| Cholesky {Matrix} | R Documentation |
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.
Cholesky(A, perm = TRUE, LDL = !super, super = FALSE, Imult = 0, ...)
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 TRUE. |
super |
logical scalar indicating is a supernodal decomposition
should be created. The alternative is a simplicial decomposition.
Default is FALSE. |
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. |
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(.).
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().
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”.
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).
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))
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 !
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))))