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#!/usr/bin/env python
""" Test functions for the sparse.linalg.eigen.lobpcg module
"""
from __future__ import division, print_function, absolute_import
import numpy as np
from numpy.testing import (run_module_suite, assert_almost_equal, assert_equal,
assert_allclose, assert_array_less, assert_)
from scipy import ones, rand, r_, diag, linalg, eye
from scipy.linalg import eig, eigh, toeplitz
import scipy.sparse
from scipy.sparse.linalg.eigen.lobpcg import lobpcg
def ElasticRod(n):
# Fixed-free elastic rod
L = 1.0
le = L/n
rho = 7.85e3
S = 1.e-4
E = 2.1e11
mass = rho*S*le/6.
k = E*S/le
A = k*(diag(r_[2.*ones(n-1),1])-diag(ones(n-1),1)-diag(ones(n-1),-1))
B = mass*(diag(r_[4.*ones(n-1),2])+diag(ones(n-1),1)+diag(ones(n-1),-1))
return A,B
def MikotaPair(n):
# Mikota pair acts as a nice test since the eigenvalues
# are the squares of the integers n, n=1,2,...
x = np.arange(1,n+1)
B = diag(1./x)
y = np.arange(n-1,0,-1)
z = np.arange(2*n-1,0,-2)
A = diag(z)-diag(y,-1)-diag(y,1)
return A,B
def compare_solutions(A,B,m):
n = A.shape[0]
np.random.seed(0)
V = rand(n,m)
X = linalg.orth(V)
eigs,vecs = lobpcg(A, X, B=B, tol=1e-5, maxiter=30)
eigs.sort()
w,v = eig(A,b=B)
w.sort()
assert_almost_equal(w[:int(m/2)],eigs[:int(m/2)],decimal=2)
def test_Small():
A,B = ElasticRod(10)
compare_solutions(A,B,10)
A,B = MikotaPair(10)
compare_solutions(A,B,10)
def test_ElasticRod():
A,B = ElasticRod(100)
compare_solutions(A,B,20)
def test_MikotaPair():
A,B = MikotaPair(100)
compare_solutions(A,B,20)
def test_trivial():
n = 5
X = ones((n, 1))
A = eye(n)
compare_solutions(A, None, n)
def test_regression():
# https://mail.scipy.org/pipermail/scipy-user/2010-October/026944.html
n = 10
X = np.ones((n, 1))
A = np.identity(n)
w, V = lobpcg(A, X)
assert_allclose(w, [1])
def test_diagonal():
# This test was moved from '__main__' in lobpcg.py.
# Coincidentally or not, this is the same eigensystem
# required to reproduce arpack bug
# http://forge.scilab.org/index.php/p/arpack-ng/issues/1397/
# even using the same n=100.
np.random.seed(1234)
# The system of interest is of size n x n.
n = 100
# We care about only m eigenpairs.
m = 4
# Define the generalized eigenvalue problem Av = cBv
# where (c, v) is a generalized eigenpair,
# and where we choose A to be the diagonal matrix whose entries are 1..n
# and where B is chosen to be the identity matrix.
vals = np.arange(1, n+1, dtype=float)
A = scipy.sparse.diags([vals], [0], (n, n))
B = scipy.sparse.eye(n)
# Let the preconditioner M be the inverse of A.
M = scipy.sparse.diags([np.reciprocal(vals)], [0], (n, n))
# Pick random initial vectors.
X = np.random.rand(n, m)
# Require that the returned eigenvectors be in the orthogonal complement
# of the first few standard basis vectors.
m_excluded = 3
Y = np.eye(n, m_excluded)
eigs, vecs = lobpcg(A, X, B, M=M, Y=Y, tol=1e-4, maxiter=40, largest=False)
assert_allclose(eigs, np.arange(1+m_excluded, 1+m_excluded+m))
_check_eigen(A, eigs, vecs, rtol=1e-3, atol=1e-3)
def _check_eigen(M, w, V, rtol=1e-8, atol=1e-14):
mult_wV = np.multiply(w, V)
dot_MV = M.dot(V)
assert_allclose(mult_wV, dot_MV, rtol=rtol, atol=atol)
def _check_fiedler(n, p):
# This is not necessarily the recommended way to find the Fiedler vector.
np.random.seed(1234)
col = np.zeros(n)
col[1] = 1
A = toeplitz(col)
D = np.diag(A.sum(axis=1))
L = D - A
# Compute the full eigendecomposition using tricks, e.g.
# http://www.cs.yale.edu/homes/spielman/561/2009/lect02-09.pdf
tmp = np.pi * np.arange(n) / n
analytic_w = 2 * (1 - np.cos(tmp))
analytic_V = np.cos(np.outer(np.arange(n) + 1/2, tmp))
_check_eigen(L, analytic_w, analytic_V)
# Compute the full eigendecomposition using eigh.
eigh_w, eigh_V = eigh(L)
_check_eigen(L, eigh_w, eigh_V)
# Check that the first eigenvalue is near zero and that the rest agree.
assert_array_less(np.abs([eigh_w[0], analytic_w[0]]), 1e-14)
assert_allclose(eigh_w[1:], analytic_w[1:])
# Check small lobpcg eigenvalues.
X = analytic_V[:, :p]
lobpcg_w, lobpcg_V = lobpcg(L, X, largest=False)
assert_equal(lobpcg_w.shape, (p,))
assert_equal(lobpcg_V.shape, (n, p))
_check_eigen(L, lobpcg_w, lobpcg_V)
assert_array_less(np.abs(np.min(lobpcg_w)), 1e-14)
assert_allclose(np.sort(lobpcg_w)[1:], analytic_w[1:p])
# Check large lobpcg eigenvalues.
X = analytic_V[:, -p:]
lobpcg_w, lobpcg_V = lobpcg(L, X, largest=True)
assert_equal(lobpcg_w.shape, (p,))
assert_equal(lobpcg_V.shape, (n, p))
_check_eigen(L, lobpcg_w, lobpcg_V)
assert_allclose(np.sort(lobpcg_w), analytic_w[-p:])
# Look for the Fiedler vector using good but not exactly correct guesses.
fiedler_guess = np.concatenate((np.ones(n//2), -np.ones(n-n//2)))
X = np.vstack((np.ones(n), fiedler_guess)).T
lobpcg_w, lobpcg_V = lobpcg(L, X, largest=False)
# Mathematically, the smaller eigenvalue should be zero
# and the larger should be the algebraic connectivity.
lobpcg_w = np.sort(lobpcg_w)
assert_allclose(lobpcg_w, analytic_w[:2], atol=1e-14)
def test_fiedler_small_8():
# This triggers the dense path because 8 < 2*5.
_check_fiedler(8, 2)
def test_fiedler_large_12():
# This does not trigger the dense path, because 2*5 <= 12.
_check_fiedler(12, 2)
if __name__ == "__main__":
run_module_suite()
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