R: Inverse from Cholesky Factor

chol2inv-methods {Matrix}

R Documentation

Inverse from Cholesky Factor

Description

Given formally upper and lower triangular matrices U and L, compute (U' U)^{-1} and (L L')^{-1}, respectively.

This function can be seen as way to compute the inverse of a symmetric positive definite matrix given its Cholesky factor. Equivalently, it can be seen as a way to compute (X' X)^{-1} given the R part of the QR factorization of X, if R is constrained to have positive diagonal entries.

Usage

chol2inv(x, ...)
## S4 method for signature 'dtrMatrix'
chol2inv(x, ...)
## S4 method for signature 'dtCMatrix'
chol2inv(x, ...)
## S4 method for signature 'generalMatrix'
chol2inv(x, uplo = "U", ...)

Arguments

`x`	a square matrix or `Matrix`, typically the result of a call to `chol`. If `x` is square but not (formally) triangular, then only the upper or lower triangle is considered, depending on optional argument `uplo` if `x` is a `Matrix`.
`uplo`	a string, either `"U"` or `"L"`, indicating which triangle of `x` contains the Cholesky factor. The default is `"U"`, to be consistent with `chol2inv` from base.
`...`	further arguments passed to or from methods.

Value

A matrix, symmetricMatrix, or diagonalMatrix representing the inverse of the positive definite matrix whose Cholesky factor is x. The result is a traditional matrix if x is a traditional matrix, dense if x is dense, and sparse if x is sparse.

Examples

(A <- Matrix(cbind(c(1, 1, 1), c(1, 2, 4), c(1, 4, 16))))
(R <- chol(A))
(L <- t(R))
(R2i <- chol2inv(R))
(L2i <- chol2inv(R))
stopifnot(exprs = {
    all.equal(R2i, tcrossprod(solve(R)))
    all.equal(L2i,  crossprod(solve(L)))
    all.equal(as(R2i %*% A, "matrix"), diag(3L)) # the identity 
    all.equal(as(L2i %*% A, "matrix"), diag(3L)) # ditto
})