mgcv (version 1.8-29)

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

A

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. NULL or <1 imply that it should be estimated.

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 function uses SVD, or a pivoted Choleski routine. It is primarily of use for turning penalized regression problems into ordinary regression problems.

Examples

# NOT RUN {
  require(mgcv)
  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")  
# }