# [R] Surprising result from integrate

Daniel Hoppe hoppe at sitewaerts.de
Fri Aug 23 19:09:50 CEST 2002

Hi all,

sorry for this extensive question, but I think that I'm missing something
fundamental.

I stepped into a surprising result with the integrate function and I would
be glad if someone could put some light onto this. I try to integrate over
an s-shaped growth-function. The results from this calculation seem to be
correct for small values of t. Just for fun I tried some large values and
suddenly one part of the formula got close to zero (i2, it is not expected
to do this). I tried different numbers and found that for t=upper=59954
everything is fine and for t=upper=59955 integrate wouldn't return the
correct result anymore (see below). I'm now wondering if I'm misinterpreting
how integrate is supposed to be used. Details can be found below.

Thanks for your thoughts and best regards,

Daniel

This is the output of the function calls:

> bergernasr.clv.continuous(10)
112.1139 with absolute error < 6.2e-13
111.1077 with absolute error < 6.2e-13
[1] 243.2216

> bergernasr.clv.continuous(59954)
112.1139 with absolute error < 6.2e-13
150.2340 with absolute error < 3e-07
[1] 282.3478

> bergernasr.clv.continuous(59955)
112.1139 with absolute error < 6.2e-13
6.137708e-05 with absolute error < 0.00012
[1] 132.1139

These are the functions required to see this effect.

bergernasr.clv.continuous.f1 <- function(t, h, v, ret, d)
{

(h*t^2 + v) * (ret / ( 1 + d ))^t
}

bergernasr.clv.continuous.f2 <- function (t, g,h, v, ret, N, d)
{
(h*g^2 + v + N*(1-exp(-t+g))) * (ret / ( 1 + d ))^t
}

bergernasr.clv.continuous <- function(t)
{
g <- 5         # Wendepunkt
v <- 20        # Sockelbetrag
h <- 4         # Umsatzwachstum
N <- 80        # Wachstum ab Wendepunkt
r <- .9        # Retention rate
d <- .2        # Discount rate

i1 <- integrate(bergernasr.clv.continuous.f1, h=h,v=v,ret=r,d=d,
lower=0, upper=5)
i2 <- integrate(bergernasr.clv.continuous.f2, g=g, h=h,v=v,ret=r,N=N,
d=d, lower=g, upper=t)
print(i1)
print(i2)
return (
v
+ i1[[1]]
+ i2[[1]]
)
}

For better readability the function in Mathtype / Tex format:

% MathType!MTEF!2!1!+-
% feaafaart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb
% a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr
% GccaWGKbWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaey4kaSYaa8qC
% IipakiabgwSixpaabmaabaWaaSaaaeaacaWGYbaabaGaaGymaiabgU
% amizamaabmaabaGaamiDaaGaayjkaiaawMcaaaWcbaGaaGimaaqaai
$CLV = v + \int\limits_0^g {\left( {ht^2 + v} \right) \cdot \left( {\frac{r}{{1 + d}}} \right)^t d\left( t \right) + \int\limits_g^n {\left( {hg^2 + v + N \cdot (1 - e^{ - t + g} )} \right)} \cdot \left( {\frac{r}{{1 + d}}} \right)^t d\left( t \right)}$

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