def eval_evec(self, d, typ, k, which):
a = d['mat'].astype(typ)
# get exact eigenvalues
exact_eval = d['eval'].astype(typ.upper())
ind = self.sort_choose(exact_eval, typ, k, which)
exact_eval = exact_eval[ind]
if verbose >= 3:
print "exact"
print exact_eval
# compute eigenvalues
eval, evec = eigen(a, k, which=which)
ind = self.sort_choose(eval, typ, k, which)
eval = eval[ind]
evec = evec[:, ind]
if verbose >= 3:
print eval
# check eigenvalues
# check eigenvectors A*evec=eval*evec
for i in range(k):
assert_almost_equal_cc(eval[i],
exact_eval[i],
decimal=_ndigits[typ])
assert_array_almost_equal_cc(dot(a, evec[:, i]),
eval[i] * evec[:, i],
decimal=_ndigits[typ])
def eval_evec(self,d,typ,k,which):
a=d['mat'].astype(typ)
# get exact eigenvalues
exact_eval=d['eval'].astype(typ.upper())
ind=self.sort_choose(exact_eval,typ,k,which)
exact_eval=exact_eval[ind]
if verbose >= 3:
print "exact"
print exact_eval
# compute eigenvalues
eval,evec=eigen(a,k,which=which)
ind=self.sort_choose(eval,typ,k,which)
eval=eval[ind]
evec=evec[:,ind]
if verbose >= 3:
print eval
# check eigenvalues
# check eigenvectors A*evec=eval*evec
for i in range(k):
assert_almost_equal_cc(eval[i],exact_eval[i],decimal=_ndigits[typ])
assert_array_almost_equal_cc(dot(a,evec[:,i]),
eval[i]*evec[:,i],
decimal=_ndigits[typ])
def calculate(self, number=inf):
m0=self.m0; k0=self.k0
number = min(number, self.totalnumber)
logging.info( "Center solver. Finding modes, m0=%d, wl=%.3g" % (m0, self.wl) )
#Create new matrix if there is no existing one
if self.si is None:
self.si = ShiftInvertBlock(overwrite=self.overwrite)
self.create()
#Time mode solve
tick = timer.timer()
tick.start()
#Array to store solved modes
evrange = array(self.bracket)**2*self.k0**2
#Update equation with new BCs and update LU decomp
self.update_lambda(self.evapprox, not self.overwrite)
self.si.set_shift(self.matrix, self.evapprox+0j)
#Find modes of linearized system
if use_arpack:
cev = eigen(self.si, k=self.totalnumber, which='LM', return_eigenvectors=False)
cev = self.si.eigenvalue_transform(cev)
else:
cev, cvecs = eigs(self.si, self.totalnumber, tol=1e-6)
#Filter cev within range
if self.ignore_outside_interval:
cev = filter(lambda x: min(evrange)<=real(x)<=max(evrange), cev)
#Refine modes
for ii in range(len(cev)):
evsearch = cev[ii]
#If the matrix is overwritten we must recreate it
if self.overwrite: self.generate()
self.si.set_shift(self.matrix, evsearch+0j)
#Calculate next mode
mode, evals = self.calculate_one(evsearch, self.searchnumber)
#Add mode to list
if (mode.residue<self.abc_convergence) or self.add_if_unconverged:
self.modes += [ mode ]
avtime = tick.lap()/(ii+1)
logging.info( "Mode #%d [%d/%.3gs], neff=%s, res: %.2e" % \
(self.numbersolved, mode.iterations, avtime, mode.neff, mode.residue) )
self.add_time(tick.lap())
#Clean up if calculation is finished!
if self.isfinished(): self.finalize()
return self.modes
def eigs_sparse(M, k):
#M = M.astype(float)
#return scipy.linalg.eig(M)
try:
return scipy.sparse.linalg.eigs(M, k=k, which='LR')
except AttributeError:
# Older scipy: should be removed once possible
import scipy.sparse.linalg.eigen.arpack as arpack
return arpack.eigen(M, k=k, which='LR')
def eigs_sparse(M, k):
#M = M.astype(float)
#return scipy.linalg.eig(M)
try:
return scipy.sparse.linalg.eigs(M, k=k, which='LR')
except AttributeError:
# Older scipy: should be removed once possible
import scipy.sparse.linalg.eigen.arpack as arpack
return arpack.eigen(M, k=k, which='LR')
def calculate_one(self, evsearch, searchnum):
"""
Calculate single mode using fixed point iterations
calculate_one(evsearch, searchnum)
"""
m0 = self.m0; k0 = self.k0
wl = self.wl
bcstart, bcend = self.equation.diff.bc_extents()
coord = self.wg.get_coord(self.base_shape, border=1)
#Create new mode
mode = self.mode_class(coord=coord, symmetry=self.wg.symmetry, m0=m0, wl=wl, evalue=evsearch, wg=self.wg)
mode.right = ones(mode.shape, dtype=self.dtype)
mode.discard_vectors = self.discard_vectors
residue = inf; numit=0; evtol=1e-10
while residue>self.abc_convergence and numit<self.abc_iterations:
numit+=1
#Update equation with new BCs and update LU decomp
self.update_lambda(mode.evalue)
if self.equation.coord.rmin==0:
self.si.update(self.matrix, uprows=bcend)
else:
self.si.set_shift(self.matrix, complex(mode.evalue))
#Solve linear eigenproblem
if use_arpack:
evals, revecs = eigen(self.si, k=searchnum, \
which='LM', return_eigenvectors=True)
evals = self.si.eigenvalue_transform(evals)
else:
evals, revecs = eigs(self.si, searchnum, tol=evtol)
revecs = revecs.T
#Locate closest eigenvalue
itrack = absolute(evals-mode.evalue).argmin()
mode.evalue = evals[itrack]
mode.right = asarray(revecs)[:,itrack]
#Residue and convergence
evconverge = abs(mode.evalue - evals[itrack])
residue = mode.residue = self.residue(mode)
mode.convergence += [ residue ]
mode.track += [ mode.evalue ]
logging.debug( "[%d] neff: %s conv: %.3g, res:%.3g" % (numit, mode.neff, evconverge, mode.residue) )
mode.iterations = numit
#Discard vectors at this stage, to free up memory if we can
mode.compress()
return mode, evals
def eval_evec(self,d,typ,k,which):
a=d['mat'].astype(typ)
# get exact eigenvalues
exact_eval=d['eval'].astype(typ)
ind=self.sort_choose(exact_eval,typ,k,which)
exact_eval=exact_eval[ind]
# compute eigenvalues
eval,evec=eigen(a,k,which=which)
ind=self.sort_choose(eval,typ,k,which)
eval=eval[ind]
evec=evec[:,ind]
# check eigenvalues
assert_array_almost_equal(eval,exact_eval,decimal=_ndigits[typ])
# check eigenvectors A*evec=eval*evec
for i in range(k):
assert_array_almost_equal(dot(a,evec[:,i]),
eval[i]*evec[:,i],
decimal=_ndigits[typ])
def eval_evec(self, d, typ, k, which):
a = d['mat'].astype(typ)
# get exact eigenvalues
exact_eval = d['eval'].astype(typ)
ind = self.sort_choose(exact_eval, typ, k, which)
exact_eval = exact_eval[ind]
# compute eigenvalues
eval, evec = eigen(a, k, which=which)
ind = self.sort_choose(eval, typ, k, which)
eval = eval[ind]
evec = evec[:, ind]
# check eigenvalues
assert_array_almost_equal(eval, exact_eval, decimal=_ndigits[typ])
# check eigenvectors A*evec=eval*evec
for i in range(k):
assert_array_almost_equal(dot(a, evec[:, i]),
eval[i] * evec[:, i],
decimal=_ndigits[typ])
def estimate_condition_number(P_kd,
P_gk,
S_dd,
N,
alpha,
compute_exact = False):
P_kd = as_Lens3D_matrix(P_kd)
P_gk = as_Lens3D_matrix(P_gk)
S_dd = as_Lens3D_matrix(S_dd)
N = as_Lens3D_matrix(N)
P_gk_cross = P_gk.conj_transpose()
P_kd_T = P_kd.transpose()
def matvec(v):
v0 = P_gk_cross.view_as_Lens3D_vec(v)
v2 = P_gk_cross.matvec( v0 )
v2 = P_kd_T.matvec( v2 )
v2 = S_dd.matvec( v2 )
v2 = P_kd.matvec( v2 )
v2 = P_gk.matvec( v2 )
v1 = N.matvec(v0)
v1 *= alpha
ret = numpy.zeros(v.shape,dtype=complex)
ret += v1.vec
ret += v2.vec
return P_gk_cross.view_as_same_type( ret , v )
M = LinearOperator(P_gk.shape,
matvec=matvec,
dtype=complex)
#compute the exact condition number
if compute_exact:
v = numpy.random.random(M.shape[1])
t0 = time()
v2 = M.matvec(v)
t = time()-t0
print " - constructing matrix representation (est. %s)" \
% printtime(t*M.shape[0])
t0 = time()
M_rep = get_mat_rep(M)
print " time to get mat rep: %.2g sec" % (time()-t0)
print " - computing SVD"
t0 = time()
sig = numpy.linalg.svd(M_rep,compute_uv=False)
print " time for SVD: %.2g sec" % (time()-t0)
print 'true condition number: %.2e / %.2e = %.2e' \
% (sig[0],sig[-1],
sig[0]/sig[-1])
#estimate condition number, assuming the noiseless matrix
# is rank-deficient. This will be true if there are more
# source lens-planes than mass lens-planes
eval_max,evec_max = arpack.eigen(M,1)
print 'estimated condition number: %.2e / %.2e = %.2e' \
% ( abs(eval_max[0]) , numpy.min(N.data),
abs(eval_max[0]) / numpy.min(N.data) )
#Labs = abs(Ldense)
I = numpy.argsort(Ldense)
II = I[0:13]
Ldensef = Ldense[II]
for i in II:
eval = Ldense[i]
evec = Xdense[:,i]
print eval
print "computing eigenvalues..."
st = time()
fac = (1/dx)**2
#Evals,Evecs = arpack.eigen(-M, k=16, which="SM", tol=1e-15, maxiter=200000, ncv=128)
Evals,Evecs = arpack.eigen(-M, k=16, which="SM", maxiter=200000, ncv=128)
print "found eigenvalues in time = " + str(time() - st)
# gives very poor results (as expected, our matrix is not symmetric)
#L,X = arpack.eigen_symmetric(M, k=50, which="SM")
#for i,lam in enumerate(L):
#print i,lam
#evec = X[:,i]
pylab.figure(1)
pylab.clf()
pylab.plot(real(Evals),imag(Evals),'rx')
pylab.title('spectrum')
def calculate_delta_arpack(gamma,
P_gk,
P_kd,
N_angular,
N_los,
border_size,
border_noise,
sig_cuts):
"""
gamma should be a vector of length (Nx*Ny*Nz)
N_angular is a vector of length (Nx*Ny)
N_los is a vector of length Nz
sigma gives the percentage of variance to cut out of the inversion
"""
Nx = gamma.Nx
Ny = gamma.Ny
Nz1 = P_kd.Nz_in
Nz2 = P_kd.Nz_out
#for all the lens-plane operations, we'll need two versions of
# Lens3D matrices, with Nz1 and Nz2 source planes.
P_gk1 = Lens3D_lp_conv(Nz1,Nx,Ny,
P_gk.dx_,P_gk.dy_,
func=P_gk.func_[0],
func_ft=P_gk.func_ft_[0])
P_gk2 = P_gk
#get a single convolution plane
P_gk_one = Lens3D_lp_conv(1,Nx,Ny,
P_gk.dx_,P_gk.dy_,
func=P_gk.func_[0],
func_ft=P_gk.func_ft_[0])
P_gk_one_H = P_gk_one.H
N_a = Lens3D_vector(1,Nx,Ny,N_angular)
N_a.set_border(border_size,border_noise)
N1I = 1./N_a.vec
N2I = 1./N_los
NI = Lens3D_vector(Nz2,Nx,Ny,numpy.outer(N2I,N1I))
#use arpack to find singular vectors
num_sing_vals = (Nx-2*border_size)*(Ny-2*border_size)+1
use_arpack = True
if use_arpack:
#define the function to send to ARPACK
def matvec(v):
matvec.N += 1
if matvec.N % 100 == 0: print matvec.N
vv = v*N1I
#vv = P_gk_one.view_as_Lens3D_vec(vv)
vv = P_gk_one_H.matvec(vv)
vv = P_gk_one.matvec(vv)
#vv = P_gk_one.view_as_same_type(vv,v)
vv *= N1I
return vv
matvec.N = 0
DMMD = LinearOperator(shape = (Nx*Ny,Nx*Ny),
matvec = matvec,
dtype = complex)
print "ARPACK: finding %i singular values of [%ix%i] matrix" \
% (num_sing_vals,DMMD.shape[0],DMMD.shape[1])
t = time()
S1_2,U1 = arpack.eigen(DMMD,num_sing_vals,which='LR')
t = time()-t
print " finished in %i iterations" % matvec.N
print " (time: %.3g sec: %.2g sec per iteration)" \
% (t,t*1./matvec.N)
exit()
S1 = numpy.sqrt(S1_2.real)
else:
P_gk_r = P_gk.as_Lens3D_lp_mat()
U1,S1,V1 = numpy.linalg.svd(N1I[:,None]*P_gk_r.mat_list_[0],
full_matrices=0)
S1 = S1[:num_sing_vals]
U1 = U1[:,:num_sing_vals]
compare_to_direct = False
if compare_to_direct:
print "compare to direct svd"
print "computing svds"
S1.sort()
pylab.plot( S1[::-1] )
P_gk_r = P_gk.as_Lens3D_lp_mat()
U1,S1,V1 = numpy.linalg.svd(N1I[:,None]*P_gk_r.mat_list_[0],
full_matrices=0)
pylab.plot( S1[:num_sing_vals] )
pylab.show()
exit()
U2,S2,V2 = numpy.linalg.svd(N2I[:,None]*P_kd.data_,
full_matrices=0)
Nz1 = P_kd.Nz_in
Nz2 = P_kd.Nz_out
U_gk1 = Lens3D_lp_mat(Nz1, num_sing_vals,1,Nx, Ny,
mat_list=U1)
U_gk2 = Lens3D_lp_mat(Nz2, num_sing_vals,1,Nx, Ny,
mat_list=U1)
S_gk1 = Lens3D_lp_diag(Nz1, num_sing_vals, 1,
S1)
S_gk2 = Lens3D_lp_diag(Nz2, num_sing_vals, 1,
S1)
#V_gk will be S_gk^-1 * U_gk^H * N1I * N1I * P_gk
V_gk = Lens3D_multi(Lens3D_lp_diag(Nz1, num_sing_vals, 1, 1./S1),
U_gk1.H,
Lens3D_lp_diag(Nz1,Nx,Ny,N1I),
P_gk1)
U_kd = Lens3D_los_mat(Nz1, Nx, Ny, Nz2, U2)
S_kd = Lens3D_diag(Nz1, Nx, Ny,
numpy.outer(S2,numpy.ones(Nx*Ny)).ravel() )
V_kd = Lens3D_los_mat(Nz1, Nx, Ny, Nz1, V2)
U = Lens3D_multi(U_kd,U_gk1)
S = Lens3D_diag(Nz1, num_sing_vals,1,
numpy.outer(S2,S1).ravel() )
V = Lens3D_multi(V_gk,V_kd)
i_sort = numpy.argsort(S.data_)
sig = S.data_[i_sort]
sig *= sig
sig_cumsum = sig.cumsum()
sig_sum = sig.sum()
N_tot = len(sig)
N_cuts = []
v_cuts = []
deltas = []
for sig_cut in sig_cuts:
N_cut = sig_cumsum.searchsorted(sig_sum*sig_cut)
v_cut = numpy.sqrt( sig[N_cut] )
SI = Lens3D_diag(Nzd,Nx,Ny,
1./S.data_)
SI.data_[i_sort[:N_cut]] = 0
#compute delta
print "computing delta"
v1 = Lens3D_vector(Nzg,Nx,Ny,gamma.vec*NI.vec)
v1 = U.H.matvec(v1)
v1 = SI.matvec(v1)
delta = V.H.matvec(v1)
N_cuts.append(N_cut)
v_cuts.append(v_cut)
deltas.append(delta)
return deltas,N_cuts,v_cuts,S1,S2
from numpy import *
from math import atan2
from pylab import *
from solver1 import *
from solver2 import *
import scipy.sparse.linalg.eigen.arpack as arp
import pickle
#Load mesh
fin = open("mesh.pkl", "rb")
mesh = pickle.load(fin)
pts = pickle.load(fin)
boundary = pickle.load(fin)
fin.close()
#Construct matrix
U, A, b = fe_solve(mesh, pts[:,0], pts[:,1], lambda x,y:0, boundary, zeros((len(pts))))
#Compute top 4 Eigen values of A
w, v = arp.eigen(A, 4)
#Plot results
for k in range(4):
clf()
plot_mesh(mesh, abs(v[:,k]))
savefig("prob4_eig" + str(k) + ".png")
show()
def calculate(self, number=inf):
m0 = self.m0
k0 = self.k0
number = min(number, self.totalnumber)
logging.info("Center solver. Finding modes, m0=%d, wl=%.3g" %
(m0, self.wl))
#Create new matrix if there is no existing one
if self.si is None:
self.si = ShiftInvertBlock(overwrite=self.overwrite)
self.create()
#Time mode solve
tick = timer.timer()
tick.start()
#Array to store solved modes
evrange = array(self.bracket)**2 * self.k0**2
#Update equation with new BCs and update LU decomp
self.update_lambda(self.evapprox, not self.overwrite)
self.si.set_shift(self.matrix, self.evapprox + 0j)
#Find modes of linearized system
if use_arpack:
cev = eigen(self.si,
k=self.totalnumber,
which='LM',
return_eigenvectors=False)
cev = self.si.eigenvalue_transform(cev)
else:
cev, cvecs = eigs(self.si, self.totalnumber, tol=1e-6)
#Filter cev within range
if self.ignore_outside_interval:
cev = filter(lambda x: min(evrange) <= real(x) <= max(evrange),
cev)
#Refine modes
for ii in range(len(cev)):
evsearch = cev[ii]
#If the matrix is overwritten we must recreate it
if self.overwrite: self.generate()
self.si.set_shift(self.matrix, evsearch + 0j)
#Calculate next mode
mode, evals = self.calculate_one(evsearch, self.searchnumber)
#Add mode to list
if (mode.residue <
self.abc_convergence) or self.add_if_unconverged:
self.modes += [mode]
avtime = tick.lap() / (ii + 1)
logging.info( "Mode #%d [%d/%.3gs], neff=%s, res: %.2e" % \
(self.numbersolved, mode.iterations, avtime, mode.neff, mode.residue) )
self.add_time(tick.lap())
#Clean up if calculation is finished!
if self.isfinished(): self.finalize()
return self.modes
def calculate_one(self, evsearch, searchnum):
"""
Calculate single mode using fixed point iterations
calculate_one(evsearch, searchnum)
"""
m0 = self.m0
k0 = self.k0
wl = self.wl
bcstart, bcend = self.equation.diff.bc_extents()
coord = self.wg.get_coord(self.base_shape, border=1)
#Create new mode
mode = self.mode_class(coord=coord,
symmetry=self.wg.symmetry,
m0=m0,
wl=wl,
evalue=evsearch,
wg=self.wg)
mode.right = ones(mode.shape, dtype=self.dtype)
mode.discard_vectors = self.discard_vectors
residue = inf
numit = 0
evtol = 1e-10
while residue > self.abc_convergence and numit < self.abc_iterations:
numit += 1
#Update equation with new BCs and update LU decomp
self.update_lambda(mode.evalue)
if self.equation.coord.rmin == 0:
self.si.update(self.matrix, uprows=bcend)
else:
self.si.set_shift(self.matrix, complex(mode.evalue))
#Solve linear eigenproblem
if use_arpack:
evals, revecs = eigen(self.si, k=searchnum, \
which='LM', return_eigenvectors=True)
evals = self.si.eigenvalue_transform(evals)
else:
evals, revecs = eigs(self.si, searchnum, tol=evtol)
revecs = revecs.T
#Locate closest eigenvalue
itrack = absolute(evals - mode.evalue).argmin()
mode.evalue = evals[itrack]
mode.right = asarray(revecs)[:, itrack]
#Residue and convergence
evconverge = abs(mode.evalue - evals[itrack])
residue = mode.residue = self.residue(mode)
mode.convergence += [residue]
mode.track += [mode.evalue]
logging.debug("[%d] neff: %s conv: %.3g, res:%.3g" %
(numit, mode.neff, evconverge, mode.residue))
mode.iterations = numit
#Discard vectors at this stage, to free up memory if we can
mode.compress()
return mode, evals