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Adaptive numerical limit (and residue) estimation

by

John D'Errico (view profile)

 

26 May 2008 (Updated )

Numerical extrapolation of a limit (with an error estimate) from only function values

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Description

LIMEST will find the limit of a general function (specified only for evaluation) at a given point. You might think of limest like quad, but for limits.
While to me this seems to appear more often as a homework exercise than anything else, it was an interesting problem to solve as robustly as possible for a general case.

As an example, I'll use a moderately difficult one that is simple to analyze, but more difficult to deal with numerically.

fun = @(x) (exp(x)-1-x)./x.^2;

This function cannot be evaluated in MATLAB at x = 0, returning a NaN. While a Taylor series expansion shows the limit to be 1/2, the brute force evaluation of fun anywhere near zero results in numerical trash because of the two orders of cancellation.

fun(0)
ans =
   NaN

fun(1e-15)
ans =
           110223024625156

fun(1e-10)
ans =
          827.403709626582

fun(1e-5)
ans =
         0.500000696482408

fun(1e-2)
ans =
         0.501670841679489

Limest computes the limit, also returning an approximate error estimate.

[lim,err] = limest(fun,0)
lim =
         0.499999999681485
err =
      2.20308196660258e-09

I've now added the residueEst tool, for computation of the residue of a function at a known pole. For example, here is a function with a first order pole at z = 0
 
[r,e]=residueEst(@(z) 1./(1-exp(2*z)),0)
r =
        -0.5
e =
     4.5382e-12

Again, both an estimate of the residue, as well as an uncertainty around that estimate are provided. Next, consider a function with a second order pole around z = pi.
 
[r,e]=residueEst(@(z) 1./(sin(z).^2),pi,'poleorder',2)
 
r =
           1
e =
     2.6336e-11

See the included demos for many other examples of use.

Acknowledgements

Adaptive Robust Numerical Differentiation inspired this file.

MATLAB release MATLAB 7.4 (R2007a)
MATLAB Search Path
/
/LIMEST
/LIMEST/html
Other requirements Moderately older versions of Matlab should be able to use this utility with an inline function.
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Comments and Ratings (3)
13 Mar 2015 adrian zizo

nice worke

Comment only
30 Aug 2008 C Schwalm

Well done! And useful...

04 Jun 2008 Bill McKeeman

Nice work.

Updates
29 May 2008

Improved some error messages

13 Mar 2015

Included residueEst

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