[Rd] An example of very slow computation
Michael Lachmann
lachmann at eva.mpg.de
Thu Aug 18 00:30:32 CEST 2011
Just a small addition:
If you replace below
> sol<-function(t)100-sum(expm(A*t)%*%x0)
by
sol<-function(t){A at x=A at x*t;100-sum(expm(A)@x * x0)}
(ugly! But avoiding the conversions and generics)
The time on my machine drop further down to 0.3 seconds. (from the original 13 seconds, and then from the 1.5 seconds change mentioned below.
So, overall a ~40 fold improvement, though on my machine, the initial ratio was ~3200 times slower, so a ~80 fold slowdown is still present.
Michael
On 17 Aug 2011, at 11:27PM, Michael Lachmann wrote:
>
> On 17 Aug 2011, at 7:08PM, <cberry at tajo.ucsd.edu> <cberry at tajo.ucsd.edu> wrote:
>
>> 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.
>>
>> 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.
>>
>
> You are right!
>
> I was testing running nlogL below 100 times. expm() is then called 2500 times.
> The total running time on my machine was 13 seconds.
>
> If you replace in nlogL the part:
> --
> 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)
> --
> with:
> --
> A<-rbind(
> c(-k[1], k[2]),
> c( k[1], -(k[2]+k[3]))
> )
> A<-as(A,"dgeMatrix") # <--- this is the difference
> sol<-function(t)100-sum(expm(A*t)%*%x0)
> --
>
> this time drops to 1.5 seconds (!).
>
> At that point, expm() takes up much less time than, for example, calculating A*t in sol(), and the sum() - I think because conversions have to be done.
>
> Thus, if m is a 2x2 dgeMatrix, then
>> system.time({for(i in 1:2500) m*3.2})
> user system elapsed
> 0.425 0.004 0.579
> (i.e. 1/3 of the total time for nlogL() above)
>
> whereas if mm=as.matrix(m), then
>> system.time({for(i in 1:2500) mm*3.2})
> user system elapsed
> 0.004 0.000 0.005
>
> (!!)
>
> and, similarly:
> --
>> system.time({for(i in 1:2500) sum(m)})
> user system elapsed
> 0.399 0.002 0.494
>> system.time({for(i in 1:2500) sum(mm)})
> user system elapsed
> 0.002 0.000 0.028
> --
> whereas
>> system.time({for(i in 1:2500) expm(m)})
> user system elapsed
> 0.106 0.001 0.118
>
> to sum it up, of 13 seconds, 11.5 were spent on conversions to dgeMatrix
> 0.5 are spent on multiplying a dgeMatrix by a double
> 0.5 are spent on summing a dgeMatrix
> and 0.1 are actually spent in expm.
>
> Michael
>
> P.S. You could have used Rprof() to see these times, only that interpreting summaryRprof() is a bit hard. (Is there something that does summaryRprof(), but also shows the call graph?)
>
>
>
>
>
>> 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)
>>> # 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")
>>>
>>
>> --
>> Charles C. Berry cberry at tajo.ucsd.edu
>>
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>>
>>
>>
>
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