# Re: [R] integrate() problem {was "mathematica -> r ..."}

From: Martin Maechler <maechler_at_stat.math.ethz.ch>
Date: Tue 08 Aug 2006 - 17:55:50 EST

>>>>> "Leo" == Leo G³rtler <leog@anicca-vijja.de>
>>>>> on Tue, 08 Aug 2006 00:13:19 +0200 writes:

Leo> Dear R-list,
Leo> I try to transform a mathematica script to R.

```    Leo> #######relevant part of the Mathematica script
Leo> (* p_sv *)
Leo> dd = NN (DsD - DD^2);
Leo> lownum = NN (L-DD)^2;
Leo> upnum  = NN (H-DD)^2;
Leo> low = lownum/(2s^2);
Leo> up  = upnum/(2s^2);
Leo> psv = NIntegrate[1/(s^NN) Exp[-dd/(2s^2)]
Leo>      (Gamma[1/2,0,up] + Gamma[1/2,0,low]),{s,sL,sH}, MinRecursion-> 3];
Leo> PSV = psv/Sqrt[2NN];
Leo> Print["------------- Results ------------------------------------"];
Leo> Print[" "];
Leo> Print["p(sv|D_1D_2I)   = const. ",N[PSV,6]];
Leo> ########

```

Leo> # R part
Leo> library(fOptions)

```    Leo> ###raw values for reproduction
Leo> NN <- 58
Leo> dd <- 0.411769
Leo> lownum <- 20.81512
Leo> upnum <- 6.741643
Leo> sL <- 0.029
Leo> sH <- 0.092
Leo> ###

Leo> integpsv <- function(s) { 1 / (s^NN) * exp(-dd / (2 * s^2)) *
Leo>    ( (igamma((upnum/(2*s^2)),1/2) - igamma(0,1/2) ) +
Leo>    (igamma((lownum/(2*s^2)),1/2) - igamma(0,1/2) ) )
Leo> }
Leo> psv <- integrate(integpsv, lower=sL, upper=sH)
Leo> PSV <- psv\$value / sqrt(2*NN)
Leo> print("------------- Results ------------------------------------\n")
```
Leo> print(paste("p(sv|D_1D_2I) = const. ",PSV, sep=""))

Leo> The results of variable "PSV" are not the same.

Leo> In mathematica -> PSV ~ 2.67223e+47     Leo> with rounding errors due to the initial values, in R -> PSV ~ 1.5e+47

Leo> I am not that familiar with gamma functions and integration, thus I     Leo> assume there the source of the problem can be located.

Yes.
A few remarks

1. No need to use package "fOptions" and igamma(); standard R's pgamma() is all you need {igamma() has added value only for *complex* arguments!}
2. igamma(0, 1/2) == pgamma(0, 1/2) == 0 , so you can really drop them from your integrand.

integpsv <- function(s) {
1 / (s^NN) * exp(-dd / (2 * s^2)) *
( pgamma(upnum/(2*s^2), 1/2) + pgamma(lownum/(2*s^2), 1/2) ) }

3) But then the problem could really be with the algorithm used in

integrate(), and indeed if you plot your integrand

plot(integpsv, from= sL, to = sH)

you see that indeed your integrand looks ``almost    constant'' in the left half --- whereas that is actually not    true but the range of the integrand varies so dramatically    that it ``looks like'' constant 0 upto about x= .06.

However, if I experiment, using integrate() in two parts, or using many other numerical integration approximators,
all methods give ( your 'psv', not PSV )

integrate(integpsv, lower=sL, upper=sH)

Could it be that you are not using the same definition of incomplete gamma in Mathematica and R ?

Martin Maechler, ETH Zurich

Leo> Thanks for helping me to adjust the sript.

Leo> best wishes
Leo> leo

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