def __init__( self, A, factorize=True, **kwargs ):
"""
Create a PyMa57Context object representing a context to solve
the square symmetric linear system of equations
A x = b.
A should be given in ll_mat format and should be symmetric.
The system will first be analyzed and factorized, for later
solution. Residuals will be computed dynamically if requested.
The factorization is a multi-frontal variant of the Bunch-Parlett
factorization, i.e.
A = L B Lt
where L is unit lower triangular, and B is symmetric and block diagonal
with either 1x1 or 2x2 blocks.
A special option is available is the matrix A is known to be symmetric
(positive or negative) definite, or symmetric quasi-definite (sqd).
SQD matrices have the general form
[ E Gt ]
[ G -F ]
where both E and F are positive definite. As a special case, positive
definite matrices and negative definite matrices are sqd. SQD matrices
can be factorized with potentially much sparser factors. Moreover, the
matrix B reduces to a diagonal matrix.
Currently accepted keyword arguments are:
sqd Flag indicating symmetric quasi-definite matrix (default: False)
Example:
>>> import pyma57
>>> import numpy
>>> P = pyma57.PyMa57Context(A)
>>> P.solve(rhs, get_resid=True)
>>> print numpy.linalg.norm(P.residual)
Pyma57 relies on the sparse direct multifrontal code MA57
from the Harwell Subroutine Library.
From the MA57 spec sheet, 'In addition to being more efficient largely
through its use of the Level 3 BLAS, it has many added features. Among
these are: a fast mapping of data prior to a numerical factorization,
the ability to modify pivots if the matrix is not definite, the
efficient solution of several right-hand sides, a routine implementing
iterative refinement, and the possibility of restarting the
factorization should it run out of space.'
References
----------
1. I. S. Duff, "MA57 -- A New Code for the Solution of Sparse Symmetric
Indefinite Systems", ACM Transactions on Mathematical Software (30),
p. 118-144, 2004.
2. I. S. Duff and J. K. Reid, "The Multifrontal Solution of Indefinite
Sparse Symmetric Linear Systems", Transactions on Mathematical
Software (9), p. 302--325, 1983.
3. http://hsl.rl.ac.uk/hsl2007/hsl20074researchers.html
"""
if isinstance(A, PysparseMatrix):
thisA = A.matrix
else:
thisA = A
Sils.__init__( self, thisA, **kwargs )
# Statistics on A
self.nzFact = 0 # Number of nonzeros in factors
self.nRealFact = 0 # Storage for real data of factors
self.nIntFact = 0 # Storage for int data of factors
self.front = 0 # Largest front size
self.n2x2pivots = 0 # Number of 2x2 pivots used
self.neig = 0 # Number of negative eigenvalues detected
self.rank = 0 # Matrix rank
# Factors
#self.L = spmatrix.ll_mat( self.n, self.n, 0 )
#self.B = spmatrix.ll_mat_sym( self.n, 0 )
# Analyze and factorize matrix
self.context = _pyma57.analyze( thisA, self.sqd )
self.factorized = False
if factorize: self.factorize(thisA)
return
def __init__( self, A, **kwargs ):
"""
Create a PyMa27Context object representing a context to solve
the square symmetric linear system of equations
A x = b.
A should be given in ll_mat format and should be symmetric.
The system will first be analyzed and factorized, for later
solution. Residuals will be computed dynamically if requested.
The factorization is a multi-frontal variant of the Bunch-Parlett
factorization, i.e.
A = L B Lt
where L is unit lower triangular, and B is symmetric and block diagonal
with either 1x1 or 2x2 blocks.
A special option is available is the matrix A is known to be symmetric
(positive or negative) definite, or symmetric quasi-definite (sqd).
SQD matrices have the general form
[ E Gt ]
[ G -F ]
where both E and F are positive definite. As a special case, positive
definite matrices and negative definite matrices are sqd. SQD matrices
can be factorized with potentially much sparser factors. Moreover, the
matrix B reduces to a diagonal matrix.
Currently accepted keyword arguments are:
sqd Flag indicating symmetric quasi-definite matrix (default: False)
Example:
from nlpy.linalg import pyma27
from nlpy.tools import norms
P = pyma27.PyMa27Context( A )
P.solve( rhs, get_resid = True )
print norms.norm2( P.residual )
Pyma27 relies on the sparse direct multifrontal code MA27
from the Harwell Subroutine Library archive.
"""
if isinstance(A, PysparseMatrix):
thisA = A.matrix
else:
thisA = A
Sils.__init__( self, thisA, **kwargs )
# Statistics on A
self.rwords = 0 # Number of real words used during factorization
self.iwords = 0 # int
self.ncomp = 0 # data compresses performed in analysis
self.nrcomp = 0 # real
self.nicomp = 0 # int
self.n2x2pivots = 0 # 2x2 pivots used
self.neig = 0 # negative eigenvalues detected
# Factors
self.L = spmatrix.ll_mat( self.n, self.n, 0 )
self.B = spmatrix.ll_mat_sym( self.n, 0 )
# Analyze and factorize matrix
self.context = _pyma27.factor( thisA, self.sqd )
(self.rwords, self.iwords, self.ncomp, self.nrcomp, self.nicomp,
self.n2x2pivots, self.neig, self.rank) = self.context.stats()
self.isFullRank = (self.rank == self.n)
def __init__(self, A, **kwargs):
"""
Create a PyMa27Context object representing a context to solve
the square symmetric linear system of equations
A x = b.
A should be given in ll_mat format and should be symmetric.
The system will first be analyzed and factorized, for later
solution. Residuals will be computed dynamically if requested.
The factorization is a multi-frontal variant of the Bunch-Parlett
factorization, i.e.
A = L B Lt
where L is unit lower triangular, and B is symmetric and block diagonal
with either 1x1 or 2x2 blocks.
A special option is available is the matrix A is known to be symmetric
(positive or negative) definite, or symmetric quasi-definite (sqd).
SQD matrices have the general form
[ E Gt ]
[ G -F ]
where both E and F are positive definite. As a special case, positive
definite matrices and negative definite matrices are sqd. SQD matrices
can be factorized with potentially much sparser factors. Moreover, the
matrix B reduces to a diagonal matrix.
Currently accepted keyword arguments are:
sqd Flag indicating symmetric quasi-definite matrix (default: False)
Example:
from nlpy.linalg import pyma27
from nlpy.tools import norms
P = pyma27.PyMa27Context( A )
P.solve( rhs, get_resid = True )
print norms.norm2( P.residual )
Pyma27 relies on the sparse direct multifrontal code MA27
from the Harwell Subroutine Library archive.
"""
if isinstance(A, PysparseMatrix):
thisA = A.matrix
else:
thisA = A
Sils.__init__(self, thisA, **kwargs)
# Statistics on A
self.rwords = 0 # Number of real words used during factorization
self.iwords = 0 # int
self.ncomp = 0 # data compresses performed in analysis
self.nrcomp = 0 # real
self.nicomp = 0 # int
self.n2x2pivots = 0 # 2x2 pivots used
self.neig = 0 # negative eigenvalues detected
# Factors
self.L = spmatrix.ll_mat(self.n, self.n, 0)
self.B = spmatrix.ll_mat_sym(self.n, 0)
# Analyze and factorize matrix
self.context = _pyma27.factor(thisA, self.sqd)
(self.rwords, self.iwords, self.ncomp, self.nrcomp, self.nicomp,
self.n2x2pivots, self.neig, self.rank) = self.context.stats()
self.isFullRank = (self.rank == self.n)