Matrix (version 1.0-9)

chol: Choleski Decomposition - 'Matrix' S4 Generic

Description

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

Usage

chol(x, ...)
## S3 method for class 'dsCMatrix':
chol(x, pivot = FALSE, \dots)

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 used.

...

potentially further arguments passed to methods.

Value

• a matrix of class Cholesky, i.e., upper triangular: $R$ such that $R'R = x$.

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.

Examples

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")), tol= 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 <- Matrix(toeplitz(c(10, 0, 1, 0, 3)), sparse = TRUE)) # 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)))