Re: [R] How to double integrate a function in R
Tiago V. Pereira mbe.bio.br> writes: > I am trying to double integrate the following expression: > > # expression > (1/(2*pi))*exp(-y2/2)*sqrt((y1/(y2-y1))) > > for y2>y1>0. > > I am trying the following approach > > # first attempt > > library(cubature) > fun <- function(x) { (1/(2*pi))*exp(-x[2]/2)*sqrt((x[1]/(x[2]-x[1])))} > adaptIntegrate(fun, lower = c(0,0), upper =c(5, 6), tol=1e-8) > > However, I don't know how to constrain the integration so that y2>y1>0. > > Any ideas? > Tiago You could use integral2() in package 'pracma'. It implements the "TwoD" algorithm and has the following properties: (1) The boundaries of the second variable y can be functions of the first variable x; (2) it can handle singularities on the boundaries (to a certain extent). > library(pracma) > fun <- function(y1, y2) (1/(2*pi))*exp(-y2/2)*sqrt((y1/(y2-y1))) > integral2(fun, 0, 5, function(x) x, 6, singular=TRUE) $Q [1] 0.7706771 $error [1] 7.890093e-11 The relative error is a bit optimistic, the absolute error here is < 0.5e-6. The computation time is 0.025 seconds. Hans Werner __ 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] How to double integrate a function in R
On Jul 26, 2013, at 9:33 AM, David Winsemius wrote: > fun2 <- function(x,y) { (x0)* (1/(2*pi))*exp(-y/2)* sqrt((x/(y-x)))} > persp(outer(0:5, 0:6, fun2) ) There does seem to be some potential pathology at the edges of the range, Restricting it to x >= 0.03 removes most of that concern. fun2 <- function(x,y) { (x0)* (1/(2*pi))*exp(-y/2)* sqrt((x/(y-x)))} persp(outer(seq(0.01,5,by=.01), seq(.02,6,by=.01), fun2) ,ticktype="detailed") > fun <- function(x) { (x[1]0)* > (1/(2*pi))*exp(-x[2]/2)*if(x[1]>x[2]){0}else{ sqrt((x[1]/(x[2]-x[1])) )}} > adaptIntegrate(fun, lower = c(0.03,0.03), upper =c(5, 6), tol=1e-2 ) $integral [1] 0.7605703 $error [1] 0.00760384 $functionEvaluations [1] 190859 $returnCode [1] 0 I tried decreasing the tolerance to 1e-3 but the wait exceeds the patience I have allocated to the problem. -- David Winsemius Alameda, CA, USA __ 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] How to double integrate a function in R
On Jul 26, 2013, at 8:44 AM, Tiago V. Pereira wrote: > I am trying to double integrate the following expression: > > # expression > (1/(2*pi))*exp(-y2/2)*sqrt((y1/(y2-y1))) > > for y2>y1>0. > > I am trying the following approach > > # first attempt > > library(cubature) >fun <- function(x) { (1/(2*pi))*exp(-x[2]/2)*sqrt((x[1]/(x[2]-x[1])))} >adaptIntegrate(fun, lower = c(0,0), upper =c(5, 6), tol=1e-8) > > However, I don't know how to constrain the integration so that y2>y1>0. Generally incorporating boundaries is accomplished by multiplying the integrand with logical vectors that encapsulate what are effectively two conditions: Perhaps: fun <- function(x) { (x[1]0)* (1/(2*pi))*exp(-x[2]/2)* sqrt((x[1]/(x[2]-x[1])))} That was taking quite a long time and I interrupted it. There were quite a few warnings of the sort 1: In sqrt((x[1]/(x[2] - x[1]))) : NaNs produced 2: In sqrt((x[1]/(x[2] - x[1]))) : NaNs produced Thinking the NaNs might sabotage the integration process, I added a conditional to the section of that expression that was generating the NaNs. I don't really know whether NaN's are excluded from the summation process in adaptIntegrate: fun <- function(x) { (x[1]0)* (1/(2*pi))*exp(-x[2]/2)* if(x[1]>x[2]){ 0 }else{ sqrt((x[1]/(x[2]-x[1])) )} } adaptIntegrate(fun, lower = c(0,0), upper =c(5, 6) ) I still didn't have the patience to wait for an answer, but I did plot the function: fun2 <- function(x,y) { (x0)* (1/(2*pi))*exp(-y/2)* sqrt((x/(y-x)))} persp(outer(0:5, 0:6, fun2) ) So at least the function is finite over most of its domain. -- David Winsemius Alameda, CA, USA __ 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.