MAXIT = 800
SSOR_STEPS = 2
L = poisson2d_vec_sym_blk(N)
S = L.to_sss()
b = np.ones(N*N, 'd')
print 'Solving 2D-Laplace equation using PCG and SSOR preconditioner with variable omega'
print
print 'omega nit time resid'
print '--------------------------------'
for omega in [0.1*(i+1) for i in range(20)]:
K_ssor = precon.ssor(S, omega, SSOR_STEPS)
t1 = time.clock()
x = np.empty(N*N, 'd')
info, iter, relres = pcg(S, b, x, TOL, MAXIT, K_ssor)
elapsed_time = time.clock() - t1
r = np.zeros(N*N, 'd')
S.matvec(x, r)
r = b - r
res_nrm2 = np.linalg.norm(r)
if info == 0:
iter_str = str(iter)
else:
@Deprecated('Use pysparse.itsolvers.Gmres instead.')
def gmres(*args, **kwargs):
return krylov.gmres(*args, **kwargs)
if __name__ == '__main__':
import numpy
from pysparse.precon import precon
from pysparse.tools import poisson
n = 100
n2 = n*n
A = PysparseMatrix(matrix=poisson.poisson2d_sym(n))
b = numpy.ones(n2); b /= numpy.linalg.norm(b)
x = numpy.empty(n2)
K = precon.ssor(A.matrix.to_sss())
fmt = '%8s %7.1e %2d %4d'
def resid(A, b, x):
r = b - A*x
return numpy.linalg.norm(r)
for Solver in [Pcg, Minres, Cgs, Qmrs, Gmres, Bicgstab]:
solver = Solver(A)
solver.solve(b, x, 1.0e-6, 3*n)
print fmt % (solver.name, resid(A, b, x), solver.nofCalled,
solver.totalIterations)
solver.solve(b, x, 1.0e-6, 3*n, K)
print fmt % (solver.name, resid(A, b, x), solver.nofCalled,
solver.totalIterations)
x = np.empty(n * n, "d") info, iter, relres = gmres(L, b, x, 1e-12, 200) print "info=%d, iter=%d, relres=%e" % (info, iter, relres) print "Solve time using LL matrix: %8.2f s" % (time.clock() - t1) print "norm(x) = %g" % np.linalg.norm(x) r = np.empty(n * n, "d") A.matvec(x, r) r = b - r print "norm(b - A*x) = %g" % np.linalg.norm(r) # ----------------------------------------------------------------------------- K_ssor = precon.ssor(S, 1.0) t1 = time.clock() x = np.empty(n * n, "d") info, iter, relres = gmres(S, b, x, 1e-12, 500, K_ssor, 20) print "info=%d, iter=%d, relres=%e" % (info, iter, relres) print "Solve time using SSS matrix and SSOR preconditioner: %8.2f s" % (time.clock() - t1) print "norm(x) = %g" % np.linalg.norm(x) r = np.empty(n * n, "d") S.matvec(x, r) r = b - r print "norm(b - A*x) = %g" % np.linalg.norm(r)
def _applyToMatrix(self, A):
"""
Returns (preconditioning matrix, resulting matrix)
"""
A = A.to_sss()
return precon.ssor(A), A
x = np.empty(n*n, 'd') info, iter, relres = gmres(L, b, x, 1e-12, 200) print 'info=%d, iter=%d, relres=%e' % (info, iter, relres) print 'Solve time using LL matrix: %8.2f s' % (time.clock() - t1) print 'norm(x) = %g' % np.linalg.norm(x) r = np.empty(n*n, 'd') A.matvec(x, r) r = b - r print 'norm(b - A*x) = %g' % np.linalg.norm(r) # ----------------------------------------------------------------------------- K_ssor = precon.ssor(S, 1.0) t1 = time.clock() x = np.empty(n*n, 'd') info, iter, relres = gmres(S, b, x, 1e-12, 500, K_ssor, 20) print 'info=%d, iter=%d, relres=%e' % (info, iter, relres) print 'Solve time using SSS matrix and SSOR preconditioner: %8.2f s' % (time.clock() - t1) print 'norm(x) = %g' % np.linalg.norm(x) r = np.empty(n*n, 'd') S.matvec(x, r) r = b - r print 'norm(b - A*x) = %g' % np.linalg.norm(r)
self.dinv[i] = 1.0 / A[i, i]
def precon(self, x, y):
np.multiply(x, self.dinv, y)
def resid(A, b, x):
r = x.copy()
A.matvec(x, r)
r = b - r
return math.sqrt(np.dot(r, r))
K_diag = diag_prec(A)
K_jac = precon.jacobi(A, 1.0, 1)
K_ssor = precon.ssor(A, 1.0, 1)
# K_ilu = precon.ilutp(L)
n = L.shape[0]
b = np.arange(n).astype(np.Float)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000)
print 'pcg, K_none: ', info, iter, relres, resid(A, b, x)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000, K_diag)
print 'pcg, K_diag: ', info, iter, relres, resid(A, b, x)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000, K_jac)
print 'pcg, K_jac: ', info, iter, relres, resid(A, b, x)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000, K_ssor)
nof += 1
if nof >= nofPrimes:
break
i = i+2
return primes
n = 20000
print 'Generating first %d primes...' % n
primes = get_primes(n)
print 'Assembling coefficient matrix...'
A = spmatrix.ll_mat_sym(n, n*8)
d = 1
while d < n:
for i in range(d, n):
A[i,i-d] = 1.0
d *= 2
for i in range(n):
A[i,i] = 1.0 * primes[i]
A = A.to_sss()
K = precon.ssor(A)
print 'Solving linear system...'
b = np.zeros(n); b[0] = 1.0
x = np.empty(n)
info, iter, relres = minres(A, b, x, 1e-16, n, K)
print info, iter, relres
print '%.16e' % x[0]
def gmres(*args, **kwargs):
return krylov.gmres(*args, **kwargs)
if __name__ == '__main__':
import numpy
from pysparse.precon import precon
from pysparse.tools import poisson
n = 100
n2 = n * n
A = PysparseMatrix(matrix=poisson.poisson2d_sym(n))
b = numpy.ones(n2)
b /= numpy.linalg.norm(b)
x = numpy.empty(n2)
K = precon.ssor(A.matrix.to_sss())
fmt = '%8s %7.1e %2d %4d'
def resid(A, b, x):
r = b - A * x
return numpy.linalg.norm(r)
for Solver in [Pcg, Minres, Cgs, Qmrs, Gmres, Bicgstab]:
solver = Solver(A)
solver.solve(b, x, 1.0e-6, 3 * n)
print fmt % (solver.name, resid(
A, b, x), solver.nofCalled, solver.totalIterations)
solver.solve(b, x, 1.0e-6, 3 * n, K)
print fmt % (solver.name, resid(
A, b, x), solver.nofCalled, solver.totalIterations)
n = self.shape[0]
self.dinv = np.zeros(n, 'd')
for i in xrange(n):
self.dinv[i] = 1.0 / A[i,i]
def precon(self, x, y):
np.multiply(x, self.dinv, y)
def resid(A, b, x):
r = x.copy()
A.matvec(x, r)
r = b - r
return math.sqrt(np.dot(r, r))
K_diag = diag_prec(A)
K_jac = precon.jacobi(A, 1.0, 1)
K_ssor = precon.ssor(A, 1.0, 1)
# K_ilu = precon.ilutp(L)
n = L.shape[0];
b = np.arange(n).astype(np.Float)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000)
print 'pcg, K_none: ', info, iter, relres, resid(A, b, x)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000, K_diag)
print 'pcg, K_diag: ', info, iter, relres, resid(A, b, x)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000, K_jac)
print 'pcg, K_jac: ', info, iter, relres, resid(A, b, x)
x = np.zeros(n, 'd')
info, iter, relres = pcg(A, b, x, 1e-6, 1000, K_ssor)
break
i = i + 2
return primes
n = 20000
print 'Generating first %d primes...' % n
primes = get_primes(n)
print 'Assembling coefficient matrix...'
A = spmatrix.ll_mat_sym(n, n * 8)
d = 1
while d < n:
for i in range(d, n):
A[i, i - d] = 1.0
d *= 2
for i in range(n):
A[i, i] = 1.0 * primes[i]
A = A.to_sss()
K = precon.ssor(A)
print 'Solving linear system...'
b = np.zeros(n)
b[0] = 1.0
x = np.empty(n)
info, iter, relres = minres(A, b, x, 1e-16, n, K)
print info, iter, relres
print '%.16e' % x[0]
def test_pcg(ProblemList, tol=1.0e-6):
if len(ProblemList) == 0:
usage()
sys.exit(1)
header1 = '%10s %6s %6s ' % ('Name', 'n', 'nnz')
header2 = '%6s %8s %8s %4s %6s %6s\n' % ('iter','relres','error','info','form M','solve')
dheader1 = '%10s %6d %6d '
dheader2 = '%6d %8.1e %8.1e %4d %6.2f %6.2f\n'
lhead1 = len(header1)
lhead2 = len(header2)
lhead = lhead1 + lhead2
sys.stderr.write('-' * lhead + '\n')
sys.stderr.write(header1)
sys.stderr.write(header2)
sys.stderr.write('-' * lhead + '\n')
# Record timings for each preconditioner
timings = { 'None' : [],
'Diagonal' : [],
'SSOR' : []
}
for problem in ProblemList:
A = spmatrix.ll_mat_from_mtx(problem)
(m, n) = A.shape
if m != n: break
prob = os.path.basename(problem)
if prob[-4:] == '.mtx': prob = prob[:-4]
# Right-hand side is Ae
e = np.ones(n, 'd')
b = np.empty(n, 'd')
A.matvec(e, b)
sys.stdout.write(dheader1 % (prob, n, A.nnz))
# No preconditioner
x = np.zeros(n, 'd')
t = cputime()
info, iter, relres = pcg(A, b, x, tol, 2*n)
t_noprec = cputime() - t
err = np.linalg.norm(x-e, ord=np.Inf)
sys.stdout.write(dheader2 % (iter, relres, err, info, 0.0, t_noprec))
timings['None'].append(t_noprec)
# Diagonal preconditioner
x = np.zeros(n, 'd')
t = cputime()
M = precon.jacobi(A, 1.0, 1)
t_getM_diag = cputime() - t
t = cputime()
info, iter, relres = pcg(A, b, x, tol, 2*n, M)
t_diag = cputime() - t
err = np.linalg.norm(x-e, ord=np.Inf)
sys.stdout.write(lhead1 * ' ')
sys.stdout.write(dheader2 % (iter,relres,err,info,t_getM_diag,t_diag))
timings['Diagonal'].append(t_diag)
# SSOR preconditioner
# It appears that steps=1 and omega=1.0 are nearly optimal in all cases
x = np.zeros(n, 'd')
t = cputime()
M = precon.ssor(A.to_sss(), 1.0, 1)
t_getM_ssor = cputime() - t
t = cputime()
info, iter, relres = pcg(A, b, x, tol, 2*n, M)
t_ssor = cputime() - t
err = np.linalg.norm(x-e, ord=np.Inf)
sys.stdout.write(lhead1 * ' ')
sys.stdout.write(dheader2 % (iter,relres,err,info,t_getM_ssor,t_ssor))
timings['SSOR'].append(t_ssor)
sys.stderr.write('-' * lhead + '\n')
return timings
MAXIT = 800
SSOR_STEPS = 2
L = poisson2d_vec_sym_blk(N)
S = L.to_sss()
b = np.ones(N * N, 'd')
print 'Solving 2D-Laplace equation using PCG and SSOR preconditioner with variable omega'
print
print 'omega nit time resid'
print '--------------------------------'
for omega in [0.1 * (i + 1) for i in range(20)]:
K_ssor = precon.ssor(S, omega, SSOR_STEPS)
t1 = time.clock()
x = np.empty(N * N, 'd')
info, iter, relres = pcg(S, b, x, TOL, MAXIT, K_ssor)
elapsed_time = time.clock() - t1
r = np.zeros(N * N, 'd')
S.matvec(x, r)
r = b - r
res_nrm2 = np.linalg.norm(r)
if info == 0:
iter_str = str(iter)
else: