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Forum index » Science and Technology » Math » num-analysis
integrand numerical singular near boundary
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Peter Spellucci
science forum Guru


Joined: 29 Apr 2005
Posts: 702

PostPosted: Wed Jul 12, 2006 9:19 pm    Post subject: Re: integrand numerical singular near boundary Reply with quote

In article <44B53BD2.4A2DBC6A@axelvogt.de>,
Axel Vogt <test3@axelvogt.de> writes:
Quote:
What is an appropriate method to (automatically) care for something like
Int(t/(1-z*t),t = 0 .. 1), z = 1-1e-8*I close to the boundary?

I do not mean the specific form above (only: to integrate over an interval).
If the singularity 1/z is off the bounds one can pass to another path to
reach 0 and 1. But if 1/z is close contour integrals do not help me much.

Is there some standard method to 'zoom in around the boundary'?

quadpack as an adaptive code exatly for this. indeed it transforms
integrals over infinite intervals into ones over [0,1] with singularity
at the boundary and integrates this with adaptive refinement at the
boundaries using gauss and gauss kronrod nodes (interior nodes)
try it here:
http://numawww.mathematik.tu-darmstadt.de:8081/
-> quadratur -> uneigentliche integrale
hth
peter
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Robert B. Israel
science forum Guru


Joined: 24 Mar 2005
Posts: 2151

PostPosted: Wed Jul 12, 2006 9:19 pm    Post subject: Re: integrand numerical singular near boundary Reply with quote

In article <44B53BD2.4A2DBC6A@axelvogt.de>,
Axel Vogt <test3@axelvogt.de> wrote:
Quote:
What is an appropriate method to (automatically) care for something like
Int(t/(1-z*t),t = 0 .. 1), z = 1-1e-8*I close to the boundary?

I do not mean the specific form above (only: to integrate over an interval).
If the singularity 1/z is off the bounds one can pass to another path to
reach 0 and 1. But if 1/z is close contour integrals do not help me much.

Is there some standard method to 'zoom in around the boundary'?

One thing you might try is subtracting off the singular part and
integrating it separately.
Thus the singular part of your integrand t/(1-z t) at t=1/z is
-1/z^2 (t-1/z)^(-1), and int_0^1 (t-1/z)^(-1) dt = ln(1-z).
After subtracting the singular part, you have a nice function
with no singularity at t=1/z.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
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Axel Vogt
science forum addict


Joined: 03 May 2005
Posts: 93

PostPosted: Wed Jul 12, 2006 6:13 pm    Post subject: integrand numerical singular near boundary Reply with quote

What is an appropriate method to (automatically) care for something like
Int(t/(1-z*t),t = 0 .. 1), z = 1-1e-8*I close to the boundary?

I do not mean the specific form above (only: to integrate over an interval).
If the singularity 1/z is off the bounds one can pass to another path to
reach 0 and 1. But if 1/z is close contour integrals do not help me much.

Is there some standard method to 'zoom in around the boundary'?
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