mroot: Smallest square root of matrix

Description

Find a square root of a positive semi-definite matrix, having as few columns as possible. Uses either pivoted choleski decomposition or singular value decomposition to do this.

Usage

mroot(A,rank=NULL,method="chol")

Arguments

The positive semi-definite matrix, a square root of which is to be found.

rank

if the rank of the matrix A is known then it should be supplied.

method

"chol" to use pivoted choloeski decompositon, which is fast but tends to over-estimate rank. "svd" to use singular value decomposition, which is slow, but is the most accurate way to estimate rank.

Value

A matrix, ${\bf B}$ with as many columns as the rank of ${\bf A}$, and such that ${\bf A} = {\bf BB}^\prime$.

Details

The routine uses an LAPACK SVD routine, or the LINPACK pivoted Choleski routine. It is primarily of use for turning penalized regression problems into ordinary regression problems.

Examples

Run this code

set.seed(0)
  a <- matrix(runif(24),6,4)
  A <- a%*%t(a) ## A is +ve semi-definite, rank 4
  B <- mroot(A) ## default pivoted choleski method
  tol <- 100*.Machine$double.eps
  chol.err <- max(abs(A-B%*%t(B)));chol.err
  if (chol.err>tol) warning("mroot (chol) suspect")
  B <- mroot(A,method="svd") ## svd method
  svd.err <- max(abs(A-B%*%t(B)));svd.err
  if (svd.err>tol) warning("mroot (svd) suspect")