Matrix (version 0.999375-46)

Cholesky: Cholesky Decomposition of a Sparse Matrix

Description

Computes the Cholesky decomposition of a sparse, symmetric, positive-definite matrix. However, typically chol() should rather be used unless you are interested in the different kinds of sparse Cholesky decompositions.

Usage

Cholesky(A, perm = TRUE, LDL = !super, super = FALSE, Imult = 0, ...)

Arguments

A

sparse symmetric matrix. No missing values or IEEE special values are allowed.

perm

logical scalar indicating if a fill-reducing permutation should be computed and applied to the rows and columns of A. Default is TRUE.

LDL

logical scalar indicating if the decomposition should be computed as LDL' where L is a unit lower triangular matrix. The alternative is LL' where L is lower triangular with arbitrary diagonal elements. Default is

super

logical scalar indicating is a supernodal decomposition should be created. The alternative is a simplicial decomposition. Default is FALSE. Setting it to NA leaves the choice to a CHOL

Imult

numeric scalar which defaults to zero. The matrix that is decomposed is $A+m*I$ where $m$ is the value of Imult and I is the identity matrix of order ncol(A).

...

further arguments passed to or from other methods.

Value

• an object inheriting from either "CHMsuper", or "CHMsimpl", depending on the super argument; both classes extend "CHMfactor" which extends "MatrixFactorization".

  In other words, the result of Cholesky() is not a matrix, and if you want one, you should probably rather use chol().

Details

This is a generic function with special methods for different types of matrices. Use showMethods("Cholesky") to list all the methods for the Cholesky generic.

The method for class dsCMatrix of sparse matrices --- the only one available currently --- is based on functions from the CHOLMOD library.

Again: If you just want the Cholesky decomposition of a matrix, you should probably rather use chol(.).

References

Tim Davis (2005) {CHOLMOD}: sparse supernodal {Cholesky} factorization and update/downdate http://www.cise.ufl.edu/research/sparse/cholmod/

Timothy A. Davis (2006) Direct Methods for Sparse Linear Systems, SIAM Series Fundamentals of Algorithms.

See Also

Class definitions CHMfactor and dsCMatrix and function expand. Note the extra solve(*, system = . ) options in CHMfactor.

Note that chol() returns matrices (inheriting from "Matrix") whereas Cholesky() returns a "CHMfactor" object, and hence a typical user will rather use chol(A).

Examples

data(KNex)
mtm <- with(KNex, crossprod(mm))
str(mtm@factors) # empty list()
(C1 <- Cholesky(mtm))             # uses show(<MatrixFactorization>)
str(mtm@factors) # 'sPDCholesky' (simpl)
(Cm <- Cholesky(mtm, super = TRUE))
c(C1 = isLDL(C1), Cm = isLDL(Cm))
str(mtm@factors) # 'sPDCholesky'  *and* 'SPdCholesky'
str(cm1  <- as(C1, "sparseMatrix"))
str(cmat <- as(Cm, "sparseMatrix"))# hmm: super is *less* sparse here
cm1[1:20, 1:20]

b <- matrix(c(rep(0, 711), 1), nc = 1)
## solve(Cm, b) by default solves  Ax = b, where A = Cm'Cm !
## hence, the identical() check *should* work, but fails on some GOTOblas:
x <- solve(Cm, b)
stopifnot(identical(x, solve(Cm, b, system = "A")),
          all.equal(x, solve(mtm, b)))

Cn <- Cholesky(mtm, perm = FALSE)# no permutation -- much worse:
sizes <- c(simple = object.size(C1),
           super  = object.size(Cm),
           noPerm = object.size(Cn))
format(cbind(100 * sizes / sizes[1]), digits=4)

## Visualize the sparseness:
dq <- function(ch) paste('"',ch,'"', sep="") ## dQuote(<UTF-8>) gives bad plots
image(mtm, main=paste("crossprod(mm) : Sparse", dq(class(mtm))))
image(cm1, main= paste("as(Cholesky(crossprod(mm)),"sparseMatrix"):",
                        dq(class(cm1))))
expand(C1) ## to check printing