From: Martin Maechler <maechler_at_stat.math.ethz.ch>

Date: Thu, 01 Sep 2011 11:50:49 +0200

* >>> A principled way to solve this system of ODEs is to use the idea of
*

* >>> "fundamental matrix", which is the same idea as that of
*

* >>> diagonalization of a matrix (see Boyce and DiPrima or any ODE text).
*

* >>>
*

* >>> Here is the code for that:
*

* >>>
*

* >>>
*

>>> nlogL2 <- function(theta){ k <- exp(theta[1:3]) sigma <-

* >>> exp(theta[4]) A <- rbind( c(-k[1], k[2]), c( k[1], -(k[2]+k[3])) ) eA
*

* >>> <- eigen(A) T <- eA$vectors r <- eA$values x0 <- c(0,100) Tx0 <- T
*

* >>> %*% x0
*

* >>>
*

* >>> sol <- function(t) 100 - sum(T %*% diag(exp(r*t)) %*% Tx0) pred <-
*

* >>> sapply(dat[,1], sol) -sum(dnorm(dat[,2], mean=pred, sd=sigma,
*

* >>> log=TRUE)) } This is much faster than using expm(A*t), but much
*

* >>> slower than "by hand" calculations since we have to repeatedly do the
*

* >>> diagonalization. An obvious advantage of this fuunction is that it
*

* >>> applies to *any* linear system of ODEs for which the eigenvalues are
*

* >>> real (and negative).
*

* >>>
*

* >>> Ravi.
*

* >>>
*

* >>> -------------------------------------------------------
*

* >>> Ravi Varadhan, Ph.D.
*

>>> Assistant Professor,

* >>> Division of Geriatric Medicine and Gerontology School of Medicine Johns Hopkins University
*

* >>>
*

* >>> Ph. (410) 502-2619
*

* >>> email: rvaradhan_at_jhmi.edu
*

* >>>
*

* >>>
*

* >>> -----Original Message-----
*

* >>> From: r-devel-bounces_at_r-project.org [mailto:r-devel-bounces_at_r-project.org] On Behalf Of Ravi Varadhan
*

* >>> Sent: Wednesday, August 17, 2011 2:33 PM
*

* >>> To: 'cberry_at_tajo.ucsd.edu'; r-devel_at_stat.math.ethz.ch; 'nashjc_at_uottawa.ca'
*

* >>> Subject: Re: [Rd] An example of very slow computation
*

* >>>
*

* >>> Yes, the culprit is the evaluation of expm(A*t). This is a lazy way of solving the system of ODEs, where you save analytic effort, but you pay for it dearly in terms of computational effort!
*

* >>>
*

* >>> Even in this lazy approach, I am sure there ought to be faster ways to evaluating exponent of a matrix than that in "Matrix" package.
*

* >>>
*

* >>> Ravi.
*

* >>>
*

* >>> -------------------------------------------------------
*

* >>> Ravi Varadhan, Ph.D.
*

* >>> Assistant Professor,
*

* >>> Division of Geriatric Medicine and Gerontology School of Medicine Johns Hopkins University
*

* >>>
*

* >>> Ph. (410) 502-2619
*

* >>> email: rvaradhan_at_jhmi.edu
*

* >>>
*

* >>> -----Original Message-----
*

* >>> From: r-devel-bounces_at_r-project.org [mailto:r-devel-bounces_at_r-project.org] On Behalf Of cberry_at_tajo.ucsd.edu
*

* >>> Sent: Wednesday, August 17, 2011 1:08 PM
*

* >>> To: r-devel_at_stat.math.ethz.ch
*

* >>> Subject: Re: [Rd] An example of very slow computation
*

* >>>
*

* >>> John C Nash <nashjc_at_uottawa.ca> writes:
*

* >>>
*

* >>>
*

* >>> Looks like A*t in expm(A*t) is a "matrix".
*

* >>>
*

>>> 'getMethod("expm","matrix")' coerces it arg to a "Matrix", then calls

* >>> expm(), whose method coerces its arg to a "dMatrix" and calls expm(),
*

* >>> whose method coerces its arg to a 'dgeMatrix' and calls expm(), whose
*

* >>> method finally calls '.Call(dgeMatrix_exp, x)'
*

* >>>
*

* >>> Whew!
*

* >>>
*

* >>> The time difference between 'expm( diag(10)+1 )' and 'expm( as( diag(10)+1,
*

* >>> "dgeMatrix" ))' is a factor of 10 on my box.
*

* >>>
*

* >>> Dunno 'bout the other factor of 100.
*

* >>>
*

* >>> Chuck
*

* >>>
*

* >>>
*

* >>>
*

* >>> --
*

* >>> Charles C. Berry cberry_at_tajo.ucsd.edu
*

* >>>
*

* >>> ______________________________________________
*

>>> R-devel@r-project.org mailing list

* >>> https://stat.ethz.ch/mailman/listinfo/r-devel
*

https://stat.ethz.ch/mailman/listinfo/r-devel Received on Thu 01 Sep 2011 - 09:53:27 GMT

Date: Thu, 01 Sep 2011 11:50:49 +0200

>>>>> Michael Lachmann <lachmann_at_eva.mpg.de> >>>>> on Fri, 19 Aug 2011 02:08:48 +0200 writes:

> On my trials, after eliminating all the extra matrix<->dgeMatrix > conversions, using expm() and the method below were equally fast.

> Michael

I'm sorry I hadn't time to get into this thread when it was
hot..

and I have told Ravi in the mean time what I would have said
*VERY* early here:

- There's an 'expm' package --- which will be mentioned on the Matrix::expm help page in the next version of Matrix --- that has better and faster algorithms (notably also some which work with *sparse* matrices!) than the one in Matrix.
- Matrix::expm() is really more reliable than one of the 19 dubious ways that Peter mentioned correctly, and, it is really typically not a good idea to use a faster but considerably less reliable method.

Reliability is *much* *much* more important
than speed, *really*

and we (R-core) have always tried to emphasize this approach
in all contexts.

3) Thanks to Ravi, an upcoming version of the 'expm' package

will also contain a (Krylov space) version of expm() which is faster for the special case of evaluating

expm(A * t) %*% v {A matrix, v vector, t scalar} and *still* numerically reliable.

Martin Maechler, ETH Zurich

>> Which is why I said it applies when the system is "diagonalizable". >> It won't work for non-diagonalizable matrix A, because T (eigenvector >> matrix) is singular. >> >> Ravi. ________________________________________ From: peter dalgaard >> [pdalgd_at_gmail.com] Sent: Thursday, August 18, 2011 6:37 PM To: Ravi >> Varadhan Cc: 'cberry_at_tajo.ucsd.edu'; r-devel_at_stat.math.ethz.ch; >> 'nashjc_at_uottawa.ca' Subject: Re: [Rd] An example of very slow >> computation >> >> On Aug 17, 2011, at 23:24 , Ravi Varadhan wrote: >>

>>> nlogL2 <- function(theta){ k <- exp(theta[1:3]) sigma <-

>> >> I believe this is method 14 of the "19 dubious ways..." (Google for >> it) and doesn't work for certain non-symmetric A matrices. >>

>>> Assistant Professor,

>>>> This message is about a curious difference in timing between two ways of computing the >>>> same function. One uses expm, so is expected to be a bit slower, but "a bit" turned out to >>>> be a factor of >1000. The code is below. We would be grateful if anyone can point out any >>>> egregious bad practice in our code, or enlighten us on why one approach is so much slower >>>> than the other.

>>> 'getMethod("expm","matrix")' coerces it arg to a "Matrix", then calls

>>>> The problem arose in an activity to develop guidelines for nonlinear >>>> modeling in ecology (at NCEAS, Santa Barbara, with worldwide participants), and we will be >>>> trying to include suggestions of how to prepare problems like this for efficient and >>>> effective solution. The code for nlogL was the "original" from the worker who supplied the >>>> problem. >>>> >>>> Best, >>>> >>>> John Nash >>>> >>>> -------------------------------------------------------------------------------------- >>>> >>>> cat("mineral-timing.R == benchmark MIN functions in R\n") >>>> # J C Nash July 31, 2011 >>>> >>>> require("microbenchmark") >>>> require("numDeriv") >>>> library(Matrix) >>>> library(optimx) >>>> # dat<-read.table('min.dat', skip=3, header=FALSE) >>>> # t<-dat[,1] >>>> t <- c(0.77, 1.69, 2.69, 3.67, 4.69, 5.71, 7.94, 9.67, 11.77, 17.77, >>>> 23.77, 32.77, 40.73, 47.75, 54.90, 62.81, 72.88, 98.77, 125.92, 160.19, >>>> 191.15, 223.78, 287.70, 340.01, 340.95, 342.01) >>>> >>>> # y<-dat[,2] # ?? tidy up >>>> y<- c(1.396, 3.784, 5.948, 7.717, 9.077, 10.100, 11.263, 11.856, 12.251, 12.699, >>>> 12.869, 13.048, 13.222, 13.347, 13.507, 13.628, 13.804, 14.087, 14.185, 14.351, >>>> 14.458, 14.756, 15.262, 15.703, 15.703, 15.703) >>>> >>>> >>>> ones<-rep(1,length(t)) >>>> theta<-c(-2,-2,-2,-2) >>>> >>>> >>>> nlogL<-function(theta){ >>>> k<-exp(theta[1:3]) >>>> sigma<-exp(theta[4]) >>>> A<-rbind( >>>> c(-k[1], k[2]), >>>> c( k[1], -(k[2]+k[3])) >>>> ) >>>> x0<-c(0,100) >>>> sol<-function(t)100-sum(expm(A*t)%*%x0) >>>> pred<-sapply(dat[,1],sol) >>>> -sum(dnorm(dat[,2],mean=pred,sd=sigma, log=TRUE)) >>>> } >>>> >>>> getpred<-function(theta, t){ >>>> k<-exp(theta[1:3]) >>>> sigma<-exp(theta[4]) >>>> A<-rbind( >>>> c(-k[1], k[2]), >>>> c( k[1], -(k[2]+k[3])) >>>> ) >>>> x0<-c(0,100) >>>> sol<-function(tt)100-sum(expm(A*tt)%*%x0) >>>> pred<-sapply(t,sol) >>>> } >>>> >>>> Mpred <- function(theta) { >>>> # WARNING: assumes t global >>>> kvec<-exp(theta[1:3]) >>>> k1<-kvec[1] >>>> k2<-kvec[2] >>>> k3<-kvec[3] >>>> # MIN problem terbuthylazene disappearance >>>> z<-k1+k2+k3 >>>> y<-z*z-4*k1*k3 >>>> l1<-0.5*(-z+sqrt(y)) >>>> l2<-0.5*(-z-sqrt(y)) >>>> val<-100*(1-((k1+k2+l2)*exp(l2*t)-(k1+k2+l1)*exp(l1*t))/(l2-l1)) >>>> } # val should be a vector if t is a vector >>>> >>>> negll <- function(theta){ >>>> # non expm version JN 110731 >>>> pred<-Mpred(theta) >>>> sigma<-exp(theta[4]) >>>> -sum(dnorm(dat[,2],mean=pred,sd=sigma, log=TRUE)) >>>> } >>>> >>>> theta<-rep(-2,4) >>>> fand<-nlogL(theta) >>>> fsim<-negll(theta) >>>> cat("Check fn vals: expm =",fand," simple=",fsim," diff=",fand-fsim,"\n") >>>> >>>> cat("time the function in expm form\n") >>>> tnlogL<-microbenchmark(nlogL(theta), times=100L) >>>> tnlogL >>>> >>>> cat("time the function in simpler form\n") >>>> tnegll<-microbenchmark(negll(theta), times=100L) >>>> tnegll >>>> >>>> ftimes<-data.frame(texpm=tnlogL$time, tsimp=tnegll$time) >>>> # ftimes >>>> >>>> >>>> boxplot(log(ftimes)) >>>> title("Log times in nanoseconds for matrix exponential and simple MIN fn") >>>>

>>> R-devel@r-project.org mailing list

>> >> -- >> Peter Dalgaard, Professor, >> Center for Statistics, Copenhagen Business School >> Solbjerg Plads 3, 2000 Frederiksberg, Denmark >> Phone: (+45)38153501 >> Email: pd.mes_at_cbs.dk Priv: PDalgd_at_gmail.com >> "Døden skal tape!" --- Nordahl Grieg ______________________________________________R-devel_at_r-project.org mailing list

https://stat.ethz.ch/mailman/listinfo/r-devel Received on Thu 01 Sep 2011 - 09:53:27 GMT

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