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rational_krylov_routine.py
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/
rational_krylov_routine.py
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import numpy as np
import scipy.sparse as sp
import math
import scipy.sparse.linalg as sla
from numpy.random import randn as nprand
import scipy.linalg as la
from scipy.io import loadmat, savemat
from time import time
import pickle
import pyamg
from scipy.spatial import ConvexHull
def gram_schmidt_orth(u, v):
v = v.reshape(-1, 1)
v_norm = la.norm(v)
v -= u.dot(u.T.conjugate().dot(v))
if la.norm(v) < 0.3 * v_norm:
v /= la.norm(v)
v -= u.dot(u.T.conjugate().dot(v))
v /= la.norm(v)
v -= u.dot(u.T.conjugate().dot(v))
v /= la.norm(v)
return v
def eliminate_nonfinite(v):
nonfinite_flag = ~np.isfinite(v)
cnt = np.sum(nonfinite_flag)
if cnt > 0:
print "Nonfinite components eliminated."
v[nonfinite_flag] = np.zeros(cnt)
return
def get_rand_pole(shifts):
s = shifts[0] + (shifts[1] - shifts[0]) * rand(1)[0]
return s
def pyamg_solver_shifting(solver, shift, matrix_hierarchy):
for i in xrange(len(solver.levels)):
solver.levels[i].A = matrix_hierarchy['a'][i] + shift * matrix_hierarchy['1'][i]
def copy_hierarchy(solver):
matrix_hierarchy = {'a': [], '1': []}
#matrix_hierarchy = {'a': [a], '1': [np.eye(a.shape[0])]}
for i in xrange(len(solver.levels)):
matrix_hierarchy['a'].append(solver.levels[i].A.copy())
matrix_hierarchy['1'].append(sp.eye(solver.levels[i].A.shape[0], format='csr'))
return matrix_hierarchy
def time_count_start(time_list):
time_list.append(time())
return
def time_count_finish(time_list):
time_list[-1] = time() - time_list[-1]
return
def direct_solver(a, id_n, s, w, solver_timing, solver_tol=1e-14):
time_count_start(solver_timing)
v = sla.spsolve(a - s * id_n, w, tol = solver_tol).reshape(-1, 1)
time_count_finish(solver_timing)
return v
def pyamg_solver(amg_hierarchy, orig_hierarchy, s, w, solver_timing, solver_tol=1e-14):
solver_timing.append(time())
pyamg_solver_shifting(amg_hierarchy, -s, orig_hierarchy)
v = amg_hierarchy.solve(w, cycle = 'W', accel = 'gmres', tol = solver_tol, maxiter = 300).reshape(-1, 1)
#v = amg_hierarchy.solve(w, tol = solver_tol).reshape(-1, 1)
solver_timing[-1] = time() - solver_timing[-1]
return v
def compute_lyapunov_solution(b):
rhs = np.zeros(b.shape)
rhs[0, 0] = 1.
z_i = la.solve_lyapunov(b, rhs)
return z_i
def mode_solver(mode, solver_data, shift, w):
#safeguard stratagy with positive shifts:
if shift.real > 0:
shift = -shift.real + 1j * shift.imag
#safeguard strategy with complex shifts:
if math.fabs(shift.imag) < 1e-10 * math.fabs(shift.real) and la.norm(w.imag) < 1e-10 * la.norm(w):
shift, w = shift.real, w.real
time_count_start(solver_data['solver_timing'])
if (mode == 'pyamg' or mode == 'pyamgE'): #and not np.any(solver_data['is_solver_direct']):
#print 'Try pyamg'
amg_hierarchy, orig_hierarchy = solver_data['amg_hierarchy'], solver_data['orig_hierarchy']
pyamg_solver_shifting(amg_hierarchy, shift, orig_hierarchy)
v = amg_hierarchy.solve(w, cycle = 'W', accel = 'gmres', tol = solver_data['solver_tol'], maxiter = 300).reshape(-1, 1)
a, id_n = solver_data['a'], solver_data['id_n']
if la.norm(a.dot(v) + shift * v - w) > solver_data['solver_tol'] * la.norm(w) * 1e5:
print 'Fail, direct solver is used.', shift, la.norm(a.dot(v) + shift * v - w)
v = sla.spsolve(a + shift * id_n, w).reshape(-1, 1)
solver_data['is_solver_direct'].append(True)
else:
solver_data['is_solver_direct'].append(False)
elif mode == 'splu':
v = solver_data['lu'].solve(w).reshape(-1, 1)
solver_data['is_solver_direct'].append(False)
else:
a, id_n = solver_data['a'], solver_data['id_n']
v = sla.spsolve(a + shift * id_n, w).reshape(-1, 1)
solver_data['is_solver_direct'].append(True)
time_count_finish(solver_data['solver_timing'])
return v
def mode_solver_initialisation(mode, solver_data, a, solver_tol):
solver_data['solver_timing'], solver_data['solver_tol'] = [], solver_tol
solver_data['is_solver_direct'] = []
id_n = sp.eye(a.shape[0], format='csr')
solver_data['a'] = a
solver_data['id_n'] = id_n
if mode == 'pyamg':
amg_hierarchy = pyamg.ruge_stuben_solver(a)
orig_hierarchy = copy_hierarchy(amg_hierarchy)
solver_data['amg_hierarchy'], solver_data['orig_hierarchy'] = amg_hierarchy, orig_hierarchy
elif mode == 'pyamgE':
B = np.ones((a.shape[0],1), dtype=a.dtype);
amg_hierarchy = pyamg.rootnode_solver(a, max_levels = 15, max_coarse = 300, coarse_solver = 'pinv', presmoother = ('block_gauss_seidel', {'sweep': 'symmetric', 'iterations': 1}), postsmoother = ('block_gauss_seidel', {'sweep': 'symmetric', 'iterations': 1}), BH = B.copy(),
strength = ('evolution', {'epsilon': 4.0, 'k': 2, 'proj_type': 'l2'}), aggregate ='standard', smooth = ('energy', {'weighting': 'local', 'krylov': 'gmres', 'degree': 1, 'maxiter': 2}), improve_candidates = [('block_gauss_seidel', {'sweep': 'symmetric', 'iterations': 4}), None, None, None, None, None, None, None, None, None, None, None, None, None, None])
orig_hierarchy = copy_hierarchy(amg_hierarchy)
solver_data['amg_hierarchy'], solver_data['orig_hierarchy'] = amg_hierarchy, orig_hierarchy
elif mode == 'splu':
solver_data['lu'] = sla.splu(a.tocsc())
return
def update_b_alr_way(au, u, b):
k = u.shape[1] - 2
b[k, k-1] = u[:, -2].dot(au[:, k-1])
b[k+1, k-1] = u[:, -1].dot(au[:, k-1])
b[:k+2, k] = u.T.dot(au[:, k:k+1]).flatten()
b[:k+2, k+1] = u.T.dot(au[:, k+1:k+2]).flatten()
return b[:k+2, :k+2]
def update_b_full_way(au, u, b, double_flag=True, k=0):
if k==0: k = u.shape[1]
if double_flag:
update_angle_in_b(u, au, b, k-2)
update_angle_in_b(u, au, b, k-1)
else:
update_angle_in_b(u, au, b, k-1)
#if k % 5 == 0:
# b_true = u.T.conjugate().dot(au[:, :u.shape[1]])
# resnorm = la.norm(b[:k, :k] - b_true)
# if resnorm > 1e-8 * la.norm(b_true):
# print 'Bad B', resnorm
# b[:k, :k] = b_true
return b[:k, :k]
def update_angle_in_b(u, au, b, start_k):
if la.norm(b[start_k, :start_k+1]) == 0:
b[start_k, :start_k+1] = u[:, start_k].T.conjugate().dot(au[:, :start_k+1])
if la.norm(b[:start_k, start_k]) == 0:
b[:start_k, start_k] = u[:, :start_k].T.conjugate().dot(au[:, start_k])
return
def update_res_au(res_au, u, au, b, double_flag=True, alr_way=False):
k = u.shape[1]
res_au[:, :k] = au[:, :k] - u.dot(b[:k, :k])
if alr_way and k > 0 and la.norm(res_au[:, :k-1]) > 1e-10:
res_au[:, :k] -= u.dot(u.T.dot(res_au[:, :k]))
if la.norm(res_au[:, :k-1]) > 1e-6:
#print 'Bad res_AU', la.norm(res_au[:, :k-1])
pass
return res_au
if double_flag:
res_au[:, k-2:k] = au[:, k-2:k] - u[:, :k].dot(b[:k, k-2:k])
res_au[:, :k-2] -= u[:, -2:].dot(b[k-2:k, :k-2])
#res_au[:, k-2:k] = au[:, k-2:k] - u[:, :k-2].dot(b[:k-2, k-2:k])
#res_au[:, :k] -= u[:, -2:].dot(b[k-2:k, :k])
else:
res_au[:, k-1:k] = au[:, k-1:k] - u.dot(b[:k, k-1:k])
res_au[:, :k-1] -= u[:, -1:].dot(b[k-1:k, :k-1])
#res_au[:, k-2:k] -= u.dot(u.T.dot(res_au[:, k-2:k]))
#res_au[:, :k] -= u[:, -2:-1].dot(u[:, -2:-1].T.dot(res_au[:, :k]))
#res_au[:, :k] -= u[:, -1:].dot(u[:, -1:].T.dot(res_au[:, :k]))
return res_au