2795 lines
104 KiB
Python
2795 lines
104 KiB
Python
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import itertools
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import platform
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import sys
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import numpy as np
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from numpy.testing import (assert_equal, assert_almost_equal,
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assert_array_almost_equal, assert_array_equal,
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assert_, assert_allclose)
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import pytest
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from pytest import raises as assert_raises
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from scipy.linalg import (eig, eigvals, lu, svd, svdvals, cholesky, qr,
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schur, rsf2csf, lu_solve, lu_factor, solve, diagsvd,
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hessenberg, rq, eig_banded, eigvals_banded, eigh,
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eigvalsh, qr_multiply, qz, orth, ordqz,
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subspace_angles, hadamard, eigvalsh_tridiagonal,
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eigh_tridiagonal, null_space, cdf2rdf, LinAlgError)
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from scipy.linalg.lapack import (dgbtrf, dgbtrs, zgbtrf, zgbtrs, dsbev,
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dsbevd, dsbevx, zhbevd, zhbevx)
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from scipy.linalg._misc import norm
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from scipy.linalg._decomp_qz import _select_function
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from scipy.stats import ortho_group
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from numpy import (array, diag, full, linalg, argsort, zeros, arange,
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float32, complex64, ravel, sqrt, iscomplex, shape, sort,
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sign, asarray, isfinite, ndarray, eye,)
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from scipy.linalg._testutils import assert_no_overwrite
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from scipy.sparse._sputils import matrix
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from scipy._lib._testutils import check_free_memory
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from scipy.linalg.blas import HAS_ILP64
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try:
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from scipy.__config__ import CONFIG
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except ImportError:
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CONFIG = None
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def _random_hermitian_matrix(n, posdef=False, dtype=float):
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"Generate random sym/hermitian array of the given size n"
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if dtype in COMPLEX_DTYPES:
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A = np.random.rand(n, n) + np.random.rand(n, n)*1.0j
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A = (A + A.conj().T)/2
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else:
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A = np.random.rand(n, n)
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A = (A + A.T)/2
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if posdef:
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A += sqrt(2*n)*np.eye(n)
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return A.astype(dtype)
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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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# XXX: This function should not be defined here, but somewhere in
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# scipy.linalg namespace
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def symrand(dim_or_eigv, rng):
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"""Return a random symmetric (Hermitian) matrix.
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If 'dim_or_eigv' is an integer N, return a NxN matrix, with eigenvalues
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uniformly distributed on (-1,1).
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If 'dim_or_eigv' is 1-D real array 'a', return a matrix whose
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eigenvalues are 'a'.
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"""
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if isinstance(dim_or_eigv, int):
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dim = dim_or_eigv
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d = rng.random(dim)*2 - 1
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elif (isinstance(dim_or_eigv, ndarray) and
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len(dim_or_eigv.shape) == 1):
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dim = dim_or_eigv.shape[0]
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d = dim_or_eigv
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else:
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raise TypeError("input type not supported.")
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v = ortho_group.rvs(dim)
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h = v.T.conj() @ diag(d) @ v
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# to avoid roundoff errors, symmetrize the matrix (again)
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h = 0.5*(h.T+h)
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return h
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class TestEigVals:
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def test_simple(self):
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a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
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w = eigvals(a)
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exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
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assert_array_almost_equal(w, exact_w)
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def test_simple_tr(self):
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a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6]], 'd').T
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a = a.copy()
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a = a.T
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w = eigvals(a)
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exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
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assert_array_almost_equal(w, exact_w)
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def test_simple_complex(self):
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a = [[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]]
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w = eigvals(a)
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exact_w = [(9+1j+sqrt(92+6j))/2,
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0,
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(9+1j-sqrt(92+6j))/2]
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assert_array_almost_equal(w, exact_w)
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def test_finite(self):
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a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
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w = eigvals(a, check_finite=False)
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exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
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assert_array_almost_equal(w, exact_w)
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class TestEig:
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def test_simple(self):
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a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6]])
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w, v = eig(a)
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exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
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v0 = array([1, 1, (1+sqrt(93)/3)/2])
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v1 = array([3., 0, -1])
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v2 = array([1, 1, (1-sqrt(93)/3)/2])
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v0 = v0 / norm(v0)
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v1 = v1 / norm(v1)
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v2 = v2 / norm(v2)
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assert_array_almost_equal(w, exact_w)
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assert_array_almost_equal(v0, v[:, 0]*sign(v[0, 0]))
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assert_array_almost_equal(v1, v[:, 1]*sign(v[0, 1]))
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assert_array_almost_equal(v2, v[:, 2]*sign(v[0, 2]))
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for i in range(3):
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assert_array_almost_equal(a @ v[:, i], w[i]*v[:, i])
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w, v = eig(a, left=1, right=0)
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for i in range(3):
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assert_array_almost_equal(a.T @ v[:, i], w[i]*v[:, i])
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def test_simple_complex_eig(self):
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a = array([[1, 2], [-2, 1]])
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w, vl, vr = eig(a, left=1, right=1)
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assert_array_almost_equal(w, array([1+2j, 1-2j]))
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for i in range(2):
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assert_array_almost_equal(a @ vr[:, i], w[i]*vr[:, i])
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for i in range(2):
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assert_array_almost_equal(a.conj().T @ vl[:, i],
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w[i].conj()*vl[:, i])
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def test_simple_complex(self):
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a = array([[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]])
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w, vl, vr = eig(a, left=1, right=1)
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for i in range(3):
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assert_array_almost_equal(a @ vr[:, i], w[i]*vr[:, i])
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for i in range(3):
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assert_array_almost_equal(a.conj().T @ vl[:, i],
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w[i].conj()*vl[:, i])
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def test_gh_3054(self):
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a = [[1]]
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b = [[0]]
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w, vr = eig(a, b, homogeneous_eigvals=True)
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assert_allclose(w[1, 0], 0)
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assert_(w[0, 0] != 0)
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assert_allclose(vr, 1)
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w, vr = eig(a, b)
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assert_equal(w, np.inf)
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assert_allclose(vr, 1)
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def _check_gen_eig(self, A, B, atol_homog=1e-13, rtol_homog=1e-13):
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if B is not None:
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A, B = asarray(A), asarray(B)
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B0 = B
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else:
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A = asarray(A)
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B0 = B
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B = np.eye(*A.shape)
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msg = f"\n{A!r}\n{B!r}"
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# Eigenvalues in homogeneous coordinates
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w, vr = eig(A, B0, homogeneous_eigvals=True)
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wt = eigvals(A, B0, homogeneous_eigvals=True)
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val1 = A @ vr * w[1, :]
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val2 = B @ vr * w[0, :]
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for i in range(val1.shape[1]):
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assert_allclose(val1[:, i], val2[:, i],
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rtol=rtol_homog, atol=atol_homog, err_msg=msg)
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if B0 is None:
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assert_allclose(w[1, :], 1)
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assert_allclose(wt[1, :], 1)
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perm = np.lexsort(w)
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permt = np.lexsort(wt)
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assert_allclose(w[:, perm], wt[:, permt], atol=1e-7, rtol=1e-7,
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err_msg=msg)
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length = np.empty(len(vr))
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for i in range(len(vr)):
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length[i] = norm(vr[:, i])
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assert_allclose(length, np.ones(length.size), err_msg=msg,
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atol=1e-7, rtol=1e-7)
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# Convert homogeneous coordinates
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beta_nonzero = (w[1, :] != 0)
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wh = w[0, beta_nonzero] / w[1, beta_nonzero]
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# Eigenvalues in standard coordinates
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w, vr = eig(A, B0)
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wt = eigvals(A, B0)
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val1 = A @ vr
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val2 = B @ vr * w
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res = val1 - val2
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for i in range(res.shape[1]):
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if np.all(isfinite(res[:, i])):
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assert_allclose(res[:, i], 0,
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rtol=1e-13, atol=1e-13, err_msg=msg)
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# try to consistently order eigenvalues, including complex conjugate pairs
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w_fin = w[isfinite(w)]
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wt_fin = wt[isfinite(wt)]
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# prune noise in the real parts
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w_fin = -1j * np.real_if_close(1j*w_fin, tol=1e-10)
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wt_fin = -1j * np.real_if_close(1j*wt_fin, tol=1e-10)
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perm = argsort(w_fin)
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permt = argsort(wt_fin)
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assert_allclose(w_fin[perm], wt_fin[permt],
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atol=1e-7, rtol=1e-7, err_msg=msg)
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length = np.empty(len(vr))
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for i in range(len(vr)):
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length[i] = norm(vr[:, i])
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assert_allclose(length, np.ones(length.size), err_msg=msg)
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# Compare homogeneous and nonhomogeneous versions
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assert_allclose(sort(wh), sort(w[np.isfinite(w)]))
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def test_singular(self):
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# Example taken from
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# https://web.archive.org/web/20040903121217/http://www.cs.umu.se/research/nla/singular_pairs/guptri/matlab.html
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A = array([[22, 34, 31, 31, 17],
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[45, 45, 42, 19, 29],
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[39, 47, 49, 26, 34],
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[27, 31, 26, 21, 15],
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[38, 44, 44, 24, 30]])
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B = array([[13, 26, 25, 17, 24],
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[31, 46, 40, 26, 37],
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[26, 40, 19, 25, 25],
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[16, 25, 27, 14, 23],
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[24, 35, 18, 21, 22]])
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with np.errstate(all='ignore'):
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self._check_gen_eig(A, B, atol_homog=5e-13)
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def test_falker(self):
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# Test matrices giving some Nan generalized eigenvalues.
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M = diag(array([1, 0, 3]))
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K = array(([2, -1, -1], [-1, 2, -1], [-1, -1, 2]))
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D = array(([1, -1, 0], [-1, 1, 0], [0, 0, 0]))
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Z = zeros((3, 3))
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I3 = eye(3)
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A = np.block([[I3, Z], [Z, -K]])
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B = np.block([[Z, I3], [M, D]])
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with np.errstate(all='ignore'):
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self._check_gen_eig(A, B)
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def test_bad_geneig(self):
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# Ticket #709 (strange return values from DGGEV)
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def matrices(omega):
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c1 = -9 + omega**2
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c2 = 2*omega
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A = [[1, 0, 0, 0],
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[0, 1, 0, 0],
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[0, 0, c1, 0],
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[0, 0, 0, c1]]
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B = [[0, 0, 1, 0],
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[0, 0, 0, 1],
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[1, 0, 0, -c2],
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[0, 1, c2, 0]]
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return A, B
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# With a buggy LAPACK, this can fail for different omega on different
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# machines -- so we need to test several values
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with np.errstate(all='ignore'):
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for k in range(100):
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A, B = matrices(omega=k*5./100)
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self._check_gen_eig(A, B)
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def test_make_eigvals(self):
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# Step through all paths in _make_eigvals
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# Real eigenvalues
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rng = np.random.RandomState(1234)
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A = symrand(3, rng)
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self._check_gen_eig(A, None)
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B = symrand(3, rng)
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self._check_gen_eig(A, B)
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# Complex eigenvalues
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A = rng.random((3, 3)) + 1j*rng.random((3, 3))
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self._check_gen_eig(A, None)
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B = rng.random((3, 3)) + 1j*rng.random((3, 3))
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self._check_gen_eig(A, B)
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def test_check_finite(self):
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a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
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w, v = eig(a, check_finite=False)
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exact_w = [(9+sqrt(93))/2, 0, (9-sqrt(93))/2]
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v0 = array([1, 1, (1+sqrt(93)/3)/2])
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v1 = array([3., 0, -1])
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v2 = array([1, 1, (1-sqrt(93)/3)/2])
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v0 = v0 / norm(v0)
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v1 = v1 / norm(v1)
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v2 = v2 / norm(v2)
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assert_array_almost_equal(w, exact_w)
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assert_array_almost_equal(v0, v[:, 0]*sign(v[0, 0]))
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assert_array_almost_equal(v1, v[:, 1]*sign(v[0, 1]))
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|
assert_array_almost_equal(v2, v[:, 2]*sign(v[0, 2]))
|
||
|
|
for i in range(3):
|
||
|
|
assert_array_almost_equal(a @ v[:, i], w[i]*v[:, i])
|
||
|
|
|
||
|
|
def test_not_square_error(self):
|
||
|
|
"""Check that passing a non-square array raises a ValueError."""
|
||
|
|
A = np.arange(6).reshape(3, 2)
|
||
|
|
assert_raises(ValueError, eig, A)
|
||
|
|
|
||
|
|
def test_shape_mismatch(self):
|
||
|
|
"""Check that passing arrays of with different shapes
|
||
|
|
raises a ValueError."""
|
||
|
|
A = eye(2)
|
||
|
|
B = np.arange(9.0).reshape(3, 3)
|
||
|
|
assert_raises(ValueError, eig, A, B)
|
||
|
|
assert_raises(ValueError, eig, B, A)
|
||
|
|
|
||
|
|
def test_gh_11577(self):
|
||
|
|
# https://github.com/scipy/scipy/issues/11577
|
||
|
|
# `A - lambda B` should have 4 and 8 among the eigenvalues, and this
|
||
|
|
# was apparently broken on some platforms
|
||
|
|
A = np.array([[12.0, 28.0, 76.0, 220.0],
|
||
|
|
[16.0, 32.0, 80.0, 224.0],
|
||
|
|
[24.0, 40.0, 88.0, 232.0],
|
||
|
|
[40.0, 56.0, 104.0, 248.0]], dtype='float64')
|
||
|
|
B = np.array([[2.0, 4.0, 10.0, 28.0],
|
||
|
|
[3.0, 5.0, 11.0, 29.0],
|
||
|
|
[5.0, 7.0, 13.0, 31.0],
|
||
|
|
[9.0, 11.0, 17.0, 35.0]], dtype='float64')
|
||
|
|
|
||
|
|
D, V = eig(A, B)
|
||
|
|
|
||
|
|
# The problem is ill-conditioned, and two other eigenvalues
|
||
|
|
# depend on ATLAS/OpenBLAS version, compiler version etc
|
||
|
|
# see gh-11577 for discussion
|
||
|
|
#
|
||
|
|
# NB: it is tempting to use `assert_allclose(D[:2], [4, 8])` instead but
|
||
|
|
# the ordering of eigenvalues also comes out different on different
|
||
|
|
# systems depending on who knows what.
|
||
|
|
with np.testing.suppress_warnings() as sup:
|
||
|
|
# isclose chokes on inf/nan values
|
||
|
|
sup.filter(RuntimeWarning, "invalid value encountered in multiply")
|
||
|
|
assert np.isclose(D, 4.0, atol=1e-14).any()
|
||
|
|
assert np.isclose(D, 8.0, atol=1e-14).any()
|
||
|
|
|
||
|
|
|
||
|
|
class TestEigBanded:
|
||
|
|
def setup_method(self):
|
||
|
|
self.create_bandmat()
|
||
|
|
|
||
|
|
def create_bandmat(self):
|
||
|
|
"""Create the full matrix `self.fullmat` and
|
||
|
|
the corresponding band matrix `self.bandmat`."""
|
||
|
|
N = 10
|
||
|
|
self.KL = 2 # number of subdiagonals (below the diagonal)
|
||
|
|
self.KU = 2 # number of superdiagonals (above the diagonal)
|
||
|
|
|
||
|
|
# symmetric band matrix
|
||
|
|
self.sym_mat = (diag(full(N, 1.0))
|
||
|
|
+ diag(full(N-1, -1.0), -1) + diag(full(N-1, -1.0), 1)
|
||
|
|
+ diag(full(N-2, -2.0), -2) + diag(full(N-2, -2.0), 2))
|
||
|
|
|
||
|
|
# hermitian band matrix
|
||
|
|
self.herm_mat = (diag(full(N, -1.0))
|
||
|
|
+ 1j*diag(full(N-1, 1.0), -1)
|
||
|
|
- 1j*diag(full(N-1, 1.0), 1)
|
||
|
|
+ diag(full(N-2, -2.0), -2)
|
||
|
|
+ diag(full(N-2, -2.0), 2))
|
||
|
|
|
||
|
|
# general real band matrix
|
||
|
|
self.real_mat = (diag(full(N, 1.0))
|
||
|
|
+ diag(full(N-1, -1.0), -1) + diag(full(N-1, -3.0), 1)
|
||
|
|
+ diag(full(N-2, 2.0), -2) + diag(full(N-2, -2.0), 2))
|
||
|
|
|
||
|
|
# general complex band matrix
|
||
|
|
self.comp_mat = (1j*diag(full(N, 1.0))
|
||
|
|
+ diag(full(N-1, -1.0), -1)
|
||
|
|
+ 1j*diag(full(N-1, -3.0), 1)
|
||
|
|
+ diag(full(N-2, 2.0), -2)
|
||
|
|
+ diag(full(N-2, -2.0), 2))
|
||
|
|
|
||
|
|
# Eigenvalues and -vectors from linalg.eig
|
||
|
|
ew, ev = linalg.eig(self.sym_mat)
|
||
|
|
ew = ew.real
|
||
|
|
args = argsort(ew)
|
||
|
|
self.w_sym_lin = ew[args]
|
||
|
|
self.evec_sym_lin = ev[:, args]
|
||
|
|
|
||
|
|
ew, ev = linalg.eig(self.herm_mat)
|
||
|
|
ew = ew.real
|
||
|
|
args = argsort(ew)
|
||
|
|
self.w_herm_lin = ew[args]
|
||
|
|
self.evec_herm_lin = ev[:, args]
|
||
|
|
|
||
|
|
# Extract upper bands from symmetric and hermitian band matrices
|
||
|
|
# (for use in dsbevd, dsbevx, zhbevd, zhbevx
|
||
|
|
# and their single precision versions)
|
||
|
|
LDAB = self.KU + 1
|
||
|
|
self.bandmat_sym = zeros((LDAB, N), dtype=float)
|
||
|
|
self.bandmat_herm = zeros((LDAB, N), dtype=complex)
|
||
|
|
for i in range(LDAB):
|
||
|
|
self.bandmat_sym[LDAB-i-1, i:N] = diag(self.sym_mat, i)
|
||
|
|
self.bandmat_herm[LDAB-i-1, i:N] = diag(self.herm_mat, i)
|
||
|
|
|
||
|
|
# Extract bands from general real and complex band matrix
|
||
|
|
# (for use in dgbtrf, dgbtrs and their single precision versions)
|
||
|
|
LDAB = 2*self.KL + self.KU + 1
|
||
|
|
self.bandmat_real = zeros((LDAB, N), dtype=float)
|
||
|
|
self.bandmat_real[2*self.KL, :] = diag(self.real_mat) # diagonal
|
||
|
|
for i in range(self.KL):
|
||
|
|
# superdiagonals
|
||
|
|
self.bandmat_real[2*self.KL-1-i, i+1:N] = diag(self.real_mat, i+1)
|
||
|
|
# subdiagonals
|
||
|
|
self.bandmat_real[2*self.KL+1+i, 0:N-1-i] = diag(self.real_mat,
|
||
|
|
-i-1)
|
||
|
|
|
||
|
|
self.bandmat_comp = zeros((LDAB, N), dtype=complex)
|
||
|
|
self.bandmat_comp[2*self.KL, :] = diag(self.comp_mat) # diagonal
|
||
|
|
for i in range(self.KL):
|
||
|
|
# superdiagonals
|
||
|
|
self.bandmat_comp[2*self.KL-1-i, i+1:N] = diag(self.comp_mat, i+1)
|
||
|
|
# subdiagonals
|
||
|
|
self.bandmat_comp[2*self.KL+1+i, 0:N-1-i] = diag(self.comp_mat,
|
||
|
|
-i-1)
|
||
|
|
|
||
|
|
# absolute value for linear equation system A*x = b
|
||
|
|
self.b = 1.0*arange(N)
|
||
|
|
self.bc = self.b * (1 + 1j)
|
||
|
|
|
||
|
|
#####################################################################
|
||
|
|
|
||
|
|
def test_dsbev(self):
|
||
|
|
"""Compare dsbev eigenvalues and eigenvectors with
|
||
|
|
the result of linalg.eig."""
|
||
|
|
w, evec, info = dsbev(self.bandmat_sym, compute_v=1)
|
||
|
|
evec_ = evec[:, argsort(w)]
|
||
|
|
assert_array_almost_equal(sort(w), self.w_sym_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin))
|
||
|
|
|
||
|
|
def test_dsbevd(self):
|
||
|
|
"""Compare dsbevd eigenvalues and eigenvectors with
|
||
|
|
the result of linalg.eig."""
|
||
|
|
w, evec, info = dsbevd(self.bandmat_sym, compute_v=1)
|
||
|
|
evec_ = evec[:, argsort(w)]
|
||
|
|
assert_array_almost_equal(sort(w), self.w_sym_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin))
|
||
|
|
|
||
|
|
def test_dsbevx(self):
|
||
|
|
"""Compare dsbevx eigenvalues and eigenvectors
|
||
|
|
with the result of linalg.eig."""
|
||
|
|
N, N = shape(self.sym_mat)
|
||
|
|
# Achtung: Argumente 0.0,0.0,range?
|
||
|
|
w, evec, num, ifail, info = dsbevx(self.bandmat_sym, 0.0, 0.0, 1, N,
|
||
|
|
compute_v=1, range=2)
|
||
|
|
evec_ = evec[:, argsort(w)]
|
||
|
|
assert_array_almost_equal(sort(w), self.w_sym_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_), abs(self.evec_sym_lin))
|
||
|
|
|
||
|
|
def test_zhbevd(self):
|
||
|
|
"""Compare zhbevd eigenvalues and eigenvectors
|
||
|
|
with the result of linalg.eig."""
|
||
|
|
w, evec, info = zhbevd(self.bandmat_herm, compute_v=1)
|
||
|
|
evec_ = evec[:, argsort(w)]
|
||
|
|
assert_array_almost_equal(sort(w), self.w_herm_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_), abs(self.evec_herm_lin))
|
||
|
|
|
||
|
|
def test_zhbevx(self):
|
||
|
|
"""Compare zhbevx eigenvalues and eigenvectors
|
||
|
|
with the result of linalg.eig."""
|
||
|
|
N, N = shape(self.herm_mat)
|
||
|
|
# Achtung: Argumente 0.0,0.0,range?
|
||
|
|
w, evec, num, ifail, info = zhbevx(self.bandmat_herm, 0.0, 0.0, 1, N,
|
||
|
|
compute_v=1, range=2)
|
||
|
|
evec_ = evec[:, argsort(w)]
|
||
|
|
assert_array_almost_equal(sort(w), self.w_herm_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_), abs(self.evec_herm_lin))
|
||
|
|
|
||
|
|
def test_eigvals_banded(self):
|
||
|
|
"""Compare eigenvalues of eigvals_banded with those of linalg.eig."""
|
||
|
|
w_sym = eigvals_banded(self.bandmat_sym)
|
||
|
|
w_sym = w_sym.real
|
||
|
|
assert_array_almost_equal(sort(w_sym), self.w_sym_lin)
|
||
|
|
|
||
|
|
w_herm = eigvals_banded(self.bandmat_herm)
|
||
|
|
w_herm = w_herm.real
|
||
|
|
assert_array_almost_equal(sort(w_herm), self.w_herm_lin)
|
||
|
|
|
||
|
|
# extracting eigenvalues with respect to an index range
|
||
|
|
ind1 = 2
|
||
|
|
ind2 = np.longlong(6)
|
||
|
|
w_sym_ind = eigvals_banded(self.bandmat_sym,
|
||
|
|
select='i', select_range=(ind1, ind2))
|
||
|
|
assert_array_almost_equal(sort(w_sym_ind),
|
||
|
|
self.w_sym_lin[ind1:ind2+1])
|
||
|
|
w_herm_ind = eigvals_banded(self.bandmat_herm,
|
||
|
|
select='i', select_range=(ind1, ind2))
|
||
|
|
assert_array_almost_equal(sort(w_herm_ind),
|
||
|
|
self.w_herm_lin[ind1:ind2+1])
|
||
|
|
|
||
|
|
# extracting eigenvalues with respect to a value range
|
||
|
|
v_lower = self.w_sym_lin[ind1] - 1.0e-5
|
||
|
|
v_upper = self.w_sym_lin[ind2] + 1.0e-5
|
||
|
|
w_sym_val = eigvals_banded(self.bandmat_sym,
|
||
|
|
select='v', select_range=(v_lower, v_upper))
|
||
|
|
assert_array_almost_equal(sort(w_sym_val),
|
||
|
|
self.w_sym_lin[ind1:ind2+1])
|
||
|
|
|
||
|
|
v_lower = self.w_herm_lin[ind1] - 1.0e-5
|
||
|
|
v_upper = self.w_herm_lin[ind2] + 1.0e-5
|
||
|
|
w_herm_val = eigvals_banded(self.bandmat_herm,
|
||
|
|
select='v',
|
||
|
|
select_range=(v_lower, v_upper))
|
||
|
|
assert_array_almost_equal(sort(w_herm_val),
|
||
|
|
self.w_herm_lin[ind1:ind2+1])
|
||
|
|
|
||
|
|
w_sym = eigvals_banded(self.bandmat_sym, check_finite=False)
|
||
|
|
w_sym = w_sym.real
|
||
|
|
assert_array_almost_equal(sort(w_sym), self.w_sym_lin)
|
||
|
|
|
||
|
|
def test_eig_banded(self):
|
||
|
|
"""Compare eigenvalues and eigenvectors of eig_banded
|
||
|
|
with those of linalg.eig. """
|
||
|
|
w_sym, evec_sym = eig_banded(self.bandmat_sym)
|
||
|
|
evec_sym_ = evec_sym[:, argsort(w_sym.real)]
|
||
|
|
assert_array_almost_equal(sort(w_sym), self.w_sym_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_sym_), abs(self.evec_sym_lin))
|
||
|
|
|
||
|
|
w_herm, evec_herm = eig_banded(self.bandmat_herm)
|
||
|
|
evec_herm_ = evec_herm[:, argsort(w_herm.real)]
|
||
|
|
assert_array_almost_equal(sort(w_herm), self.w_herm_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_herm_), abs(self.evec_herm_lin))
|
||
|
|
|
||
|
|
# extracting eigenvalues with respect to an index range
|
||
|
|
ind1 = 2
|
||
|
|
ind2 = 6
|
||
|
|
w_sym_ind, evec_sym_ind = eig_banded(self.bandmat_sym,
|
||
|
|
select='i',
|
||
|
|
select_range=(ind1, ind2))
|
||
|
|
assert_array_almost_equal(sort(w_sym_ind),
|
||
|
|
self.w_sym_lin[ind1:ind2+1])
|
||
|
|
assert_array_almost_equal(abs(evec_sym_ind),
|
||
|
|
abs(self.evec_sym_lin[:, ind1:ind2+1]))
|
||
|
|
|
||
|
|
w_herm_ind, evec_herm_ind = eig_banded(self.bandmat_herm,
|
||
|
|
select='i',
|
||
|
|
select_range=(ind1, ind2))
|
||
|
|
assert_array_almost_equal(sort(w_herm_ind),
|
||
|
|
self.w_herm_lin[ind1:ind2+1])
|
||
|
|
assert_array_almost_equal(abs(evec_herm_ind),
|
||
|
|
abs(self.evec_herm_lin[:, ind1:ind2+1]))
|
||
|
|
|
||
|
|
# extracting eigenvalues with respect to a value range
|
||
|
|
v_lower = self.w_sym_lin[ind1] - 1.0e-5
|
||
|
|
v_upper = self.w_sym_lin[ind2] + 1.0e-5
|
||
|
|
w_sym_val, evec_sym_val = eig_banded(self.bandmat_sym,
|
||
|
|
select='v',
|
||
|
|
select_range=(v_lower, v_upper))
|
||
|
|
assert_array_almost_equal(sort(w_sym_val),
|
||
|
|
self.w_sym_lin[ind1:ind2+1])
|
||
|
|
assert_array_almost_equal(abs(evec_sym_val),
|
||
|
|
abs(self.evec_sym_lin[:, ind1:ind2+1]))
|
||
|
|
|
||
|
|
v_lower = self.w_herm_lin[ind1] - 1.0e-5
|
||
|
|
v_upper = self.w_herm_lin[ind2] + 1.0e-5
|
||
|
|
w_herm_val, evec_herm_val = eig_banded(self.bandmat_herm,
|
||
|
|
select='v',
|
||
|
|
select_range=(v_lower, v_upper))
|
||
|
|
assert_array_almost_equal(sort(w_herm_val),
|
||
|
|
self.w_herm_lin[ind1:ind2+1])
|
||
|
|
assert_array_almost_equal(abs(evec_herm_val),
|
||
|
|
abs(self.evec_herm_lin[:, ind1:ind2+1]))
|
||
|
|
|
||
|
|
w_sym, evec_sym = eig_banded(self.bandmat_sym, check_finite=False)
|
||
|
|
evec_sym_ = evec_sym[:, argsort(w_sym.real)]
|
||
|
|
assert_array_almost_equal(sort(w_sym), self.w_sym_lin)
|
||
|
|
assert_array_almost_equal(abs(evec_sym_), abs(self.evec_sym_lin))
|
||
|
|
|
||
|
|
def test_dgbtrf(self):
|
||
|
|
"""Compare dgbtrf LU factorisation with the LU factorisation result
|
||
|
|
of linalg.lu."""
|
||
|
|
M, N = shape(self.real_mat)
|
||
|
|
lu_symm_band, ipiv, info = dgbtrf(self.bandmat_real, self.KL, self.KU)
|
||
|
|
|
||
|
|
# extract matrix u from lu_symm_band
|
||
|
|
u = diag(lu_symm_band[2*self.KL, :])
|
||
|
|
for i in range(self.KL + self.KU):
|
||
|
|
u += diag(lu_symm_band[2*self.KL-1-i, i+1:N], i+1)
|
||
|
|
|
||
|
|
p_lin, l_lin, u_lin = lu(self.real_mat, permute_l=0)
|
||
|
|
assert_array_almost_equal(u, u_lin)
|
||
|
|
|
||
|
|
def test_zgbtrf(self):
|
||
|
|
"""Compare zgbtrf LU factorisation with the LU factorisation result
|
||
|
|
of linalg.lu."""
|
||
|
|
M, N = shape(self.comp_mat)
|
||
|
|
lu_symm_band, ipiv, info = zgbtrf(self.bandmat_comp, self.KL, self.KU)
|
||
|
|
|
||
|
|
# extract matrix u from lu_symm_band
|
||
|
|
u = diag(lu_symm_band[2*self.KL, :])
|
||
|
|
for i in range(self.KL + self.KU):
|
||
|
|
u += diag(lu_symm_band[2*self.KL-1-i, i+1:N], i+1)
|
||
|
|
|
||
|
|
p_lin, l_lin, u_lin = lu(self.comp_mat, permute_l=0)
|
||
|
|
assert_array_almost_equal(u, u_lin)
|
||
|
|
|
||
|
|
def test_dgbtrs(self):
|
||
|
|
"""Compare dgbtrs solutions for linear equation system A*x = b
|
||
|
|
with solutions of linalg.solve."""
|
||
|
|
|
||
|
|
lu_symm_band, ipiv, info = dgbtrf(self.bandmat_real, self.KL, self.KU)
|
||
|
|
y, info = dgbtrs(lu_symm_band, self.KL, self.KU, self.b, ipiv)
|
||
|
|
|
||
|
|
y_lin = linalg.solve(self.real_mat, self.b)
|
||
|
|
assert_array_almost_equal(y, y_lin)
|
||
|
|
|
||
|
|
def test_zgbtrs(self):
|
||
|
|
"""Compare zgbtrs solutions for linear equation system A*x = b
|
||
|
|
with solutions of linalg.solve."""
|
||
|
|
|
||
|
|
lu_symm_band, ipiv, info = zgbtrf(self.bandmat_comp, self.KL, self.KU)
|
||
|
|
y, info = zgbtrs(lu_symm_band, self.KL, self.KU, self.bc, ipiv)
|
||
|
|
|
||
|
|
y_lin = linalg.solve(self.comp_mat, self.bc)
|
||
|
|
assert_array_almost_equal(y, y_lin)
|
||
|
|
|
||
|
|
|
||
|
|
class TestEigTridiagonal:
|
||
|
|
def setup_method(self):
|
||
|
|
self.create_trimat()
|
||
|
|
|
||
|
|
def create_trimat(self):
|
||
|
|
"""Create the full matrix `self.fullmat`, `self.d`, and `self.e`."""
|
||
|
|
N = 10
|
||
|
|
|
||
|
|
# symmetric band matrix
|
||
|
|
self.d = full(N, 1.0)
|
||
|
|
self.e = full(N-1, -1.0)
|
||
|
|
self.full_mat = (diag(self.d) + diag(self.e, -1) + diag(self.e, 1))
|
||
|
|
|
||
|
|
ew, ev = linalg.eig(self.full_mat)
|
||
|
|
ew = ew.real
|
||
|
|
args = argsort(ew)
|
||
|
|
self.w = ew[args]
|
||
|
|
self.evec = ev[:, args]
|
||
|
|
|
||
|
|
def test_degenerate(self):
|
||
|
|
"""Test error conditions."""
|
||
|
|
# Wrong sizes
|
||
|
|
assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e[:-1])
|
||
|
|
# Must be real
|
||
|
|
assert_raises(TypeError, eigvalsh_tridiagonal, self.d, self.e * 1j)
|
||
|
|
# Bad driver
|
||
|
|
assert_raises(TypeError, eigvalsh_tridiagonal, self.d, self.e,
|
||
|
|
lapack_driver=1.)
|
||
|
|
assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e,
|
||
|
|
lapack_driver='foo')
|
||
|
|
# Bad bounds
|
||
|
|
assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e,
|
||
|
|
select='i', select_range=(0, -1))
|
||
|
|
|
||
|
|
def test_eigvalsh_tridiagonal(self):
|
||
|
|
"""Compare eigenvalues of eigvalsh_tridiagonal with those of eig."""
|
||
|
|
# can't use ?STERF with subselection
|
||
|
|
for driver in ('sterf', 'stev', 'stebz', 'stemr', 'auto'):
|
||
|
|
w = eigvalsh_tridiagonal(self.d, self.e, lapack_driver=driver)
|
||
|
|
assert_array_almost_equal(sort(w), self.w)
|
||
|
|
|
||
|
|
for driver in ('sterf', 'stev'):
|
||
|
|
assert_raises(ValueError, eigvalsh_tridiagonal, self.d, self.e,
|
||
|
|
lapack_driver='stev', select='i',
|
||
|
|
select_range=(0, 1))
|
||
|
|
for driver in ('stebz', 'stemr', 'auto'):
|
||
|
|
# extracting eigenvalues with respect to the full index range
|
||
|
|
w_ind = eigvalsh_tridiagonal(
|
||
|
|
self.d, self.e, select='i', select_range=(0, len(self.d)-1),
|
||
|
|
lapack_driver=driver)
|
||
|
|
assert_array_almost_equal(sort(w_ind), self.w)
|
||
|
|
|
||
|
|
# extracting eigenvalues with respect to an index range
|
||
|
|
ind1 = 2
|
||
|
|
ind2 = 6
|
||
|
|
w_ind = eigvalsh_tridiagonal(
|
||
|
|
self.d, self.e, select='i', select_range=(ind1, ind2),
|
||
|
|
lapack_driver=driver)
|
||
|
|
assert_array_almost_equal(sort(w_ind), self.w[ind1:ind2+1])
|
||
|
|
|
||
|
|
# extracting eigenvalues with respect to a value range
|
||
|
|
v_lower = self.w[ind1] - 1.0e-5
|
||
|
|
v_upper = self.w[ind2] + 1.0e-5
|
||
|
|
w_val = eigvalsh_tridiagonal(
|
||
|
|
self.d, self.e, select='v', select_range=(v_lower, v_upper),
|
||
|
|
lapack_driver=driver)
|
||
|
|
assert_array_almost_equal(sort(w_val), self.w[ind1:ind2+1])
|
||
|
|
|
||
|
|
def test_eigh_tridiagonal(self):
|
||
|
|
"""Compare eigenvalues and eigenvectors of eigh_tridiagonal
|
||
|
|
with those of eig. """
|
||
|
|
# can't use ?STERF when eigenvectors are requested
|
||
|
|
assert_raises(ValueError, eigh_tridiagonal, self.d, self.e,
|
||
|
|
lapack_driver='sterf')
|
||
|
|
for driver in ('stebz', 'stev', 'stemr', 'auto'):
|
||
|
|
w, evec = eigh_tridiagonal(self.d, self.e, lapack_driver=driver)
|
||
|
|
evec_ = evec[:, argsort(w)]
|
||
|
|
assert_array_almost_equal(sort(w), self.w)
|
||
|
|
assert_array_almost_equal(abs(evec_), abs(self.evec))
|
||
|
|
|
||
|
|
assert_raises(ValueError, eigh_tridiagonal, self.d, self.e,
|
||
|
|
lapack_driver='stev', select='i', select_range=(0, 1))
|
||
|
|
for driver in ('stebz', 'stemr', 'auto'):
|
||
|
|
# extracting eigenvalues with respect to an index range
|
||
|
|
ind1 = 0
|
||
|
|
ind2 = len(self.d)-1
|
||
|
|
w, evec = eigh_tridiagonal(
|
||
|
|
self.d, self.e, select='i', select_range=(ind1, ind2),
|
||
|
|
lapack_driver=driver)
|
||
|
|
assert_array_almost_equal(sort(w), self.w)
|
||
|
|
assert_array_almost_equal(abs(evec), abs(self.evec))
|
||
|
|
ind1 = 2
|
||
|
|
ind2 = 6
|
||
|
|
w, evec = eigh_tridiagonal(
|
||
|
|
self.d, self.e, select='i', select_range=(ind1, ind2),
|
||
|
|
lapack_driver=driver)
|
||
|
|
assert_array_almost_equal(sort(w), self.w[ind1:ind2+1])
|
||
|
|
assert_array_almost_equal(abs(evec),
|
||
|
|
abs(self.evec[:, ind1:ind2+1]))
|
||
|
|
|
||
|
|
# extracting eigenvalues with respect to a value range
|
||
|
|
v_lower = self.w[ind1] - 1.0e-5
|
||
|
|
v_upper = self.w[ind2] + 1.0e-5
|
||
|
|
w, evec = eigh_tridiagonal(
|
||
|
|
self.d, self.e, select='v', select_range=(v_lower, v_upper),
|
||
|
|
lapack_driver=driver)
|
||
|
|
assert_array_almost_equal(sort(w), self.w[ind1:ind2+1])
|
||
|
|
assert_array_almost_equal(abs(evec),
|
||
|
|
abs(self.evec[:, ind1:ind2+1]))
|
||
|
|
|
||
|
|
def test_eigh_tridiagonal_1x1(self):
|
||
|
|
"""See gh-20075"""
|
||
|
|
a = np.array([-2.0])
|
||
|
|
b = np.array([])
|
||
|
|
x = eigh_tridiagonal(a, b, eigvals_only=True)
|
||
|
|
assert x.ndim == 1
|
||
|
|
assert_allclose(x, a)
|
||
|
|
x, V = eigh_tridiagonal(a, b, select="i", select_range=(0, 0))
|
||
|
|
assert x.ndim == 1
|
||
|
|
assert V.ndim == 2
|
||
|
|
assert_allclose(x, a)
|
||
|
|
assert_allclose(V, array([[1.]]))
|
||
|
|
|
||
|
|
x, V = eigh_tridiagonal(a, b, select="v", select_range=(-2, 0))
|
||
|
|
assert x.size == 0
|
||
|
|
assert x.shape == (0,)
|
||
|
|
assert V.shape == (1, 0)
|
||
|
|
|
||
|
|
|
||
|
|
class TestEigh:
|
||
|
|
def setup_class(self):
|
||
|
|
np.random.seed(1234)
|
||
|
|
|
||
|
|
def test_wrong_inputs(self):
|
||
|
|
# Nonsquare a
|
||
|
|
assert_raises(ValueError, eigh, np.ones([1, 2]))
|
||
|
|
# Nonsquare b
|
||
|
|
assert_raises(ValueError, eigh, np.ones([2, 2]), np.ones([2, 1]))
|
||
|
|
# Incompatible a, b sizes
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([2, 2]))
|
||
|
|
# Wrong type parameter for generalized problem
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
type=4)
|
||
|
|
# Both value and index subsets requested
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
subset_by_value=[1, 2], subset_by_index=[2, 4])
|
||
|
|
with np.testing.suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning, "Keyword argument 'eigvals")
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
subset_by_value=[1, 2], eigvals=[2, 4])
|
||
|
|
# Invalid upper index spec
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
subset_by_index=[0, 4])
|
||
|
|
with np.testing.suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning, "Keyword argument 'eigvals")
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
eigvals=[0, 4])
|
||
|
|
# Invalid lower index
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
subset_by_index=[-2, 2])
|
||
|
|
with np.testing.suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning, "Keyword argument 'eigvals")
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
eigvals=[-2, 2])
|
||
|
|
# Invalid index spec #2
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
subset_by_index=[2, 0])
|
||
|
|
with np.testing.suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning, "Keyword argument 'eigvals")
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
subset_by_index=[2, 0])
|
||
|
|
# Invalid value spec
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
subset_by_value=[2, 0])
|
||
|
|
# Invalid driver name
|
||
|
|
assert_raises(ValueError, eigh, np.ones([2, 2]), driver='wrong')
|
||
|
|
# Generalized driver selection without b
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), None, driver='gvx')
|
||
|
|
# Standard driver with b
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
driver='evr')
|
||
|
|
# Subset request from invalid driver
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
driver='gvd', subset_by_index=[1, 2])
|
||
|
|
assert_raises(ValueError, eigh, np.ones([3, 3]), np.ones([3, 3]),
|
||
|
|
driver='gvd', subset_by_index=[1, 2])
|
||
|
|
|
||
|
|
def test_nonpositive_b(self):
|
||
|
|
assert_raises(LinAlgError, eigh, np.ones([3, 3]), np.ones([3, 3]))
|
||
|
|
|
||
|
|
# index based subsets are done in the legacy test_eigh()
|
||
|
|
def test_value_subsets(self):
|
||
|
|
for ind, dt in enumerate(DTYPES):
|
||
|
|
|
||
|
|
a = _random_hermitian_matrix(20, dtype=dt)
|
||
|
|
w, v = eigh(a, subset_by_value=[-2, 2])
|
||
|
|
assert_equal(v.shape[1], len(w))
|
||
|
|
assert all((w > -2) & (w < 2))
|
||
|
|
|
||
|
|
b = _random_hermitian_matrix(20, posdef=True, dtype=dt)
|
||
|
|
w, v = eigh(a, b, subset_by_value=[-2, 2])
|
||
|
|
assert_equal(v.shape[1], len(w))
|
||
|
|
assert all((w > -2) & (w < 2))
|
||
|
|
|
||
|
|
def test_eigh_integer(self):
|
||
|
|
a = array([[1, 2], [2, 7]])
|
||
|
|
b = array([[3, 1], [1, 5]])
|
||
|
|
w, z = eigh(a)
|
||
|
|
w, z = eigh(a, b)
|
||
|
|
|
||
|
|
def test_eigh_of_sparse(self):
|
||
|
|
# This tests the rejection of inputs that eigh cannot currently handle.
|
||
|
|
import scipy.sparse
|
||
|
|
a = scipy.sparse.identity(2).tocsc()
|
||
|
|
b = np.atleast_2d(a)
|
||
|
|
assert_raises(ValueError, eigh, a)
|
||
|
|
assert_raises(ValueError, eigh, b)
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('dtype_', DTYPES)
|
||
|
|
@pytest.mark.parametrize('driver', ("ev", "evd", "evr", "evx"))
|
||
|
|
def test_various_drivers_standard(self, driver, dtype_):
|
||
|
|
a = _random_hermitian_matrix(n=20, dtype=dtype_)
|
||
|
|
w, v = eigh(a, driver=driver)
|
||
|
|
assert_allclose(a @ v - (v * w), 0.,
|
||
|
|
atol=1000*np.finfo(dtype_).eps,
|
||
|
|
rtol=0.)
|
||
|
|
|
||
|
|
@pytest.mark.parametrize('type', (1, 2, 3))
|
||
|
|
@pytest.mark.parametrize('driver', ("gv", "gvd", "gvx"))
|
||
|
|
def test_various_drivers_generalized(self, driver, type):
|
||
|
|
atol = np.spacing(5000.)
|
||
|
|
a = _random_hermitian_matrix(20)
|
||
|
|
b = _random_hermitian_matrix(20, posdef=True)
|
||
|
|
w, v = eigh(a=a, b=b, driver=driver, type=type)
|
||
|
|
if type == 1:
|
||
|
|
assert_allclose(a @ v - w*(b @ v), 0., atol=atol, rtol=0.)
|
||
|
|
elif type == 2:
|
||
|
|
assert_allclose(a @ b @ v - v * w, 0., atol=atol, rtol=0.)
|
||
|
|
else:
|
||
|
|
assert_allclose(b @ a @ v - v * w, 0., atol=atol, rtol=0.)
|
||
|
|
|
||
|
|
def test_eigvalsh_new_args(self):
|
||
|
|
a = _random_hermitian_matrix(5)
|
||
|
|
w = eigvalsh(a, subset_by_index=[1, 2])
|
||
|
|
assert_equal(len(w), 2)
|
||
|
|
|
||
|
|
w2 = eigvalsh(a, subset_by_index=[1, 2])
|
||
|
|
assert_equal(len(w2), 2)
|
||
|
|
assert_allclose(w, w2)
|
||
|
|
|
||
|
|
b = np.diag([1, 1.2, 1.3, 1.5, 2])
|
||
|
|
w3 = eigvalsh(b, subset_by_value=[1, 1.4])
|
||
|
|
assert_equal(len(w3), 2)
|
||
|
|
assert_allclose(w3, np.array([1.2, 1.3]))
|
||
|
|
|
||
|
|
@pytest.mark.parametrize("method", [eigh, eigvalsh])
|
||
|
|
def test_deprecation_warnings(self, method):
|
||
|
|
with pytest.warns(DeprecationWarning,
|
||
|
|
match="Keyword argument 'turbo'"):
|
||
|
|
method(np.zeros((2, 2)), turbo=True)
|
||
|
|
with pytest.warns(DeprecationWarning,
|
||
|
|
match="Keyword argument 'eigvals'"):
|
||
|
|
method(np.zeros((2, 2)), eigvals=[0, 1])
|
||
|
|
with pytest.deprecated_call(match="use keyword arguments"):
|
||
|
|
method(np.zeros((2,2)), np.eye(2, 2), True)
|
||
|
|
|
||
|
|
def test_deprecation_results(self):
|
||
|
|
a = _random_hermitian_matrix(3)
|
||
|
|
b = _random_hermitian_matrix(3, posdef=True)
|
||
|
|
|
||
|
|
# check turbo gives same result as driver='gvd'
|
||
|
|
with np.testing.suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning, "Keyword argument 'turbo'")
|
||
|
|
w_dep, v_dep = eigh(a, b, turbo=True)
|
||
|
|
w, v = eigh(a, b, driver='gvd')
|
||
|
|
assert_allclose(w_dep, w)
|
||
|
|
assert_allclose(v_dep, v)
|
||
|
|
|
||
|
|
# check eigvals gives the same result as subset_by_index
|
||
|
|
with np.testing.suppress_warnings() as sup:
|
||
|
|
sup.filter(DeprecationWarning, "Keyword argument 'eigvals'")
|
||
|
|
w_dep, v_dep = eigh(a, eigvals=[0, 1])
|
||
|
|
w, v = eigh(a, subset_by_index=[0, 1])
|
||
|
|
assert_allclose(w_dep, w)
|
||
|
|
assert_allclose(v_dep, v)
|
||
|
|
|
||
|
|
|
||
|
|
class TestSVD_GESDD:
|
||
|
|
lapack_driver = 'gesdd'
|
||
|
|
|
||
|
|
def test_degenerate(self):
|
||
|
|
assert_raises(TypeError, svd, [[1.]], lapack_driver=1.)
|
||
|
|
assert_raises(ValueError, svd, [[1.]], lapack_driver='foo')
|
||
|
|
|
||
|
|
def test_simple(self):
|
||
|
|
a = [[1, 2, 3], [1, 20, 3], [2, 5, 6]]
|
||
|
|
for full_matrices in (True, False):
|
||
|
|
u, s, vh = svd(a, full_matrices=full_matrices,
|
||
|
|
lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.T @ u, eye(3))
|
||
|
|
assert_array_almost_equal(vh.T @ vh, eye(3))
|
||
|
|
sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_simple_singular(self):
|
||
|
|
a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
|
||
|
|
for full_matrices in (True, False):
|
||
|
|
u, s, vh = svd(a, full_matrices=full_matrices,
|
||
|
|
lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.T @ u, eye(3))
|
||
|
|
assert_array_almost_equal(vh.T @ vh, eye(3))
|
||
|
|
sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_simple_underdet(self):
|
||
|
|
a = [[1, 2, 3], [4, 5, 6]]
|
||
|
|
for full_matrices in (True, False):
|
||
|
|
u, s, vh = svd(a, full_matrices=full_matrices,
|
||
|
|
lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.T @ u, eye(u.shape[0]))
|
||
|
|
sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_simple_overdet(self):
|
||
|
|
a = [[1, 2], [4, 5], [3, 4]]
|
||
|
|
for full_matrices in (True, False):
|
||
|
|
u, s, vh = svd(a, full_matrices=full_matrices,
|
||
|
|
lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.T @ u, eye(u.shape[1]))
|
||
|
|
assert_array_almost_equal(vh.T @ vh, eye(2))
|
||
|
|
sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_random(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
m = 15
|
||
|
|
for i in range(3):
|
||
|
|
for a in [rng.random([n, m]), rng.random([m, n])]:
|
||
|
|
for full_matrices in (True, False):
|
||
|
|
u, s, vh = svd(a, full_matrices=full_matrices,
|
||
|
|
lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.T @ u, eye(u.shape[1]))
|
||
|
|
assert_array_almost_equal(vh @ vh.T, eye(vh.shape[0]))
|
||
|
|
sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_simple_complex(self):
|
||
|
|
a = [[1, 2, 3], [1, 2j, 3], [2, 5, 6]]
|
||
|
|
for full_matrices in (True, False):
|
||
|
|
u, s, vh = svd(a, full_matrices=full_matrices,
|
||
|
|
lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.conj().T @ u, eye(u.shape[1]))
|
||
|
|
assert_array_almost_equal(vh.conj().T @ vh, eye(vh.shape[0]))
|
||
|
|
sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_random_complex(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
m = 15
|
||
|
|
for i in range(3):
|
||
|
|
for full_matrices in (True, False):
|
||
|
|
for a in [rng.random([n, m]), rng.random([m, n])]:
|
||
|
|
a = a + 1j*rng.random(list(a.shape))
|
||
|
|
u, s, vh = svd(a, full_matrices=full_matrices,
|
||
|
|
lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.conj().T @ u,
|
||
|
|
eye(u.shape[1]))
|
||
|
|
# This fails when [m,n]
|
||
|
|
# assert_array_almost_equal(vh.conj().T @ vh,
|
||
|
|
# eye(len(vh),dtype=vh.dtype.char))
|
||
|
|
sigma = zeros((u.shape[1], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_crash_1580(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
sizes = [(13, 23), (30, 50), (60, 100)]
|
||
|
|
for sz in sizes:
|
||
|
|
for dt in [np.float32, np.float64, np.complex64, np.complex128]:
|
||
|
|
a = rng.rand(*sz).astype(dt)
|
||
|
|
# should not crash
|
||
|
|
svd(a, lapack_driver=self.lapack_driver)
|
||
|
|
|
||
|
|
def test_check_finite(self):
|
||
|
|
a = [[1, 2, 3], [1, 20, 3], [2, 5, 6]]
|
||
|
|
u, s, vh = svd(a, check_finite=False, lapack_driver=self.lapack_driver)
|
||
|
|
assert_array_almost_equal(u.T @ u, eye(3))
|
||
|
|
assert_array_almost_equal(vh.T @ vh, eye(3))
|
||
|
|
sigma = zeros((u.shape[0], vh.shape[0]), s.dtype.char)
|
||
|
|
for i in range(len(s)):
|
||
|
|
sigma[i, i] = s[i]
|
||
|
|
assert_array_almost_equal(u @ sigma @ vh, a)
|
||
|
|
|
||
|
|
def test_gh_5039(self):
|
||
|
|
# This is a smoke test for https://github.com/scipy/scipy/issues/5039
|
||
|
|
#
|
||
|
|
# The following is reported to raise "ValueError: On entry to DGESDD
|
||
|
|
# parameter number 12 had an illegal value".
|
||
|
|
# `interp1d([1,2,3,4], [1,2,3,4], kind='cubic')`
|
||
|
|
# This is reported to only show up on LAPACK 3.0.3.
|
||
|
|
#
|
||
|
|
# The matrix below is taken from the call to
|
||
|
|
# `B = _fitpack._bsplmat(order, xk)` in interpolate._find_smoothest
|
||
|
|
b = np.array(
|
||
|
|
[[0.16666667, 0.66666667, 0.16666667, 0., 0., 0.],
|
||
|
|
[0., 0.16666667, 0.66666667, 0.16666667, 0., 0.],
|
||
|
|
[0., 0., 0.16666667, 0.66666667, 0.16666667, 0.],
|
||
|
|
[0., 0., 0., 0.16666667, 0.66666667, 0.16666667]])
|
||
|
|
svd(b, lapack_driver=self.lapack_driver)
|
||
|
|
|
||
|
|
@pytest.mark.skipif(not HAS_ILP64, reason="64-bit LAPACK required")
|
||
|
|
@pytest.mark.slow
|
||
|
|
def test_large_matrix(self):
|
||
|
|
check_free_memory(free_mb=17000)
|
||
|
|
A = np.zeros([1, 2**31], dtype=np.float32)
|
||
|
|
A[0, -1] = 1
|
||
|
|
u, s, vh = svd(A, full_matrices=False)
|
||
|
|
assert_allclose(s[0], 1.0)
|
||
|
|
assert_allclose(u[0, 0] * vh[0, -1], 1.0)
|
||
|
|
|
||
|
|
|
||
|
|
class TestSVD_GESVD(TestSVD_GESDD):
|
||
|
|
lapack_driver = 'gesvd'
|
||
|
|
|
||
|
|
|
||
|
|
def test_svd_gesdd_nofegfault():
|
||
|
|
# svd(a) with {U,VT}.size > INT_MAX does not segfault
|
||
|
|
# cf https://github.com/scipy/scipy/issues/14001
|
||
|
|
df=np.ones((4799, 53130), dtype=np.float64)
|
||
|
|
with assert_raises(ValueError):
|
||
|
|
svd(df)
|
||
|
|
|
||
|
|
|
||
|
|
class TestSVDVals:
|
||
|
|
|
||
|
|
def test_empty(self):
|
||
|
|
for a in [[]], np.empty((2, 0)), np.ones((0, 3)):
|
||
|
|
s = svdvals(a)
|
||
|
|
assert_equal(s, np.empty(0))
|
||
|
|
|
||
|
|
def test_simple(self):
|
||
|
|
a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
|
||
|
|
s = svdvals(a)
|
||
|
|
assert_(len(s) == 3)
|
||
|
|
assert_(s[0] >= s[1] >= s[2])
|
||
|
|
|
||
|
|
def test_simple_underdet(self):
|
||
|
|
a = [[1, 2, 3], [4, 5, 6]]
|
||
|
|
s = svdvals(a)
|
||
|
|
assert_(len(s) == 2)
|
||
|
|
assert_(s[0] >= s[1])
|
||
|
|
|
||
|
|
def test_simple_overdet(self):
|
||
|
|
a = [[1, 2], [4, 5], [3, 4]]
|
||
|
|
s = svdvals(a)
|
||
|
|
assert_(len(s) == 2)
|
||
|
|
assert_(s[0] >= s[1])
|
||
|
|
|
||
|
|
def test_simple_complex(self):
|
||
|
|
a = [[1, 2, 3], [1, 20, 3j], [2, 5, 6]]
|
||
|
|
s = svdvals(a)
|
||
|
|
assert_(len(s) == 3)
|
||
|
|
assert_(s[0] >= s[1] >= s[2])
|
||
|
|
|
||
|
|
def test_simple_underdet_complex(self):
|
||
|
|
a = [[1, 2, 3], [4, 5j, 6]]
|
||
|
|
s = svdvals(a)
|
||
|
|
assert_(len(s) == 2)
|
||
|
|
assert_(s[0] >= s[1])
|
||
|
|
|
||
|
|
def test_simple_overdet_complex(self):
|
||
|
|
a = [[1, 2], [4, 5], [3j, 4]]
|
||
|
|
s = svdvals(a)
|
||
|
|
assert_(len(s) == 2)
|
||
|
|
assert_(s[0] >= s[1])
|
||
|
|
|
||
|
|
def test_check_finite(self):
|
||
|
|
a = [[1, 2, 3], [1, 2, 3], [2, 5, 6]]
|
||
|
|
s = svdvals(a, check_finite=False)
|
||
|
|
assert_(len(s) == 3)
|
||
|
|
assert_(s[0] >= s[1] >= s[2])
|
||
|
|
|
||
|
|
@pytest.mark.slow
|
||
|
|
def test_crash_2609(self):
|
||
|
|
np.random.seed(1234)
|
||
|
|
a = np.random.rand(1500, 2800)
|
||
|
|
# Shouldn't crash:
|
||
|
|
svdvals(a)
|
||
|
|
|
||
|
|
|
||
|
|
class TestDiagSVD:
|
||
|
|
|
||
|
|
def test_simple(self):
|
||
|
|
assert_array_almost_equal(diagsvd([1, 0, 0], 3, 3),
|
||
|
|
[[1, 0, 0], [0, 0, 0], [0, 0, 0]])
|
||
|
|
|
||
|
|
|
||
|
|
class TestQR:
|
||
|
|
def test_simple(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_simple_left(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
q, r = qr(a)
|
||
|
|
c = [1, 2, 3]
|
||
|
|
qc, r2 = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
qc, r2 = qr_multiply(a, eye(3), "left")
|
||
|
|
assert_array_almost_equal(q, qc)
|
||
|
|
|
||
|
|
def test_simple_right(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
q, r = qr(a)
|
||
|
|
c = [1, 2, 3]
|
||
|
|
qc, r2 = qr_multiply(a, c)
|
||
|
|
assert_array_almost_equal(c @ q, qc)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
qc, r = qr_multiply(a, eye(3))
|
||
|
|
assert_array_almost_equal(q, qc)
|
||
|
|
|
||
|
|
def test_simple_pivoting(self):
|
||
|
|
a = np.asarray([[8, 2, 3], [2, 9, 3], [5, 3, 6]])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_simple_left_pivoting(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
q, r, jpvt = qr(a, pivoting=True)
|
||
|
|
c = [1, 2, 3]
|
||
|
|
qc, r, jpvt = qr_multiply(a, c, "left", True)
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
|
||
|
|
def test_simple_right_pivoting(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
q, r, jpvt = qr(a, pivoting=True)
|
||
|
|
c = [1, 2, 3]
|
||
|
|
qc, r, jpvt = qr_multiply(a, c, pivoting=True)
|
||
|
|
assert_array_almost_equal(c @ q, qc)
|
||
|
|
|
||
|
|
def test_simple_trap(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3]]
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_simple_trap_pivoting(self):
|
||
|
|
a = np.asarray([[8, 2, 3], [2, 9, 3]])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_simple_tall(self):
|
||
|
|
# full version
|
||
|
|
a = [[8, 2], [2, 9], [5, 3]]
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_simple_tall_pivoting(self):
|
||
|
|
# full version pivoting
|
||
|
|
a = np.asarray([[8, 2], [2, 9], [5, 3]])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_simple_tall_e(self):
|
||
|
|
# economy version
|
||
|
|
a = [[8, 2], [2, 9], [5, 3]]
|
||
|
|
q, r = qr(a, mode='economic')
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
assert_equal(q.shape, (3, 2))
|
||
|
|
assert_equal(r.shape, (2, 2))
|
||
|
|
|
||
|
|
def test_simple_tall_e_pivoting(self):
|
||
|
|
# economy version pivoting
|
||
|
|
a = np.asarray([[8, 2], [2, 9], [5, 3]])
|
||
|
|
q, r, p = qr(a, pivoting=True, mode='economic')
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p], mode='economic')
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_simple_tall_left(self):
|
||
|
|
a = [[8, 2], [2, 9], [5, 3]]
|
||
|
|
q, r = qr(a, mode="economic")
|
||
|
|
c = [1, 2]
|
||
|
|
qc, r2 = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
c = array([1, 2, 0])
|
||
|
|
qc, r2 = qr_multiply(a, c, "left", overwrite_c=True)
|
||
|
|
assert_array_almost_equal(q @ c[:2], qc)
|
||
|
|
qc, r = qr_multiply(a, eye(2), "left")
|
||
|
|
assert_array_almost_equal(qc, q)
|
||
|
|
|
||
|
|
def test_simple_tall_left_pivoting(self):
|
||
|
|
a = [[8, 2], [2, 9], [5, 3]]
|
||
|
|
q, r, jpvt = qr(a, mode="economic", pivoting=True)
|
||
|
|
c = [1, 2]
|
||
|
|
qc, r, kpvt = qr_multiply(a, c, "left", True)
|
||
|
|
assert_array_equal(jpvt, kpvt)
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
qc, r, jpvt = qr_multiply(a, eye(2), "left", True)
|
||
|
|
assert_array_almost_equal(qc, q)
|
||
|
|
|
||
|
|
def test_simple_tall_right(self):
|
||
|
|
a = [[8, 2], [2, 9], [5, 3]]
|
||
|
|
q, r = qr(a, mode="economic")
|
||
|
|
c = [1, 2, 3]
|
||
|
|
cq, r2 = qr_multiply(a, c)
|
||
|
|
assert_array_almost_equal(c @ q, cq)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
cq, r = qr_multiply(a, eye(3))
|
||
|
|
assert_array_almost_equal(cq, q)
|
||
|
|
|
||
|
|
def test_simple_tall_right_pivoting(self):
|
||
|
|
a = [[8, 2], [2, 9], [5, 3]]
|
||
|
|
q, r, jpvt = qr(a, pivoting=True, mode="economic")
|
||
|
|
c = [1, 2, 3]
|
||
|
|
cq, r, jpvt = qr_multiply(a, c, pivoting=True)
|
||
|
|
assert_array_almost_equal(c @ q, cq)
|
||
|
|
cq, r, jpvt = qr_multiply(a, eye(3), pivoting=True)
|
||
|
|
assert_array_almost_equal(cq, q)
|
||
|
|
|
||
|
|
def test_simple_fat(self):
|
||
|
|
# full version
|
||
|
|
a = [[8, 2, 5], [2, 9, 3]]
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
assert_equal(q.shape, (2, 2))
|
||
|
|
assert_equal(r.shape, (2, 3))
|
||
|
|
|
||
|
|
def test_simple_fat_pivoting(self):
|
||
|
|
# full version pivoting
|
||
|
|
a = np.asarray([[8, 2, 5], [2, 9, 3]])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
assert_equal(q.shape, (2, 2))
|
||
|
|
assert_equal(r.shape, (2, 3))
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_simple_fat_e(self):
|
||
|
|
# economy version
|
||
|
|
a = [[8, 2, 3], [2, 9, 5]]
|
||
|
|
q, r = qr(a, mode='economic')
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
assert_equal(q.shape, (2, 2))
|
||
|
|
assert_equal(r.shape, (2, 3))
|
||
|
|
|
||
|
|
def test_simple_fat_e_pivoting(self):
|
||
|
|
# economy version pivoting
|
||
|
|
a = np.asarray([[8, 2, 3], [2, 9, 5]])
|
||
|
|
q, r, p = qr(a, pivoting=True, mode='economic')
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
assert_equal(q.shape, (2, 2))
|
||
|
|
assert_equal(r.shape, (2, 3))
|
||
|
|
q2, r2 = qr(a[:, p], mode='economic')
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_simple_fat_left(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 5]]
|
||
|
|
q, r = qr(a, mode="economic")
|
||
|
|
c = [1, 2]
|
||
|
|
qc, r2 = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
qc, r = qr_multiply(a, eye(2), "left")
|
||
|
|
assert_array_almost_equal(qc, q)
|
||
|
|
|
||
|
|
def test_simple_fat_left_pivoting(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 5]]
|
||
|
|
q, r, jpvt = qr(a, mode="economic", pivoting=True)
|
||
|
|
c = [1, 2]
|
||
|
|
qc, r, jpvt = qr_multiply(a, c, "left", True)
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
qc, r, jpvt = qr_multiply(a, eye(2), "left", True)
|
||
|
|
assert_array_almost_equal(qc, q)
|
||
|
|
|
||
|
|
def test_simple_fat_right(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 5]]
|
||
|
|
q, r = qr(a, mode="economic")
|
||
|
|
c = [1, 2]
|
||
|
|
cq, r2 = qr_multiply(a, c)
|
||
|
|
assert_array_almost_equal(c @ q, cq)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
cq, r = qr_multiply(a, eye(2))
|
||
|
|
assert_array_almost_equal(cq, q)
|
||
|
|
|
||
|
|
def test_simple_fat_right_pivoting(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 5]]
|
||
|
|
q, r, jpvt = qr(a, pivoting=True, mode="economic")
|
||
|
|
c = [1, 2]
|
||
|
|
cq, r, jpvt = qr_multiply(a, c, pivoting=True)
|
||
|
|
assert_array_almost_equal(c @ q, cq)
|
||
|
|
cq, r, jpvt = qr_multiply(a, eye(2), pivoting=True)
|
||
|
|
assert_array_almost_equal(cq, q)
|
||
|
|
|
||
|
|
def test_simple_complex(self):
|
||
|
|
a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.conj().T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_simple_complex_left(self):
|
||
|
|
a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
|
||
|
|
q, r = qr(a)
|
||
|
|
c = [1, 2, 3+4j]
|
||
|
|
qc, r = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
qc, r = qr_multiply(a, eye(3), "left")
|
||
|
|
assert_array_almost_equal(q, qc)
|
||
|
|
|
||
|
|
def test_simple_complex_right(self):
|
||
|
|
a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
|
||
|
|
q, r = qr(a)
|
||
|
|
c = [1, 2, 3+4j]
|
||
|
|
qc, r = qr_multiply(a, c)
|
||
|
|
assert_array_almost_equal(c @ q, qc)
|
||
|
|
qc, r = qr_multiply(a, eye(3))
|
||
|
|
assert_array_almost_equal(q, qc)
|
||
|
|
|
||
|
|
def test_simple_tall_complex_left(self):
|
||
|
|
a = [[8, 2+3j], [2, 9], [5+7j, 3]]
|
||
|
|
q, r = qr(a, mode="economic")
|
||
|
|
c = [1, 2+2j]
|
||
|
|
qc, r2 = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
c = array([1, 2, 0])
|
||
|
|
qc, r2 = qr_multiply(a, c, "left", overwrite_c=True)
|
||
|
|
assert_array_almost_equal(q @ c[:2], qc)
|
||
|
|
qc, r = qr_multiply(a, eye(2), "left")
|
||
|
|
assert_array_almost_equal(qc, q)
|
||
|
|
|
||
|
|
def test_simple_complex_left_conjugate(self):
|
||
|
|
a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
|
||
|
|
q, r = qr(a)
|
||
|
|
c = [1, 2, 3+4j]
|
||
|
|
qc, r = qr_multiply(a, c, "left", conjugate=True)
|
||
|
|
assert_array_almost_equal(q.conj() @ c, qc)
|
||
|
|
|
||
|
|
def test_simple_complex_tall_left_conjugate(self):
|
||
|
|
a = [[3, 3+4j], [5, 2+2j], [3, 2]]
|
||
|
|
q, r = qr(a, mode='economic')
|
||
|
|
c = [1, 3+4j]
|
||
|
|
qc, r = qr_multiply(a, c, "left", conjugate=True)
|
||
|
|
assert_array_almost_equal(q.conj() @ c, qc)
|
||
|
|
|
||
|
|
def test_simple_complex_right_conjugate(self):
|
||
|
|
a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
|
||
|
|
q, r = qr(a)
|
||
|
|
c = np.array([1, 2, 3+4j])
|
||
|
|
qc, r = qr_multiply(a, c, conjugate=True)
|
||
|
|
assert_array_almost_equal(c @ q.conj(), qc)
|
||
|
|
|
||
|
|
def test_simple_complex_pivoting(self):
|
||
|
|
a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.conj().T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_simple_complex_left_pivoting(self):
|
||
|
|
a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]])
|
||
|
|
q, r, jpvt = qr(a, pivoting=True)
|
||
|
|
c = [1, 2, 3+4j]
|
||
|
|
qc, r, jpvt = qr_multiply(a, c, "left", True)
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
|
||
|
|
def test_simple_complex_right_pivoting(self):
|
||
|
|
a = array([[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]])
|
||
|
|
q, r, jpvt = qr(a, pivoting=True)
|
||
|
|
c = [1, 2, 3+4j]
|
||
|
|
qc, r, jpvt = qr_multiply(a, c, pivoting=True)
|
||
|
|
assert_array_almost_equal(c @ q, qc)
|
||
|
|
|
||
|
|
def test_random(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(n))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_random_left(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
c = rng.random([n])
|
||
|
|
qc, r = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
qc, r = qr_multiply(a, eye(n), "left")
|
||
|
|
assert_array_almost_equal(q, qc)
|
||
|
|
|
||
|
|
def test_random_right(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
c = rng.random([n])
|
||
|
|
cq, r = qr_multiply(a, c)
|
||
|
|
assert_array_almost_equal(c @ q, cq)
|
||
|
|
cq, r = qr_multiply(a, eye(n))
|
||
|
|
assert_array_almost_equal(q, cq)
|
||
|
|
|
||
|
|
def test_random_pivoting(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(n))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_random_tall(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
# full version
|
||
|
|
m = 200
|
||
|
|
n = 100
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(m))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_random_tall_left(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
# full version
|
||
|
|
m = 200
|
||
|
|
n = 100
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r = qr(a, mode="economic")
|
||
|
|
c = rng.random([n])
|
||
|
|
qc, r = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
qc, r = qr_multiply(a, eye(n), "left")
|
||
|
|
assert_array_almost_equal(qc, q)
|
||
|
|
|
||
|
|
def test_random_tall_right(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
# full version
|
||
|
|
m = 200
|
||
|
|
n = 100
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r = qr(a, mode="economic")
|
||
|
|
c = rng.random([m])
|
||
|
|
cq, r = qr_multiply(a, c)
|
||
|
|
assert_array_almost_equal(c @ q, cq)
|
||
|
|
cq, r = qr_multiply(a, eye(m))
|
||
|
|
assert_array_almost_equal(cq, q)
|
||
|
|
|
||
|
|
def test_random_tall_pivoting(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
# full version pivoting
|
||
|
|
m = 200
|
||
|
|
n = 100
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(m))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_random_tall_e(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
# economy version
|
||
|
|
m = 200
|
||
|
|
n = 100
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r = qr(a, mode='economic')
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(n))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
assert_equal(q.shape, (m, n))
|
||
|
|
assert_equal(r.shape, (n, n))
|
||
|
|
|
||
|
|
def test_random_tall_e_pivoting(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
# economy version pivoting
|
||
|
|
m = 200
|
||
|
|
n = 100
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r, p = qr(a, pivoting=True, mode='economic')
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(n))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
assert_equal(q.shape, (m, n))
|
||
|
|
assert_equal(r.shape, (n, n))
|
||
|
|
q2, r2 = qr(a[:, p], mode='economic')
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_random_trap(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
m = 100
|
||
|
|
n = 200
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(m))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_random_trap_pivoting(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
m = 100
|
||
|
|
n = 200
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(m))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_random_complex(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
assert_array_almost_equal(q.conj().T @ q, eye(n))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_random_complex_left(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
c = rng.random([n]) + 1j*rng.random([n])
|
||
|
|
qc, r = qr_multiply(a, c, "left")
|
||
|
|
assert_array_almost_equal(q @ c, qc)
|
||
|
|
qc, r = qr_multiply(a, eye(n), "left")
|
||
|
|
assert_array_almost_equal(q, qc)
|
||
|
|
|
||
|
|
def test_random_complex_right(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
q, r = qr(a)
|
||
|
|
c = rng.random([n]) + 1j*rng.random([n])
|
||
|
|
cq, r = qr_multiply(a, c)
|
||
|
|
assert_array_almost_equal(c @ q, cq)
|
||
|
|
cq, r = qr_multiply(a, eye(n))
|
||
|
|
assert_array_almost_equal(q, cq)
|
||
|
|
|
||
|
|
def test_random_complex_pivoting(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
q, r, p = qr(a, pivoting=True)
|
||
|
|
d = abs(diag(r))
|
||
|
|
assert_(np.all(d[1:] <= d[:-1]))
|
||
|
|
assert_array_almost_equal(q.conj().T @ q, eye(n))
|
||
|
|
assert_array_almost_equal(q @ r, a[:, p])
|
||
|
|
q2, r2 = qr(a[:, p])
|
||
|
|
assert_array_almost_equal(q, q2)
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_check_finite(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
q, r = qr(a, check_finite=False)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(q @ r, a)
|
||
|
|
|
||
|
|
def test_lwork(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
# Get comparison values
|
||
|
|
q, r = qr(a, lwork=None)
|
||
|
|
|
||
|
|
# Test against minimum valid lwork
|
||
|
|
q2, r2 = qr(a, lwork=3)
|
||
|
|
assert_array_almost_equal(q2, q)
|
||
|
|
assert_array_almost_equal(r2, r)
|
||
|
|
|
||
|
|
# Test against larger lwork
|
||
|
|
q3, r3 = qr(a, lwork=10)
|
||
|
|
assert_array_almost_equal(q3, q)
|
||
|
|
assert_array_almost_equal(r3, r)
|
||
|
|
|
||
|
|
# Test against explicit lwork=-1
|
||
|
|
q4, r4 = qr(a, lwork=-1)
|
||
|
|
assert_array_almost_equal(q4, q)
|
||
|
|
assert_array_almost_equal(r4, r)
|
||
|
|
|
||
|
|
# Test against invalid lwork
|
||
|
|
assert_raises(Exception, qr, (a,), {'lwork': 0})
|
||
|
|
assert_raises(Exception, qr, (a,), {'lwork': 2})
|
||
|
|
|
||
|
|
|
||
|
|
class TestRQ:
|
||
|
|
def test_simple(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q @ q.T, eye(3))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_r(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
r, q = rq(a)
|
||
|
|
r2 = rq(a, mode='r')
|
||
|
|
assert_array_almost_equal(r, r2)
|
||
|
|
|
||
|
|
def test_random(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n])
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q @ q.T, eye(n))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_simple_trap(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3]]
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(3))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_simple_tall(self):
|
||
|
|
a = [[8, 2], [2, 9], [5, 3]]
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q.T @ q, eye(2))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_simple_fat(self):
|
||
|
|
a = [[8, 2, 5], [2, 9, 3]]
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q @ q.T, eye(3))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_simple_complex(self):
|
||
|
|
a = [[3, 3+4j, 5], [5, 2, 2+7j], [3, 2, 7]]
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q @ q.conj().T, eye(3))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_random_tall(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
m = 200
|
||
|
|
n = 100
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q @ q.T, eye(n))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_random_trap(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
m = 100
|
||
|
|
n = 200
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q @ q.T, eye(n))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_random_trap_economic(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
m = 100
|
||
|
|
n = 200
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n])
|
||
|
|
r, q = rq(a, mode='economic')
|
||
|
|
assert_array_almost_equal(q @ q.T, eye(m))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
assert_equal(q.shape, (m, n))
|
||
|
|
assert_equal(r.shape, (m, m))
|
||
|
|
|
||
|
|
def test_random_complex(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
r, q = rq(a)
|
||
|
|
assert_array_almost_equal(q @ q.conj().T, eye(n))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
def test_random_complex_economic(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
m = 100
|
||
|
|
n = 200
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([m, n]) + 1j*rng.random([m, n])
|
||
|
|
r, q = rq(a, mode='economic')
|
||
|
|
assert_array_almost_equal(q @ q.conj().T, eye(m))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
assert_equal(q.shape, (m, n))
|
||
|
|
assert_equal(r.shape, (m, m))
|
||
|
|
|
||
|
|
def test_check_finite(self):
|
||
|
|
a = [[8, 2, 3], [2, 9, 3], [5, 3, 6]]
|
||
|
|
r, q = rq(a, check_finite=False)
|
||
|
|
assert_array_almost_equal(q @ q.T, eye(3))
|
||
|
|
assert_array_almost_equal(r @ q, a)
|
||
|
|
|
||
|
|
|
||
|
|
class TestSchur:
|
||
|
|
|
||
|
|
def check_schur(self, a, t, u, rtol, atol):
|
||
|
|
# Check that the Schur decomposition is correct.
|
||
|
|
assert_allclose(u @ t @ u.conj().T, a, rtol=rtol, atol=atol,
|
||
|
|
err_msg="Schur decomposition does not match 'a'")
|
||
|
|
# The expected value of u @ u.H - I is all zeros, so test
|
||
|
|
# with absolute tolerance only.
|
||
|
|
assert_allclose(u @ u.conj().T - np.eye(len(u)), 0, rtol=0, atol=atol,
|
||
|
|
err_msg="u is not unitary")
|
||
|
|
|
||
|
|
def test_simple(self):
|
||
|
|
a = [[8, 12, 3], [2, 9, 3], [10, 3, 6]]
|
||
|
|
t, z = schur(a)
|
||
|
|
self.check_schur(a, t, z, rtol=1e-14, atol=5e-15)
|
||
|
|
tc, zc = schur(a, 'complex')
|
||
|
|
assert_(np.any(ravel(iscomplex(zc))) and np.any(ravel(iscomplex(tc))))
|
||
|
|
self.check_schur(a, tc, zc, rtol=1e-14, atol=5e-15)
|
||
|
|
tc2, zc2 = rsf2csf(tc, zc)
|
||
|
|
self.check_schur(a, tc2, zc2, rtol=1e-14, atol=5e-15)
|
||
|
|
|
||
|
|
@pytest.mark.parametrize(
|
||
|
|
'sort, expected_diag',
|
||
|
|
[('lhp', [-np.sqrt(2), -0.5, np.sqrt(2), 0.5]),
|
||
|
|
('rhp', [np.sqrt(2), 0.5, -np.sqrt(2), -0.5]),
|
||
|
|
('iuc', [-0.5, 0.5, np.sqrt(2), -np.sqrt(2)]),
|
||
|
|
('ouc', [np.sqrt(2), -np.sqrt(2), -0.5, 0.5]),
|
||
|
|
(lambda x: x >= 0.0, [np.sqrt(2), 0.5, -np.sqrt(2), -0.5])]
|
||
|
|
)
|
||
|
|
def test_sort(self, sort, expected_diag):
|
||
|
|
# The exact eigenvalues of this matrix are
|
||
|
|
# -sqrt(2), sqrt(2), -1/2, 1/2.
|
||
|
|
a = [[4., 3., 1., -1.],
|
||
|
|
[-4.5, -3.5, -1., 1.],
|
||
|
|
[9., 6., -4., 4.5],
|
||
|
|
[6., 4., -3., 3.5]]
|
||
|
|
t, u, sdim = schur(a, sort=sort)
|
||
|
|
self.check_schur(a, t, u, rtol=1e-14, atol=5e-15)
|
||
|
|
assert_allclose(np.diag(t), expected_diag, rtol=1e-12)
|
||
|
|
assert_equal(2, sdim)
|
||
|
|
|
||
|
|
def test_sort_errors(self):
|
||
|
|
a = [[4., 3., 1., -1.],
|
||
|
|
[-4.5, -3.5, -1., 1.],
|
||
|
|
[9., 6., -4., 4.5],
|
||
|
|
[6., 4., -3., 3.5]]
|
||
|
|
assert_raises(ValueError, schur, a, sort='unsupported')
|
||
|
|
assert_raises(ValueError, schur, a, sort=1)
|
||
|
|
|
||
|
|
def test_check_finite(self):
|
||
|
|
a = [[8, 12, 3], [2, 9, 3], [10, 3, 6]]
|
||
|
|
t, z = schur(a, check_finite=False)
|
||
|
|
assert_array_almost_equal(z @ t @ z.conj().T, a)
|
||
|
|
|
||
|
|
|
||
|
|
class TestHessenberg:
|
||
|
|
|
||
|
|
def test_simple(self):
|
||
|
|
a = [[-149, -50, -154],
|
||
|
|
[537, 180, 546],
|
||
|
|
[-27, -9, -25]]
|
||
|
|
h1 = [[-149.0000, 42.2037, -156.3165],
|
||
|
|
[-537.6783, 152.5511, -554.9272],
|
||
|
|
[0, 0.0728, 2.4489]]
|
||
|
|
h, q = hessenberg(a, calc_q=1)
|
||
|
|
assert_array_almost_equal(q.T @ a @ q, h)
|
||
|
|
assert_array_almost_equal(h, h1, decimal=4)
|
||
|
|
|
||
|
|
def test_simple_complex(self):
|
||
|
|
a = [[-149, -50, -154],
|
||
|
|
[537, 180j, 546],
|
||
|
|
[-27j, -9, -25]]
|
||
|
|
h, q = hessenberg(a, calc_q=1)
|
||
|
|
assert_array_almost_equal(q.conj().T @ a @ q, h)
|
||
|
|
|
||
|
|
def test_simple2(self):
|
||
|
|
a = [[1, 2, 3, 4, 5, 6, 7],
|
||
|
|
[0, 2, 3, 4, 6, 7, 2],
|
||
|
|
[0, 2, 2, 3, 0, 3, 2],
|
||
|
|
[0, 0, 2, 8, 0, 0, 2],
|
||
|
|
[0, 3, 1, 2, 0, 1, 2],
|
||
|
|
[0, 1, 2, 3, 0, 1, 0],
|
||
|
|
[0, 0, 0, 0, 0, 1, 2]]
|
||
|
|
h, q = hessenberg(a, calc_q=1)
|
||
|
|
assert_array_almost_equal(q.T @ a @ q, h)
|
||
|
|
|
||
|
|
def test_simple3(self):
|
||
|
|
a = np.eye(3)
|
||
|
|
a[-1, 0] = 2
|
||
|
|
h, q = hessenberg(a, calc_q=1)
|
||
|
|
assert_array_almost_equal(q.T @ a @ q, h)
|
||
|
|
|
||
|
|
def test_random(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n])
|
||
|
|
h, q = hessenberg(a, calc_q=1)
|
||
|
|
assert_array_almost_equal(q.T @ a @ q, h)
|
||
|
|
|
||
|
|
def test_random_complex(self):
|
||
|
|
rng = np.random.RandomState(1234)
|
||
|
|
n = 20
|
||
|
|
for k in range(2):
|
||
|
|
a = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
h, q = hessenberg(a, calc_q=1)
|
||
|
|
assert_array_almost_equal(q.conj().T @ a @ q, h)
|
||
|
|
|
||
|
|
def test_check_finite(self):
|
||
|
|
a = [[-149, -50, -154],
|
||
|
|
[537, 180, 546],
|
||
|
|
[-27, -9, -25]]
|
||
|
|
h1 = [[-149.0000, 42.2037, -156.3165],
|
||
|
|
[-537.6783, 152.5511, -554.9272],
|
||
|
|
[0, 0.0728, 2.4489]]
|
||
|
|
h, q = hessenberg(a, calc_q=1, check_finite=False)
|
||
|
|
assert_array_almost_equal(q.T @ a @ q, h)
|
||
|
|
assert_array_almost_equal(h, h1, decimal=4)
|
||
|
|
|
||
|
|
def test_2x2(self):
|
||
|
|
a = [[2, 1], [7, 12]]
|
||
|
|
|
||
|
|
h, q = hessenberg(a, calc_q=1)
|
||
|
|
assert_array_almost_equal(q, np.eye(2))
|
||
|
|
assert_array_almost_equal(h, a)
|
||
|
|
|
||
|
|
b = [[2-7j, 1+2j], [7+3j, 12-2j]]
|
||
|
|
h2, q2 = hessenberg(b, calc_q=1)
|
||
|
|
assert_array_almost_equal(q2, np.eye(2))
|
||
|
|
assert_array_almost_equal(h2, b)
|
||
|
|
|
||
|
|
|
||
|
|
blas_provider = blas_version = None
|
||
|
|
if CONFIG is not None:
|
||
|
|
blas_provider = CONFIG['Build Dependencies']['blas']['name']
|
||
|
|
blas_version = CONFIG['Build Dependencies']['blas']['version']
|
||
|
|
|
||
|
|
|
||
|
|
class TestQZ:
|
||
|
|
@pytest.mark.xfail(
|
||
|
|
sys.platform == 'darwin' and
|
||
|
|
blas_provider == 'openblas' and
|
||
|
|
blas_version < "0.3.21.dev",
|
||
|
|
reason="gges[float32] broken for OpenBLAS on macOS, see gh-16949"
|
||
|
|
)
|
||
|
|
def test_qz_single(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
n = 5
|
||
|
|
A = rng.random([n, n]).astype(float32)
|
||
|
|
B = rng.random([n, n]).astype(float32)
|
||
|
|
AA, BB, Q, Z = qz(A, B)
|
||
|
|
assert_array_almost_equal(Q @ AA @ Z.T, A, decimal=5)
|
||
|
|
assert_array_almost_equal(Q @ BB @ Z.T, B, decimal=5)
|
||
|
|
assert_array_almost_equal(Q @ Q.T, eye(n), decimal=5)
|
||
|
|
assert_array_almost_equal(Z @ Z.T, eye(n), decimal=5)
|
||
|
|
assert_(np.all(diag(BB) >= 0))
|
||
|
|
|
||
|
|
def test_qz_double(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
n = 5
|
||
|
|
A = rng.random([n, n])
|
||
|
|
B = rng.random([n, n])
|
||
|
|
AA, BB, Q, Z = qz(A, B)
|
||
|
|
assert_array_almost_equal(Q @ AA @ Z.T, A)
|
||
|
|
assert_array_almost_equal(Q @ BB @ Z.T, B)
|
||
|
|
assert_array_almost_equal(Q @ Q.T, eye(n))
|
||
|
|
assert_array_almost_equal(Z @ Z.T, eye(n))
|
||
|
|
assert_(np.all(diag(BB) >= 0))
|
||
|
|
|
||
|
|
def test_qz_complex(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
n = 5
|
||
|
|
A = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
B = rng.random([n, n]) + 1j*rng.random([n, n])
|
||
|
|
AA, BB, Q, Z = qz(A, B)
|
||
|
|
assert_array_almost_equal(Q @ AA @ Z.conj().T, A)
|
||
|
|
assert_array_almost_equal(Q @ BB @ Z.conj().T, B)
|
||
|
|
assert_array_almost_equal(Q @ Q.conj().T, eye(n))
|
||
|
|
assert_array_almost_equal(Z @ Z.conj().T, eye(n))
|
||
|
|
assert_(np.all(diag(BB) >= 0))
|
||
|
|
assert_(np.all(diag(BB).imag == 0))
|
||
|
|
|
||
|
|
def test_qz_complex64(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
n = 5
|
||
|
|
A = (rng.random([n, n]) + 1j*rng.random([n, n])).astype(complex64)
|
||
|
|
B = (rng.random([n, n]) + 1j*rng.random([n, n])).astype(complex64)
|
||
|
|
AA, BB, Q, Z = qz(A, B)
|
||
|
|
assert_array_almost_equal(Q @ AA @ Z.conj().T, A, decimal=5)
|
||
|
|
assert_array_almost_equal(Q @ BB @ Z.conj().T, B, decimal=5)
|
||
|
|
assert_array_almost_equal(Q @ Q.conj().T, eye(n), decimal=5)
|
||
|
|
assert_array_almost_equal(Z @ Z.conj().T, eye(n), decimal=5)
|
||
|
|
assert_(np.all(diag(BB) >= 0))
|
||
|
|
assert_(np.all(diag(BB).imag == 0))
|
||
|
|
|
||
|
|
def test_qz_double_complex(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
n = 5
|
||
|
|
A = rng.random([n, n])
|
||
|
|
B = rng.random([n, n])
|
||
|
|
AA, BB, Q, Z = qz(A, B, output='complex')
|
||
|
|
aa = Q @ AA @ Z.conj().T
|
||
|
|
assert_array_almost_equal(aa.real, A)
|
||
|
|
assert_array_almost_equal(aa.imag, 0)
|
||
|
|
bb = Q @ BB @ Z.conj().T
|
||
|
|
assert_array_almost_equal(bb.real, B)
|
||
|
|
assert_array_almost_equal(bb.imag, 0)
|
||
|
|
assert_array_almost_equal(Q @ Q.conj().T, eye(n))
|
||
|
|
assert_array_almost_equal(Z @ Z.conj().T, eye(n))
|
||
|
|
assert_(np.all(diag(BB) >= 0))
|
||
|
|
|
||
|
|
def test_qz_double_sort(self):
|
||
|
|
# from https://www.nag.com/lapack-ex/node119.html
|
||
|
|
# NOTE: These matrices may be ill-conditioned and lead to a
|
||
|
|
# seg fault on certain python versions when compiled with
|
||
|
|
# sse2 or sse3 older ATLAS/LAPACK binaries for windows
|
||
|
|
# A = np.array([[3.9, 12.5, -34.5, -0.5],
|
||
|
|
# [ 4.3, 21.5, -47.5, 7.5],
|
||
|
|
# [ 4.3, 21.5, -43.5, 3.5],
|
||
|
|
# [ 4.4, 26.0, -46.0, 6.0 ]])
|
||
|
|
|
||
|
|
# B = np.array([[ 1.0, 2.0, -3.0, 1.0],
|
||
|
|
# [1.0, 3.0, -5.0, 4.0],
|
||
|
|
# [1.0, 3.0, -4.0, 3.0],
|
||
|
|
# [1.0, 3.0, -4.0, 4.0]])
|
||
|
|
A = np.array([[3.9, 12.5, -34.5, 2.5],
|
||
|
|
[4.3, 21.5, -47.5, 7.5],
|
||
|
|
[4.3, 1.5, -43.5, 3.5],
|
||
|
|
[4.4, 6.0, -46.0, 6.0]])
|
||
|
|
|
||
|
|
B = np.array([[1.0, 1.0, -3.0, 1.0],
|
||
|
|
[1.0, 3.0, -5.0, 4.4],
|
||
|
|
[1.0, 2.0, -4.0, 1.0],
|
||
|
|
[1.2, 3.0, -4.0, 4.0]])
|
||
|
|
|
||
|
|
assert_raises(ValueError, qz, A, B, sort=lambda ar, ai, beta: ai == 0)
|
||
|
|
if False:
|
||
|
|
AA, BB, Q, Z, sdim = qz(A, B, sort=lambda ar, ai, beta: ai == 0)
|
||
|
|
# assert_(sdim == 2)
|
||
|
|
assert_(sdim == 4)
|
||
|
|
assert_array_almost_equal(Q @ AA @ Z.T, A)
|
||
|
|
assert_array_almost_equal(Q @ BB @ Z.T, B)
|
||
|
|
|
||
|
|
# test absolute values bc the sign is ambiguous and
|
||
|
|
# might be platform dependent
|
||
|
|
assert_array_almost_equal(np.abs(AA), np.abs(np.array(
|
||
|
|
[[35.7864, -80.9061, -12.0629, -9.498],
|
||
|
|
[0., 2.7638, -2.3505, 7.3256],
|
||
|
|
[0., 0., 0.6258, -0.0398],
|
||
|
|
[0., 0., 0., -12.8217]])), 4)
|
||
|
|
assert_array_almost_equal(np.abs(BB), np.abs(np.array(
|
||
|
|
[[4.5324, -8.7878, 3.2357, -3.5526],
|
||
|
|
[0., 1.4314, -2.1894, 0.9709],
|
||
|
|
[0., 0., 1.3126, -0.3468],
|
||
|
|
[0., 0., 0., 0.559]])), 4)
|
||
|
|
assert_array_almost_equal(np.abs(Q), np.abs(np.array(
|
||
|
|
[[-0.4193, -0.605, -0.1894, -0.6498],
|
||
|
|
[-0.5495, 0.6987, 0.2654, -0.3734],
|
||
|
|
[-0.4973, -0.3682, 0.6194, 0.4832],
|
||
|
|
[-0.5243, 0.1008, -0.7142, 0.4526]])), 4)
|
||
|
|
assert_array_almost_equal(np.abs(Z), np.abs(np.array(
|
||
|
|
[[-0.9471, -0.2971, -0.1217, 0.0055],
|
||
|
|
[-0.0367, 0.1209, 0.0358, 0.9913],
|
||
|
|
[0.3171, -0.9041, -0.2547, 0.1312],
|
||
|
|
[0.0346, 0.2824, -0.9587, 0.0014]])), 4)
|
||
|
|
|
||
|
|
# test absolute values bc the sign is ambiguous and might be platform
|
||
|
|
# dependent
|
||
|
|
# assert_array_almost_equal(abs(AA), abs(np.array([
|
||
|
|
# [3.8009, -69.4505, 50.3135, -43.2884],
|
||
|
|
# [0.0000, 9.2033, -0.2001, 5.9881],
|
||
|
|
# [0.0000, 0.0000, 1.4279, 4.4453],
|
||
|
|
# [0.0000, 0.0000, 0.9019, -1.1962]])), 4)
|
||
|
|
# assert_array_almost_equal(abs(BB), abs(np.array([
|
||
|
|
# [1.9005, -10.2285, 0.8658, -5.2134],
|
||
|
|
# [0.0000, 2.3008, 0.7915, 0.4262],
|
||
|
|
# [0.0000, 0.0000, 0.8101, 0.0000],
|
||
|
|
# [0.0000, 0.0000, 0.0000, -0.2823]])), 4)
|
||
|
|
# assert_array_almost_equal(abs(Q), abs(np.array([
|
||
|
|
# [0.4642, 0.7886, 0.2915, -0.2786],
|
||
|
|
# [0.5002, -0.5986, 0.5638, -0.2713],
|
||
|
|
# [0.5002, 0.0154, -0.0107, 0.8657],
|
||
|
|
# [0.5331, -0.1395, -0.7727, -0.3151]])), 4)
|
||
|
|
# assert_array_almost_equal(dot(Q,Q.T), eye(4))
|
||
|
|
# assert_array_almost_equal(abs(Z), abs(np.array([
|
||
|
|
# [0.9961, -0.0014, 0.0887, -0.0026],
|
||
|
|
# [0.0057, -0.0404, -0.0938, -0.9948],
|
||
|
|
# [0.0626, 0.7194, -0.6908, 0.0363],
|
||
|
|
# [0.0626, -0.6934, -0.7114, 0.0956]])), 4)
|
||
|
|
# assert_array_almost_equal(dot(Z,Z.T), eye(4))
|
||
|
|
|
||
|
|
# def test_qz_complex_sort(self):
|
||
|
|
# cA = np.array([
|
||
|
|
# [-21.10+22.50*1j, 53.50+-50.50*1j, -34.50+127.50*1j, 7.50+ 0.50*1j],
|
||
|
|
# [-0.46+ -7.78*1j, -3.50+-37.50*1j, -15.50+ 58.50*1j,-10.50+ -1.50*1j],
|
||
|
|
# [ 4.30+ -5.50*1j, 39.70+-17.10*1j, -68.50+ 12.50*1j, -7.50+ -3.50*1j],
|
||
|
|
# [ 5.50+ 4.40*1j, 14.40+ 43.30*1j, -32.50+-46.00*1j,-19.00+-32.50*1j]])
|
||
|
|
|
||
|
|
# cB = np.array([
|
||
|
|
# [1.00+ -5.00*1j, 1.60+ 1.20*1j,-3.00+ 0.00*1j, 0.00+ -1.00*1j],
|
||
|
|
# [0.80+ -0.60*1j, 3.00+ -5.00*1j,-4.00+ 3.00*1j,-2.40+ -3.20*1j],
|
||
|
|
# [1.00+ 0.00*1j, 2.40+ 1.80*1j,-4.00+ -5.00*1j, 0.00+ -3.00*1j],
|
||
|
|
# [0.00+ 1.00*1j,-1.80+ 2.40*1j, 0.00+ -4.00*1j, 4.00+ -5.00*1j]])
|
||
|
|
|
||
|
|
# AAS,BBS,QS,ZS,sdim = qz(cA,cB,sort='lhp')
|
||
|
|
|
||
|
|
# eigenvalues = diag(AAS)/diag(BBS)
|
||
|
|
# assert_(np.all(np.real(eigenvalues[:sdim] < 0)))
|
||
|
|
# assert_(np.all(np.real(eigenvalues[sdim:] > 0)))
|
||
|
|
|
||
|
|
def test_check_finite(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
n = 5
|
||
|
|
A = rng.random([n, n])
|
||
|
|
B = rng.random([n, n])
|
||
|
|
AA, BB, Q, Z = qz(A, B, check_finite=False)
|
||
|
|
assert_array_almost_equal(Q @ AA @ Z.T, A)
|
||
|
|
assert_array_almost_equal(Q @ BB @ Z.T, B)
|
||
|
|
assert_array_almost_equal(Q @ Q.T, eye(n))
|
||
|
|
assert_array_almost_equal(Z @ Z.T, eye(n))
|
||
|
|
assert_(np.all(diag(BB) >= 0))
|
||
|
|
|
||
|
|
|
||
|
|
class TestOrdQZ:
|
||
|
|
@classmethod
|
||
|
|
def setup_class(cls):
|
||
|
|
# https://www.nag.com/lapack-ex/node119.html
|
||
|
|
A1 = np.array([[-21.10 - 22.50j, 53.5 - 50.5j, -34.5 + 127.5j,
|
||
|
|
7.5 + 0.5j],
|
||
|
|
[-0.46 - 7.78j, -3.5 - 37.5j, -15.5 + 58.5j,
|
||
|
|
-10.5 - 1.5j],
|
||
|
|
[4.30 - 5.50j, 39.7 - 17.1j, -68.5 + 12.5j,
|
||
|
|
-7.5 - 3.5j],
|
||
|
|
[5.50 + 4.40j, 14.4 + 43.3j, -32.5 - 46.0j,
|
||
|
|
-19.0 - 32.5j]])
|
||
|
|
|
||
|
|
B1 = np.array([[1.0 - 5.0j, 1.6 + 1.2j, -3 + 0j, 0.0 - 1.0j],
|
||
|
|
[0.8 - 0.6j, .0 - 5.0j, -4 + 3j, -2.4 - 3.2j],
|
||
|
|
[1.0 + 0.0j, 2.4 + 1.8j, -4 - 5j, 0.0 - 3.0j],
|
||
|
|
[0.0 + 1.0j, -1.8 + 2.4j, 0 - 4j, 4.0 - 5.0j]])
|
||
|
|
|
||
|
|
# https://www.nag.com/numeric/fl/nagdoc_fl23/xhtml/F08/f08yuf.xml
|
||
|
|
A2 = np.array([[3.9, 12.5, -34.5, -0.5],
|
||
|
|
[4.3, 21.5, -47.5, 7.5],
|
||
|
|
[4.3, 21.5, -43.5, 3.5],
|
||
|
|
[4.4, 26.0, -46.0, 6.0]])
|
||
|
|
|
||
|
|
B2 = np.array([[1, 2, -3, 1],
|
||
|
|
[1, 3, -5, 4],
|
||
|
|
[1, 3, -4, 3],
|
||
|
|
[1, 3, -4, 4]])
|
||
|
|
|
||
|
|
# example with the eigenvalues
|
||
|
|
# -0.33891648, 1.61217396+0.74013521j, 1.61217396-0.74013521j,
|
||
|
|
# 0.61244091
|
||
|
|
# thus featuring:
|
||
|
|
# * one complex conjugate eigenvalue pair,
|
||
|
|
# * one eigenvalue in the lhp
|
||
|
|
# * 2 eigenvalues in the unit circle
|
||
|
|
# * 2 non-real eigenvalues
|
||
|
|
A3 = np.array([[5., 1., 3., 3.],
|
||
|
|
[4., 4., 2., 7.],
|
||
|
|
[7., 4., 1., 3.],
|
||
|
|
[0., 4., 8., 7.]])
|
||
|
|
B3 = np.array([[8., 10., 6., 10.],
|
||
|
|
[7., 7., 2., 9.],
|
||
|
|
[9., 1., 6., 6.],
|
||
|
|
[5., 1., 4., 7.]])
|
||
|
|
|
||
|
|
# example with infinite eigenvalues
|
||
|
|
A4 = np.eye(2)
|
||
|
|
B4 = np.diag([0, 1])
|
||
|
|
|
||
|
|
# example with (alpha, beta) = (0, 0)
|
||
|
|
A5 = np.diag([1, 0])
|
||
|
|
|
||
|
|
cls.A = [A1, A2, A3, A4, A5]
|
||
|
|
cls.B = [B1, B2, B3, B4, A5]
|
||
|
|
|
||
|
|
def qz_decomp(self, sort):
|
||
|
|
with np.errstate(all='raise'):
|
||
|
|
ret = [ordqz(Ai, Bi, sort=sort) for Ai, Bi in zip(self.A, self.B)]
|
||
|
|
return tuple(ret)
|
||
|
|
|
||
|
|
def check(self, A, B, sort, AA, BB, alpha, beta, Q, Z):
|
||
|
|
Id = np.eye(*A.shape)
|
||
|
|
# make sure Q and Z are orthogonal
|
||
|
|
assert_array_almost_equal(Q @ Q.T.conj(), Id)
|
||
|
|
assert_array_almost_equal(Z @ Z.T.conj(), Id)
|
||
|
|
# check factorization
|
||
|
|
assert_array_almost_equal(Q @ AA, A @ Z)
|
||
|
|
assert_array_almost_equal(Q @ BB, B @ Z)
|
||
|
|
# check shape of AA and BB
|
||
|
|
assert_array_equal(np.tril(AA, -2), np.zeros(AA.shape))
|
||
|
|
assert_array_equal(np.tril(BB, -1), np.zeros(BB.shape))
|
||
|
|
# check eigenvalues
|
||
|
|
for i in range(A.shape[0]):
|
||
|
|
# does the current diagonal element belong to a 2-by-2 block
|
||
|
|
# that was already checked?
|
||
|
|
if i > 0 and A[i, i - 1] != 0:
|
||
|
|
continue
|
||
|
|
# take care of 2-by-2 blocks
|
||
|
|
if i < AA.shape[0] - 1 and AA[i + 1, i] != 0:
|
||
|
|
evals, _ = eig(AA[i:i + 2, i:i + 2], BB[i:i + 2, i:i + 2])
|
||
|
|
# make sure the pair of complex conjugate eigenvalues
|
||
|
|
# is ordered consistently (positive imaginary part first)
|
||
|
|
if evals[0].imag < 0:
|
||
|
|
evals = evals[[1, 0]]
|
||
|
|
tmp = alpha[i:i + 2]/beta[i:i + 2]
|
||
|
|
if tmp[0].imag < 0:
|
||
|
|
tmp = tmp[[1, 0]]
|
||
|
|
assert_array_almost_equal(evals, tmp)
|
||
|
|
else:
|
||
|
|
if alpha[i] == 0 and beta[i] == 0:
|
||
|
|
assert_equal(AA[i, i], 0)
|
||
|
|
assert_equal(BB[i, i], 0)
|
||
|
|
elif beta[i] == 0:
|
||
|
|
assert_equal(BB[i, i], 0)
|
||
|
|
else:
|
||
|
|
assert_almost_equal(AA[i, i]/BB[i, i], alpha[i]/beta[i])
|
||
|
|
sortfun = _select_function(sort)
|
||
|
|
lastsort = True
|
||
|
|
for i in range(A.shape[0]):
|
||
|
|
cursort = sortfun(np.array([alpha[i]]), np.array([beta[i]]))
|
||
|
|
# once the sorting criterion was not matched all subsequent
|
||
|
|
# eigenvalues also shouldn't match
|
||
|
|
if not lastsort:
|
||
|
|
assert not cursort
|
||
|
|
lastsort = cursort
|
||
|
|
|
||
|
|
def check_all(self, sort):
|
||
|
|
ret = self.qz_decomp(sort)
|
||
|
|
|
||
|
|
for reti, Ai, Bi in zip(ret, self.A, self.B):
|
||
|
|
self.check(Ai, Bi, sort, *reti)
|
||
|
|
|
||
|
|
def test_lhp(self):
|
||
|
|
self.check_all('lhp')
|
||
|
|
|
||
|
|
def test_rhp(self):
|
||
|
|
self.check_all('rhp')
|
||
|
|
|
||
|
|
def test_iuc(self):
|
||
|
|
self.check_all('iuc')
|
||
|
|
|
||
|
|
def test_ouc(self):
|
||
|
|
self.check_all('ouc')
|
||
|
|
|
||
|
|
def test_ref(self):
|
||
|
|
# real eigenvalues first (top-left corner)
|
||
|
|
def sort(x, y):
|
||
|
|
out = np.empty_like(x, dtype=bool)
|
||
|
|
nonzero = (y != 0)
|
||
|
|
out[~nonzero] = False
|
||
|
|
out[nonzero] = (x[nonzero]/y[nonzero]).imag == 0
|
||
|
|
return out
|
||
|
|
|
||
|
|
self.check_all(sort)
|
||
|
|
|
||
|
|
def test_cef(self):
|
||
|
|
# complex eigenvalues first (top-left corner)
|
||
|
|
def sort(x, y):
|
||
|
|
out = np.empty_like(x, dtype=bool)
|
||
|
|
nonzero = (y != 0)
|
||
|
|
out[~nonzero] = False
|
||
|
|
out[nonzero] = (x[nonzero]/y[nonzero]).imag != 0
|
||
|
|
return out
|
||
|
|
|
||
|
|
self.check_all(sort)
|
||
|
|
|
||
|
|
def test_diff_input_types(self):
|
||
|
|
ret = ordqz(self.A[1], self.B[2], sort='lhp')
|
||
|
|
self.check(self.A[1], self.B[2], 'lhp', *ret)
|
||
|
|
|
||
|
|
ret = ordqz(self.B[2], self.A[1], sort='lhp')
|
||
|
|
self.check(self.B[2], self.A[1], 'lhp', *ret)
|
||
|
|
|
||
|
|
def test_sort_explicit(self):
|
||
|
|
# Test order of the eigenvalues in the 2 x 2 case where we can
|
||
|
|
# explicitly compute the solution
|
||
|
|
A1 = np.eye(2)
|
||
|
|
B1 = np.diag([-2, 0.5])
|
||
|
|
expected1 = [('lhp', [-0.5, 2]),
|
||
|
|
('rhp', [2, -0.5]),
|
||
|
|
('iuc', [-0.5, 2]),
|
||
|
|
('ouc', [2, -0.5])]
|
||
|
|
A2 = np.eye(2)
|
||
|
|
B2 = np.diag([-2 + 1j, 0.5 + 0.5j])
|
||
|
|
expected2 = [('lhp', [1/(-2 + 1j), 1/(0.5 + 0.5j)]),
|
||
|
|
('rhp', [1/(0.5 + 0.5j), 1/(-2 + 1j)]),
|
||
|
|
('iuc', [1/(-2 + 1j), 1/(0.5 + 0.5j)]),
|
||
|
|
('ouc', [1/(0.5 + 0.5j), 1/(-2 + 1j)])]
|
||
|
|
# 'lhp' is ambiguous so don't test it
|
||
|
|
A3 = np.eye(2)
|
||
|
|
B3 = np.diag([2, 0])
|
||
|
|
expected3 = [('rhp', [0.5, np.inf]),
|
||
|
|
('iuc', [0.5, np.inf]),
|
||
|
|
('ouc', [np.inf, 0.5])]
|
||
|
|
# 'rhp' is ambiguous so don't test it
|
||
|
|
A4 = np.eye(2)
|
||
|
|
B4 = np.diag([-2, 0])
|
||
|
|
expected4 = [('lhp', [-0.5, np.inf]),
|
||
|
|
('iuc', [-0.5, np.inf]),
|
||
|
|
('ouc', [np.inf, -0.5])]
|
||
|
|
A5 = np.diag([0, 1])
|
||
|
|
B5 = np.diag([0, 0.5])
|
||
|
|
# 'lhp' and 'iuc' are ambiguous so don't test them
|
||
|
|
expected5 = [('rhp', [2, np.nan]),
|
||
|
|
('ouc', [2, np.nan])]
|
||
|
|
|
||
|
|
A = [A1, A2, A3, A4, A5]
|
||
|
|
B = [B1, B2, B3, B4, B5]
|
||
|
|
expected = [expected1, expected2, expected3, expected4, expected5]
|
||
|
|
for Ai, Bi, expectedi in zip(A, B, expected):
|
||
|
|
for sortstr, expected_eigvals in expectedi:
|
||
|
|
_, _, alpha, beta, _, _ = ordqz(Ai, Bi, sort=sortstr)
|
||
|
|
azero = (alpha == 0)
|
||
|
|
bzero = (beta == 0)
|
||
|
|
x = np.empty_like(alpha)
|
||
|
|
x[azero & bzero] = np.nan
|
||
|
|
x[~azero & bzero] = np.inf
|
||
|
|
x[~bzero] = alpha[~bzero]/beta[~bzero]
|
||
|
|
assert_allclose(expected_eigvals, x)
|
||
|
|
|
||
|
|
|
||
|
|
class TestOrdQZWorkspaceSize:
|
||
|
|
def test_decompose(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
N = 202
|
||
|
|
# raises error if lwork parameter to dtrsen is too small
|
||
|
|
for ddtype in [np.float32, np.float64]:
|
||
|
|
A = rng.random((N, N)).astype(ddtype)
|
||
|
|
B = rng.random((N, N)).astype(ddtype)
|
||
|
|
# sort = lambda ar, ai, b: ar**2 + ai**2 < b**2
|
||
|
|
_ = ordqz(A, B, sort=lambda alpha, beta: alpha < beta,
|
||
|
|
output='real')
|
||
|
|
|
||
|
|
for ddtype in [np.complex128, np.complex64]:
|
||
|
|
A = rng.random((N, N)).astype(ddtype)
|
||
|
|
B = rng.random((N, N)).astype(ddtype)
|
||
|
|
_ = ordqz(A, B, sort=lambda alpha, beta: alpha < beta,
|
||
|
|
output='complex')
|
||
|
|
|
||
|
|
@pytest.mark.slow
|
||
|
|
def test_decompose_ouc(self):
|
||
|
|
rng = np.random.RandomState(12345)
|
||
|
|
N = 202
|
||
|
|
# segfaults if lwork parameter to dtrsen is too small
|
||
|
|
for ddtype in [np.float32, np.float64, np.complex128, np.complex64]:
|
||
|
|
A = rng.random((N, N)).astype(ddtype)
|
||
|
|
B = rng.random((N, N)).astype(ddtype)
|
||
|
|
S, T, alpha, beta, U, V = ordqz(A, B, sort='ouc')
|
||
|
|
|
||
|
|
|
||
|
|
class TestDatacopied:
|
||
|
|
|
||
|
|
def test_datacopied(self):
|
||
|
|
from scipy.linalg._decomp import _datacopied
|
||
|
|
|
||
|
|
M = matrix([[0, 1], [2, 3]])
|
||
|
|
A = asarray(M)
|
||
|
|
L = M.tolist()
|
||
|
|
M2 = M.copy()
|
||
|
|
|
||
|
|
class Fake1:
|
||
|
|
def __array__(self, dtype=None, copy=None):
|
||
|
|
return A
|
||
|
|
|
||
|
|
class Fake2:
|
||
|
|
__array_interface__ = A.__array_interface__
|
||
|
|
|
||
|
|
F1 = Fake1()
|
||
|
|
F2 = Fake2()
|
||
|
|
|
||
|
|
for item, status in [(M, False), (A, False), (L, True),
|
||
|
|
(M2, False), (F1, False), (F2, False)]:
|
||
|
|
arr = asarray(item)
|
||
|
|
assert_equal(_datacopied(arr, item), status,
|
||
|
|
err_msg=repr(item))
|
||
|
|
|
||
|
|
|
||
|
|
def test_aligned_mem_float():
|
||
|
|
"""Check linalg works with non-aligned memory (float32)"""
|
||
|
|
# Allocate 402 bytes of memory (allocated on boundary)
|
||
|
|
a = arange(402, dtype=np.uint8)
|
||
|
|
|
||
|
|
# Create an array with boundary offset 4
|
||
|
|
z = np.frombuffer(a.data, offset=2, count=100, dtype=float32)
|
||
|
|
z.shape = 10, 10
|
||
|
|
|
||
|
|
eig(z, overwrite_a=True)
|
||
|
|
eig(z.T, overwrite_a=True)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.skipif(platform.machine() == 'ppc64le',
|
||
|
|
reason="crashes on ppc64le")
|
||
|
|
def test_aligned_mem():
|
||
|
|
"""Check linalg works with non-aligned memory (float64)"""
|
||
|
|
# Allocate 804 bytes of memory (allocated on boundary)
|
||
|
|
a = arange(804, dtype=np.uint8)
|
||
|
|
|
||
|
|
# Create an array with boundary offset 4
|
||
|
|
z = np.frombuffer(a.data, offset=4, count=100, dtype=float)
|
||
|
|
z.shape = 10, 10
|
||
|
|
|
||
|
|
eig(z, overwrite_a=True)
|
||
|
|
eig(z.T, overwrite_a=True)
|
||
|
|
|
||
|
|
|
||
|
|
def test_aligned_mem_complex():
|
||
|
|
"""Check that complex objects don't need to be completely aligned"""
|
||
|
|
# Allocate 1608 bytes of memory (allocated on boundary)
|
||
|
|
a = zeros(1608, dtype=np.uint8)
|
||
|
|
|
||
|
|
# Create an array with boundary offset 8
|
||
|
|
z = np.frombuffer(a.data, offset=8, count=100, dtype=complex)
|
||
|
|
z.shape = 10, 10
|
||
|
|
|
||
|
|
eig(z, overwrite_a=True)
|
||
|
|
# This does not need special handling
|
||
|
|
eig(z.T, overwrite_a=True)
|
||
|
|
|
||
|
|
|
||
|
|
def check_lapack_misaligned(func, args, kwargs):
|
||
|
|
args = list(args)
|
||
|
|
for i in range(len(args)):
|
||
|
|
a = args[:]
|
||
|
|
if isinstance(a[i], np.ndarray):
|
||
|
|
# Try misaligning a[i]
|
||
|
|
aa = np.zeros(a[i].size*a[i].dtype.itemsize+8, dtype=np.uint8)
|
||
|
|
aa = np.frombuffer(aa.data, offset=4, count=a[i].size,
|
||
|
|
dtype=a[i].dtype)
|
||
|
|
aa.shape = a[i].shape
|
||
|
|
aa[...] = a[i]
|
||
|
|
a[i] = aa
|
||
|
|
func(*a, **kwargs)
|
||
|
|
if len(a[i].shape) > 1:
|
||
|
|
a[i] = a[i].T
|
||
|
|
func(*a, **kwargs)
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.xfail(run=False,
|
||
|
|
reason="Ticket #1152, triggers a segfault in rare cases.")
|
||
|
|
def test_lapack_misaligned():
|
||
|
|
M = np.eye(10, dtype=float)
|
||
|
|
R = np.arange(100)
|
||
|
|
R.shape = 10, 10
|
||
|
|
S = np.arange(20000, dtype=np.uint8)
|
||
|
|
S = np.frombuffer(S.data, offset=4, count=100, dtype=float)
|
||
|
|
S.shape = 10, 10
|
||
|
|
b = np.ones(10)
|
||
|
|
LU, piv = lu_factor(S)
|
||
|
|
for (func, args, kwargs) in [
|
||
|
|
(eig, (S,), dict(overwrite_a=True)), # crash
|
||
|
|
(eigvals, (S,), dict(overwrite_a=True)), # no crash
|
||
|
|
(lu, (S,), dict(overwrite_a=True)), # no crash
|
||
|
|
(lu_factor, (S,), dict(overwrite_a=True)), # no crash
|
||
|
|
(lu_solve, ((LU, piv), b), dict(overwrite_b=True)),
|
||
|
|
(solve, (S, b), dict(overwrite_a=True, overwrite_b=True)),
|
||
|
|
(svd, (M,), dict(overwrite_a=True)), # no crash
|
||
|
|
(svd, (R,), dict(overwrite_a=True)), # no crash
|
||
|
|
(svd, (S,), dict(overwrite_a=True)), # crash
|
||
|
|
(svdvals, (S,), dict()), # no crash
|
||
|
|
(svdvals, (S,), dict(overwrite_a=True)), # crash
|
||
|
|
(cholesky, (M,), dict(overwrite_a=True)), # no crash
|
||
|
|
(qr, (S,), dict(overwrite_a=True)), # crash
|
||
|
|
(rq, (S,), dict(overwrite_a=True)), # crash
|
||
|
|
(hessenberg, (S,), dict(overwrite_a=True)), # crash
|
||
|
|
(schur, (S,), dict(overwrite_a=True)), # crash
|
||
|
|
]:
|
||
|
|
check_lapack_misaligned(func, args, kwargs)
|
||
|
|
# not properly tested
|
||
|
|
# cholesky, rsf2csf, lu_solve, solve, eig_banded, eigvals_banded, eigh, diagsvd
|
||
|
|
|
||
|
|
|
||
|
|
class TestOverwrite:
|
||
|
|
def test_eig(self):
|
||
|
|
assert_no_overwrite(eig, [(3, 3)])
|
||
|
|
assert_no_overwrite(eig, [(3, 3), (3, 3)])
|
||
|
|
|
||
|
|
def test_eigh(self):
|
||
|
|
assert_no_overwrite(eigh, [(3, 3)])
|
||
|
|
assert_no_overwrite(eigh, [(3, 3), (3, 3)])
|
||
|
|
|
||
|
|
def test_eig_banded(self):
|
||
|
|
assert_no_overwrite(eig_banded, [(3, 2)])
|
||
|
|
|
||
|
|
def test_eigvals(self):
|
||
|
|
assert_no_overwrite(eigvals, [(3, 3)])
|
||
|
|
|
||
|
|
def test_eigvalsh(self):
|
||
|
|
assert_no_overwrite(eigvalsh, [(3, 3)])
|
||
|
|
|
||
|
|
def test_eigvals_banded(self):
|
||
|
|
assert_no_overwrite(eigvals_banded, [(3, 2)])
|
||
|
|
|
||
|
|
def test_hessenberg(self):
|
||
|
|
assert_no_overwrite(hessenberg, [(3, 3)])
|
||
|
|
|
||
|
|
def test_lu_factor(self):
|
||
|
|
assert_no_overwrite(lu_factor, [(3, 3)])
|
||
|
|
|
||
|
|
def test_lu_solve(self):
|
||
|
|
x = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 8]])
|
||
|
|
xlu = lu_factor(x)
|
||
|
|
assert_no_overwrite(lambda b: lu_solve(xlu, b), [(3,)])
|
||
|
|
|
||
|
|
def test_lu(self):
|
||
|
|
assert_no_overwrite(lu, [(3, 3)])
|
||
|
|
|
||
|
|
def test_qr(self):
|
||
|
|
assert_no_overwrite(qr, [(3, 3)])
|
||
|
|
|
||
|
|
def test_rq(self):
|
||
|
|
assert_no_overwrite(rq, [(3, 3)])
|
||
|
|
|
||
|
|
def test_schur(self):
|
||
|
|
assert_no_overwrite(schur, [(3, 3)])
|
||
|
|
|
||
|
|
def test_schur_complex(self):
|
||
|
|
assert_no_overwrite(lambda a: schur(a, 'complex'), [(3, 3)],
|
||
|
|
dtypes=[np.float32, np.float64])
|
||
|
|
|
||
|
|
def test_svd(self):
|
||
|
|
assert_no_overwrite(svd, [(3, 3)])
|
||
|
|
assert_no_overwrite(lambda a: svd(a, lapack_driver='gesvd'), [(3, 3)])
|
||
|
|
|
||
|
|
def test_svdvals(self):
|
||
|
|
assert_no_overwrite(svdvals, [(3, 3)])
|
||
|
|
|
||
|
|
|
||
|
|
def _check_orth(n, dtype, skip_big=False):
|
||
|
|
X = np.ones((n, 2), dtype=float).astype(dtype)
|
||
|
|
|
||
|
|
eps = np.finfo(dtype).eps
|
||
|
|
tol = 1000 * eps
|
||
|
|
|
||
|
|
Y = orth(X)
|
||
|
|
assert_equal(Y.shape, (n, 1))
|
||
|
|
assert_allclose(Y, Y.mean(), atol=tol)
|
||
|
|
|
||
|
|
Y = orth(X.T)
|
||
|
|
assert_equal(Y.shape, (2, 1))
|
||
|
|
assert_allclose(Y, Y.mean(), atol=tol)
|
||
|
|
|
||
|
|
if n > 5 and not skip_big:
|
||
|
|
np.random.seed(1)
|
||
|
|
X = np.random.rand(n, 5) @ np.random.rand(5, n)
|
||
|
|
X = X + 1e-4 * np.random.rand(n, 1) @ np.random.rand(1, n)
|
||
|
|
X = X.astype(dtype)
|
||
|
|
|
||
|
|
Y = orth(X, rcond=1e-3)
|
||
|
|
assert_equal(Y.shape, (n, 5))
|
||
|
|
|
||
|
|
Y = orth(X, rcond=1e-6)
|
||
|
|
assert_equal(Y.shape, (n, 5 + 1))
|
||
|
|
|
||
|
|
|
||
|
|
@pytest.mark.slow
|
||
|
|
@pytest.mark.skipif(np.dtype(np.intp).itemsize < 8,
|
||
|
|
reason="test only on 64-bit, else too slow")
|
||
|
|
def test_orth_memory_efficiency():
|
||
|
|
# Pick n so that 16*n bytes is reasonable but 8*n*n bytes is unreasonable.
|
||
|
|
# Keep in mind that @pytest.mark.slow tests are likely to be running
|
||
|
|
# under configurations that support 4Gb+ memory for tests related to
|
||
|
|
# 32 bit overflow.
|
||
|
|
n = 10*1000*1000
|
||
|
|
try:
|
||
|
|
_check_orth(n, np.float64, skip_big=True)
|
||
|
|
except MemoryError as e:
|
||
|
|
raise AssertionError(
|
||
|
|
'memory error perhaps caused by orth regression'
|
||
|
|
) from e
|
||
|
|
|
||
|
|
|
||
|
|
def test_orth():
|
||
|
|
dtypes = [np.float32, np.float64, np.complex64, np.complex128]
|
||
|
|
sizes = [1, 2, 3, 10, 100]
|
||
|
|
for dt, n in itertools.product(dtypes, sizes):
|
||
|
|
_check_orth(n, dt)
|
||
|
|
|
||
|
|
|
||
|
|
def test_null_space():
|
||
|
|
np.random.seed(1)
|
||
|
|
|
||
|
|
dtypes = [np.float32, np.float64, np.complex64, np.complex128]
|
||
|
|
sizes = [1, 2, 3, 10, 100]
|
||
|
|
|
||
|
|
for dt, n in itertools.product(dtypes, sizes):
|
||
|
|
X = np.ones((2, n), dtype=dt)
|
||
|
|
|
||
|
|
eps = np.finfo(dt).eps
|
||
|
|
tol = 1000 * eps
|
||
|
|
|
||
|
|
Y = null_space(X)
|
||
|
|
assert_equal(Y.shape, (n, n-1))
|
||
|
|
assert_allclose(X @ Y, 0, atol=tol)
|
||
|
|
|
||
|
|
Y = null_space(X.T)
|
||
|
|
assert_equal(Y.shape, (2, 1))
|
||
|
|
assert_allclose(X.T @ Y, 0, atol=tol)
|
||
|
|
|
||
|
|
X = np.random.randn(1 + n//2, n)
|
||
|
|
Y = null_space(X)
|
||
|
|
assert_equal(Y.shape, (n, n - 1 - n//2))
|
||
|
|
assert_allclose(X @ Y, 0, atol=tol)
|
||
|
|
|
||
|
|
if n > 5:
|
||
|
|
np.random.seed(1)
|
||
|
|
X = np.random.rand(n, 5) @ np.random.rand(5, n)
|
||
|
|
X = X + 1e-4 * np.random.rand(n, 1) @ np.random.rand(1, n)
|
||
|
|
X = X.astype(dt)
|
||
|
|
|
||
|
|
Y = null_space(X, rcond=1e-3)
|
||
|
|
assert_equal(Y.shape, (n, n - 5))
|
||
|
|
|
||
|
|
Y = null_space(X, rcond=1e-6)
|
||
|
|
assert_equal(Y.shape, (n, n - 6))
|
||
|
|
|
||
|
|
|
||
|
|
def test_subspace_angles():
|
||
|
|
H = hadamard(8, float)
|
||
|
|
A = H[:, :3]
|
||
|
|
B = H[:, 3:]
|
||
|
|
assert_allclose(subspace_angles(A, B), [np.pi / 2.] * 3, atol=1e-14)
|
||
|
|
assert_allclose(subspace_angles(B, A), [np.pi / 2.] * 3, atol=1e-14)
|
||
|
|
for x in (A, B):
|
||
|
|
assert_allclose(subspace_angles(x, x), np.zeros(x.shape[1]),
|
||
|
|
atol=1e-14)
|
||
|
|
# From MATLAB function "subspace", which effectively only returns the
|
||
|
|
# last value that we calculate
|
||
|
|
x = np.array(
|
||
|
|
[[0.537667139546100, 0.318765239858981, 3.578396939725760, 0.725404224946106], # noqa: E501
|
||
|
|
[1.833885014595086, -1.307688296305273, 2.769437029884877, -0.063054873189656], # noqa: E501
|
||
|
|
[-2.258846861003648, -0.433592022305684, -1.349886940156521, 0.714742903826096], # noqa: E501
|
||
|
|
[0.862173320368121, 0.342624466538650, 3.034923466331855, -0.204966058299775]]) # noqa: E501
|
||
|
|
expected = 1.481454682101605
|
||
|
|
assert_allclose(subspace_angles(x[:, :2], x[:, 2:])[0], expected,
|
||
|
|
rtol=1e-12)
|
||
|
|
assert_allclose(subspace_angles(x[:, 2:], x[:, :2])[0], expected,
|
||
|
|
rtol=1e-12)
|
||
|
|
expected = 0.746361174247302
|
||
|
|
assert_allclose(subspace_angles(x[:, :2], x[:, [2]]), expected, rtol=1e-12)
|
||
|
|
assert_allclose(subspace_angles(x[:, [2]], x[:, :2]), expected, rtol=1e-12)
|
||
|
|
expected = 0.487163718534313
|
||
|
|
assert_allclose(subspace_angles(x[:, :3], x[:, [3]]), expected, rtol=1e-12)
|
||
|
|
assert_allclose(subspace_angles(x[:, [3]], x[:, :3]), expected, rtol=1e-12)
|
||
|
|
expected = 0.328950515907756
|
||
|
|
assert_allclose(subspace_angles(x[:, :2], x[:, 1:]), [expected, 0],
|
||
|
|
atol=1e-12)
|
||
|
|
# Degenerate conditions
|
||
|
|
assert_raises(ValueError, subspace_angles, x[0], x)
|
||
|
|
assert_raises(ValueError, subspace_angles, x, x[0])
|
||
|
|
assert_raises(ValueError, subspace_angles, x[:-1], x)
|
||
|
|
|
||
|
|
# Test branch if mask.any is True:
|
||
|
|
A = np.array([[1, 0, 0],
|
||
|
|
[0, 1, 0],
|
||
|
|
[0, 0, 1],
|
||
|
|
[0, 0, 0],
|
||
|
|
[0, 0, 0]])
|
||
|
|
B = np.array([[1, 0, 0],
|
||
|
|
[0, 1, 0],
|
||
|
|
[0, 0, 0],
|
||
|
|
[0, 0, 0],
|
||
|
|
[0, 0, 1]])
|
||
|
|
expected = np.array([np.pi/2, 0, 0])
|
||
|
|
assert_allclose(subspace_angles(A, B), expected, rtol=1e-12)
|
||
|
|
|
||
|
|
# Complex
|
||
|
|
# second column in "b" does not affect result, just there so that
|
||
|
|
# b can have more cols than a, and vice-versa (both conditional code paths)
|
||
|
|
a = [[1 + 1j], [0]]
|
||
|
|
b = [[1 - 1j, 0], [0, 1]]
|
||
|
|
assert_allclose(subspace_angles(a, b), 0., atol=1e-14)
|
||
|
|
assert_allclose(subspace_angles(b, a), 0., atol=1e-14)
|
||
|
|
|
||
|
|
|
||
|
|
class TestCDF2RDF:
|
||
|
|
|
||
|
|
def matmul(self, a, b):
|
||
|
|
return np.einsum('...ij,...jk->...ik', a, b)
|
||
|
|
|
||
|
|
def assert_eig_valid(self, w, v, x):
|
||
|
|
assert_array_almost_equal(
|
||
|
|
self.matmul(v, w),
|
||
|
|
self.matmul(x, v)
|
||
|
|
)
|
||
|
|
|
||
|
|
def test_single_array0x0real(self):
|
||
|
|
# eig doesn't support 0x0 in old versions of numpy
|
||
|
|
X = np.empty((0, 0))
|
||
|
|
w, v = np.empty(0), np.empty((0, 0))
|
||
|
|
wr, vr = cdf2rdf(w, v)
|
||
|
|
self.assert_eig_valid(wr, vr, X)
|
||
|
|
|
||
|
|
def test_single_array2x2_real(self):
|
||
|
|
X = np.array([[1, 2], [3, -1]])
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
wr, vr = cdf2rdf(w, v)
|
||
|
|
self.assert_eig_valid(wr, vr, X)
|
||
|
|
|
||
|
|
def test_single_array2x2_complex(self):
|
||
|
|
X = np.array([[1, 2], [-2, 1]])
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
wr, vr = cdf2rdf(w, v)
|
||
|
|
self.assert_eig_valid(wr, vr, X)
|
||
|
|
|
||
|
|
def test_single_array3x3_real(self):
|
||
|
|
X = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6]])
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
wr, vr = cdf2rdf(w, v)
|
||
|
|
self.assert_eig_valid(wr, vr, X)
|
||
|
|
|
||
|
|
def test_single_array3x3_complex(self):
|
||
|
|
X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]])
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
wr, vr = cdf2rdf(w, v)
|
||
|
|
self.assert_eig_valid(wr, vr, X)
|
||
|
|
|
||
|
|
def test_random_1d_stacked_arrays(self):
|
||
|
|
# cannot test M == 0 due to bug in old numpy
|
||
|
|
for M in range(1, 7):
|
||
|
|
np.random.seed(999999999)
|
||
|
|
X = np.random.rand(100, M, M)
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
wr, vr = cdf2rdf(w, v)
|
||
|
|
self.assert_eig_valid(wr, vr, X)
|
||
|
|
|
||
|
|
def test_random_2d_stacked_arrays(self):
|
||
|
|
# cannot test M == 0 due to bug in old numpy
|
||
|
|
for M in range(1, 7):
|
||
|
|
X = np.random.rand(10, 10, M, M)
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
wr, vr = cdf2rdf(w, v)
|
||
|
|
self.assert_eig_valid(wr, vr, X)
|
||
|
|
|
||
|
|
def test_low_dimensionality_error(self):
|
||
|
|
w, v = np.empty(()), np.array((2,))
|
||
|
|
assert_raises(ValueError, cdf2rdf, w, v)
|
||
|
|
|
||
|
|
def test_not_square_error(self):
|
||
|
|
# Check that passing a non-square array raises a ValueError.
|
||
|
|
w, v = np.arange(3), np.arange(6).reshape(3, 2)
|
||
|
|
assert_raises(ValueError, cdf2rdf, w, v)
|
||
|
|
|
||
|
|
def test_swapped_v_w_error(self):
|
||
|
|
# Check that exchanging places of w and v raises ValueError.
|
||
|
|
X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]])
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
assert_raises(ValueError, cdf2rdf, v, w)
|
||
|
|
|
||
|
|
def test_non_associated_error(self):
|
||
|
|
# Check that passing non-associated eigenvectors raises a ValueError.
|
||
|
|
w, v = np.arange(3), np.arange(16).reshape(4, 4)
|
||
|
|
assert_raises(ValueError, cdf2rdf, w, v)
|
||
|
|
|
||
|
|
def test_not_conjugate_pairs(self):
|
||
|
|
# Check that passing non-conjugate pairs raises a ValueError.
|
||
|
|
X = np.array([[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]])
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
assert_raises(ValueError, cdf2rdf, w, v)
|
||
|
|
|
||
|
|
# different arrays in the stack, so not conjugate
|
||
|
|
X = np.array([
|
||
|
|
[[1, 2, 3], [1, 2, 3], [2, 5, 6+1j]],
|
||
|
|
[[1, 2, 3], [1, 2, 3], [2, 5, 6-1j]],
|
||
|
|
])
|
||
|
|
w, v = np.linalg.eig(X)
|
||
|
|
assert_raises(ValueError, cdf2rdf, w, v)
|