chol: The Cholesky Decomposition - 'Matrix' S4 Generic and Methods

Description

Compute the Cholesky factorization of a real symmetric positive definite square matrix.

Usage

chol(x, ...)
# S4 method for dsyMatrix
chol(x, ...)
# S4 method for dspMatrix
chol(x, ...)
# S4 method for dsCMatrix
chol(x, pivot = FALSE, ...)
# S4 method for dsRMatrix
chol(x, pivot = FALSE, cache = TRUE, ...)
# S4 method for dsTMatrix
chol(x, pivot = FALSE, cache = TRUE, ...)

Value

a matrix of class Cholesky, i.e., upper triangular: $R$ such that $R^{'} R = x$ (if

pivot=FALSE) or

$P^{'} R^{'} R P = x$ (if

pivot=TRUE and $P$ is the corresponding permutation matrix).

Arguments

x: a (sparse or dense) square matrix, here inheriting from class Matrix; if x is not symmetric positive definite, then an error is signalled.
pivot: logical indicating if pivoting is to be used. Currently, this is not made use of for dense matrices.
cache: logical indicating if the result should be cached in x@factors; note that this argument is experimental and only available for certain classes inheriting from compMatrix.
...: potentially further arguments passed to methods.

Methods

Use showMethods(chol) to see all; some are worth mentioning here:

chol: signature(x = "dpoMatrix"): Returns (and stores) the Cholesky decomposition of x, via LAPACK routines dlacpy and dpotrf.
chol: signature(x = "dppMatrix"): Returns (and stores) the Cholesky decomposition of x, via LAPACK routine dpptrf.
chol: signature(x = "dsyMatrix"): works via "dpoMatrix", see class dpoMatrix.
chol: signature(x = "dspMatrix"): works via "dppMatrix", see class dppMatrix.
chol: signature(x = "dsCMatrix"): Returns (and stores) the Cholesky decomposition of x. If pivot is TRUE, then the Approximate Minimal Degree (AMD) algorithm is used to create a reordering of the rows and columns of x so as to reduce fill-in.
chol: signature(x = "dsRMatrix"): works via "dsCMatrix", see class dsCMatrix.
chol: signature(x = "dsTMatrix"): works via "dsCMatrix", see class dsCMatrix.

Details

Note that these Cholesky factorizations are typically cached with x currently, and these caches are available in x@factors, which may be useful for the sparse case when pivot = TRUE, where the permutation can be retrieved; see also the examples.

However, this should not be considered part of the API and made use of. Rather consider Cholesky() in such situations, since chol(x, pivot=TRUE) uses the same algorithm (but not the same return value!) as Cholesky(x, LDL=FALSE) and chol(x) corresponds to Cholesky(x, perm=FALSE, LDL=FALSE).

References

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

Tim Davis (1996), An approximate minimal degree ordering algorithm, SIAM J. Matrix Analysis and Applications, 17, 4, 886--905.

Examples

Run this code

showMethods(chol, inherited = FALSE) # show different methods

sy2 <- new("dsyMatrix", Dim = as.integer(c(2,2)), x = c(14, NA,32,77))
(c2 <- chol(sy2))#-> "Cholesky" matrix
stopifnot(all.equal(c2, chol(as(sy2, "dpoMatrix")), tolerance= 1e-13))
str(c2)

## An example where chol() can't work
(sy3 <- new("dsyMatrix", Dim = as.integer(c(2,2)), x = c(14, -1, 2, -7)))
try(chol(sy3)) # error, since it is not positive definite

## A sparse example --- exemplifying 'pivot'
(mm <- toeplitz(as(c(10, 0, 1, 0, 3), "sparseVector"))) # 5 x 5
(R <- chol(mm)) ## default:  pivot = FALSE
R2 <- chol(mm, pivot=FALSE)
stopifnot( identical(R, R2), all.equal(crossprod(R), mm) )
(R. <- chol(mm, pivot=TRUE))# nice band structure,
## but of course crossprod(R.) is *NOT* equal to mm
## --> see Cholesky() and its examples, for the pivot structure & factorization
stopifnot(all.equal(sqrt(det(mm)), det(R)),
          all.equal(prod(diag(R)), det(R)),
          all.equal(prod(diag(R.)), det(R)))

## a second, even sparser example:
(M2 <- toeplitz(as(c(1,.5, rep(0,12), -.1), "sparseVector")))
c2 <- chol(M2)
C2 <- chol(M2, pivot=TRUE)
## For the experts, check the caching of the factorizations:
ff <- M2@factors[["spdCholesky"]]
FF <- M2@factors[["sPdCholesky"]]
L1 <- as(ff, "Matrix")# pivot=FALSE: no perm.
L2 <- as(FF, "Matrix"); P2 <- as(FF, "pMatrix")
stopifnot(identical(t(L1), c2),
          all.equal(t(L2), C2, tolerance=0),#-- why not identical()?
          all.equal(M2, tcrossprod(L1)),             # M = LL'
          all.equal(M2, crossprod(crossprod(L2, P2)))# M = P'L L'P
         )

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