Very nice, Duncan.
Here is a little function called loch() that implements your idea for the
Lochesky factorization:
loch <- function(mat) {
n <- ncol(mat)
rev <- diag(1, n)[, n: 1]
rev %*% chol(rev %*% mat %*% rev) %*% rev
}
x=matrix(c(5,1,2,1,3,1,2,1,4),3,3)
L <- loch(x)
all.equal(x, t(L) %*% L)
A <- matrix(rnorm(36), 6, 6)
A <- A %*% t(A)
L <- loch(x)
all.equal(x, t(L) %*% L)
Ravi.
____________________________________________________________________
Ravi Varadhan, Ph.D.
Assistant Professor,
Division of Geriatric Medicine and Gerontology
School of Medicine
Johns Hopkins University
Ph. (410) 502-2619
email: rvarad...@jhmi.edu
----- Original Message -----
From: Duncan Murdoch <murd...@stats.uwo.ca>
Date: Friday, March 27, 2009 1:29 pm
Subject: Re: [R] about the Choleski factorization
To: 93354...@nccu.edu.tw
Cc: r-help <r-help@r-project.org>
> On 3/27/2009 11:46 AM, 93354504 wrote:
> > Hi there,
> >
> > Given a positive definite symmetric matrix, I can use chol(x) to
> obtain U where U is upper triangular
> > and x=U'U. For example,
> >
> > x=matrix(c(5,1,2,1,3,1,2,1,4),3,3)
> > U=chol(x)
> > U
> > # [,1] [,2] [,3]
> > #[1,] 2.236068 0.4472136 0.8944272
> > #[2,] 0.000000 1.6733201 0.3585686
> > #[3,] 0.000000 0.0000000 1.7525492
> > t(U)%*%U # this is exactly x
> >
> > Does anyone know how to obtain L such that L is lower triangular
> and x=L'L? Thank you.
> >
> > Alex
> >
> > ______________________________________________
> > R-help@r-project.org mailing list
> >
> > PLEASE do read the posting guide
> > and provide commented, minimal, self-contained, reproducible code.
>
> > rev <- matrix(c(0,0,1,0,1,0,1,0,0),3,3)
> > rev
> [,1] [,2] [,3]
> [1,] 0 0 1
> [2,] 0 1 0
> [3,] 1 0 0
>
> (the matrix that reverses the row and column order when you pre and
> post
> multiply it).
>
> Then
>
> L <- rev %*% chol(rev %*% x %*% rev) %*% rev
>
> is what you want, i.e. you reverse the row and column order of the
> Choleski square root of the reversed x:
>
> > x
> [,1] [,2] [,3]
> [1,] 5 1 2
> [2,] 1 3 1
> [3,] 2 1 4
>
> > L <- rev %*% chol(rev %*% x %*% rev) %*% rev
> > L
> [,1] [,2] [,3]
> [1,] 1.9771421 0.000000 0
> [2,] 0.3015113 1.658312 0
> [3,] 1.0000000 0.500000 2
> > t(L) %*% L
> [,1] [,2] [,3]
> [1,] 5 1 2
> [2,] 1 3 1
> [3,] 2 1 4
>
> Duncan Murdoch
>
> ______________________________________________
> R-help@r-project.org mailing list
>
> PLEASE do read the posting guide
> and provide commented, minimal, self-contained, reproducible code.
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