should the matrix be permuted, and if, with what
algorithm, see ‘Details’ below.
method
Currently, only NgPeyton is implemented.
memory
Parameters specific to the method, see ‘Details’ below.
eps
threshold to test symmetry. Defaults to .Spam$eps.
...
further arguments passed to or from other methods.
object
an object from a previous call to chol.
Details
chol performs a Cholesky decomposition of a symmetric
positive definite sparse matrix x of class
spam. Currently, there is only the block sparse Cholesky
algorithm of Ng and Peyton (1993) implemented (method="NgPeyton").
To pivot/permute the matrix, you can choose between the multiple minimum
degree (pivot="MMD") or reverse Cuthill-Mckee (pivot="RCM")
from George and Lui (1981). It is also possible to furnish a specific
permutation in which case pivot is a vector. For compatibility
reasons, pivot can also take a logical in which for FALSE
no permutation is done and for TRUE is equivalent to
MMD.
Often the sparsity structure is fixed and does not change, but the
entries do. In those cases, we can update the Cholesky factor with
update.spam.chol.NgPeyton by suppling a Cholesky factor and the
updated matrix. Notice that the structure is effectively object <-
update(object, x). The update feature without assignement has been disabled.
The option cholupdatesingular determines how singular matrices
are handled by update. The function hands back an error
("error"), a warning ("warning") or the value NULL
("null").
The Cholesky decompositions requires parameters, linked to memory
allocation. If the default values are too small the Fortran routine
returns an error to R, which allocates more space and calls the Fortran
routine again. The user can also pass better estimates of the allocation
sizes to chol with the argument memory=list(nnzR=...,
nnzcolindices=...). The minimal sizes for a fixed sparsity
structure can be obtained from a summary call, see ‘Examples’.
The output of chol can be used with forwardsolve and
backsolve to solve a system of linear equations.
Notice that the Cholesky factorization of the package SparseM is also
based on the algorithm of Ng and Peyton (1993). Whereas the Cholesky
routine of the package Matrix are based on
CHOLMOD by Timothy A. Davis (C code).
Value
The function returns the Cholesky factor in an object of class
spam.chol.method. Recall that the latter is the Cholesky
factor of a reordered matrix x, see also ordering.
Note
Although the symmetric structure of x is needed, only the upper
diagonal entries are used. By default, the code does check for
symmetry (contrarily to base:::chol). However,
depending on the matrix size, this is a time consuming test.
A test is ignored if
spam.options( "cholsymmetrycheck") is set to FALSE.
If a permutation is supplied with pivot,
spam.options( "cholpivotcheck") determines if the permutation is
tested for validity (defaults to TRUE).
Author(s)
Reinhard Furrer, based on Ng and Peyton (1993) Fortran routines
References
Ng, E. G. and Peyton, B. W. (1993) Block sparse Cholesky algorithms on
advanced uniprocessor computers, SIAM J. Sci. Comput., 14,
1034–1056.
Gilbert, J. R., Ng, E. G. and Peyton, B. W. (1994) An efficient
algorithm to compute row and column counts for sparse Cholesky
factorization, SIAM J. Matrix Anal. Appl., 15,
1075–1091.
George, A. and Liu, J. (1981)
Computer Solution of Large Sparse Positive Definite Systems,
Prentice Hall.
See Also
det.spam, solve.spam,
forwardsolve.spam, backsolve.spam and ordering.
Examples
# generate multivariate normals:
set.seed(13)
n <- 25 # dimension
N <- 1000 # sample size
Sigma <- .25^abs(outer(1:n,1:n,"-"))
Sigma <- as.spam( Sigma, eps=1e-4)
cholS <- chol( Sigma)
# cholS is the upper triangular part of the permutated matrix Sigma
iord <- ordering(cholS, inv=TRUE)
R <- as.spam(cholS)
mvsample <- ( array(rnorm(N*n),c(N,n)) %*% R)[,iord]
# It is often better to order the sample than the matrix
# R itself.
# 'mvsample' is of class 'spam'. We need to transform it to a
# regular matrix, as there is no method 'var' for 'spam' (should there?).
norm( var( as.matrix( mvsample)) - Sigma, type='m')
norm( t(R) %*% R - Sigma)
# To speed up factorizations, memory allocations can be optimized:
opt <- summary(cholS)
# here, some elements of Sigma may be changed...
cholS <- chol( Sigma, memory=list(nnzR=opt$nnzR,nnzcolindices=opt$nnzc))