EVOLUTION-MANAGER

Edit File: test_decomp.py

""" Test functions for linalg.decomp module """ from __future__ import division, print_function, absolute_import __usage__ = """ Build linalg: python setup_linalg.py build Run tests if scipy is installed: python -c 'import scipy;scipy.linalg.test()' """ import itertools import platform import numpy as np from numpy.testing import (assert_equal, assert_almost_equal, assert_array_almost_equal, assert_array_equal, assert_, assert_allclose) import pytest from pytest import raises as assert_raises from scipy._lib.six import xrange from scipy.linalg import (eig, eigvals, lu, svd, svdvals, cholesky, qr, schur, rsf2csf, lu_solve, lu_factor, solve, diagsvd, hessenberg, rq, eig_banded, eigvals_banded, eigh, eigvalsh, qr_multiply, qz, orth, ordqz, subspace_angles, hadamard, eigvalsh_tridiagonal, eigh_tridiagonal, null_space, cdf2rdf) from scipy.linalg.lapack import dgbtrf, dgbtrs, zgbtrf, zgbtrs, \ dsbev, dsbevd, dsbevx, zhbevd, zhbevx from scipy.linalg.misc import norm from scipy.linalg._decomp_qz import _select_function from numpy import array, transpose, diag, ones, full, linalg, \ argsort, zeros, arange, float32, complex64, dot, conj, identity, \ ravel, sqrt, iscomplex, shape, sort, conjugate, sign, \ asarray, isfinite, ndarray, outer, eye, dtype, empty,\ triu, tril from numpy.random import normal, seed, random from scipy.linalg._testutils import assert_no_overwrite from scipy.sparse.sputils import bmat, matrix # digit precision to use in asserts for different types DIGITS = {'d':11, 'D':11, 'f':4, 'F':4} def clear_fuss(ar, fuss_binary_bits=7): """Clears trailing `fuss_binary_bits` of mantissa of a floating number""" x = np.asanyarray(ar) if np.iscomplexobj(x): return clear_fuss(x.real) + 1j * clear_fuss(x.imag) significant_binary_bits = np.finfo(x.dtype).nmant x_mant, x_exp = np.frexp(x) f = 2.0**(significant_binary_bits - fuss_binary_bits) x_mant *= f np.rint(x_mant, out=x_mant) x_mant /= f return np.ldexp(x_mant, x_exp) # XXX: This function should be available through numpy.testing def assert_dtype_equal(act, des): if isinstance(act, ndarray): act = act.dtype else: act = dtype(act) if isinstance(des, ndarray): des = des.dtype else: des = dtype(des) assert_(act == des, 'dtype mismatch: "%s" (should be "%s") ' % (act, des)) # XXX: This function should not be defined here, but somewhere in # scipy.linalg namespace def symrand(dim_or_eigv): """Return a random symmetric (Hermitian) matrix. If 'dim_or_eigv' is an integer N, return a NxN matrix, with eigenvalues uniformly distributed on (-1,1). If 'dim_or_eigv' is 1-D real array 'a', return a matrix whose eigenvalues are 'a'. """ if isinstance(dim_or_eigv, int): dim = dim_or_eigv d = random(dim)*2 - 1 elif (isinstance(dim_or_eigv, ndarray) and len(dim_or_eigv.shape) == 1): dim = dim_or_eigv.shape[0] d = dim_or_eigv else: raise TypeError("input type not supported.") v = random_rot(dim) h = dot(dot(v.T.conj(), diag(d)), v) # to avoid roundoff errors, symmetrize the matrix (again) h = 0.5*(h.T+h) return h # XXX: This function should not be defined here, but somewhere in # scipy.linalg namespace def random_rot(dim): """Return a random rotation matrix, drawn from the Haar distribution (the only uniform distribution on SO(n)). The algorithm is described in the paper Stewart, G.W., 'The efficient generation of random orthogonal matrices with an application to condition estimators', SIAM Journal on Numerical Analysis, 17(3), pp. 403-409, 1980. For more information see https://en.wikipedia.org/wiki/Orthogonal_matrix#Randomization""" H = eye(dim) D = ones((dim,)) for n in range(1, dim): x = normal(size=(dim-n+1,)) D[n-1] = sign(x[0]) x[0] -= D[n-1]*sqrt((x*x).sum()) # Householder transformation Hx = eye(dim-n+1) - 2.*outer(x, x)/(x*x).sum() mat = eye(dim) mat[n-1:,n-1:] = Hx H = dot(H, mat) # Fix the last sign such that the determinant is 1 D[-1] = -D.prod() H = (D*H.T).T return H class TestEigVals(object): def test_simple(self): a = [[1,2,3],[1,2,3],[2,5,6]] w = eigvals(a) exact_w = [(9+sqrt(93))/2,0,(9-sqrt(93))/2] assert_array_almost_equal(w,exact_w) def test_simple_tr(self): a = array([[1,2,3],[1,2,3],[2,5,6]],'d') a = transpose(a).copy() a = transpose(a) w = eigvals(a) exact_w = [(9+sqrt(93))/2,0,(9-sqrt(93))/2] assert_array_almost_equal(w,exact_w) def test_simple_complex(self): a = [[1,2,3],[1,2,3],[2,5,6+1j]] w = eigvals(a) exact_w = [(9+1j+sqrt(92+6j))/2, 0, (9+1j-sqrt(92+6j))/2] assert_array_almost_equal(w,exact_w) def test_finite(self): a = [[1,2,3],[1,2,3],[2,5,6]] w = eigvals(a, check_finite=False) exact_w = [(9+sqrt(93))/2,0,(9-sqrt(93))/2] assert_array_almost_equal(w,exact_w) class TestEig(object): def test_simple(self): a = [[1,2,3],[1,2,3],[2,5,6]] w,v = eig(a) exact_w = [(9+sqrt(93))/2,0,(9-sqrt(93))/2] v0 = array([1,1,(1+sqrt(93)/3)/2]) v1 = array([3.,0,-1]) v2 = array([1,1,(1-sqrt(93)/3)/2]) v0 = v0 / sqrt(dot(v0,transpose(v0))) v1 = v1 / sqrt(dot(v1,transpose(v1))) v2 = v2 / sqrt(dot(v2,transpose(v2))) assert_array_almost_equal(w,exact_w) assert_array_almost_equal(v0,v[:,0]*sign(v[0,0])) assert_array_almost_equal(v1,v[:,1]*sign(v[0,1])) assert_array_almost_equal(v2,v[:,2]*sign(v[0,2])) for i in range(3): assert_array_almost_equal(dot(a,v[:,i]),w[i]*v[:,i]) w,v = eig(a,left=1,right=0) for i in range(3): assert_array_almost_equal(dot(transpose(a),v[:,i]),w[i]*v[:,i]) def test_simple_complex_eig(self): a = [[1,2],[-2,1]] w,vl,vr = eig(a,left=1,right=1) assert_array_almost_equal(w, array([1+2j, 1-2j])) for i in range(2): assert_array_almost_equal(dot(a,vr[:,i]),w[i]*vr[:,i]) for i in range(2): assert_array_almost_equal(dot(conjugate(transpose(a)),vl[:,i]), conjugate(w[i])*vl[:,i]) def test_simple_complex(self): a = [[1,2,3],[1,2,3],[2,5,6+1j]] w,vl,vr = eig(a,left=1,right=1) for i in range(3): assert_array_almost_equal(dot(a,vr[:,i]),w[i]*vr[:,i]) for i in range(3): assert_array_almost_equal(dot(conjugate(transpose(a)),vl[:,i]), conjugate(w[i])*vl[:,i]) def test_gh_3054(self): a = [[1]] b = [[0]] w, vr = eig(a, b, homogeneous_eigvals=True) assert_allclose(w[1,0], 0) assert_(w[0,0] != 0) assert_allclose(vr, 1) w, vr = eig(a, b) assert_equal(w, np.inf) assert_allclose(vr, 1) def _check_gen_eig(self, A, B): if B is not None: A, B = asarray(A), asarray(B) B0 = B else: A = asarray(A) B0 = B B = np.eye(*A.shape) msg = "\n%r\n%r" % (A, B) # Eigenvalues in homogeneous coordinates w, vr = eig(A, B0, homogeneous_eigvals=True) wt = eigvals(A, B0, homogeneous_eigvals=True) val1 = dot(A, vr) * w[1,:] val2 = dot(B, vr) * w[0,:] for i in range(val1.shape[1]): assert_allclose(val1[:,i], val2[:,i], rtol=1e-13, atol=1e-13, err_msg=msg) if B0 is None: assert_allclose(w[1,:], 1) assert_allclose(wt[1,:], 1) perm = np.lexsort(w) permt = np.lexsort(wt) assert_allclose(w[:,perm], wt[:,permt], atol=1e-7, rtol=1e-7, err_msg=msg) length = np.empty(len(vr)) for i in xrange(len(vr)): length[i] = norm(vr[:,i]) assert_allclose(length, np.ones(length.size), err_msg=msg, atol=1e-7, rtol=1e-7) # Convert homogeneous coordinates beta_nonzero = (w[1,:] != 0) wh = w[0,beta_nonzero] / w[1,beta_nonzero] # Eigenvalues in standard coordinates w, vr = eig(A, B0) wt = eigvals(A, B0) val1 = dot(A, vr) val2 = dot(B, vr) * w res = val1 - val2 for i in range(res.shape[1]): if np.all(isfinite(res[:,i])): assert_allclose(res[:,i], 0, rtol=1e-13, atol=1e-13, err_msg=msg) w_fin = w[isfinite(w)] wt_fin = wt[isfinite(wt)] perm = argsort(clear_fuss(w_fin)) permt = argsort(clear_fuss(wt_fin)) assert_allclose(w[perm], wt[permt], atol=1e-7, rtol=1e-7, err_msg=msg) length = np.empty(len(vr)) for i in xrange(len(vr)): length[i] = norm(vr[:,i]) assert_allclose(length, np.ones(length.size), err_msg=msg) # Compare homogeneous and nonhomogeneous versions assert_allclose(sort(wh), sort(w[np.isfinite(w)])) @pytest.mark.xfail(reason="See gh-2254") def test_singular(self): # Example taken from # https://web.archive.org/web/20040903121217/http://www.cs.umu.se/research/nla/singular_pairs/guptri/matlab.html A = array(([22,34,31,31,17], [45,45,42,19,29], [39,47,49,26,34], [27,31,26,21,15], [38,44,44,24,30])) B = array(([13,26,25,17,24], [31,46,40,26,37], [26,40,19,25,25], [16,25,27,14,23], [24,35,18,21,22])) olderr = np.seterr(all='ignore') try: self._check_gen_eig(A, B) finally: np.seterr(**olderr) def test_falker(self): # Test matrices giving some Nan generalized eigenvalues. M = diag(array(([1,0,3]))) K = array(([2,-1,-1],[-1,2,-1],[-1,-1,2])) D = array(([1,-1,0],[-1,1,0],[0,0,0])) Z = zeros((3,3)) I3 = identity(3) A = bmat([[I3, Z], [Z, -K]]) B = bmat([[Z, I3], [M, D]]) olderr = np.seterr(all='ignore') try: self._check_gen_eig(A, B) finally: np.seterr(**olderr) def test_bad_geneig(self): # Ticket #709 (strange return values from DGGEV) def matrices(omega): c1 = -9 + omega**2 c2 = 2*omega A = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, c1, 0], [0, 0, 0, c1]] B = [[0, 0, 1, 0], [0, 0, 0, 1], [1, 0, 0, -c2], [0, 1, c2, 0]] return A, B # With a buggy LAPACK, this can fail for different omega on different # machines -- so we need to test several values olderr = np.seterr(all='ignore') try: for k in xrange(100): A, B = matrices(omega=k*5./100) self._check_gen_eig(A, B) finally: np.seterr(**olderr) def test_make_eigvals(self): # Step through all paths in _make_eigvals seed(1234) # Real eigenvalues A = symrand(3) self._check_gen_eig(A, None) B = symrand(3) self._check_gen_eig(A, B) # Complex eigenvalues A = random((3, 3)) + 1j*random((3, 3)) self._check_gen_eig(A, None) B = random((3, 3)) + 1j*random((3, 3)) self._check_gen_eig(A, B) def test_check_finite(self): a = [[1,2,3],[1,2,3],[2,5,6]] w,v = eig(a, check_finite=False) exact_w = [(9+sqrt(93))/2,0,(9-sqrt(93))/2] v0 = array([1,1,(1+sqrt(93)/3)/2]) v1 = array([3.,0,-1]) v2 = array([1,1,(1-sqrt(93)/3)/2]) v0 = v0 / sqrt(dot(v0,transpose(v0))) v1 = v1 / sqrt(dot(v1,transpose(v1))) v2 = v2 / sqrt(dot(v2,transpose(v2))) assert_array_almost_equal(w,exact_w) assert_array_almost_equal(v0,v[:,0]*sign(v[0,0])) assert_array_almost_equal(v1,v[:,1]*sign(v[0,1])) assert_array_almost_equal(v2,v[:,2]*sign(v[0,2])) for i in range(3): assert_array_almost_equal(dot(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 = identity(2) B = np.arange(9.0).reshape(3,3) assert_raises(ValueError, eig, A, B) assert_raises(ValueError, eig, B, A) class TestEigBanded(object): 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 xrange(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 xrange(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 xrange(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 xrange(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 xrange(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(object): 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(): DIM = 6 v = {'dim': (DIM,), 'dtype': ('f','d','F','D'), 'overwrite': (True, False), 'lower': (True, False), 'turbo': (True, False), 'eigvals': (None, (2, DIM-2))} for dim in v['dim']: for typ in v['dtype']: for overwrite in v['overwrite']: for turbo in v['turbo']: for eigenvalues in v['eigvals']: for lower in v['lower']: eigenhproblem_standard( 'ordinary', dim, typ, overwrite, lower, turbo, eigenvalues) eigenhproblem_general( 'general ', dim, typ, overwrite, lower, turbo, eigenvalues) def test_eigh_of_sparse(): # 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) def _complex_symrand(dim, dtype): a1, a2 = symrand(dim), symrand(dim) # add antisymmetric matrix as imag part a = a1 + 1j*(triu(a2)-tril(a2)) return a.astype(dtype) def eigenhproblem_standard(desc, dim, dtype, overwrite, lower, turbo, eigenvalues): """Solve a standard eigenvalue problem.""" if iscomplex(empty(1, dtype=dtype)): a = _complex_symrand(dim, dtype) else: a = symrand(dim).astype(dtype) if overwrite: a_c = a.copy() else: a_c = a w, z = eigh(a, overwrite_a=overwrite, lower=lower, eigvals=eigenvalues) assert_dtype_equal(z.dtype, dtype) w = w.astype(dtype) diag_ = diag(dot(z.T.conj(), dot(a_c, z))).real assert_array_almost_equal(diag_, w, DIGITS[dtype]) def eigenhproblem_general(desc, dim, dtype, overwrite, lower, turbo, eigenvalues): """Solve a generalized eigenvalue problem.""" if iscomplex(empty(1, dtype=dtype)): a = _complex_symrand(dim, dtype) b = _complex_symrand(dim, dtype)+diag([2.1]*dim).astype(dtype) else: a = symrand(dim).astype(dtype) b = symrand(dim).astype(dtype)+diag([2.1]*dim).astype(dtype) if overwrite: a_c, b_c = a.copy(), b.copy() else: a_c, b_c = a, b w, z = eigh(a, b, overwrite_a=overwrite, lower=lower, overwrite_b=overwrite, turbo=turbo, eigvals=eigenvalues) assert_dtype_equal(z.dtype, dtype) w = w.astype(dtype) diag1_ = diag(dot(z.T.conj(), dot(a_c, z))).real assert_array_almost_equal(diag1_, w, DIGITS[dtype]) diag2_ = diag(dot(z.T.conj(), dot(b_c, z))).real assert_array_almost_equal(diag2_, ones(diag2_.shape[0]), DIGITS[dtype]) def test_eigh_integer(): a = array([[1,2],[2,7]]) b = array([[3,1],[1,5]]) w,z = eigh(a) w,z = eigh(a,b) class TestLU(object): def setup_method(self): self.a = array([[1,2,3],[1,2,3],[2,5,6]]) self.ca = array([[1,2,3],[1,2,3],[2,5j,6]]) # Those matrices are more robust to detect problems in permutation # matrices than the ones above self.b = array([[1,2,3],[4,5,6],[7,8,9]]) self.cb = array([[1j,2j,3j],[4j,5j,6j],[7j,8j,9j]]) # Reectangular matrices self.hrect = array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]]) self.chrect = 1.j * array([[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 12, 12]]) self.vrect = array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]]) self.cvrect = 1.j * array([[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 12, 12]]) # Medium sizes matrices self.med = random((30, 40)) self.cmed = random((30, 40)) + 1.j * random((30, 40)) def _test_common(self, data): p,l,u = lu(data) assert_array_almost_equal(dot(dot(p,l),u),data) pl,u = lu(data,permute_l=1) assert_array_almost_equal(dot(pl,u),data) # Simple tests def test_simple(self): self._test_common(self.a) def test_simple_complex(self): self._test_common(self.ca) def test_simple2(self): self._test_common(self.b) def test_simple2_complex(self): self._test_common(self.cb) # rectangular matrices tests def test_hrectangular(self): self._test_common(self.hrect) def test_vrectangular(self): self._test_common(self.vrect) def test_hrectangular_complex(self): self._test_common(self.chrect) def test_vrectangular_complex(self): self._test_common(self.cvrect) # Bigger matrices def test_medium1(self): """Check lu decomposition on medium size, rectangular matrix.""" self._test_common(self.med) def test_medium1_complex(self): """Check lu decomposition on medium size, rectangular matrix.""" self._test_common(self.cmed) def test_check_finite(self): p, l, u = lu(self.a, check_finite=False) assert_array_almost_equal(dot(dot(p,l),u), self.a) def test_simple_known(self): # Ticket #1458 for order in ['C', 'F']: A = np.array([[2, 1],[0, 1.]], order=order) LU, P = lu_factor(A) assert_array_almost_equal(LU, np.array([[2, 1], [0, 1]])) assert_array_equal(P, np.array([0, 1])) class TestLUSingle(TestLU): """LU testers for single precision, real and double""" def setup_method(self): TestLU.setup_method(self) self.a = self.a.astype(float32) self.ca = self.ca.astype(complex64) self.b = self.b.astype(float32) self.cb = self.cb.astype(complex64) self.hrect = self.hrect.astype(float32) self.chrect = self.hrect.astype(complex64) self.vrect = self.vrect.astype(float32) self.cvrect = self.vrect.astype(complex64) self.med = self.vrect.astype(float32) self.cmed = self.vrect.astype(complex64) class TestLUSolve(object): def setup_method(self): seed(1234) def test_lu(self): a0 = random((10,10)) b = random((10,)) for order in ['C', 'F']: a = np.array(a0, order=order) x1 = solve(a,b) lu_a = lu_factor(a) x2 = lu_solve(lu_a,b) assert_array_almost_equal(x1,x2) def test_check_finite(self): a = random((10,10)) b = random((10,)) x1 = solve(a,b) lu_a = lu_factor(a, check_finite=False) x2 = lu_solve(lu_a,b, check_finite=False) assert_array_almost_equal(x1,x2) class TestSVD_GESDD(object): def setup_method(self): self.lapack_driver = 'gesdd' seed(1234) 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(dot(transpose(u),u),identity(3)) assert_array_almost_equal(dot(transpose(vh),vh),identity(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(dot(dot(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(dot(transpose(u),u),identity(3)) assert_array_almost_equal(dot(transpose(vh),vh),identity(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(dot(dot(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(dot(transpose(u),u),identity(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(dot(dot(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(dot(transpose(u),u), identity(u.shape[1])) assert_array_almost_equal(dot(transpose(vh),vh),identity(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(dot(dot(u,sigma),vh),a) def test_random(self): n = 20 m = 15 for i in range(3): for a in [random([n,m]),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(dot(transpose(u),u),identity(u.shape[1])) assert_array_almost_equal(dot(vh, transpose(vh)),identity(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(dot(dot(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(dot(conj(transpose(u)),u),identity(u.shape[1])) assert_array_almost_equal(dot(conj(transpose(vh)),vh),identity(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(dot(dot(u,sigma),vh),a) def test_random_complex(self): n = 20 m = 15 for i in range(3): for full_matrices in (True, False): for a in [random([n,m]),random([m,n])]: a = a + 1j*random(list(a.shape)) u,s,vh = svd(a, full_matrices=full_matrices, lapack_driver=self.lapack_driver) assert_array_almost_equal(dot(conj(transpose(u)),u),identity(u.shape[1])) # This fails when [m,n] # assert_array_almost_equal(dot(conj(transpose(vh)),vh),identity(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(dot(dot(u,sigma),vh),a) def test_crash_1580(self): sizes = [(13, 23), (30, 50), (60, 100)] np.random.seed(1234) for sz in sizes: for dt in [np.float32, np.float64, np.complex64, np.complex128]: a = np.random.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(dot(transpose(u),u),identity(3)) assert_array_almost_equal(dot(transpose(vh),vh),identity(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(dot(dot(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) class TestSVD_GESVD(TestSVD_GESDD): def setup_method(self): self.lapack_driver = 'gesvd' seed(1234) class TestSVDVals(object): 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(object): 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(object): def setup_method(self): seed(1234) def test_simple(self): a = [[8,2,3],[2,9,3],[5,3,6]] q,r = qr(a) assert_array_almost_equal(dot(transpose(q),q),identity(3)) assert_array_almost_equal(dot(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(dot(q, c), qc) assert_array_almost_equal(r, r2) qc,r2 = qr_multiply(a, identity(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(dot(c, q), qc) assert_array_almost_equal(r, r2) qc,r = qr_multiply(a, identity(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(dot(transpose(q),q),identity(3)) assert_array_almost_equal(dot(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(dot(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(dot(c, q), qc) def test_simple_trap(self): a = [[8,2,3],[2,9,3]] q,r = qr(a) assert_array_almost_equal(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(3)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(3)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(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(dot(q, c[:2]), qc) qc,r = qr_multiply(a, identity(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(dot(q, c), qc) qc,r,jpvt = qr_multiply(a, identity(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(dot(c, q), cq) assert_array_almost_equal(r, r2) cq,r = qr_multiply(a, identity(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(dot(c, q), cq) cq,r,jpvt = qr_multiply(a, identity(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(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(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(dot(q, c), qc) assert_array_almost_equal(r, r2) qc,r = qr_multiply(a, identity(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(dot(q, c), qc) qc,r,jpvt = qr_multiply(a, identity(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(dot(c, q), cq) assert_array_almost_equal(r, r2) cq,r = qr_multiply(a, identity(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(dot(c, q), cq) cq,r,jpvt = qr_multiply(a, identity(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(dot(conj(transpose(q)),q),identity(3)) assert_array_almost_equal(dot(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(dot(q, c), qc) qc,r = qr_multiply(a, identity(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(dot(c, q), qc) qc,r = qr_multiply(a, identity(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(dot(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(dot(q, c[:2]), qc) qc,r = qr_multiply(a, identity(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(dot(q.conjugate(), 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(dot(q.conjugate(), 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 = [1, 2, 3+4j] qc,r = qr_multiply(a, c, conjugate=True) assert_array_almost_equal(dot(c, q.conjugate()), qc) def test_simple_complex_pivoting(self): a = np.asarray([[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(dot(conj(transpose(q)),q),identity(3)) assert_array_almost_equal(dot(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 = np.asarray([[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(dot(q, c), qc) def test_simple_complex_right_pivoting(self): a = np.asarray([[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(dot(c, q), qc) def test_random(self): n = 20 for k in range(2): a = random([n,n]) q,r = qr(a) assert_array_almost_equal(dot(transpose(q),q),identity(n)) assert_array_almost_equal(dot(q,r),a) def test_random_left(self): n = 20 for k in range(2): a = random([n,n]) q,r = qr(a) c = random([n]) qc,r = qr_multiply(a, c, "left") assert_array_almost_equal(dot(q, c), qc) qc,r = qr_multiply(a, identity(n), "left") assert_array_almost_equal(q, qc) def test_random_right(self): n = 20 for k in range(2): a = random([n,n]) q,r = qr(a) c = random([n]) cq,r = qr_multiply(a, c) assert_array_almost_equal(dot(c, q), cq) cq,r = qr_multiply(a, identity(n)) assert_array_almost_equal(q, cq) def test_random_pivoting(self): n = 20 for k in range(2): a = 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(dot(transpose(q),q),identity(n)) assert_array_almost_equal(dot(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): # full version m = 200 n = 100 for k in range(2): a = random([m,n]) q,r = qr(a) assert_array_almost_equal(dot(transpose(q),q),identity(m)) assert_array_almost_equal(dot(q,r),a) def test_random_tall_left(self): # full version m = 200 n = 100 for k in range(2): a = random([m,n]) q,r = qr(a, mode="economic") c = random([n]) qc,r = qr_multiply(a, c, "left") assert_array_almost_equal(dot(q, c), qc) qc,r = qr_multiply(a, identity(n), "left") assert_array_almost_equal(qc, q) def test_random_tall_right(self): # full version m = 200 n = 100 for k in range(2): a = random([m,n]) q,r = qr(a, mode="economic") c = random([m]) cq,r = qr_multiply(a, c) assert_array_almost_equal(dot(c, q), cq) cq,r = qr_multiply(a, identity(m)) assert_array_almost_equal(cq, q) def test_random_tall_pivoting(self): # full version pivoting m = 200 n = 100 for k in range(2): a = 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(dot(transpose(q),q),identity(m)) assert_array_almost_equal(dot(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): # economy version m = 200 n = 100 for k in range(2): a = random([m,n]) q,r = qr(a, mode='economic') assert_array_almost_equal(dot(transpose(q),q),identity(n)) assert_array_almost_equal(dot(q,r),a) assert_equal(q.shape, (m,n)) assert_equal(r.shape, (n,n)) def test_random_tall_e_pivoting(self): # economy version pivoting m = 200 n = 100 for k in range(2): a = 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(dot(transpose(q),q),identity(n)) assert_array_almost_equal(dot(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): m = 100 n = 200 for k in range(2): a = random([m,n]) q,r = qr(a) assert_array_almost_equal(dot(transpose(q),q),identity(m)) assert_array_almost_equal(dot(q,r),a) def test_random_trap_pivoting(self): m = 100 n = 200 for k in range(2): a = 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(dot(transpose(q),q),identity(m)) assert_array_almost_equal(dot(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): n = 20 for k in range(2): a = random([n,n])+1j*random([n,n]) q,r = qr(a) assert_array_almost_equal(dot(conj(transpose(q)),q),identity(n)) assert_array_almost_equal(dot(q,r),a) def test_random_complex_left(self): n = 20 for k in range(2): a = random([n,n])+1j*random([n,n]) q,r = qr(a) c = random([n])+1j*random([n]) qc,r = qr_multiply(a, c, "left") assert_array_almost_equal(dot(q, c), qc) qc,r = qr_multiply(a, identity(n), "left") assert_array_almost_equal(q, qc) def test_random_complex_right(self): n = 20 for k in range(2): a = random([n,n])+1j*random([n,n]) q,r = qr(a) c = random([n])+1j*random([n]) cq,r = qr_multiply(a, c) assert_array_almost_equal(dot(c, q), cq) cq,r = qr_multiply(a, identity(n)) assert_array_almost_equal(q, cq) def test_random_complex_pivoting(self): n = 20 for k in range(2): a = random([n,n])+1j*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(dot(conj(transpose(q)),q),identity(n)) assert_array_almost_equal(dot(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(dot(transpose(q),q),identity(3)) assert_array_almost_equal(dot(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(object): def setup_method(self): seed(1234) def test_simple(self): a = [[8,2,3],[2,9,3],[5,3,6]] r,q = rq(a) assert_array_almost_equal(dot(q, transpose(q)),identity(3)) assert_array_almost_equal(dot(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): n = 20 for k in range(2): a = random([n,n]) r,q = rq(a) assert_array_almost_equal(dot(q, transpose(q)),identity(n)) assert_array_almost_equal(dot(r,q),a) def test_simple_trap(self): a = [[8,2,3],[2,9,3]] r,q = rq(a) assert_array_almost_equal(dot(transpose(q),q),identity(3)) assert_array_almost_equal(dot(r,q),a) def test_simple_tall(self): a = [[8,2],[2,9],[5,3]] r,q = rq(a) assert_array_almost_equal(dot(transpose(q),q),identity(2)) assert_array_almost_equal(dot(r,q),a) def test_simple_fat(self): a = [[8,2,5],[2,9,3]] r,q = rq(a) assert_array_almost_equal(dot(transpose(q),q),identity(3)) assert_array_almost_equal(dot(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(dot(q, conj(transpose(q))),identity(3)) assert_array_almost_equal(dot(r,q),a) def test_random_tall(self): m = 200 n = 100 for k in range(2): a = random([m,n]) r,q = rq(a) assert_array_almost_equal(dot(q, transpose(q)),identity(n)) assert_array_almost_equal(dot(r,q),a) def test_random_trap(self): m = 100 n = 200 for k in range(2): a = random([m,n]) r,q = rq(a) assert_array_almost_equal(dot(q, transpose(q)),identity(n)) assert_array_almost_equal(dot(r,q),a) def test_random_trap_economic(self): m = 100 n = 200 for k in range(2): a = random([m,n]) r,q = rq(a, mode='economic') assert_array_almost_equal(dot(q,transpose(q)),identity(m)) assert_array_almost_equal(dot(r,q),a) assert_equal(q.shape, (m, n)) assert_equal(r.shape, (m, m)) def test_random_complex(self): n = 20 for k in range(2): a = random([n,n])+1j*random([n,n]) r,q = rq(a) assert_array_almost_equal(dot(q, conj(transpose(q))),identity(n)) assert_array_almost_equal(dot(r,q),a) def test_random_complex_economic(self): m = 100 n = 200 for k in range(2): a = random([m,n])+1j*random([m,n]) r,q = rq(a, mode='economic') assert_array_almost_equal(dot(q,conj(transpose(q))),identity(m)) assert_array_almost_equal(dot(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(dot(q, transpose(q)),identity(3)) assert_array_almost_equal(dot(r,q),a) transp = transpose class TestSchur(object): def test_simple(self): a = [[8,12,3],[2,9,3],[10,3,6]] t,z = schur(a) assert_array_almost_equal(dot(dot(z,t),transp(conj(z))),a) tc,zc = schur(a,'complex') assert_(np.any(ravel(iscomplex(zc))) and np.any(ravel(iscomplex(tc)))) assert_array_almost_equal(dot(dot(zc,tc),transp(conj(zc))),a) tc2,zc2 = rsf2csf(tc,zc) assert_array_almost_equal(dot(dot(zc2,tc2),transp(conj(zc2))),a) def test_sort(self): a = [[4.,3.,1.,-1.],[-4.5,-3.5,-1.,1.],[9.,6.,-4.,4.5],[6.,4.,-3.,3.5]] s,u,sdim = schur(a,sort='lhp') assert_array_almost_equal([[0.1134,0.5436,0.8316,0.], [-0.1134,-0.8245,0.5544,0.], [-0.8213,0.1308,0.0265,-0.5547], [-0.5475,0.0872,0.0177,0.8321]], u,3) assert_array_almost_equal([[-1.4142,0.1456,-11.5816,-7.7174], [0.,-0.5000,9.4472,-0.7184], [0.,0.,1.4142,-0.1456], [0.,0.,0.,0.5]], s,3) assert_equal(2,sdim) s,u,sdim = schur(a,sort='rhp') assert_array_almost_equal([[0.4862,-0.4930,0.1434,-0.7071], [-0.4862,0.4930,-0.1434,-0.7071], [0.6042,0.3944,-0.6924,0.], [0.4028,0.5986,0.6924,0.]], u,3) assert_array_almost_equal([[1.4142,-0.9270,4.5368,-14.4130], [0.,0.5,6.5809,-3.1870], [0.,0.,-1.4142,0.9270], [0.,0.,0.,-0.5]], s,3) assert_equal(2,sdim) s,u,sdim = schur(a,sort='iuc') assert_array_almost_equal([[0.5547,0.,-0.5721,-0.6042], [-0.8321,0.,-0.3814,-0.4028], [0.,0.7071,-0.5134,0.4862], [0.,0.7071,0.5134,-0.4862]], u,3) assert_array_almost_equal([[-0.5000,0.0000,-6.5809,-4.0974], [0.,0.5000,-3.3191,-14.4130], [0.,0.,1.4142,2.1573], [0.,0.,0.,-1.4142]], s,3) assert_equal(2,sdim) s,u,sdim = schur(a,sort='ouc') assert_array_almost_equal([[0.4862,-0.5134,0.7071,0.], [-0.4862,0.5134,0.7071,0.], [0.6042,0.5721,0.,-0.5547], [0.4028,0.3814,0.,0.8321]], u,3) assert_array_almost_equal([[1.4142,-2.1573,14.4130,4.0974], [0.,-1.4142,3.3191,6.5809], [0.,0.,-0.5000,0.], [0.,0.,0.,0.5000]], s,3) assert_equal(2,sdim) rhp_function = lambda x: x >= 0.0 s,u,sdim = schur(a,sort=rhp_function) assert_array_almost_equal([[0.4862,-0.4930,0.1434,-0.7071], [-0.4862,0.4930,-0.1434,-0.7071], [0.6042,0.3944,-0.6924,0.], [0.4028,0.5986,0.6924,0.]], u,3) assert_array_almost_equal([[1.4142,-0.9270,4.5368,-14.4130], [0.,0.5,6.5809,-3.1870], [0.,0.,-1.4142,0.9270], [0.,0.,0.,-0.5]], s,3) 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(dot(dot(z,t),transp(conj(z))),a) class TestHessenberg(object): 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(dot(transp(q),dot(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) h1 = dot(transp(conj(q)),dot(a,q)) assert_array_almost_equal(h1,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(dot(transp(q),dot(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(dot(transp(q), dot(a, q)), h) def test_random(self): n = 20 for k in range(2): a = random([n,n]) h,q = hessenberg(a,calc_q=1) assert_array_almost_equal(dot(transp(q),dot(a,q)),h) def test_random_complex(self): n = 20 for k in range(2): a = random([n,n])+1j*random([n,n]) h,q = hessenberg(a,calc_q=1) h1 = dot(transp(conj(q)),dot(a,q)) assert_array_almost_equal(h1,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(dot(transp(q),dot(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) class TestQZ(object): def setup_method(self): seed(12345) def test_qz_single(self): n = 5 A = random([n,n]).astype(float32) B = random([n,n]).astype(float32) AA,BB,Q,Z = qz(A,B) assert_array_almost_equal(dot(dot(Q,AA),Z.T), A, decimal=5) assert_array_almost_equal(dot(dot(Q,BB),Z.T), B, decimal=5) assert_array_almost_equal(dot(Q,Q.T), eye(n), decimal=5) assert_array_almost_equal(dot(Z,Z.T), eye(n), decimal=5) assert_(np.all(diag(BB) >= 0)) def test_qz_double(self): n = 5 A = random([n,n]) B = random([n,n]) AA,BB,Q,Z = qz(A,B) assert_array_almost_equal(dot(dot(Q,AA),Z.T), A) assert_array_almost_equal(dot(dot(Q,BB),Z.T), B) assert_array_almost_equal(dot(Q,Q.T), eye(n)) assert_array_almost_equal(dot(Z,Z.T), eye(n)) assert_(np.all(diag(BB) >= 0)) def test_qz_complex(self): n = 5 A = random([n,n]) + 1j*random([n,n]) B = random([n,n]) + 1j*random([n,n]) AA,BB,Q,Z = qz(A,B) assert_array_almost_equal(dot(dot(Q,AA),Z.conjugate().T), A) assert_array_almost_equal(dot(dot(Q,BB),Z.conjugate().T), B) assert_array_almost_equal(dot(Q,Q.conjugate().T), eye(n)) assert_array_almost_equal(dot(Z,Z.conjugate().T), eye(n)) assert_(np.all(diag(BB) >= 0)) assert_(np.all(diag(BB).imag == 0)) def test_qz_complex64(self): n = 5 A = (random([n,n]) + 1j*random([n,n])).astype(complex64) B = (random([n,n]) + 1j*random([n,n])).astype(complex64) AA,BB,Q,Z = qz(A,B) assert_array_almost_equal(dot(dot(Q,AA),Z.conjugate().T), A, decimal=5) assert_array_almost_equal(dot(dot(Q,BB),Z.conjugate().T), B, decimal=5) assert_array_almost_equal(dot(Q,Q.conjugate().T), eye(n), decimal=5) assert_array_almost_equal(dot(Z,Z.conjugate().T), eye(n), decimal=5) assert_(np.all(diag(BB) >= 0)) assert_(np.all(diag(BB).imag == 0)) def test_qz_double_complex(self): n = 5 A = random([n,n]) B = random([n,n]) AA,BB,Q,Z = qz(A,B, output='complex') aa = dot(dot(Q,AA),Z.conjugate().T) assert_array_almost_equal(aa.real, A) assert_array_almost_equal(aa.imag, 0) bb = dot(dot(Q,BB),Z.conjugate().T) assert_array_almost_equal(bb.real, B) assert_array_almost_equal(bb.imag, 0) assert_array_almost_equal(dot(Q,Q.conjugate().T), eye(n)) assert_array_almost_equal(dot(Z,Z.conjugate().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]]) sort = lambda ar,ai,beta: ai == 0 assert_raises(ValueError, qz, A, B, sort=sort) if False: AA,BB,Q,Z,sdim = qz(A,B,sort=sort) # assert_(sdim == 2) assert_(sdim == 4) assert_array_almost_equal(dot(dot(Q,AA),Z.T), A) assert_array_almost_equal(dot(dot(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): n = 5 A = random([n,n]) B = random([n,n]) AA,BB,Q,Z = qz(A,B,check_finite=False) assert_array_almost_equal(dot(dot(Q,AA),Z.T), A) assert_array_almost_equal(dot(dot(Q,BB),Z.T), B) assert_array_almost_equal(dot(Q,Q.T), eye(n)) assert_array_almost_equal(dot(Z,Z.T), eye(n)) assert_(np.all(diag(BB) >= 0)) def _make_pos(X): # the decompositions can have different signs than verified results return np.sign(X)*X class TestOrdQZ(object): @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]) B5 = np.diag([1, 0]) cls.A = [A1, A2, A3, A4, A5] cls.B = [B1, B2, B3, B4, A5] def qz_decomp(self, sort): try: olderr = np.seterr('raise') ret = [ordqz(Ai, Bi, sort=sort) for Ai, Bi in zip(self.A, self.B)] finally: np.seterr(**olderr) 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.dot(Q.T.conj()), Id) assert_array_almost_equal(Z.dot(Z.T.conj()), Id) # check factorization assert_array_almost_equal(Q.dot(AA), A.dot(Z)) assert_array_almost_equal(Q.dot(BB), B.dot(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(object): def setup_method(self): seed(12345) def test_decompose(self): N = 202 # raises error if lwork parameter to dtrsen is too small for ddtype in [np.float32, np.float64]: A = random((N,N)).astype(ddtype) B = random((N,N)).astype(ddtype) # sort = lambda alphar, alphai, beta: alphar**2 + alphai**2< beta**2 sort = lambda alpha, beta: alpha < beta [S,T,alpha,beta,U,V] = ordqz(A,B,sort=sort, output='real') for ddtype in [np.complex, np.complex64]: A = random((N,N)).astype(ddtype) B = random((N,N)).astype(ddtype) sort = lambda alpha, beta: alpha < beta [S,T,alpha,beta,U,V] = ordqz(A,B,sort=sort, output='complex') @pytest.mark.slow def test_decompose_ouc(self): N = 202 # segfaults if lwork parameter to dtrsen is too small for ddtype in [np.float32, np.float64, np.complex, np.complex64]: A = random((N,N)).astype(ddtype) B = random((N,N)).astype(ddtype) [S,T,alpha,beta,U,V] = ordqz(A,B,sort='ouc') class TestDatacopied(object): 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): 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""" # 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.xfail(platform.machine() == 'ppc64le', reason="fails on ppc64le") def test_aligned_mem(): """Check linalg works with non-aligned memory""" # 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(object): 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).dot(np.random.rand(5, n)) X = X + 1e-4 * np.random.rand(n, 1).dot(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: raise AssertionError('memory error perhaps caused by orth regression') 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.dot(Y), 0, atol=tol) Y = null_space(X.T) assert_equal(Y.shape, (2, 1)) assert_allclose(X.T.dot(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.dot(Y), 0, atol=tol) if n > 5: np.random.seed(1) X = np.random.rand(n, 5).dot(np.random.rand(5, n)) X = X + 1e-4 * np.random.rand(n, 1).dot(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(object): 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): 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)