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"""Tests for the linalg.isolve.lgmres module
"""
from __future__ import division, print_function, absolute_import
from numpy.testing import assert_, assert_allclose, assert_equal
import numpy as np
from numpy import zeros, array, allclose
from scipy.linalg import norm
from scipy.sparse import csr_matrix, eye, rand
from scipy.sparse.linalg.interface import LinearOperator
from scipy.sparse.linalg import splu
from scipy.sparse.linalg.isolve import lgmres, gmres
from scipy._lib._numpy_compat import suppress_warnings
Am = csr_matrix(array([[-2,1,0,0,0,9],
[1,-2,1,0,5,0],
[0,1,-2,1,0,0],
[0,0,1,-2,1,0],
[0,3,0,1,-2,1],
[1,0,0,0,1,-2]]))
b = array([1,2,3,4,5,6])
count = [0]
def matvec(v):
count[0] += 1
return Am*v
A = LinearOperator(matvec=matvec, shape=Am.shape, dtype=Am.dtype)
def do_solve(**kw):
count[0] = 0
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x0, flag = lgmres(A, b, x0=zeros(A.shape[0]), inner_m=6, tol=1e-14, **kw)
count_0 = count[0]
assert_(allclose(A*x0, b, rtol=1e-12, atol=1e-12), norm(A*x0-b))
return x0, count_0
class TestLGMRES(object):
def test_preconditioner(self):
# Check that preconditioning works
pc = splu(Am.tocsc())
M = LinearOperator(matvec=pc.solve, shape=A.shape, dtype=A.dtype)
x0, count_0 = do_solve()
x1, count_1 = do_solve(M=M)
assert_(count_1 == 3)
assert_(count_1 < count_0/2)
assert_(allclose(x1, x0, rtol=1e-14))
def test_outer_v(self):
# Check that the augmentation vectors behave as expected
outer_v = []
x0, count_0 = do_solve(outer_k=6, outer_v=outer_v)
assert_(len(outer_v) > 0)
assert_(len(outer_v) <= 6)
x1, count_1 = do_solve(outer_k=6, outer_v=outer_v, prepend_outer_v=True)
assert_(count_1 == 2, count_1)
assert_(count_1 < count_0/2)
assert_(allclose(x1, x0, rtol=1e-14))
# ---
outer_v = []
x0, count_0 = do_solve(outer_k=6, outer_v=outer_v, store_outer_Av=False)
assert_(array([v[1] is None for v in outer_v]).all())
assert_(len(outer_v) > 0)
assert_(len(outer_v) <= 6)
x1, count_1 = do_solve(outer_k=6, outer_v=outer_v, prepend_outer_v=True)
assert_(count_1 == 3, count_1)
assert_(count_1 < count_0/2)
assert_(allclose(x1, x0, rtol=1e-14))
def test_arnoldi(self):
np.random.rand(1234)
A = eye(10000) + rand(10000,10000,density=1e-4)
b = np.random.rand(10000)
# The inner arnoldi should be equivalent to gmres
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x0, flag0 = lgmres(A, b, x0=zeros(A.shape[0]), inner_m=15, maxiter=1)
x1, flag1 = gmres(A, b, x0=zeros(A.shape[0]), restart=15, maxiter=1)
assert_equal(flag0, 1)
assert_equal(flag1, 1)
assert_(np.linalg.norm(A.dot(x0) - b) > 1e-3)
assert_allclose(x0, x1)
def test_cornercase(self):
np.random.seed(1234)
# Rounding error may prevent convergence with tol=0 --- ensure
# that the return values in this case are correct, and no
# exceptions are raised
for n in [3, 5, 10, 100]:
A = 2*eye(n)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
b = np.ones(n)
x, info = lgmres(A, b, maxiter=10)
assert_equal(info, 0)
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
x, info = lgmres(A, b, tol=0, maxiter=10)
if info == 0:
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
b = np.random.rand(n)
x, info = lgmres(A, b, maxiter=10)
assert_equal(info, 0)
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
x, info = lgmres(A, b, tol=0, maxiter=10)
if info == 0:
assert_allclose(A.dot(x) - b, 0, atol=1e-14)
def test_nans(self):
A = eye(3, format='lil')
A[1,1] = np.nan
b = np.ones(3)
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
x, info = lgmres(A, b, tol=0, maxiter=10)
assert_equal(info, 1)
def test_breakdown_with_outer_v(self):
A = np.array([[1, 2], [3, 4]], dtype=float)
b = np.array([1, 2])
x = np.linalg.solve(A, b)
v0 = np.array([1, 0])
# The inner iteration should converge to the correct solution,
# since it's in the outer vector list
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b, outer_v=[(v0, None), (x, None)], maxiter=1)
assert_allclose(xp, x, atol=1e-12)
def test_breakdown_underdetermined(self):
# Should find LSQ solution in the Krylov span in one inner
# iteration, despite solver breakdown from nilpotent A.
A = np.array([[0, 1, 1, 1],
[0, 0, 1, 1],
[0, 0, 0, 1],
[0, 0, 0, 0]], dtype=float)
bs = [
np.array([1, 1, 1, 1]),
np.array([1, 1, 1, 0]),
np.array([1, 1, 0, 0]),
np.array([1, 0, 0, 0]),
]
for b in bs:
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b, maxiter=1)
resp = np.linalg.norm(A.dot(xp) - b)
K = np.c_[b, A.dot(b), A.dot(A.dot(b)), A.dot(A.dot(A.dot(b)))]
y, _, _, _ = np.linalg.lstsq(A.dot(K), b, rcond=-1)
x = K.dot(y)
res = np.linalg.norm(A.dot(x) - b)
assert_allclose(resp, res, err_msg=repr(b))
def test_denormals(self):
# Check that no warnings are emitted if the matrix contains
# numbers for which 1/x has no float representation, and that
# the solver behaves properly.
A = np.array([[1, 2], [3, 4]], dtype=float)
A *= 100 * np.nextafter(0, 1)
b = np.array([1, 1])
with suppress_warnings() as sup:
sup.filter(DeprecationWarning, ".*called without specifying.*")
xp, info = lgmres(A, b)
if info == 0:
assert_allclose(A.dot(xp), b)
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