# [Rd] An example of very slow computation

Michael Lachmann lachmann at eva.mpg.de
Fri Aug 19 02:08:48 CEST 2011

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

Michael

On 19 Aug 2011, at 1:32AM, Ravi Varadhan wrote:

> 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
> 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:
>
>> 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).
>
> I believe this is method 14 of the "19 dubious ways..." (Google for it) and doesn't work for certain non-symmetric A matrices.
>
>>
>> Ravi.
>>
>> -------------------------------------------------------
>> Assistant Professor,
>> Division of Geriatric Medicine and Gerontology School of Medicine Johns Hopkins University
>>
>> Ph. (410) 502-2619
>>
>>
>> -----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.
>>
>> -------------------------------------------------------
>> Assistant Professor,
>> Division of Geriatric Medicine and Gerontology School of Medicine Johns Hopkins University
>>
>> Ph. (410) 502-2619
>>
>> -----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:
>>
>>> 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.
>>
>> 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
>>
>>
>>
>>
>>> 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)
>>> # 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")
>>>
>>
>> --
>> Charles C. Berry                         cberry at tajo.ucsd.edu
>>
>> ______________________________________________
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>> https://stat.ethz.ch/mailman/listinfo/r-devel
>>
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>>
>> ______________________________________________
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>> https://stat.ethz.ch/mailman/listinfo/r-devel
>
> --
> 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
>
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```