[R] about the Choleski factorization
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,] 0.00 1.6733201 0.3585686 #[3,] 0.00 0.000 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 https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] about the Choleski factorization
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,] 0.00 1.6733201 0.3585686 #[3,] 0.00 0.000 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 https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html 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,]001 [2,]010 [3,]100 (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,]512 [2,]131 [3,]214 L - rev %*% chol(rev %*% x %*% rev) %*% rev L [,1] [,2] [,3] [1,] 1.9771421 0.000 [2,] 0.3015113 1.6583120 [3,] 1.000 0.502 t(L) %*% L [,1] [,2] [,3] [1,]512 [2,]131 [3,]214 Duncan Murdoch __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] about the Choleski factorization
You want a factorizzation of the form: A = L' L. Am I right (we may name this a Lochesky factorization)? By convention, Cholesky factorization is of the form A = L L', where L is a lower triangular matrix, and L', its transpose, is upper traingular. So, all numerical routines compute L according to this definition. R gives you U = L', which is obviously upper triangular. If you want to use a different definition: A = L' L, that is fine mathematically. Although there is no easy way to transform the result of existing routines to get what you want, you can actually derive an algorithm to convert the standard factorization to the form you want. Rather than go to this trouble, you might as well just code it up from scratch. The big question of course is why do you want the Lochesky factorization? It doesn't do anything special that the traditional Cholesky factorization can do for a symmetric, positive-definite matrix (mainly, solve a system of equations). 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: 93354504 93354...@nccu.edu.tw Date: Friday, March 27, 2009 11:58 am Subject: [R] about the Choleski factorization To: r-help r-help@r-project.org 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,] 0.00 1.6733201 0.3585686 #[3,] 0.00 0.000 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. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] about the Choleski factorization
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: 93354504 93354...@nccu.edu.tw
Date: Friday, March 27, 2009 11:58 am
Subject: [R] about the Choleski factorization
To: r-help r-help@r-project.org
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,] 0.00 1.6733201 0.3585686
#[3,] 0.00 0.000 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.
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] about the Choleski factorization
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,] 0.00 1.6733201 0.3585686
#[3,] 0.00 0.000 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,]001
[2,]010
[3,]100
(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,]512
[2,]131
[3,]214
L - rev %*% chol(rev %*% x %*% rev) %*% rev
L
[,1] [,2] [,3]
[1,] 1.9771421 0.000
[2,] 0.3015113 1.6583120
[3,] 1.000 0.502
t(L) %*% L
[,1] [,2] [,3]
[1,]512
[2,]131
[3,]214
Duncan Murdoch
__
R-help@r-project.org mailing list
PLEASE do read the posting guide
and provide commented, minimal, self-contained, reproducible code.
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] about the Choleski factorization
Duncan Murdoch wrote: 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,] 0.00 1.6733201 0.3585686 #[3,] 0.00 0.000 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 https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html 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,]001 [2,]010 [3,]100 (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,]512 [2,]131 [3,]214 L - rev %*% chol(rev %*% x %*% rev) %*% rev L [,1] [,2] [,3] [1,] 1.9771421 0.000 [2,] 0.3015113 1.6583120 [3,] 1.000 0.502 Or just r-3:1 chol(x[r,r])[r,r] [,1] [,2] [,3] [1,] 1.9771421 0.000 [2,] 0.3015113 1.6583120 [3,] 1.000 0.502 (It is after all, just a matter of starting from the other end). -- O__ Peter Dalgaard Øster Farimagsgade 5, Entr.B c/ /'_ --- Dept. of Biostatistics PO Box 2099, 1014 Cph. K (*) \(*) -- University of Copenhagen Denmark Ph: (+45) 35327918 ~~ - (p.dalga...@biostat.ku.dk) FAX: (+45) 35327907 __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
