def _diagonalize(self):
if self.sparse is not None:
from MMTK_sparseev import sparseMatrixEV
eigenvalues, eigenvectors = sparseMatrixEV(self.fc, self.nmodes)
self.array = eigenvectors[:self.nmodes]
return eigenvalues
# Calculate eigenvalues and eigenvectors of self.array
if symeig is not None:
_symmetrize(self.array)
ev, modes = symeig(self.array, overwrite=True)
self.array = N.transpose(modes)
elif dsyevd is None:
ev, modes = Heigenvectors(self.array)
ev = ev.real
modes = modes.real
self.array = modes
else:
ev = N.zeros((self.nmodes,), N.Float)
work = N.zeros((1,), N.Float)
iwork = N.zeros((1,), int_type)
results = dsyevd('V', 'L', self.nmodes, self.array, self.nmodes,
ev, work, -1, iwork, -1, 0)
lwork = int(work[0])
liwork = iwork[0]
work = N.zeros((lwork,), N.Float)
iwork = N.zeros((liwork,), int_type)
results = dsyevd('V', 'L', self.nmodes, self.array, self.nmodes,
ev, work, lwork, iwork, liwork, 0)
if results['info'] > 0:
raise ValueError('Eigenvalue calculation did not converge')
if self.basis is not None:
self.array = N.dot(self.array, self.basis)
return ev
def _diagonalize(self):
if self.sparse is not None:
from MMTK_sparseev import sparseMatrixEV
eigenvalues, eigenvectors = sparseMatrixEV(self.fc, self.nmodes)
self.array = eigenvectors[:self.nmodes]
return eigenvalues
# Calculate eigenvalues and eigenvectors of self.array
if symeig is not None:
_symmetrize(self.array)
ev, modes = symeig(self.array, overwrite=True)
self.array = N.transpose(modes)
elif dsyevd is None:
ev, modes = Heigenvectors(self.array)
ev = ev.real
modes = modes.real
self.array = modes
else:
ev = N.zeros((self.nmodes, ), N.Float)
work = N.zeros((1, ), N.Float)
iwork = N.zeros((1, ), int_type)
results = dsyevd('V', 'L', self.nmodes, self.array, self.nmodes,
ev, work, -1, iwork, -1, 0)
lwork = int(work[0])
liwork = iwork[0]
work = N.zeros((lwork, ), N.Float)
iwork = N.zeros((liwork, ), int_type)
results = dsyevd('V', 'L', self.nmodes, self.array, self.nmodes,
ev, work, lwork, iwork, liwork, 0)
if results['info'] > 0:
raise ValueError('Eigenvalue calculation did not converge')
if self.basis is not None:
self.array = N.dot(self.array, self.basis)
return ev
tt = time.clock()
(LorU, lower) = linalg.cho_factor(A, lower=0, overwrite_a=0)
eigs, vecs = lobpcg.lobpcg(X,
A,
B,
operatorT=precond,
residualTolerance=1e-4,
maxIterations=40)
data1.append(time.clock() - tt)
eigs = sort(eigs)
print
print 'Results by LOBPCG'
print
print n, eigs
tt = time.clock()
w, v = symeig(A, B, range=(1, m))
data2.append(time.clock() - tt)
print
print 'Results by symeig'
print
print n, w
xlabel(r'Size $n$')
ylabel(r'Elapsed time $t$')
plot(N, data1, label='LOBPCG')
plot(N, data2, label='SYMEIG')
legend()
show()
def test2(n): x = arange(1,n+1) B = diag(1./x) y = arange(n-1,0,-1) z = arange(2*n-1,0,-2) A = diag(z)-diag(y,-1)-diag(y,1) return A,B n = 100 # Dimension A,B = test1(n) # Fixed-free elastic rod A,B = test2(n) # Mikota pair acts as a nice test since the eigenvalues are the squares of the integers n, n=1,2,... m = 20 V = rand(n,m) X = linalg.orth(V) eigs,vecs = lobpcg.lobpcg(X,A,B) eigs = sort(eigs) w,v=symeig(A,B) plot(arange(0,len(w[:m])),w[:m],'bx',label='Results by symeig') plot(arange(0,len(eigs)),eigs,'r+',label='Results by lobpcg') legend() xlabel(r'Eigenvalue $i$') ylabel(r'$\lambda_i$') show()
print('******', n)
A,B = test(n) # Mikota pair
X = rand(n,m)
X = linalg.orth(X)
tt = time.clock()
(LorU, lower) = linalg.cho_factor(A, lower=0, overwrite_a=0)
eigs,vecs = lobpcg.lobpcg(X,A,B,operatorT=precond,
residualTolerance=1e-4, maxIterations=40)
data1.append(time.clock()-tt)
eigs = sort(eigs)
print()
print('Results by LOBPCG')
print()
print(n,eigs)
tt = time.clock()
w,v = symeig(A,B,range=(1,m))
data2.append(time.clock()-tt)
print()
print('Results by symeig')
print()
print(n, w)
xlabel(r'Size $n$')
ylabel(r'Elapsed time $t$')
plot(N,data1,label='LOBPCG')
plot(N,data2,label='SYMEIG')
legend()
show()
def Eigensystem(Heff, Olap):
""" General eigenvalue problem solved"""
w, Z = symeig.symeig(Heff, Olap,
type=1) # symmetric generalized eigenvalue problem
Energy = w[:2]
return (Energy, Z[:, :2])
def Eigensystem(Heff, Olap):
""" General eigenvalue problem solved"""
w,Z = symeig.symeig(Heff, Olap, type=1) # symmetric generalized eigenvalue problem
Energy = w[:2]
return (Energy, Z[:,:2])
def lobpcg( blockVectorX, operatorA,
operatorB = None, operatorT = None, blockVectorY = None,
residualTolerance = None, maxIterations = 20,
largest = True, verbosityLevel = 0,
retLambdaHistory = False, retResidualNormsHistory = False ):
"""
LOBPCG solves symmetric partial eigenproblems using preconditioning.
Required input:
blockVectorX - initial approximation to eigenvectors, full or sparse matrix
n-by-blockSize
operatorA - the operator of the problem, can be given as a matrix or as an
M-file
Optional input:
operatorB - the second operator, if solving a generalized eigenproblem; by
default, or if empty, operatorB = I.
operatorT - preconditioner; by default, operatorT = I.
Optional constraints input:
blockVectorY - n-by-sizeY matrix of constraints, sizeY < n. The iterations
will be performed in the (operatorB-) orthogonal complement of the
column-space of blockVectorY. blockVectorY must be full rank.
Optional scalar input parameters:
residualTolerance - tolerance, by default, residualTolerance=n*sqrt(eps)
maxIterations - max number of iterations, by default, maxIterations =
min(n,20)
largest - when true, solve for the largest eigenvalues, otherwise for the
smallest
verbosityLevel - by default, verbosityLevel = 0.
retLambdaHistory - return eigenvalue history
retResidualNormsHistory - return history of residual norms
Output:
blockVectorX and lambda are computed blockSize eigenpairs, where
blockSize=size(blockVectorX,2) for the initial guess blockVectorX if it is
full rank.
If both retLambdaHistory and retResidualNormsHistory are True, the
return tuple has the flollowing order:
lambda, blockVectorX, lambda history, residual norms history
"""
failureFlag = True
if blockVectorY is not None:
sizeY = blockVectorY.shape[1]
else:
sizeY = 0
# Block size.
n, sizeX = blockVectorX.shape
if sizeX > n:
raise ValueError,\
'the first input argument blockVectorX must be tall, not fat' +\
' (%d, %d)' % blockVectorX.shape
if n < 1:
raise ValueError,\
'the matrix size is wrong (%d)' % n
operatorA = makeOperator( operatorA, (n, n) )
if operatorB is not None:
operatorB = makeOperator( operatorB, (n, n) )
if (n - sizeY) < (5 * sizeX):
print 'The problem size is too small, compared to the block size, for LOBPCG to run.'
print 'Trying to use symeig instead, without preconditioning.'
if blockVectorY is not None:
print 'symeig does not support constraints'
raise ValueError
if largest:
lohi = (n - sizeX, n)
else:
lohi = (1, sizeX)
if operatorA.kind == 'function':
print 'symeig does not support matrix A given by function'
if operatorB is not None:
if operatorB.kind == 'function':
print 'symeig does not support matrix B given by function'
_lambda, eigBlockVector = symeig( operatorA.asMatrix(),
operatorB.asMatrix(),
range = lohi )
else:
_lambda, eigBlockVector = symeig( operatorA.asMatrix(),
range = lohi )
return _lambda, eigBlockVector
if operatorT is not None:
operatorT = makeOperator( operatorT, (n, n) )
## if n != operatorA.shape[0]:
## aux = 'The size (%d, %d) of operatorA is not the same as\n'+\
## '%d - the number of rows of blockVectorX' % operatorA.shape + (n,)
## raise ValueError, aux
## if operatorA.shape[0] != operatorA.shape[1]:
## raise ValueError, 'operatorA must be a square matrix (%d, %d)' %\
## operatorA.shape
if residualTolerance is None:
residualTolerance = nm.sqrt( 1e-15 ) * n
maxIterations = min( n, maxIterations )
if verbosityLevel:
aux = "Solving "
if operatorB is None:
aux += "standard"
else:
aux += "generalized"
aux += " eigenvalue problem with"
if operatorT is None:
aux += "out"
aux += " preconditioning\n\n"
aux += "matrix size %d\n" % n
aux += "block size %d\n\n" % sizeX
if blockVectorY is None:
aux += "No constraints\n\n"
else:
if sizeY > 1:
aux += "%d constraints\n\n" % sizeY
else:
aux += "%d constraint\n\n" % sizeY
print aux
##
# Apply constraints to X.
if blockVectorY is not None:
if operatorB is not None:
blockVectorBY = operatorB( blockVectorY )
else:
blockVectorBY = blockVectorY
# gramYBY is a dense array.
gramYBY = sc.dot( blockVectorY.T, blockVectorBY )
try:
# gramYBY is a Cholesky factor from now on...
gramYBY = la.cho_factor( gramYBY )
except:
print 'cannot handle linear dependent constraints'
raise
applyConstraints( blockVectorX, gramYBY, blockVectorBY, blockVectorY )
##
# B-orthonormalize X.
blockVectorX, blockVectorBX = b_orthonormalize( operatorB, blockVectorX )
##
# Compute the initial Ritz vectors: solve the eigenproblem.
blockVectorAX = operatorA( blockVectorX )
gramXAX = sc.dot( blockVectorX.T, blockVectorAX )
# gramXBX is X^T * X.
gramXBX = sc.dot( blockVectorX.T, blockVectorX )
_lambda, eigBlockVector = symeig( gramXAX )
ii = nm.argsort( _lambda )[:sizeX]
if largest:
ii = ii[::-1]
_lambda = _lambda[ii]
eigBlockVector = nm.asarray( eigBlockVector[:,ii] )
# pause()
blockVectorX = sc.dot( blockVectorX, eigBlockVector )
blockVectorAX = sc.dot( blockVectorAX, eigBlockVector )
if operatorB is not None:
blockVectorBX = sc.dot( blockVectorBX, eigBlockVector )
##
# Active index set.
activeMask = nm.ones( (sizeX,), dtype = nm.bool )
lambdaHistory = [_lambda]
residualNormsHistory = []
previousBlockSize = sizeX
ident = nm.eye( sizeX, dtype = operatorA.dtype )
ident0 = nm.eye( sizeX, dtype = operatorA.dtype )
##
# Main iteration loop.
for iterationNumber in xrange( maxIterations ):
if verbosityLevel > 0:
print 'iteration %d' % iterationNumber
aux = blockVectorBX * _lambda[nm.newaxis,:]
blockVectorR = blockVectorAX - aux
aux = nm.sum( blockVectorR.conjugate() * blockVectorR, 0 )
residualNorms = nm.sqrt( aux )
## if iterationNumber == 2:
## print blockVectorAX
## print blockVectorBX
## print blockVectorR
## pause()
residualNormsHistory.append( residualNorms )
ii = nm.where( residualNorms > residualTolerance, True, False )
activeMask = activeMask & ii
if verbosityLevel > 2:
print activeMask
currentBlockSize = activeMask.sum()
if currentBlockSize != previousBlockSize:
previousBlockSize = currentBlockSize
ident = nm.eye( currentBlockSize, dtype = operatorA.dtype )
if currentBlockSize == 0:
failureFlag = False # All eigenpairs converged.
break
if verbosityLevel > 0:
print 'current block size:', currentBlockSize
print 'eigenvalue:', _lambda
print 'residual norms:', residualNorms
if verbosityLevel > 10:
print eigBlockVector
activeBlockVectorR = as2d( blockVectorR[:,activeMask] )
if iterationNumber > 0:
activeBlockVectorP = as2d( blockVectorP[:,activeMask] )
activeBlockVectorAP = as2d( blockVectorAP[:,activeMask] )
activeBlockVectorBP = as2d( blockVectorBP[:,activeMask] )
# print activeBlockVectorR
if operatorT is not None:
##
# Apply preconditioner T to the active residuals.
activeBlockVectorR = operatorT( activeBlockVectorR )
# assert nm.all( blockVectorR == activeBlockVectorR )
##
# Apply constraints to the preconditioned residuals.
if blockVectorY is not None:
applyConstraints( activeBlockVectorR,
gramYBY, blockVectorBY, blockVectorY )
# assert nm.all( blockVectorR == activeBlockVectorR )
##
# B-orthonormalize the preconditioned residuals.
# print activeBlockVectorR
aux = b_orthonormalize( operatorB, activeBlockVectorR )
activeBlockVectorR, activeBlockVectorBR = aux
# print activeBlockVectorR
activeBlockVectorAR = operatorA( activeBlockVectorR )
if iterationNumber > 0:
aux = b_orthonormalize( operatorB, activeBlockVectorP,
activeBlockVectorBP, retInvR = True )
activeBlockVectorP, activeBlockVectorBP, invR = aux
activeBlockVectorAP = sc.dot( activeBlockVectorAP, invR )
##
# Perform the Rayleigh Ritz Procedure:
# Compute symmetric Gram matrices:
xaw = sc.dot( blockVectorX.T, activeBlockVectorAR )
waw = sc.dot( activeBlockVectorR.T, activeBlockVectorAR )
xbw = sc.dot( blockVectorX.T, activeBlockVectorBR )
if iterationNumber > 0:
xap = sc.dot( blockVectorX.T, activeBlockVectorAP )
wap = sc.dot( activeBlockVectorR.T, activeBlockVectorAP )
pap = sc.dot( activeBlockVectorP.T, activeBlockVectorAP )
xbp = sc.dot( blockVectorX.T, activeBlockVectorBP )
wbp = sc.dot( activeBlockVectorR.T, activeBlockVectorBP )
gramA = nm.bmat( [[nm.diag( _lambda ), xaw, xap],
[xaw.T, waw, wap],
[xap.T, wap.T, pap]] )
try:
gramB = nm.bmat( [[ident0, xbw, xbp],
[xbw.T, ident, wbp],
[xbp.T, wbp.T, ident]] )
except:
print ident
print xbw
raise
else:
gramA = nm.bmat( [[nm.diag( _lambda ), xaw],
[xaw.T, waw]] )
gramB = nm.bmat( [[ident0, xbw],
[xbw.T, ident0]] )
try:
assert nm.allclose( gramA.T, gramA )
except:
print gramA.T - gramA
raise
try:
assert nm.allclose( gramB.T, gramB )
except:
print gramB.T - gramB
raise
## print nm.diag( _lambda )
## print xaw
## print waw
## print xbw
## try:
## print xap
## print wap
## print pap
## print xbp
## print wbp
## except:
## pass
## pause()
if verbosityLevel > 10:
save( gramA, 'gramA' )
save( gramB, 'gramB' )
##
# Solve the generalized eigenvalue problem.
# _lambda, eigBlockVector = la.eig( gramA, gramB )
_lambda, eigBlockVector = symeig( gramA, gramB )
ii = nm.argsort( _lambda )[:sizeX]
if largest:
ii = ii[::-1]
if verbosityLevel > 10:
print ii
_lambda = _lambda[ii].astype( nm.float64 )
eigBlockVector = nm.asarray( eigBlockVector[:,ii].astype( nm.float64 ) )
lambdaHistory.append( _lambda )
if verbosityLevel > 10:
print 'lambda:', _lambda
## # Normalize eigenvectors!
## aux = nm.sum( eigBlockVector.conjugate() * eigBlockVector, 0 )
## eigVecNorms = nm.sqrt( aux )
## eigBlockVector = eigBlockVector / eigVecNorms[nm.newaxis,:]
# eigBlockVector, aux = b_orthonormalize( operatorB, eigBlockVector )
if verbosityLevel > 10:
print eigBlockVector
pause()
##
# Compute Ritz vectors.
if iterationNumber > 0:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:sizeX+currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX+currentBlockSize:]
pp = sc.dot( activeBlockVectorR, eigBlockVectorR )\
+ sc.dot( activeBlockVectorP, eigBlockVectorP )
app = sc.dot( activeBlockVectorAR, eigBlockVectorR )\
+ sc.dot( activeBlockVectorAP, eigBlockVectorP )
bpp = sc.dot( activeBlockVectorBR, eigBlockVectorR )\
+ sc.dot( activeBlockVectorBP, eigBlockVectorP )
else:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp = sc.dot( activeBlockVectorR, eigBlockVectorR )
app = sc.dot( activeBlockVectorAR, eigBlockVectorR )
bpp = sc.dot( activeBlockVectorBR, eigBlockVectorR )
if verbosityLevel > 10:
print pp
print app
print bpp
pause()
# print pp.shape, app.shape, bpp.shape
blockVectorX = sc.dot( blockVectorX, eigBlockVectorX ) + pp
blockVectorAX = sc.dot( blockVectorAX, eigBlockVectorX ) + app
blockVectorBX = sc.dot( blockVectorBX, eigBlockVectorX ) + bpp
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
aux = blockVectorBX * _lambda[nm.newaxis,:]
blockVectorR = blockVectorAX - aux
aux = nm.sum( blockVectorR.conjugate() * blockVectorR, 0 )
residualNorms = nm.sqrt( aux )
if verbosityLevel > 0:
print 'final eigenvalue:', _lambda
print 'final residual norms:', residualNorms
if retLambdaHistory:
if retResidualNormsHistory:
return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
else:
return _lambda, blockVectorX, lambdaHistory
else:
if retResidualNormsHistory:
return _lambda, blockVectorX, residualNormsHistory
else:
return _lambda, blockVectorX
