Choleski Decomposition - 'Matrix' S4 Generic and Methods

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Description

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

Usage

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chol(x, ...)
## S4 method for signature 'dsCMatrix'
chol(x, pivot = FALSE, ...)
## S4 method for signature 'dsparseMatrix'
chol(x, pivot = FALSE, cache = TRUE, ...)

Arguments

x

a (sparse or dense) square matrix, here inheriting from class Matrix; if x is not positive definite, 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 cashed in x@factors; note that this argument is experimental and only available for some sparse matrices.

...

potentially further arguments passed to methods.

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).

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).

Methods

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

chol

signature(x = "dgeMatrix"): works via "dpoMatrix", see class dpoMatrix.

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 via LAPACK routine dpptrf.

chol

signature(x = "dsCMatrix", pivot = "logical"): Returns (and stores) the Cholesky decomposition of x. If pivot is true, 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.

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.

See Also

The default from base, chol; for more flexibility (but not returning a matrix!) Cholesky.

Examples

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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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