Matrix (version 0.999375-2)

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 = TRUE, super = FALSE, ...)

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.

...

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 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()
Cholesky(mtm)             # uses show(<MatrixFactorization>)
str(mtm@factors) # 'sPDCholesky' (simpl)
(Cm <- Cholesky(mtm, super = TRUE))
str(mtm@factors) # 'sPDCholesky'  *and*  'SPDCholesky'
str(cmat <- as(Cm, "sparseMatrix"))
cmat[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 !
stopifnot(identical(solve(Cm, b), solve(Cm, b, system = "A")),
          all.equal(solve(mtm, b),
                    solve(Cm, b)))