination occurs we set the appropriate
|
||
|
|
# exit status.
|
||
|
|
lsvec = jobu < 2 # Calculate left singular vectors
|
||
|
|
rsvec = jobv < 2 # Calculate right singular vectors
|
||
|
|
l2tran = (jobt == 1) and (m == n)
|
||
|
|
is_complex = np.iscomplexobj(A)
|
||
|
|
|
||
|
|
invalid_real_jobv = (jobv == 1) and (not lsvec) and (not is_complex)
|
||
|
|
invalid_cplx_jobu = (jobu == 2) and not (rsvec and l2tran) and is_complex
|
||
|
|
invalid_cplx_jobv = (jobv == 2) and not (lsvec and l2tran) and is_complex
|
||
|
|
|
||
|
|
# Set the exit status to the expected value.
|
||
|
|
# Here we only check for invalid combinations, not individual
|
||
|
|
# parameters.
|
||
|
|
if invalid_cplx_jobu:
|
||
|
|
exit_status = -2
|
||
|
|
elif invalid_real_jobv or invalid_cplx_jobv:
|
||
|
|
exit_status = -3
|
||
|
|
else:
|
||
|
|
exit_status = 0
|
||
|
|
|
||
|
|
if (jobu > 1) and (jobv == 1):
|
||
|
|
assert_raises(Exception, gejsv, A, joba, jobu, jobv, jobr, jobt, jobp)
|
||
|
|
else:
|
||
|
|
sva, u, v, work, iwork, info = gejsv(A,
|
||
|
|
joba=joba,
|
||
|
|
jobu=jobu,
|
||
|
|
jobv=jobv,
|
||
|
|
jobr=jobr,
|
||
|
|
jobt=jobt,
|
||
|
|
jobp=jobp)
|
||
|
|
|
||
|
|
# Check that ?gejsv exited successfully/as expected
|
||
|
|
assert_equal(info, exit_status)
|
||
|
|
|
||
|
|
# If exit_status is non-zero the combination of jobs is invalid.
|
||
|
|
# We test this above but no calculations are performed.
|
||
|
|
if not exit_status:
|
||
|
|
|
||
|
|
# Check the returned singular values
|
||
|
|
sigma = (work[0] / work[1]) * sva[:n]
|
||
|
|
assert_allclose(sigma, svd(A, compute_uv=False), atol=atol)
|
||
|
|
|
||
|
|
if jobu == 1:
|
||
|
|
# If JOBU = 'F', then u contains the M-by-M matrix of
|
||
|
|
# the left singular vectors, including an ONB of the orthogonal
|
||
|
|
# complement of the Range(A)
|
||
|
|
# However, to recalculate A we are concerned about the
|
||
|
|
# first n singular values and so can ignore the latter.
|
||
|
|
# TODO: Add a test for ONB?
|
||
|
|
u = u[:, :n]
|
||
|
|
|
||
|
|
if lsvec and rsvec:
|
||
|
|
assert_allclose(u @ np.diag(sigma) @ v.conj().T, A, atol=atol)
|
||
|
|
if lsvec:
|
||
|
|
assert_allclose(u.conj().T @ u, np.identity(n), atol=atol)
|
||
|
|
if rsvec:
|
||
|
|
assert_allclose(v.conj().T @ v, np.identity(n), atol=atol)
|
||
|
|
|
||
|
|
assert_equal(iwork[0], np.linalg.matrix_rank(A))
|
||
|
|
assert_equal(iwork[1], np.count_nonzero(sigma))
|
||
|
|
# iwork[2] is non-zero if requested accuracy is not warranted for
|
||
|
|
# the data. This should never occur for these tests.
|
||
|
|
assert_equal(iwork[2], 0)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', REAL_DTYPES)
|
||
|
|
def test_gejsv_edge_arguments(dtype):
|
||
|
|
"""Test edge arguments return expected status"""
|
||
|
|
gejsv = get_lapack_funcs('gejsv', dtype=dtype)
|
||
|
|
|
||
|
|
# scalar A
|
||
|
|
sva, u, v, work, iwork, info = gejsv(1.)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_equal(u.shape, (1, 1))
|
||
|
|
assert_equal(v.shape, (1, 1))
|
||
|
|
assert_equal(sva, np.array([1.], dtype=dtype))
|
||
|
|
|
||
|
|
# 1d A
|
||
|
|
A = np.ones((1,), dtype=dtype)
|
||
|
|
sva, u, v, work, iwork, info = gejsv(A)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_equal(u.shape, (1, 1))
|
||
|
|
assert_equal(v.shape, (1, 1))
|
||
|
|
assert_equal(sva, np.array([1.], dtype=dtype))
|
||
|
|
|
||
|
|
# 2d empty A
|
||
|
|
A = np.ones((1, 0), dtype=dtype)
|
||
|
|
sva, u, v, work, iwork, info = gejsv(A)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_equal(u.shape, (1, 0))
|
||
|
|
assert_equal(v.shape, (1, 0))
|
||
|
|
assert_equal(sva, np.array([], dtype=dtype))
|
||
|
|
|
||
|
|
# make sure "overwrite_a" is respected - user reported in gh-13191
|
||
|
|
A = np.sin(np.arange(100).reshape(10, 10)).astype(dtype)
|
||
|
|
A = np.asfortranarray(A + A.T) # make it symmetric and column major
|
||
|
|
Ac = A.copy('A')
|
||
|
|
_ = gejsv(A)
|
||
|
|
assert_allclose(A, Ac)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize(('kwargs'),
|
||
|
|
({'joba': 9},
|
||
|
|
{'jobu': 9},
|
||
|
|
{'jobv': 9},
|
||
|
|
{'jobr': 9},
|
||
|
|
{'jobt': 9},
|
||
|
|
{'jobp': 9})
|
||
|
|
)
|
||
|
|
def test_gejsv_invalid_job_arguments(kwargs):
|
||
|
|
"""Test invalid job arguments raise an Exception"""
|
||
|
|
A = np.ones((2, 2), dtype=float)
|
||
|
|
gejsv = get_lapack_funcs('gejsv', dtype=float)
|
||
|
|
assert_raises(Exception, gejsv, A, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("A,sva_expect,u_expect,v_expect",
|
||
|
|
[(np.array([[2.27, -1.54, 1.15, -1.94],
|
||
|
|
[0.28, -1.67, 0.94, -0.78],
|
||
|
|
[-0.48, -3.09, 0.99, -0.21],
|
||
|
|
[1.07, 1.22, 0.79, 0.63],
|
||
|
|
[-2.35, 2.93, -1.45, 2.30],
|
||
|
|
[0.62, -7.39, 1.03, -2.57]]),
|
||
|
|
np.array([9.9966, 3.6831, 1.3569, 0.5000]),
|
||
|
|
np.array([[0.2774, -0.6003, -0.1277, 0.1323],
|
||
|
|
[0.2020, -0.0301, 0.2805, 0.7034],
|
||
|
|
[0.2918, 0.3348, 0.6453, 0.1906],
|
||
|
|
[-0.0938, -0.3699, 0.6781, -0.5399],
|
||
|
|
[-0.4213, 0.5266, 0.0413, -0.0575],
|
||
|
|
[0.7816, 0.3353, -0.1645, -0.3957]]),
|
||
|
|
np.array([[0.1921, -0.8030, 0.0041, -0.5642],
|
||
|
|
[-0.8794, -0.3926, -0.0752, 0.2587],
|
||
|
|
[0.2140, -0.2980, 0.7827, 0.5027],
|
||
|
|
[-0.3795, 0.3351, 0.6178, -0.6017]]))])
|
||
|
|
def test_gejsv_NAG(A, sva_expect, u_expect, v_expect):
|
||
|
|
"""
|
||
|
|
This test implements the example found in the NAG manual, f08khf.
|
||
|
|
An example was not found for the complex case.
|
||
|
|
"""
|
||
|
|
# NAG manual provides accuracy up to 4 decimals
|
||
|
|
atol = 1e-4
|
||
|
|
gejsv = get_lapack_funcs('gejsv', dtype=A.dtype)
|
||
|
|
|
||
|
|
sva, u, v, work, iwork, info = gejsv(A)
|
||
|
|
|
||
|
|
assert_allclose(sva_expect, sva, atol=atol)
|
||
|
|
assert_allclose(u_expect, u, atol=atol)
|
||
|
|
assert_allclose(v_expect, v, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype", DTYPES)
|
||
|
|
def test_gttrf_gttrs(dtype):
|
||
|
|
# The test uses ?gttrf and ?gttrs to solve a random system for each dtype,
|
||
|
|
# tests that the output of ?gttrf define LU matricies, that input
|
||
|
|
# parameters are unmodified, transposal options function correctly, that
|
||
|
|
# incompatible matrix shapes raise an error, and singular matrices return
|
||
|
|
# non zero info.
|
||
|
|
|
||
|
|
seed(42)
|
||
|
|
n = 10
|
||
|
|
atol = 100 * np.finfo(dtype).eps
|
||
|
|
|
||
|
|
# create the matrix in accordance with the data type
|
||
|
|
du = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
d = generate_random_dtype_array((n,), dtype=dtype)
|
||
|
|
dl = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
|
||
|
|
diag_cpy = [dl.copy(), d.copy(), du.copy()]
|
||
|
|
|
||
|
|
A = np.diag(d) + np.diag(dl, -1) + np.diag(du, 1)
|
||
|
|
x = np.random.rand(n)
|
||
|
|
b = A @ x
|
||
|
|
|
||
|
|
gttrf, gttrs = get_lapack_funcs(('gttrf', 'gttrs'), dtype=dtype)
|
||
|
|
|
||
|
|
_dl, _d, _du, du2, ipiv, info = gttrf(dl, d, du)
|
||
|
|
# test to assure that the inputs of ?gttrf are unmodified
|
||
|
|
assert_array_equal(dl, diag_cpy[0])
|
||
|
|
assert_array_equal(d, diag_cpy[1])
|
||
|
|
assert_array_equal(du, diag_cpy[2])
|
||
|
|
|
||
|
|
# generate L and U factors from ?gttrf return values
|
||
|
|
# L/U are lower/upper triangular by construction (initially and at end)
|
||
|
|
U = np.diag(_d, 0) + np.diag(_du, 1) + np.diag(du2, 2)
|
||
|
|
L = np.eye(n, dtype=dtype)
|
||
|
|
|
||
|
|
for i, m in enumerate(_dl):
|
||
|
|
# L is given in a factored form.
|
||
|
|
# See
|
||
|
|
# www.hpcavf.uclan.ac.uk/softwaredoc/sgi_scsl_html/sgi_html/ch03.html
|
||
|
|
piv = ipiv[i] - 1
|
||
|
|
# right multiply by permutation matrix
|
||
|
|
L[:, [i, piv]] = L[:, [piv, i]]
|
||
|
|
# right multiply by Li, rank-one modification of identity
|
||
|
|
L[:, i] += L[:, i+1]*m
|
||
|
|
|
||
|
|
# one last permutation
|
||
|
|
i, piv = -1, ipiv[-1] - 1
|
||
|
|
# right multiply by final permutation matrix
|
||
|
|
L[:, [i, piv]] = L[:, [piv, i]]
|
||
|
|
|
||
|
|
# check that the outputs of ?gttrf define an LU decomposition of A
|
||
|
|
assert_allclose(A, L @ U, atol=atol)
|
||
|
|
|
||
|
|
b_cpy = b.copy()
|
||
|
|
x_gttrs, info = gttrs(_dl, _d, _du, du2, ipiv, b)
|
||
|
|
# test that the inputs of ?gttrs are unmodified
|
||
|
|
assert_array_equal(b, b_cpy)
|
||
|
|
# test that the result of ?gttrs matches the expected input
|
||
|
|
assert_allclose(x, x_gttrs, atol=atol)
|
||
|
|
|
||
|
|
# test that ?gttrf and ?gttrs work with transposal options
|
||
|
|
if dtype in REAL_DTYPES:
|
||
|
|
trans = "T"
|
||
|
|
b_trans = A.T @ x
|
||
|
|
else:
|
||
|
|
trans = "C"
|
||
|
|
b_trans = A.conj().T @ x
|
||
|
|
|
||
|
|
x_gttrs, info = gttrs(_dl, _d, _du, du2, ipiv, b_trans, trans=trans)
|
||
|
|
assert_allclose(x, x_gttrs, atol=atol)
|
||
|
|
|
||
|
|
# test that ValueError is raised with incompatible matrix shapes
|
||
|
|
with assert_raises(ValueError):
|
||
|
|
gttrf(dl[:-1], d, du)
|
||
|
|
with assert_raises(ValueError):
|
||
|
|
gttrf(dl, d[:-1], du)
|
||
|
|
with assert_raises(ValueError):
|
||
|
|
gttrf(dl, d, du[:-1])
|
||
|
|
|
||
|
|
# test that matrix of size n=2 raises exception
|
||
|
|
with assert_raises(Exception):
|
||
|
|
gttrf(dl[0], d[:1], du[0])
|
||
|
|
|
||
|
|
# test that singular (row of all zeroes) matrix fails via info
|
||
|
|
du[0] = 0
|
||
|
|
d[0] = 0
|
||
|
|
__dl, __d, __du, _du2, _ipiv, _info = gttrf(dl, d, du)
|
||
|
|
np.testing.assert_(__d[info - 1] == 0,
|
||
|
|
"?gttrf: _d[info-1] is {}, not the illegal value :0."
|
||
|
|
.format(__d[info - 1]))
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("du, d, dl, du_exp, d_exp, du2_exp, ipiv_exp, b, x",
|
||
|
|
[(np.array([2.1, -1.0, 1.9, 8.0]),
|
||
|
|
np.array([3.0, 2.3, -5.0, -.9, 7.1]),
|
||
|
|
np.array([3.4, 3.6, 7.0, -6.0]),
|
||
|
|
np.array([2.3, -5, -.9, 7.1]),
|
||
|
|
np.array([3.4, 3.6, 7, -6, -1.015373]),
|
||
|
|
np.array([-1, 1.9, 8]),
|
||
|
|
np.array([2, 3, 4, 5, 5]),
|
||
|
|
np.array([[2.7, 6.6],
|
||
|
|
[-0.5, 10.8],
|
||
|
|
[2.6, -3.2],
|
||
|
|
[0.6, -11.2],
|
||
|
|
[2.7, 19.1]
|
||
|
|
]),
|
||
|
|
np.array([[-4, 5],
|
||
|
|
[7, -4],
|
||
|
|
[3, -3],
|
||
|
|
[-4, -2],
|
||
|
|
[-3, 1]])),
|
||
|
|
(
|
||
|
|
np.array([2 - 1j, 2 + 1j, -1 + 1j, 1 - 1j]),
|
||
|
|
np.array([-1.3 + 1.3j, -1.3 + 1.3j,
|
||
|
|
-1.3 + 3.3j, - .3 + 4.3j,
|
||
|
|
-3.3 + 1.3j]),
|
||
|
|
np.array([1 - 2j, 1 + 1j, 2 - 3j, 1 + 1j]),
|
||
|
|
# du exp
|
||
|
|
np.array([-1.3 + 1.3j, -1.3 + 3.3j,
|
||
|
|
-0.3 + 4.3j, -3.3 + 1.3j]),
|
||
|
|
np.array([1 - 2j, 1 + 1j, 2 - 3j, 1 + 1j,
|
||
|
|
-1.3399 + 0.2875j]),
|
||
|
|
np.array([2 + 1j, -1 + 1j, 1 - 1j]),
|
||
|
|
np.array([2, 3, 4, 5, 5]),
|
||
|
|
np.array([[2.4 - 5j, 2.7 + 6.9j],
|
||
|
|
[3.4 + 18.2j, - 6.9 - 5.3j],
|
||
|
|
[-14.7 + 9.7j, - 6 - .6j],
|
||
|
|
[31.9 - 7.7j, -3.9 + 9.3j],
|
||
|
|
[-1 + 1.6j, -3 + 12.2j]]),
|
||
|
|
np.array([[1 + 1j, 2 - 1j],
|
||
|
|
[3 - 1j, 1 + 2j],
|
||
|
|
[4 + 5j, -1 + 1j],
|
||
|
|
[-1 - 2j, 2 + 1j],
|
||
|
|
[1 - 1j, 2 - 2j]])
|
||
|
|
)])
|
||
|
|
def test_gttrf_gttrs_NAG_f07cdf_f07cef_f07crf_f07csf(du, d, dl, du_exp, d_exp,
|
||
|
|
du2_exp, ipiv_exp, b, x):
|
||
|
|
# test to assure that wrapper is consistent with NAG Library Manual Mark 26
|
||
|
|
# example problems: f07cdf and f07cef (real)
|
||
|
|
# examples: f07crf and f07csf (complex)
|
||
|
|
# (Links may expire, so search for "NAG Library Manual Mark 26" online)
|
||
|
|
|
||
|
|
gttrf, gttrs = get_lapack_funcs(('gttrf', "gttrs"), (du[0], du[0]))
|
||
|
|
|
||
|
|
_dl, _d, _du, du2, ipiv, info = gttrf(dl, d, du)
|
||
|
|
assert_allclose(du2, du2_exp)
|
||
|
|
assert_allclose(_du, du_exp)
|
||
|
|
assert_allclose(_d, d_exp, atol=1e-4) # NAG examples provide 4 decimals.
|
||
|
|
assert_allclose(ipiv, ipiv_exp)
|
||
|
|
|
||
|
|
x_gttrs, info = gttrs(_dl, _d, _du, du2, ipiv, b)
|
||
|
|
|
||
|
|
assert_allclose(x_gttrs, x)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
@pytest.mark.parametrize('shape', [(3, 7), (7, 3), (2**18, 2**18)])
|
||
|
|
def test_geqrfp_lwork(dtype, shape):
|
||
|
|
geqrfp_lwork = get_lapack_funcs(('geqrfp_lwork'), dtype=dtype)
|
||
|
|
m, n = shape
|
||
|
|
lwork, info = geqrfp_lwork(m=m, n=n)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("ddtype,dtype",
|
||
|
|
zip(REAL_DTYPES + REAL_DTYPES, DTYPES))
|
||
|
|
def test_pttrf_pttrs(ddtype, dtype):
|
||
|
|
seed(42)
|
||
|
|
# set test tolerance appropriate for dtype
|
||
|
|
atol = 100*np.finfo(dtype).eps
|
||
|
|
# n is the length diagonal of A
|
||
|
|
n = 10
|
||
|
|
# create diagonals according to size and dtype
|
||
|
|
|
||
|
|
# diagonal d should always be real.
|
||
|
|
# add 4 to d so it will be dominant for all dtypes
|
||
|
|
d = generate_random_dtype_array((n,), ddtype) + 4
|
||
|
|
# diagonal e may be real or complex.
|
||
|
|
e = generate_random_dtype_array((n-1,), dtype)
|
||
|
|
|
||
|
|
# assemble diagonals together into matrix
|
||
|
|
A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1)
|
||
|
|
# store a copy of diagonals to later verify
|
||
|
|
diag_cpy = [d.copy(), e.copy()]
|
||
|
|
|
||
|
|
pttrf = get_lapack_funcs('pttrf', dtype=dtype)
|
||
|
|
|
||
|
|
_d, _e, info = pttrf(d, e)
|
||
|
|
# test to assure that the inputs of ?pttrf are unmodified
|
||
|
|
assert_array_equal(d, diag_cpy[0])
|
||
|
|
assert_array_equal(e, diag_cpy[1])
|
||
|
|
assert_equal(info, 0, err_msg="pttrf: info = {}, should be 0".format(info))
|
||
|
|
|
||
|
|
# test that the factors from pttrf can be recombined to make A
|
||
|
|
L = np.diag(_e, -1) + np.diag(np.ones(n))
|
||
|
|
D = np.diag(_d)
|
||
|
|
|
||
|
|
assert_allclose(A, L@D@L.conjugate().T, atol=atol)
|
||
|
|
|
||
|
|
# generate random solution x
|
||
|
|
x = generate_random_dtype_array((n,), dtype)
|
||
|
|
# determine accompanying b to get soln x
|
||
|
|
b = A@x
|
||
|
|
|
||
|
|
# determine _x from pttrs
|
||
|
|
pttrs = get_lapack_funcs('pttrs', dtype=dtype)
|
||
|
|
_x, info = pttrs(_d, _e.conj(), b)
|
||
|
|
assert_equal(info, 0, err_msg="pttrs: info = {}, should be 0".format(info))
|
||
|
|
|
||
|
|
# test that _x from pttrs matches the expected x
|
||
|
|
assert_allclose(x, _x, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("ddtype,dtype",
|
||
|
|
zip(REAL_DTYPES + REAL_DTYPES, DTYPES))
|
||
|
|
def test_pttrf_pttrs_errors_incompatible_shape(ddtype, dtype):
|
||
|
|
n = 10
|
||
|
|
pttrf = get_lapack_funcs('pttrf', dtype=dtype)
|
||
|
|
d = generate_random_dtype_array((n,), ddtype) + 2
|
||
|
|
e = generate_random_dtype_array((n-1,), dtype)
|
||
|
|
# test that ValueError is raised with incompatible matrix shapes
|
||
|
|
assert_raises(ValueError, pttrf, d[:-1], e)
|
||
|
|
assert_raises(ValueError, pttrf, d, e[:-1])
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("ddtype,dtype",
|
||
|
|
zip(REAL_DTYPES + REAL_DTYPES, DTYPES))
|
||
|
|
def test_pttrf_pttrs_errors_singular_nonSPD(ddtype, dtype):
|
||
|
|
n = 10
|
||
|
|
pttrf = get_lapack_funcs('pttrf', dtype=dtype)
|
||
|
|
d = generate_random_dtype_array((n,), ddtype) + 2
|
||
|
|
e = generate_random_dtype_array((n-1,), dtype)
|
||
|
|
# test that singular (row of all zeroes) matrix fails via info
|
||
|
|
d[0] = 0
|
||
|
|
e[0] = 0
|
||
|
|
_d, _e, info = pttrf(d, e)
|
||
|
|
assert_equal(_d[info - 1], 0,
|
||
|
|
"?pttrf: _d[info-1] is {}, not the illegal value :0."
|
||
|
|
.format(_d[info - 1]))
|
||
|
|
|
||
|
|
# test with non-spd matrix
|
||
|
|
d = generate_random_dtype_array((n,), ddtype)
|
||
|
|
_d, _e, info = pttrf(d, e)
|
||
|
|
assert_(info != 0, "?pttrf should fail with non-spd matrix, but didn't")
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize(("d, e, d_expect, e_expect, b, x_expect"), [
|
||
|
|
(np.array([4, 10, 29, 25, 5]),
|
||
|
|
np.array([-2, -6, 15, 8]),
|
||
|
|
np.array([4, 9, 25, 16, 1]),
|
||
|
|
np.array([-.5, -.6667, .6, .5]),
|
||
|
|
np.array([[6, 10], [9, 4], [2, 9], [14, 65],
|
||
|
|
[7, 23]]),
|
||
|
|
np.array([[2.5, 2], [2, -1], [1, -3], [-1, 6],
|
||
|
|
[3, -5]])
|
||
|
|
), (
|
||
|
|
np.array([16, 41, 46, 21]),
|
||
|
|
np.array([16 + 16j, 18 - 9j, 1 - 4j]),
|
||
|
|
np.array([16, 9, 1, 4]),
|
||
|
|
np.array([1+1j, 2-1j, 1-4j]),
|
||
|
|
np.array([[64+16j, -16-32j], [93+62j, 61-66j],
|
||
|
|
[78-80j, 71-74j], [14-27j, 35+15j]]),
|
||
|
|
np.array([[2+1j, -3-2j], [1+1j, 1+1j], [1-2j, 1-2j],
|
||
|
|
[1-1j, 2+1j]])
|
||
|
|
)])
|
||
|
|
def test_pttrf_pttrs_NAG(d, e, d_expect, e_expect, b, x_expect):
|
||
|
|
# test to assure that wrapper is consistent with NAG Manual Mark 26
|
||
|
|
# example problems: f07jdf and f07jef (real)
|
||
|
|
# examples: f07jrf and f07csf (complex)
|
||
|
|
# NAG examples provide 4 decimals.
|
||
|
|
# (Links expire, so please search for "NAG Library Manual Mark 26" online)
|
||
|
|
|
||
|
|
atol = 1e-4
|
||
|
|
pttrf = get_lapack_funcs('pttrf', dtype=e[0])
|
||
|
|
_d, _e, info = pttrf(d, e)
|
||
|
|
assert_allclose(_d, d_expect, atol=atol)
|
||
|
|
assert_allclose(_e, e_expect, atol=atol)
|
||
|
|
|
||
|
|
pttrs = get_lapack_funcs('pttrs', dtype=e[0])
|
||
|
|
_x, info = pttrs(_d, _e.conj(), b)
|
||
|
|
assert_allclose(_x, x_expect, atol=atol)
|
||
|
|
|
||
|
|
# also test option `lower`
|
||
|
|
if e.dtype in COMPLEX_DTYPES:
|
||
|
|
_x, info = pttrs(_d, _e, b, lower=1)
|
||
|
|
assert_allclose(_x, x_expect, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
def pteqr_get_d_e_A_z(dtype, realtype, n, compute_z):
|
||
|
|
# used by ?pteqr tests to build parameters
|
||
|
|
# returns tuple of (d, e, A, z)
|
||
|
|
if compute_z == 1:
|
||
|
|
# build Hermitian A from Q**T * tri * Q = A by creating Q and tri
|
||
|
|
A_eig = generate_random_dtype_array((n, n), dtype)
|
||
|
|
A_eig = A_eig + np.diag(np.zeros(n) + 4*n)
|
||
|
|
A_eig = (A_eig + A_eig.conj().T) / 2
|
||
|
|
# obtain right eigenvectors (orthogonal)
|
||
|
|
vr = eigh(A_eig)[1]
|
||
|
|
# create tridiagonal matrix
|
||
|
|
d = generate_random_dtype_array((n,), realtype) + 4
|
||
|
|
e = generate_random_dtype_array((n-1,), realtype)
|
||
|
|
tri = np.diag(d) + np.diag(e, 1) + np.diag(e, -1)
|
||
|
|
# Build A using these factors that sytrd would: (Q**T * tri * Q = A)
|
||
|
|
A = vr @ tri @ vr.conj().T
|
||
|
|
# vr is orthogonal
|
||
|
|
z = vr
|
||
|
|
|
||
|
|
else:
|
||
|
|
# d and e are always real per lapack docs.
|
||
|
|
d = generate_random_dtype_array((n,), realtype)
|
||
|
|
e = generate_random_dtype_array((n-1,), realtype)
|
||
|
|
|
||
|
|
# make SPD
|
||
|
|
d = d + 4
|
||
|
|
A = np.diag(d) + np.diag(e, 1) + np.diag(e, -1)
|
||
|
|
z = np.diag(d) + np.diag(e, -1) + np.diag(e, 1)
|
||
|
|
return (d, e, A, z)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype,realtype",
|
||
|
|
zip(DTYPES, REAL_DTYPES + REAL_DTYPES))
|
||
|
|
@pytest.mark.parametrize("compute_z", range(3))
|
||
|
|
def test_pteqr(dtype, realtype, compute_z):
|
||
|
|
'''
|
||
|
|
Tests the ?pteqr lapack routine for all dtypes and compute_z parameters.
|
||
|
|
It generates random SPD matrix diagonals d and e, and then confirms
|
||
|
|
correct eigenvalues with scipy.linalg.eig. With applicable compute_z=2 it
|
||
|
|
tests that z can reform A.
|
||
|
|
'''
|
||
|
|
seed(42)
|
||
|
|
atol = 1000*np.finfo(dtype).eps
|
||
|
|
pteqr = get_lapack_funcs(('pteqr'), dtype=dtype)
|
||
|
|
|
||
|
|
n = 10
|
||
|
|
|
||
|
|
d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z)
|
||
|
|
|
||
|
|
d_pteqr, e_pteqr, z_pteqr, info = pteqr(d=d, e=e, z=z, compute_z=compute_z)
|
||
|
|
assert_equal(info, 0, "info = {}, should be 0.".format(info))
|
||
|
|
|
||
|
|
# compare the routine's eigenvalues with scipy.linalg.eig's.
|
||
|
|
assert_allclose(np.sort(eigh(A)[0]), np.sort(d_pteqr), atol=atol)
|
||
|
|
|
||
|
|
if compute_z:
|
||
|
|
# verify z_pteqr as orthogonal
|
||
|
|
assert_allclose(z_pteqr @ np.conj(z_pteqr).T, np.identity(n),
|
||
|
|
atol=atol)
|
||
|
|
# verify that z_pteqr recombines to A
|
||
|
|
assert_allclose(z_pteqr @ np.diag(d_pteqr) @ np.conj(z_pteqr).T,
|
||
|
|
A, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype,realtype",
|
||
|
|
zip(DTYPES, REAL_DTYPES + REAL_DTYPES))
|
||
|
|
@pytest.mark.parametrize("compute_z", range(3))
|
||
|
|
def test_pteqr_error_non_spd(dtype, realtype, compute_z):
|
||
|
|
seed(42)
|
||
|
|
pteqr = get_lapack_funcs(('pteqr'), dtype=dtype)
|
||
|
|
|
||
|
|
n = 10
|
||
|
|
d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z)
|
||
|
|
|
||
|
|
# test with non-spd matrix
|
||
|
|
d_pteqr, e_pteqr, z_pteqr, info = pteqr(d - 4, e, z=z, compute_z=compute_z)
|
||
|
|
assert info > 0
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype,realtype",
|
||
|
|
zip(DTYPES, REAL_DTYPES + REAL_DTYPES))
|
||
|
|
@pytest.mark.parametrize("compute_z", range(3))
|
||
|
|
def test_pteqr_raise_error_wrong_shape(dtype, realtype, compute_z):
|
||
|
|
seed(42)
|
||
|
|
pteqr = get_lapack_funcs(('pteqr'), dtype=dtype)
|
||
|
|
n = 10
|
||
|
|
d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z)
|
||
|
|
# test with incorrect/incompatible array sizes
|
||
|
|
assert_raises(ValueError, pteqr, d[:-1], e, z=z, compute_z=compute_z)
|
||
|
|
assert_raises(ValueError, pteqr, d, e[:-1], z=z, compute_z=compute_z)
|
||
|
|
if compute_z:
|
||
|
|
assert_raises(ValueError, pteqr, d, e, z=z[:-1], compute_z=compute_z)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype,realtype",
|
||
|
|
zip(DTYPES, REAL_DTYPES + REAL_DTYPES))
|
||
|
|
@pytest.mark.parametrize("compute_z", range(3))
|
||
|
|
def test_pteqr_error_singular(dtype, realtype, compute_z):
|
||
|
|
seed(42)
|
||
|
|
pteqr = get_lapack_funcs(('pteqr'), dtype=dtype)
|
||
|
|
n = 10
|
||
|
|
d, e, A, z = pteqr_get_d_e_A_z(dtype, realtype, n, compute_z)
|
||
|
|
# test with singular matrix
|
||
|
|
d[0] = 0
|
||
|
|
e[0] = 0
|
||
|
|
d_pteqr, e_pteqr, z_pteqr, info = pteqr(d, e, z=z, compute_z=compute_z)
|
||
|
|
assert info > 0
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("compute_z,d,e,d_expect,z_expect",
|
||
|
|
[(2, # "I"
|
||
|
|
np.array([4.16, 5.25, 1.09, .62]),
|
||
|
|
np.array([3.17, -.97, .55]),
|
||
|
|
np.array([8.0023, 1.9926, 1.0014, 0.1237]),
|
||
|
|
np.array([[0.6326, 0.6245, -0.4191, 0.1847],
|
||
|
|
[0.7668, -0.4270, 0.4176, -0.2352],
|
||
|
|
[-0.1082, 0.6071, 0.4594, -0.6393],
|
||
|
|
[-0.0081, 0.2432, 0.6625, 0.7084]])),
|
||
|
|
])
|
||
|
|
def test_pteqr_NAG_f08jgf(compute_z, d, e, d_expect, z_expect):
|
||
|
|
'''
|
||
|
|
Implements real (f08jgf) example from NAG Manual Mark 26.
|
||
|
|
Tests for correct outputs.
|
||
|
|
'''
|
||
|
|
# the NAG manual has 4 decimals accuracy
|
||
|
|
atol = 1e-4
|
||
|
|
pteqr = get_lapack_funcs(('pteqr'), dtype=d.dtype)
|
||
|
|
|
||
|
|
z = np.diag(d) + np.diag(e, 1) + np.diag(e, -1)
|
||
|
|
_d, _e, _z, info = pteqr(d=d, e=e, z=z, compute_z=compute_z)
|
||
|
|
assert_allclose(_d, d_expect, atol=atol)
|
||
|
|
assert_allclose(np.abs(_z), np.abs(z_expect), atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
@pytest.mark.parametrize('matrix_size', [(3, 4), (7, 6), (6, 6)])
|
||
|
|
def test_geqrfp(dtype, matrix_size):
|
||
|
|
# Tests for all dytpes, tall, wide, and square matrices.
|
||
|
|
# Using the routine with random matrix A, Q and R are obtained and then
|
||
|
|
# tested such that R is upper triangular and non-negative on the diagonal,
|
||
|
|
# and Q is an orthagonal matrix. Verifies that A=Q@R. It also
|
||
|
|
# tests against a matrix that for which the linalg.qr method returns
|
||
|
|
# negative diagonals, and for error messaging.
|
||
|
|
|
||
|
|
# set test tolerance appropriate for dtype
|
||
|
|
np.random.seed(42)
|
||
|
|
rtol = 250*np.finfo(dtype).eps
|
||
|
|
atol = 100*np.finfo(dtype).eps
|
||
|
|
# get appropriate ?geqrfp for dtype
|
||
|
|
geqrfp = get_lapack_funcs(('geqrfp'), dtype=dtype)
|
||
|
|
gqr = get_lapack_funcs(("orgqr"), dtype=dtype)
|
||
|
|
|
||
|
|
m, n = matrix_size
|
||
|
|
|
||
|
|
# create random matrix of dimentions m x n
|
||
|
|
A = generate_random_dtype_array((m, n), dtype=dtype)
|
||
|
|
# create qr matrix using geqrfp
|
||
|
|
qr_A, tau, info = geqrfp(A)
|
||
|
|
|
||
|
|
# obtain r from the upper triangular area
|
||
|
|
r = np.triu(qr_A)
|
||
|
|
|
||
|
|
# obtain q from the orgqr lapack routine
|
||
|
|
# based on linalg.qr's extraction strategy of q with orgqr
|
||
|
|
|
||
|
|
if m > n:
|
||
|
|
# this adds an extra column to the end of qr_A
|
||
|
|
# let qqr be an empty m x m matrix
|
||
|
|
qqr = np.zeros((m, m), dtype=dtype)
|
||
|
|
# set first n columns of qqr to qr_A
|
||
|
|
qqr[:, :n] = qr_A
|
||
|
|
# determine q from this qqr
|
||
|
|
# note that m is a sufficient for lwork based on LAPACK documentation
|
||
|
|
q = gqr(qqr, tau=tau, lwork=m)[0]
|
||
|
|
else:
|
||
|
|
q = gqr(qr_A[:, :m], tau=tau, lwork=m)[0]
|
||
|
|
|
||
|
|
# test that q and r still make A
|
||
|
|
assert_allclose(q@r, A, rtol=rtol)
|
||
|
|
# ensure that q is orthogonal (that q @ transposed q is the identity)
|
||
|
|
assert_allclose(np.eye(q.shape[0]), q@(q.conj().T), rtol=rtol,
|
||
|
|
atol=atol)
|
||
|
|
# ensure r is upper tri by comparing original r to r as upper triangular
|
||
|
|
assert_allclose(r, np.triu(r), rtol=rtol)
|
||
|
|
# make sure diagonals of r are positive for this random solution
|
||
|
|
assert_(np.all(np.diag(r) > np.zeros(len(np.diag(r)))))
|
||
|
|
# ensure that info is zero for this success
|
||
|
|
assert_(info == 0)
|
||
|
|
|
||
|
|
# test that this routine gives r diagonals that are positive for a
|
||
|
|
# matrix that returns negatives in the diagonal with scipy.linalg.rq
|
||
|
|
A_negative = generate_random_dtype_array((n, m), dtype=dtype) * -1
|
||
|
|
r_rq_neg, q_rq_neg = qr(A_negative)
|
||
|
|
rq_A_neg, tau_neg, info_neg = geqrfp(A_negative)
|
||
|
|
# assert that any of the entries on the diagonal from linalg.qr
|
||
|
|
# are negative and that all of geqrfp are positive.
|
||
|
|
assert_(np.any(np.diag(r_rq_neg) < 0) and
|
||
|
|
np.all(np.diag(r) > 0))
|
||
|
|
|
||
|
|
|
||
|
|
def test_geqrfp_errors_with_empty_array():
|
||
|
|
# check that empty array raises good error message
|
||
|
|
A_empty = np.array([])
|
||
|
|
geqrfp = get_lapack_funcs('geqrfp', dtype=A_empty.dtype)
|
||
|
|
assert_raises(Exception, geqrfp, A_empty)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("driver", ['ev', 'evd', 'evr', 'evx'])
|
||
|
|
@pytest.mark.parametrize("pfx", ['sy', 'he'])
|
||
|
|
def test_standard_eigh_lworks(pfx, driver):
|
||
|
|
n = 1200 # Some sufficiently big arbitrary number
|
||
|
|
dtype = REAL_DTYPES if pfx == 'sy' else COMPLEX_DTYPES
|
||
|
|
sc_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[0])
|
||
|
|
dz_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[1])
|
||
|
|
try:
|
||
|
|
_compute_lwork(sc_dlw, n, lower=1)
|
||
|
|
_compute_lwork(dz_dlw, n, lower=1)
|
||
|
|
except Exception as e:
|
||
|
|
pytest.fail("{}_lwork raised unexpected exception: {}"
|
||
|
|
"".format(pfx+driver, e))
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("driver", ['gv', 'gvx'])
|
||
|
|
@pytest.mark.parametrize("pfx", ['sy', 'he'])
|
||
|
|
def test_generalized_eigh_lworks(pfx, driver):
|
||
|
|
n = 1200 # Some sufficiently big arbitrary number
|
||
|
|
dtype = REAL_DTYPES if pfx == 'sy' else COMPLEX_DTYPES
|
||
|
|
sc_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[0])
|
||
|
|
dz_dlw = get_lapack_funcs(pfx+driver+'_lwork', dtype=dtype[1])
|
||
|
|
# Shouldn't raise any exceptions
|
||
|
|
try:
|
||
|
|
_compute_lwork(sc_dlw, n, uplo="L")
|
||
|
|
_compute_lwork(dz_dlw, n, uplo="L")
|
||
|
|
except Exception as e:
|
||
|
|
pytest.fail("{}_lwork raised unexpected exception: {}"
|
||
|
|
"".format(pfx+driver, e))
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype_", DTYPES)
|
||
|
|
@pytest.mark.parametrize("m", [1, 10, 100, 1000])
|
||
|
|
def test_orcsd_uncsd_lwork(dtype_, m):
|
||
|
|
seed(1234)
|
||
|
|
p = randint(0, m)
|
||
|
|
q = m - p
|
||
|
|
pfx = 'or' if dtype_ in REAL_DTYPES else 'un'
|
||
|
|
dlw = pfx + 'csd_lwork'
|
||
|
|
lw = get_lapack_funcs(dlw, dtype=dtype_)
|
||
|
|
lwval = _compute_lwork(lw, m, p, q)
|
||
|
|
lwval = lwval if pfx == 'un' else (lwval,)
|
||
|
|
assert all([x > 0 for x in lwval])
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype_", DTYPES)
|
||
|
|
def test_orcsd_uncsd(dtype_):
|
||
|
|
m, p, q = 250, 80, 170
|
||
|
|
|
||
|
|
pfx = 'or' if dtype_ in REAL_DTYPES else 'un'
|
||
|
|
X = ortho_group.rvs(m) if pfx == 'or' else unitary_group.rvs(m)
|
||
|
|
|
||
|
|
drv, dlw = get_lapack_funcs((pfx + 'csd', pfx + 'csd_lwork'), dtype=dtype_)
|
||
|
|
lwval = _compute_lwork(dlw, m, p, q)
|
||
|
|
lwvals = {'lwork': lwval} if pfx == 'or' else dict(zip(['lwork',
|
||
|
|
'lrwork'], lwval))
|
||
|
|
|
||
|
|
cs11, cs12, cs21, cs22, theta, u1, u2, v1t, v2t, info =\
|
||
|
|
drv(X[:p, :q], X[:p, q:], X[p:, :q], X[p:, q:], **lwvals)
|
||
|
|
|
||
|
|
assert info == 0
|
||
|
|
|
||
|
|
U = block_diag(u1, u2)
|
||
|
|
VH = block_diag(v1t, v2t)
|
||
|
|
r = min(min(p, q), min(m-p, m-q))
|
||
|
|
n11 = min(p, q) - r
|
||
|
|
n12 = min(p, m-q) - r
|
||
|
|
n21 = min(m-p, q) - r
|
||
|
|
n22 = min(m-p, m-q) - r
|
||
|
|
|
||
|
|
S = np.zeros((m, m), dtype=dtype_)
|
||
|
|
one = dtype_(1.)
|
||
|
|
for i in range(n11):
|
||
|
|
S[i, i] = one
|
||
|
|
for i in range(n22):
|
||
|
|
S[p+i, q+i] = one
|
||
|
|
for i in range(n12):
|
||
|
|
S[i+n11+r, i+n11+r+n21+n22+r] = -one
|
||
|
|
for i in range(n21):
|
||
|
|
S[p+n22+r+i, n11+r+i] = one
|
||
|
|
|
||
|
|
for i in range(r):
|
||
|
|
S[i+n11, i+n11] = np.cos(theta[i])
|
||
|
|
S[p+n22+i, i+r+n21+n22] = np.cos(theta[i])
|
||
|
|
|
||
|
|
S[i+n11, i+n11+n21+n22+r] = -np.sin(theta[i])
|
||
|
|
S[p+n22+i, i+n11] = np.sin(theta[i])
|
||
|
|
|
||
|
|
Xc = U @ S @ VH
|
||
|
|
assert_allclose(X, Xc, rtol=0., atol=1e4*np.finfo(dtype_).eps)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype", DTYPES)
|
||
|
|
@pytest.mark.parametrize("trans_bool", [False, True])
|
||
|
|
@pytest.mark.parametrize("fact", ["F", "N"])
|
||
|
|
def test_gtsvx(dtype, trans_bool, fact):
|
||
|
|
"""
|
||
|
|
These tests uses ?gtsvx to solve a random Ax=b system for each dtype.
|
||
|
|
It tests that the outputs define an LU matrix, that inputs are unmodified,
|
||
|
|
transposal options, incompatible shapes, singular matrices, and
|
||
|
|
singular factorizations. It parametrizes DTYPES and the 'fact' value along
|
||
|
|
with the fact related inputs.
|
||
|
|
"""
|
||
|
|
seed(42)
|
||
|
|
# set test tolerance appropriate for dtype
|
||
|
|
atol = 100 * np.finfo(dtype).eps
|
||
|
|
# obtain routine
|
||
|
|
gtsvx, gttrf = get_lapack_funcs(('gtsvx', 'gttrf'), dtype=dtype)
|
||
|
|
# Generate random tridiagonal matrix A
|
||
|
|
n = 10
|
||
|
|
dl = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
d = generate_random_dtype_array((n,), dtype=dtype)
|
||
|
|
du = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
A = np.diag(dl, -1) + np.diag(d) + np.diag(du, 1)
|
||
|
|
# generate random solution x
|
||
|
|
x = generate_random_dtype_array((n, 2), dtype=dtype)
|
||
|
|
# create b from x for equation Ax=b
|
||
|
|
trans = ("T" if dtype in REAL_DTYPES else "C") if trans_bool else "N"
|
||
|
|
b = (A.conj().T if trans_bool else A) @ x
|
||
|
|
|
||
|
|
# store a copy of the inputs to check they haven't been modified later
|
||
|
|
inputs_cpy = [dl.copy(), d.copy(), du.copy(), b.copy()]
|
||
|
|
|
||
|
|
# set these to None if fact = 'N', or the output of gttrf is fact = 'F'
|
||
|
|
dlf_, df_, duf_, du2f_, ipiv_, info_ = \
|
||
|
|
gttrf(dl, d, du) if fact == 'F' else [None]*6
|
||
|
|
|
||
|
|
gtsvx_out = gtsvx(dl, d, du, b, fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out
|
||
|
|
assert_(info == 0, "?gtsvx info = {}, should be zero".format(info))
|
||
|
|
|
||
|
|
# assure that inputs are unmodified
|
||
|
|
assert_array_equal(dl, inputs_cpy[0])
|
||
|
|
assert_array_equal(d, inputs_cpy[1])
|
||
|
|
assert_array_equal(du, inputs_cpy[2])
|
||
|
|
assert_array_equal(b, inputs_cpy[3])
|
||
|
|
|
||
|
|
# test that x_soln matches the expected x
|
||
|
|
assert_allclose(x, x_soln, atol=atol)
|
||
|
|
|
||
|
|
# assert that the outputs are of correct type or shape
|
||
|
|
# rcond should be a scalar
|
||
|
|
assert_(hasattr(rcond, "__len__") is not True,
|
||
|
|
"rcond should be scalar but is {}".format(rcond))
|
||
|
|
# ferr should be length of # of cols in x
|
||
|
|
assert_(ferr.shape[0] == b.shape[1], "ferr.shape is {} but shoud be {},"
|
||
|
|
.format(ferr.shape[0], b.shape[1]))
|
||
|
|
# berr should be length of # of cols in x
|
||
|
|
assert_(berr.shape[0] == b.shape[1], "berr.shape is {} but shoud be {},"
|
||
|
|
.format(berr.shape[0], b.shape[1]))
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype", DTYPES)
|
||
|
|
@pytest.mark.parametrize("trans_bool", [0, 1])
|
||
|
|
@pytest.mark.parametrize("fact", ["F", "N"])
|
||
|
|
def test_gtsvx_error_singular(dtype, trans_bool, fact):
|
||
|
|
seed(42)
|
||
|
|
# obtain routine
|
||
|
|
gtsvx, gttrf = get_lapack_funcs(('gtsvx', 'gttrf'), dtype=dtype)
|
||
|
|
# Generate random tridiagonal matrix A
|
||
|
|
n = 10
|
||
|
|
dl = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
d = generate_random_dtype_array((n,), dtype=dtype)
|
||
|
|
du = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
A = np.diag(dl, -1) + np.diag(d) + np.diag(du, 1)
|
||
|
|
# generate random solution x
|
||
|
|
x = generate_random_dtype_array((n, 2), dtype=dtype)
|
||
|
|
# create b from x for equation Ax=b
|
||
|
|
trans = "T" if dtype in REAL_DTYPES else "C"
|
||
|
|
b = (A.conj().T if trans_bool else A) @ x
|
||
|
|
|
||
|
|
# set these to None if fact = 'N', or the output of gttrf is fact = 'F'
|
||
|
|
dlf_, df_, duf_, du2f_, ipiv_, info_ = \
|
||
|
|
gttrf(dl, d, du) if fact == 'F' else [None]*6
|
||
|
|
|
||
|
|
gtsvx_out = gtsvx(dl, d, du, b, fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out
|
||
|
|
# test with singular matrix
|
||
|
|
# no need to test inputs with fact "F" since ?gttrf already does.
|
||
|
|
if fact == "N":
|
||
|
|
# Construct a singular example manually
|
||
|
|
d[-1] = 0
|
||
|
|
dl[-1] = 0
|
||
|
|
# solve using routine
|
||
|
|
gtsvx_out = gtsvx(dl, d, du, b)
|
||
|
|
dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out
|
||
|
|
# test for the singular matrix.
|
||
|
|
assert info > 0, "info should be > 0 for singular matrix"
|
||
|
|
|
||
|
|
elif fact == 'F':
|
||
|
|
# assuming that a singular factorization is input
|
||
|
|
df_[-1] = 0
|
||
|
|
duf_[-1] = 0
|
||
|
|
du2f_[-1] = 0
|
||
|
|
|
||
|
|
gtsvx_out = gtsvx(dl, d, du, b, fact=fact, dlf=dlf_, df=df_, duf=duf_,
|
||
|
|
du2=du2f_, ipiv=ipiv_)
|
||
|
|
dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out
|
||
|
|
# info should not be zero and should provide index of illegal value
|
||
|
|
assert info > 0, "info should be > 0 for singular matrix"
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype", DTYPES*2)
|
||
|
|
@pytest.mark.parametrize("trans_bool", [False, True])
|
||
|
|
@pytest.mark.parametrize("fact", ["F", "N"])
|
||
|
|
def test_gtsvx_error_incompatible_size(dtype, trans_bool, fact):
|
||
|
|
seed(42)
|
||
|
|
# obtain routine
|
||
|
|
gtsvx, gttrf = get_lapack_funcs(('gtsvx', 'gttrf'), dtype=dtype)
|
||
|
|
# Generate random tridiagonal matrix A
|
||
|
|
n = 10
|
||
|
|
dl = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
d = generate_random_dtype_array((n,), dtype=dtype)
|
||
|
|
du = generate_random_dtype_array((n-1,), dtype=dtype)
|
||
|
|
A = np.diag(dl, -1) + np.diag(d) + np.diag(du, 1)
|
||
|
|
# generate random solution x
|
||
|
|
x = generate_random_dtype_array((n, 2), dtype=dtype)
|
||
|
|
# create b from x for equation Ax=b
|
||
|
|
trans = "T" if dtype in REAL_DTYPES else "C"
|
||
|
|
b = (A.conj().T if trans_bool else A) @ x
|
||
|
|
|
||
|
|
# set these to None if fact = 'N', or the output of gttrf is fact = 'F'
|
||
|
|
dlf_, df_, duf_, du2f_, ipiv_, info_ = \
|
||
|
|
gttrf(dl, d, du) if fact == 'F' else [None]*6
|
||
|
|
|
||
|
|
if fact == "N":
|
||
|
|
assert_raises(ValueError, gtsvx, dl[:-1], d, du, b,
|
||
|
|
fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
assert_raises(ValueError, gtsvx, dl, d[:-1], du, b,
|
||
|
|
fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
assert_raises(ValueError, gtsvx, dl, d, du[:-1], b,
|
||
|
|
fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
assert_raises(Exception, gtsvx, dl, d, du, b[:-1],
|
||
|
|
fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
else:
|
||
|
|
assert_raises(ValueError, gtsvx, dl, d, du, b,
|
||
|
|
fact=fact, trans=trans, dlf=dlf_[:-1], df=df_,
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
assert_raises(ValueError, gtsvx, dl, d, du, b,
|
||
|
|
fact=fact, trans=trans, dlf=dlf_, df=df_[:-1],
|
||
|
|
duf=duf_, du2=du2f_, ipiv=ipiv_)
|
||
|
|
assert_raises(ValueError, gtsvx, dl, d, du, b,
|
||
|
|
fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_[:-1], du2=du2f_, ipiv=ipiv_)
|
||
|
|
assert_raises(ValueError, gtsvx, dl, d, du, b,
|
||
|
|
fact=fact, trans=trans, dlf=dlf_, df=df_,
|
||
|
|
duf=duf_, du2=du2f_[:-1], ipiv=ipiv_)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("du,d,dl,b,x",
|
||
|
|
[(np.array([2.1, -1.0, 1.9, 8.0]),
|
||
|
|
np.array([3.0, 2.3, -5.0, -0.9, 7.1]),
|
||
|
|
np.array([3.4, 3.6, 7.0, -6.0]),
|
||
|
|
np.array([[2.7, 6.6], [-.5, 10.8], [2.6, -3.2],
|
||
|
|
[.6, -11.2], [2.7, 19.1]]),
|
||
|
|
np.array([[-4, 5], [7, -4], [3, -3], [-4, -2],
|
||
|
|
[-3, 1]])),
|
||
|
|
(np.array([2 - 1j, 2 + 1j, -1 + 1j, 1 - 1j]),
|
||
|
|
np.array([-1.3 + 1.3j, -1.3 + 1.3j, -1.3 + 3.3j,
|
||
|
|
-.3 + 4.3j, -3.3 + 1.3j]),
|
||
|
|
np.array([1 - 2j, 1 + 1j, 2 - 3j, 1 + 1j]),
|
||
|
|
np.array([[2.4 - 5j, 2.7 + 6.9j],
|
||
|
|
[3.4 + 18.2j, -6.9 - 5.3j],
|
||
|
|
[-14.7 + 9.7j, -6 - .6j],
|
||
|
|
[31.9 - 7.7j, -3.9 + 9.3j],
|
||
|
|
[-1 + 1.6j, -3 + 12.2j]]),
|
||
|
|
np.array([[1 + 1j, 2 - 1j], [3 - 1j, 1 + 2j],
|
||
|
|
[4 + 5j, -1 + 1j], [-1 - 2j, 2 + 1j],
|
||
|
|
[1 - 1j, 2 - 2j]]))])
|
||
|
|
def test_gtsvx_NAG(du, d, dl, b, x):
|
||
|
|
# Test to ensure wrapper is consistent with NAG Manual Mark 26
|
||
|
|
# example problems: real (f07cbf) and complex (f07cpf)
|
||
|
|
gtsvx = get_lapack_funcs('gtsvx', dtype=d.dtype)
|
||
|
|
|
||
|
|
gtsvx_out = gtsvx(dl, d, du, b)
|
||
|
|
dlf, df, duf, du2f, ipiv, x_soln, rcond, ferr, berr, info = gtsvx_out
|
||
|
|
|
||
|
|
assert_array_almost_equal(x, x_soln)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype,realtype", zip(DTYPES, REAL_DTYPES
|
||
|
|
+ REAL_DTYPES))
|
||
|
|
@pytest.mark.parametrize("fact,df_de_lambda",
|
||
|
|
[("F",
|
||
|
|
lambda d, e:get_lapack_funcs('pttrf',
|
||
|
|
dtype=e.dtype)(d, e)),
|
||
|
|
("N", lambda d, e: (None, None, None))])
|
||
|
|
def test_ptsvx(dtype, realtype, fact, df_de_lambda):
|
||
|
|
'''
|
||
|
|
This tests the ?ptsvx lapack routine wrapper to solve a random system
|
||
|
|
Ax = b for all dtypes and input variations. Tests for: unmodified
|
||
|
|
input parameters, fact options, incompatible matrix shapes raise an error,
|
||
|
|
and singular matrices return info of illegal value.
|
||
|
|
'''
|
||
|
|
seed(42)
|
||
|
|
# set test tolerance appropriate for dtype
|
||
|
|
atol = 100 * np.finfo(dtype).eps
|
||
|
|
ptsvx = get_lapack_funcs('ptsvx', dtype=dtype)
|
||
|
|
n = 5
|
||
|
|
# create diagonals according to size and dtype
|
||
|
|
d = generate_random_dtype_array((n,), realtype) + 4
|
||
|
|
e = generate_random_dtype_array((n-1,), dtype)
|
||
|
|
A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1)
|
||
|
|
x_soln = generate_random_dtype_array((n, 2), dtype=dtype)
|
||
|
|
b = A @ x_soln
|
||
|
|
|
||
|
|
# use lambda to determine what df, ef are
|
||
|
|
df, ef, info = df_de_lambda(d, e)
|
||
|
|
|
||
|
|
# create copy to later test that they are unmodified
|
||
|
|
diag_cpy = [d.copy(), e.copy(), b.copy()]
|
||
|
|
|
||
|
|
# solve using routine
|
||
|
|
df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b, fact=fact,
|
||
|
|
df=df, ef=ef)
|
||
|
|
# d, e, and b should be unmodified
|
||
|
|
assert_array_equal(d, diag_cpy[0])
|
||
|
|
assert_array_equal(e, diag_cpy[1])
|
||
|
|
assert_array_equal(b, diag_cpy[2])
|
||
|
|
assert_(info == 0, "info should be 0 but is {}.".format(info))
|
||
|
|
assert_array_almost_equal(x_soln, x)
|
||
|
|
|
||
|
|
# test that the factors from ptsvx can be recombined to make A
|
||
|
|
L = np.diag(ef, -1) + np.diag(np.ones(n))
|
||
|
|
D = np.diag(df)
|
||
|
|
assert_allclose(A, L@D@(np.conj(L).T), atol=atol)
|
||
|
|
|
||
|
|
# assert that the outputs are of correct type or shape
|
||
|
|
# rcond should be a scalar
|
||
|
|
assert not hasattr(rcond, "__len__"), \
|
||
|
|
"rcond should be scalar but is {}".format(rcond)
|
||
|
|
# ferr should be length of # of cols in x
|
||
|
|
assert_(ferr.shape == (2,), "ferr.shape is {} but shoud be ({},)"
|
||
|
|
.format(ferr.shape, x_soln.shape[1]))
|
||
|
|
# berr should be length of # of cols in x
|
||
|
|
assert_(berr.shape == (2,), "berr.shape is {} but shoud be ({},)"
|
||
|
|
.format(berr.shape, x_soln.shape[1]))
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype,realtype", zip(DTYPES, REAL_DTYPES
|
||
|
|
+ REAL_DTYPES))
|
||
|
|
@pytest.mark.parametrize("fact,df_de_lambda",
|
||
|
|
[("F",
|
||
|
|
lambda d, e:get_lapack_funcs('pttrf',
|
||
|
|
dtype=e.dtype)(d, e)),
|
||
|
|
("N", lambda d, e: (None, None, None))])
|
||
|
|
def test_ptsvx_error_raise_errors(dtype, realtype, fact, df_de_lambda):
|
||
|
|
seed(42)
|
||
|
|
ptsvx = get_lapack_funcs('ptsvx', dtype=dtype)
|
||
|
|
n = 5
|
||
|
|
# create diagonals according to size and dtype
|
||
|
|
d = generate_random_dtype_array((n,), realtype) + 4
|
||
|
|
e = generate_random_dtype_array((n-1,), dtype)
|
||
|
|
A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1)
|
||
|
|
x_soln = generate_random_dtype_array((n, 2), dtype=dtype)
|
||
|
|
b = A @ x_soln
|
||
|
|
|
||
|
|
# use lambda to determine what df, ef are
|
||
|
|
df, ef, info = df_de_lambda(d, e)
|
||
|
|
|
||
|
|
# test with malformatted array sizes
|
||
|
|
assert_raises(ValueError, ptsvx, d[:-1], e, b, fact=fact, df=df, ef=ef)
|
||
|
|
assert_raises(ValueError, ptsvx, d, e[:-1], b, fact=fact, df=df, ef=ef)
|
||
|
|
assert_raises(Exception, ptsvx, d, e, b[:-1], fact=fact, df=df, ef=ef)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("dtype,realtype", zip(DTYPES, REAL_DTYPES
|
||
|
|
+ REAL_DTYPES))
|
||
|
|
@pytest.mark.parametrize("fact,df_de_lambda",
|
||
|
|
[("F",
|
||
|
|
lambda d, e:get_lapack_funcs('pttrf',
|
||
|
|
dtype=e.dtype)(d, e)),
|
||
|
|
("N", lambda d, e: (None, None, None))])
|
||
|
|
def test_ptsvx_non_SPD_singular(dtype, realtype, fact, df_de_lambda):
|
||
|
|
seed(42)
|
||
|
|
ptsvx = get_lapack_funcs('ptsvx', dtype=dtype)
|
||
|
|
n = 5
|
||
|
|
# create diagonals according to size and dtype
|
||
|
|
d = generate_random_dtype_array((n,), realtype) + 4
|
||
|
|
e = generate_random_dtype_array((n-1,), dtype)
|
||
|
|
A = np.diag(d) + np.diag(e, -1) + np.diag(np.conj(e), 1)
|
||
|
|
x_soln = generate_random_dtype_array((n, 2), dtype=dtype)
|
||
|
|
b = A @ x_soln
|
||
|
|
|
||
|
|
# use lambda to determine what df, ef are
|
||
|
|
df, ef, info = df_de_lambda(d, e)
|
||
|
|
|
||
|
|
if fact == "N":
|
||
|
|
d[3] = 0
|
||
|
|
# obtain new df, ef
|
||
|
|
df, ef, info = df_de_lambda(d, e)
|
||
|
|
# solve using routine
|
||
|
|
df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b)
|
||
|
|
# test for the singular matrix.
|
||
|
|
assert info > 0 and info <= n
|
||
|
|
|
||
|
|
# non SPD matrix
|
||
|
|
d = generate_random_dtype_array((n,), realtype)
|
||
|
|
df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b)
|
||
|
|
assert info > 0 and info <= n
|
||
|
|
else:
|
||
|
|
# assuming that someone is using a singular factorization
|
||
|
|
df, ef, info = df_de_lambda(d, e)
|
||
|
|
df[0] = 0
|
||
|
|
ef[0] = 0
|
||
|
|
df, ef, x, rcond, ferr, berr, info = ptsvx(d, e, b, fact=fact,
|
||
|
|
df=df, ef=ef)
|
||
|
|
assert info > 0
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('d,e,b,x',
|
||
|
|
[(np.array([4, 10, 29, 25, 5]),
|
||
|
|
np.array([-2, -6, 15, 8]),
|
||
|
|
np.array([[6, 10], [9, 4], [2, 9], [14, 65],
|
||
|
|
[7, 23]]),
|
||
|
|
np.array([[2.5, 2], [2, -1], [1, -3],
|
||
|
|
[-1, 6], [3, -5]])),
|
||
|
|
(np.array([16, 41, 46, 21]),
|
||
|
|
np.array([16 + 16j, 18 - 9j, 1 - 4j]),
|
||
|
|
np.array([[64 + 16j, -16 - 32j],
|
||
|
|
[93 + 62j, 61 - 66j],
|
||
|
|
[78 - 80j, 71 - 74j],
|
||
|
|
[14 - 27j, 35 + 15j]]),
|
||
|
|
np.array([[2 + 1j, -3 - 2j],
|
||
|
|
[1 + 1j, 1 + 1j],
|
||
|
|
[1 - 2j, 1 - 2j],
|
||
|
|
[1 - 1j, 2 + 1j]]))])
|
||
|
|
def test_ptsvx_NAG(d, e, b, x):
|
||
|
|
# test to assure that wrapper is consistent with NAG Manual Mark 26
|
||
|
|
# example problemss: f07jbf, f07jpf
|
||
|
|
# (Links expire, so please search for "NAG Library Manual Mark 26" online)
|
||
|
|
|
||
|
|
# obtain routine with correct type based on e.dtype
|
||
|
|
ptsvx = get_lapack_funcs('ptsvx', dtype=e.dtype)
|
||
|
|
# solve using routine
|
||
|
|
df, ef, x_ptsvx, rcond, ferr, berr, info = ptsvx(d, e, b)
|
||
|
|
# determine ptsvx's solution and x are the same.
|
||
|
|
assert_array_almost_equal(x, x_ptsvx)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('lower', [False, True])
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
def test_pptrs_pptri_pptrf_ppsv_ppcon(dtype, lower):
|
||
|
|
seed(1234)
|
||
|
|
atol = np.finfo(dtype).eps*100
|
||
|
|
# Manual conversion to/from packed format is feasible here.
|
||
|
|
n, nrhs = 10, 4
|
||
|
|
a = generate_random_dtype_array([n, n], dtype=dtype)
|
||
|
|
b = generate_random_dtype_array([n, nrhs], dtype=dtype)
|
||
|
|
|
||
|
|
a = a.conj().T + a + np.eye(n, dtype=dtype) * dtype(5.)
|
||
|
|
if lower:
|
||
|
|
inds = ([x for y in range(n) for x in range(y, n)],
|
||
|
|
[y for y in range(n) for x in range(y, n)])
|
||
|
|
else:
|
||
|
|
inds = ([x for y in range(1, n+1) for x in range(y)],
|
||
|
|
[y-1 for y in range(1, n+1) for x in range(y)])
|
||
|
|
ap = a[inds]
|
||
|
|
ppsv, pptrf, pptrs, pptri, ppcon = get_lapack_funcs(
|
||
|
|
('ppsv', 'pptrf', 'pptrs', 'pptri', 'ppcon'),
|
||
|
|
dtype=dtype,
|
||
|
|
ilp64="preferred")
|
||
|
|
|
||
|
|
ul, info = pptrf(n, ap, lower=lower)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
aul = cholesky(a, lower=lower)[inds]
|
||
|
|
assert_allclose(ul, aul, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
uli, info = pptri(n, ul, lower=lower)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
auli = inv(a)[inds]
|
||
|
|
assert_allclose(uli, auli, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
x, info = pptrs(n, ul, b, lower=lower)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
bx = solve(a, b)
|
||
|
|
assert_allclose(x, bx, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
xv, info = ppsv(n, ap, b, lower=lower)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_allclose(xv, bx, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
anorm = np.linalg.norm(a, 1)
|
||
|
|
rcond, info = ppcon(n, ap, anorm=anorm, lower=lower)
|
||
|
|
assert_equal(info, 0)
|
||
|
|
assert_(abs(1/rcond - np.linalg.cond(a, p=1))*rcond < 1)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
def test_gees_trexc(dtype):
|
||
|
|
seed(1234)
|
||
|
|
atol = np.finfo(dtype).eps*100
|
||
|
|
|
||
|
|
n = 10
|
||
|
|
a = generate_random_dtype_array([n, n], dtype=dtype)
|
||
|
|
|
||
|
|
gees, trexc = get_lapack_funcs(('gees', 'trexc'), dtype=dtype)
|
||
|
|
|
||
|
|
result = gees(lambda x: None, a, overwrite_a=False)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
t = result[0]
|
||
|
|
z = result[-3]
|
||
|
|
|
||
|
|
d2 = t[6, 6]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
result = trexc(t, z, 7, 1)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
t = result[0]
|
||
|
|
z = result[-2]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(t[0, 0], d2, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize(
|
||
|
|
"t, expect, ifst, ilst",
|
||
|
|
[(np.array([[0.80, -0.11, 0.01, 0.03],
|
||
|
|
[0.00, -0.10, 0.25, 0.35],
|
||
|
|
[0.00, -0.65, -0.10, 0.20],
|
||
|
|
[0.00, 0.00, 0.00, -0.10]]),
|
||
|
|
np.array([[-0.1000, -0.6463, 0.0874, 0.2010],
|
||
|
|
[0.2514, -0.1000, 0.0927, 0.3505],
|
||
|
|
[0.0000, 0.0000, 0.8000, -0.0117],
|
||
|
|
[0.0000, 0.0000, 0.0000, -0.1000]]),
|
||
|
|
2, 1),
|
||
|
|
(np.array([[-6.00 - 7.00j, 0.36 - 0.36j, -0.19 + 0.48j, 0.88 - 0.25j],
|
||
|
|
[0.00 + 0.00j, -5.00 + 2.00j, -0.03 - 0.72j, -0.23 + 0.13j],
|
||
|
|
[0.00 + 0.00j, 0.00 + 0.00j, 8.00 - 1.00j, 0.94 + 0.53j],
|
||
|
|
[0.00 + 0.00j, 0.00 + 0.00j, 0.00 + 0.00j, 3.00 - 4.00j]]),
|
||
|
|
np.array([[-5.0000 + 2.0000j, -0.1574 + 0.7143j,
|
||
|
|
0.1781 - 0.1913j, 0.3950 + 0.3861j],
|
||
|
|
[0.0000 + 0.0000j, 8.0000 - 1.0000j,
|
||
|
|
1.0742 + 0.1447j, 0.2515 - 0.3397j],
|
||
|
|
[0.0000 + 0.0000j, 0.0000 + 0.0000j,
|
||
|
|
3.0000 - 4.0000j, 0.2264 + 0.8962j],
|
||
|
|
[0.0000 + 0.0000j, 0.0000 + 0.0000j,
|
||
|
|
0.0000 + 0.0000j, -6.0000 - 7.0000j]]),
|
||
|
|
1, 4)])
|
||
|
|
def test_trexc_NAG(t, ifst, ilst, expect):
|
||
|
|
"""
|
||
|
|
This test implements the example found in the NAG manual,
|
||
|
|
f08qfc, f08qtc, f08qgc, f08quc.
|
||
|
|
"""
|
||
|
|
# NAG manual provides accuracy up to 4 decimals
|
||
|
|
atol = 1e-4
|
||
|
|
trexc = get_lapack_funcs('trexc', dtype=t.dtype)
|
||
|
|
|
||
|
|
result = trexc(t, t, ifst, ilst, wantq=0)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
t = result[0]
|
||
|
|
assert_allclose(expect, t, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
def test_gges_tgexc(dtype):
|
||
|
|
if dtype == np.float32 and sys.platform == 'darwin':
|
||
|
|
pytest.xfail("gges[float32] broken for OpenBLAS on macOS, see gh-16949")
|
||
|
|
|
||
|
|
seed(1234)
|
||
|
|
atol = np.finfo(dtype).eps*100
|
||
|
|
|
||
|
|
n = 10
|
||
|
|
a = generate_random_dtype_array([n, n], dtype=dtype)
|
||
|
|
b = generate_random_dtype_array([n, n], dtype=dtype)
|
||
|
|
|
||
|
|
gges, tgexc = get_lapack_funcs(('gges', 'tgexc'), dtype=dtype)
|
||
|
|
|
||
|
|
result = gges(lambda x: None, a, b, overwrite_a=False, overwrite_b=False)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
s = result[0]
|
||
|
|
t = result[1]
|
||
|
|
q = result[-4]
|
||
|
|
z = result[-3]
|
||
|
|
|
||
|
|
d1 = s[0, 0] / t[0, 0]
|
||
|
|
d2 = s[6, 6] / t[6, 6]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(s, np.triu(s), rtol=0, atol=atol)
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
result = tgexc(s, t, q, z, 7, 1)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
s = result[0]
|
||
|
|
t = result[1]
|
||
|
|
q = result[2]
|
||
|
|
z = result[3]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(s, np.triu(s), rtol=0, atol=atol)
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(s[0, 0] / t[0, 0], d2, rtol=0, atol=atol)
|
||
|
|
assert_allclose(s[1, 1] / t[1, 1], d1, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
def test_gees_trsen(dtype):
|
||
|
|
seed(1234)
|
||
|
|
atol = np.finfo(dtype).eps*100
|
||
|
|
|
||
|
|
n = 10
|
||
|
|
a = generate_random_dtype_array([n, n], dtype=dtype)
|
||
|
|
|
||
|
|
gees, trsen, trsen_lwork = get_lapack_funcs(
|
||
|
|
('gees', 'trsen', 'trsen_lwork'), dtype=dtype)
|
||
|
|
|
||
|
|
result = gees(lambda x: None, a, overwrite_a=False)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
t = result[0]
|
||
|
|
z = result[-3]
|
||
|
|
|
||
|
|
d2 = t[6, 6]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
select = np.zeros(n)
|
||
|
|
select[6] = 1
|
||
|
|
|
||
|
|
lwork = _compute_lwork(trsen_lwork, select, t)
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
result = trsen(select, t, z, lwork=lwork)
|
||
|
|
else:
|
||
|
|
result = trsen(select, t, z, lwork=lwork, liwork=lwork[1])
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
t = result[0]
|
||
|
|
z = result[1]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(z @ t @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(t[0, 0], d2, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize(
|
||
|
|
"t, q, expect, select, expect_s, expect_sep",
|
||
|
|
[(np.array([[0.7995, -0.1144, 0.0060, 0.0336],
|
||
|
|
[0.0000, -0.0994, 0.2478, 0.3474],
|
||
|
|
[0.0000, -0.6483, -0.0994, 0.2026],
|
||
|
|
[0.0000, 0.0000, 0.0000, -0.1007]]),
|
||
|
|
np.array([[0.6551, 0.1037, 0.3450, 0.6641],
|
||
|
|
[0.5236, -0.5807, -0.6141, -0.1068],
|
||
|
|
[-0.5362, -0.3073, -0.2935, 0.7293],
|
||
|
|
[0.0956, 0.7467, -0.6463, 0.1249]]),
|
||
|
|
np.array([[0.3500, 0.4500, -0.1400, -0.1700],
|
||
|
|
[0.0900, 0.0700, -0.5399, 0.3500],
|
||
|
|
[-0.4400, -0.3300, -0.0300, 0.1700],
|
||
|
|
[0.2500, -0.3200, -0.1300, 0.1100]]),
|
||
|
|
np.array([1, 0, 0, 1]),
|
||
|
|
1.75e+00, 3.22e+00),
|
||
|
|
(np.array([[-6.0004 - 6.9999j, 0.3637 - 0.3656j,
|
||
|
|
-0.1880 + 0.4787j, 0.8785 - 0.2539j],
|
||
|
|
[0.0000 + 0.0000j, -5.0000 + 2.0060j,
|
||
|
|
-0.0307 - 0.7217j, -0.2290 + 0.1313j],
|
||
|
|
[0.0000 + 0.0000j, 0.0000 + 0.0000j,
|
||
|
|
7.9982 - 0.9964j, 0.9357 + 0.5359j],
|
||
|
|
[0.0000 + 0.0000j, 0.0000 + 0.0000j,
|
||
|
|
0.0000 + 0.0000j, 3.0023 - 3.9998j]]),
|
||
|
|
np.array([[-0.8347 - 0.1364j, -0.0628 + 0.3806j,
|
||
|
|
0.2765 - 0.0846j, 0.0633 - 0.2199j],
|
||
|
|
[0.0664 - 0.2968j, 0.2365 + 0.5240j,
|
||
|
|
-0.5877 - 0.4208j, 0.0835 + 0.2183j],
|
||
|
|
[-0.0362 - 0.3215j, 0.3143 - 0.5473j,
|
||
|
|
0.0576 - 0.5736j, 0.0057 - 0.4058j],
|
||
|
|
[0.0086 + 0.2958j, -0.3416 - 0.0757j,
|
||
|
|
-0.1900 - 0.1600j, 0.8327 - 0.1868j]]),
|
||
|
|
np.array([[-3.9702 - 5.0406j, -4.1108 + 3.7002j,
|
||
|
|
-0.3403 + 1.0098j, 1.2899 - 0.8590j],
|
||
|
|
[0.3397 - 1.5006j, 1.5201 - 0.4301j,
|
||
|
|
1.8797 - 5.3804j, 3.3606 + 0.6498j],
|
||
|
|
[3.3101 - 3.8506j, 2.4996 + 3.4504j,
|
||
|
|
0.8802 - 1.0802j, 0.6401 - 1.4800j],
|
||
|
|
[-1.0999 + 0.8199j, 1.8103 - 1.5905j,
|
||
|
|
3.2502 + 1.3297j, 1.5701 - 3.4397j]]),
|
||
|
|
np.array([1, 0, 0, 1]),
|
||
|
|
1.02e+00, 1.82e-01)])
|
||
|
|
def test_trsen_NAG(t, q, select, expect, expect_s, expect_sep):
|
||
|
|
"""
|
||
|
|
This test implements the example found in the NAG manual,
|
||
|
|
f08qgc, f08quc.
|
||
|
|
"""
|
||
|
|
# NAG manual provides accuracy up to 4 and 2 decimals
|
||
|
|
atol = 1e-4
|
||
|
|
atol2 = 1e-2
|
||
|
|
trsen, trsen_lwork = get_lapack_funcs(
|
||
|
|
('trsen', 'trsen_lwork'), dtype=t.dtype)
|
||
|
|
|
||
|
|
lwork = _compute_lwork(trsen_lwork, select, t)
|
||
|
|
|
||
|
|
if t.dtype in COMPLEX_DTYPES:
|
||
|
|
result = trsen(select, t, q, lwork=lwork)
|
||
|
|
else:
|
||
|
|
result = trsen(select, t, q, lwork=lwork, liwork=lwork[1])
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
t = result[0]
|
||
|
|
q = result[1]
|
||
|
|
if t.dtype in COMPLEX_DTYPES:
|
||
|
|
s = result[4]
|
||
|
|
sep = result[5]
|
||
|
|
else:
|
||
|
|
s = result[5]
|
||
|
|
sep = result[6]
|
||
|
|
|
||
|
|
assert_allclose(expect, q @ t @ q.conj().T, atol=atol)
|
||
|
|
assert_allclose(expect_s, 1 / s, atol=atol2)
|
||
|
|
assert_allclose(expect_sep, 1 / sep, atol=atol2)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype', DTYPES)
|
||
|
|
def test_gges_tgsen(dtype):
|
||
|
|
if dtype == np.float32 and sys.platform == 'darwin':
|
||
|
|
pytest.xfail("gges[float32] broken for OpenBLAS on macOS, see gh-16949")
|
||
|
|
|
||
|
|
seed(1234)
|
||
|
|
atol = np.finfo(dtype).eps*100
|
||
|
|
|
||
|
|
n = 10
|
||
|
|
a = generate_random_dtype_array([n, n], dtype=dtype)
|
||
|
|
b = generate_random_dtype_array([n, n], dtype=dtype)
|
||
|
|
|
||
|
|
gges, tgsen, tgsen_lwork = get_lapack_funcs(
|
||
|
|
('gges', 'tgsen', 'tgsen_lwork'), dtype=dtype)
|
||
|
|
|
||
|
|
result = gges(lambda x: None, a, b, overwrite_a=False, overwrite_b=False)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
s = result[0]
|
||
|
|
t = result[1]
|
||
|
|
q = result[-4]
|
||
|
|
z = result[-3]
|
||
|
|
|
||
|
|
d1 = s[0, 0] / t[0, 0]
|
||
|
|
d2 = s[6, 6] / t[6, 6]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(s, np.triu(s), rtol=0, atol=atol)
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
select = np.zeros(n)
|
||
|
|
select[6] = 1
|
||
|
|
|
||
|
|
lwork = _compute_lwork(tgsen_lwork, select, s, t)
|
||
|
|
|
||
|
|
# off-by-one error in LAPACK, see gh-issue #13397
|
||
|
|
lwork = (lwork[0]+1, lwork[1])
|
||
|
|
|
||
|
|
result = tgsen(select, s, t, q, z, lwork=lwork)
|
||
|
|
assert_equal(result[-1], 0)
|
||
|
|
|
||
|
|
s = result[0]
|
||
|
|
t = result[1]
|
||
|
|
q = result[-7]
|
||
|
|
z = result[-6]
|
||
|
|
|
||
|
|
if dtype in COMPLEX_DTYPES:
|
||
|
|
assert_allclose(s, np.triu(s), rtol=0, atol=atol)
|
||
|
|
assert_allclose(t, np.triu(t), rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(q @ s @ z.conj().T, a, rtol=0, atol=atol)
|
||
|
|
assert_allclose(q @ t @ z.conj().T, b, rtol=0, atol=atol)
|
||
|
|
|
||
|
|
assert_allclose(s[0, 0] / t[0, 0], d2, rtol=0, atol=atol)
|
||
|
|
assert_allclose(s[1, 1] / t[1, 1], d1, rtol=0, atol=atol)
|