Dear John,
Two issues. First, the default action is stop.on.error=TRUE, so anytime the
integrate function can determine an error, it will stop. It doesn't detect an
error, so no error is produced (whether you set stop.on.error=TRUE or FALSE).
The second issue is the real problem: what automatic numerical integration can
do or not do. When the upper bound is Inf, integrate (which is based on the
QUADPACK fortran code) does a change of variable to make the region of
integration be a finite interval, then tries to evaluate that transformed
integral. When the upper bound is a large finite number, integrate tries to
evaluate the integral directly. In this case, the integrand is evaluated at
multiple x values in the interval [0,13000]. Those x values are large, and
the resulting function values are very near 0. You can verify this by putting
some trace statements into your integrand function, e.g.
> f1 <- function( x ) { y <- exp(-x); print( rbind(x,y) ); return(y) }
> integrate( f1, lower = 0, upper =13000)
The quadrature rule "sees" an integrand near 0 and returns a value for the
integral near 0. It does not detect an error, so it does not report anything
to you. It does not know how the integrand function behaves on regions where
it does not evaluate it. This is a well-known problem in numerical
integration: there is no way the integrate function can know what region to
focus on in a general problem. Using an upper bound=Inf does not guarantee
that you will get the correct value, but sometimes it works.
Hope this helps.
John
………………………………………………………………………………..
John P. Nolan
Math/Stat Dept., American University
106J Myers Hall, 4400 Massachusetts Ave, NW, Washington, DC 20016-8050
Phone: 202-885-3140 E-mail: jpno...@american.edu
Web: http://fs2.american.edu/jpnolan/www/
-----Original Message-----
From: R-devel <r-devel-boun...@r-project.org> On Behalf Of John Muschelli
Sent: Friday, March 23, 2018 6:52 PM
To: r-devel@r-project.org
Subject: [Rd] Integrate erros on certain functions
In the help for ?integrate:
>When integrating over infinite intervals do so explicitly, rather than
just using a large number as the endpoint. This increases the chance of a
correct answer – any function whose integral over an infinite interval is
finite must be near zero for most of that interval.
I understand that and there are examples such as:
## a slowly-convergent integral
integrand <- function(x) {1/((x+1)*sqrt(x))} integrate(integrand, lower = 0,
upper = Inf)
## don't do this if you really want the integral from 0 to Inf
integrate(integrand, lower = 0, upper = 1000000, stop.on.error = FALSE) #>
failed with message ‘the integral is probably divergent’
which gives an error message if stop.on.error = FALSE. But what happens on
something like the function below:
integrate(function(x) exp(-x), lower = 0, upper =Inf) #> 1 with absolute error
< 5.7e-05
integrate(function(x) exp(-x), lower = 0, upper =13000) #> 2.819306e-05 with
absolute error < 5.6e-05
*integrate(function(x) exp(-x), lower = 0, upper =13000, stop.on.error =
FALSE)#> 2.819306e-05 with absolute error < 5.6e-05*
I'm not sure this is a bug or misuse of the function, but I would assume the
last integrate to give an error if stop.on.error = FALSE.
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