[R] Incomplete gamma function (was "integrate()", {was "mathematica -> r ..."})

Martin Maechler maechler at stat.math.ethz.ch
Tue Aug 8 13:53:55 CEST 2006

```>>>>> "MM" == Martin Maechler <maechler at stat.math.ethz.ch>
>>>>>     on Tue, 8 Aug 2006 09:55:50 +0200 writes:

>>>>> "Leo" == Leo Gürtler <leog at 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.

MM> Yes.
MM> A few remarks

MM> 1) No need to use package "fOptions" and igamma();
MM> standard R's  pgamma() is all you need
MM> {igamma() has added value only for *complex* arguments!}

MM> 2) igamma(0, 1/2) == pgamma(0, 1/2) == 0 , so you can really
MM> drop them from your integrand.

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

[............]

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

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

MM> a value of   1.623779e+48   (which leads to your PSV of 1.5076e+47)

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

Offlist, Leo sent me Mathematica's definition
of
Gamma[a, z0, z1]  :=  integral_z0^z1 t^(a-1) exp(-t) dt

Now if you compare this with what  ?pgamma (not ?gamma !) tells you,
namely that R uses (Abramowitz and Stegun's definition of the
incomplete gamma function)

pgamma(x, a) =  1/ Gamma(a) * integral_0^x  t^(a-1) exp(-t) dt

which has a normalizing factor:  In your case above, it is
Gamma(1/2) with its well-known value of sqrt(pi).

And indeed, if you multiply the current result by sqrt(pi), you
get what you want -- and did get from Mathematica:

> (1.623779e+48 / sqrt(2*NN)) * sqrt(pi)
[1] 2.672224e+47

Regards,
Martin Maechler, ETH Zurich

```