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""" 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()'
Run tests if linalg is not installed:
python tests/test_decomp.py
"""
import numpy as np
from numpy.testing import (TestCase, assert_equal, assert_almost_equal,
assert_array_almost_equal, assert_array_equal,
assert_raises, assert_, assert_allclose,
run_module_suite, dec)
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)
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, sometrue, diag, ones, linalg, \
argsort, zeros, arange, float32, complex64, dot, conj, identity, \
ravel, sqrt, iscomplex, shape, sort, conjugate, bmat, sign, \
asarray, matrix, isfinite, all, ndarray, outer, eye, dtype, empty,\
triu, tril
from numpy.random import normal, seed, random
from scipy.linalg._testutils import assert_no_overwrite
# digit precision to use in asserts for different types
DIGITS = {'d':11, 'D':11, 'f':4, 'F':4}
# 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
http://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(TestCase):
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 all(isfinite(res[:,i])):
assert_allclose(res[:,i], 0, rtol=1e-13, atol=1e-13, err_msg=msg)
assert_allclose(sort(w[isfinite(w)]), sort(wt[isfinite(wt)]),
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)]))
@dec.knownfailureif(True, "See gh-2254.")
def test_singular(self):
# Example taken from
# 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))
I = identity(3)
A = bmat([[I,Z],[Z,-K]])
B = bmat([[Z,I],[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(TestCase):
def __init__(self, *args):
TestCase.__init__(self, *args)
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(1.0*ones(N))
+ diag(-1.0*ones(N-1), -1) + diag(-1.0*ones(N-1), 1)
+ diag(-2.0*ones(N-2), -2) + diag(-2.0*ones(N-2), 2))
# hermitian band matrix
self.herm_mat = (diag(-1.0*ones(N))
+ 1j*diag(1.0*ones(N-1), -1) - 1j*diag(1.0*ones(N-1), 1)
+ diag(-2.0*ones(N-2), -2) + diag(-2.0*ones(N-2), 2))
# general real band matrix
self.real_mat = (diag(1.0*ones(N))
+ diag(-1.0*ones(N-1), -1) + diag(-3.0*ones(N-1), 1)
+ diag(2.0*ones(N-2), -2) + diag(-2.0*ones(N-2), 2))
# general complex band matrix
self.comp_mat = (1j*diag(1.0*ones(N))
+ diag(-1.0*ones(N-1), -1) + 1j*diag(-3.0*ones(N-1), 1)
+ diag(2.0*ones(N-2), -2) + diag(-2.0*ones(N-2), 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 = 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)
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']:
yield (eigenhproblem_standard,
'ordinary',
dim, typ, overwrite, lower,
turbo, eigenvalues)
yield (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(TestCase):
def __init__(self, *args, **kw):
TestCase.__init__(self, *args, **kw)
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 __init__(self, *args, **kw):
TestLU.__init__(self, *args, **kw)
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(TestCase):
def setUp(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(TestCase):
def setUp(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(self):
self.lapack_driver = 'gesvd'
seed(1234)
class TestSVDVals(TestCase):
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])
@dec.slow
def test_crash_2609(self):
np.random.seed(1234)
a = np.random.rand(1500, 2800)
# Shouldn't crash:
svdvals(a)
class TestDiagSVD(TestCase):
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(TestCase):
def setUp(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_(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_(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_(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_(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_(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_(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_(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_(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_(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_(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_(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_(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(TestCase):
def setUp(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
any = sometrue
class TestSchur(TestCase):
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_(any(ravel(iscomplex(zc))) and 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(TestCase):
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(TestCase):
def setUp(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)
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_(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_(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_(all(diag(BB) >= 0))
assert_(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_(all(diag(BB) >= 0))
assert_(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_(all(diag(BB) >= 0))
def test_qz_double_sort(self):
# from http://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_(all(np.real(eigenvalues[:sdim] < 0)))
# assert_(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_(all(diag(BB) >= 0))
def _make_pos(X):
# the decompositions can have different signs than verified results
return np.sign(X)*X
class TestOrdQZ(TestCase):
@classmethod
def setupClass(cls):
# http://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]])
# http://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):
I = np.eye(*A.shape)
# make sure Q and Z are orthogonal
assert_array_almost_equal(Q.dot(Q.T.conj()), I)
assert_array_almost_equal(Z.dot(Z.T.conj()), I)
# 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(TestCase):
def setUp(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')
@dec.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(TestCase):
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)
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)
@dec.knownfailureif(True, "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
]:
yield 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):
X = np.ones((n, 2), dtype=float)
Y = orth(X)
assert_equal(Y.shape, (n, 1))
assert_allclose(Y, Y.mean(), atol=1e-10)
Y = orth(X.T)
assert_equal(Y.shape, (2, 1))
assert_allclose(Y, Y.mean())
@dec.slow
@dec.skipif(np.dtype(np.intp).itemsize < 8, "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 @dec.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)
except MemoryError:
raise AssertionError('memory error perhaps caused by orth regression')
def test_orth():
for n in 1, 2, 3, 10, 100:
_check_orth(n)
if __name__ == "__main__":
run_module_suite()