Sylvester's formula (http://en.wikipedia.org/wiki/Sylvester%27s_formula)
applies to a square matrix A = S L solve(S), where L = a diagonal matrix and S =
matrix of eigenvectors. Let "f" be an analytic function [for which f(A) is well
defined]. Then f(A) = S f(L) solve(S).
We can code this as follows:
sylvester <- function(x, f){
n <- nrow(x)
eig <- eigen(x)
vi <- solve(eig$vectors)
with(eig, (vectors * rep(f(values), each=n)) %*% vi)
}
logm <- function(x)sylvester(x, log)
Example:
A <- matrix(1:4, 2)
eA <- expm(A)
logm(eA)
With Chuck Berry's example, we get the following:
M <- matrix( c(0,1,0,0), 2 )
sylvester(M, log)
Error in solve.default(eig$vectors) :
system is computationally singular: reciprocal condition number = 1.00208e-292
This is a perfectly sensible answer in this case. We get the same result from "sylvester(M, exp)", though "expm(M)" works fine.
A better algorithm for this could be obtains by studying the code for "expm" in the Matrix package and the
references in the associated help page.
Hope this helps.
Spencer
Gabor Grothendieck wrote:
Often one uses matrix logarithms on symmetric positive definite
matrices so the assumption of being symmetric is sufficient in many
cases.
On Sat, Sep 26, 2009 at 7:28 PM, Charles C. Berry <cbe...@tajo.ucsd.edu> wrote:
On Sat, 26 Sep 2009, Gabor Grothendieck wrote:
OK. Try this:
library(Matrix)
M <- matrix(c(2, 1, 1, 2), 2); M
[,1] [,2]
[1,] 2 1
[2,] 1 2
Right. expm( M ) is diagonalizable.
But for
M <- matrix( c(0,1,0,0), 2 )
you get the wrong result.
Maybe I should have added that I do not see the machinery in R for dealing
with Jordan blocks.
HTH,
Chuck
# log of expm(M) is original matrix M
with(eigen(expm(M)), vectors %*% diag(log(values)) %*% t(vectors))
[,1] [,2]
[1,] 2 1
[2,] 1 2
On Sat, Sep 26, 2009 at 6:24 PM, Charles C. Berry <cbe...@tajo.ucsd.edu>
wrote:
On Sat, 26 Sep 2009, Gabor Grothendieck wrote:
Try:
expm( - M)
Mimosa probably meant say 'the inverse function'.
I do not see one in R.
Chuck
On Sat, Sep 26, 2009 at 5:06 PM, Mimosa Zeus <mimosa1...@yahoo.fr>
wrote:
Dear R users,
Does anyone has implemented the inverse of the matrix exponential (expm
in the package Matrix)?
In Matlab, there're logm and expm, there's only expm in R.
Cheers
Mimosa
[[alternative HTML version deleted]]
______________________________________________
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.
______________________________________________
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.
Charles C. Berry (858) 534-2098
Dept of Family/Preventive
Medicine
E mailto:cbe...@tajo.ucsd.edu UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego
92093-0901
______________________________________________
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.
Charles C. Berry (858) 534-2098
Dept of Family/Preventive
Medicine
E mailto:cbe...@tajo.ucsd.edu UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/ La Jolla, San Diego 92093-0901
______________________________________________
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.
--
Spencer Graves, PE, PhD
President and Chief Operating Officer
Structure Inspection and Monitoring, Inc.
751 Emerson Ct.
San José, CA 95126
ph: 408-655-4567
______________________________________________
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.
