Matrix (version 1.7-1)

updown-methods: Updating and Downdating Sparse Cholesky Factorizations

Description

Computes a rank-\(k\) update or downdate of a sparse Cholesky factorization $$P_{1} A P_{1}' = L_{1} D L_{1}' = L L'$$ which for some \(k\)-column matrix \(C\) is the factorization $$P_{1} (A + s C C') P_{1}' = \tilde{L}_{1} \tilde{D} \tilde{L}_{1}' = \tilde{L} \tilde{L}'$$ Here, \(s = 1\) for an update and \(s = -1\) for a downdate.

Usage

updown(update, C, L)

Value

A sparse Cholesky factorization with dimensions matching L, typically of class dCHMsimpl.

Arguments

update

a logical (TRUE or FALSE) or character ("+" or "-") indicating if L should be updated (or otherwise downdated).

C

a finite matrix or Matrix such that tcrossprod(C) has the dimensions of L.

L

an object of class dCHMsimpl or dCHMsuper specifying a sparse Cholesky factorization.

Author

Initial implementation by Nicholas Nagle, University of Tennessee.

References

Davis, T. A., Hager, W. W. (2001). Multiple-rank modifications of a sparse Cholesky factorization. SIAM Journal on Matrix Analysis and Applications, 22(4), 997-1013. tools:::Rd_expr_doi("10.1137/S0895479899357346")

See Also

Classes dCHMsimpl and dCHMsuper and their methods, notably for generic function update, which is not equivalent to updown(update = TRUE).

Generic function Cholesky.

Examples

m <- sparseMatrix(i = c(3, 1, 3:2, 2:1), p = c(0:2, 4, 4, 6), x = 1:6,
                  dimnames = list(LETTERS[1:3], letters[1:5]))
uc0 <- Cholesky(A <- crossprod(m) + Diagonal(5))
uc1 <- updown("+", Diagonal(5, 1), uc0)
uc2 <- updown("-", Diagonal(5, 1), uc1)
stopifnot(all.equal(uc0, uc2))
# \dontshow{
if(FALSE) {
## Hmm: this loses positive definiteness:
uc2 <- updown("-", Diagonal(5, 2), uc0)
image(show(as(uc0, "CsparseMatrix")))
image(show(as(uc2, "CsparseMatrix"))) # severely negative entries
}
# }