Re: Integrable singularity

From: Julian V. Noble (jvn_at_virginia.edu)
Date: 01/31/05 

Date: Mon, 31 Jan 2005 11:00:56 -0500

Madhusudan Singh wrote:
>
> Hi
>
> I wish to numerically integrate a real function with an integrable
> singularity.
>
> What are the recommended methods for accomplishing the above ?
>
> Thanks.

You need to consult my column, "Gauss-Legendre Principal Value Integration"
that appeared in Computing in Science and Engineering Jan/Feb 2000 issue.
If you can't find the article in your library you can download a *.pdf from

   http://Galileo.phys.Virginia.EDU/classes/551.jvn.fall01/Cprogs.htm

The method works as well for integrable singularities as for Cauchy
principal value type integrals.

If the singularity is something like |f(x)|^{-b}, b < 1, where f(x) has
a simple zero at, say, x=a, you can also consider factoring out the
singularity in the form |x - a|^{-b} and construction a set of orthogonal
polynomials for which that is the weight function, then use Gaussian
integration with those polynomials. Gauss-Chebyshev does that.

Since you mention a singularity at a definite point x0 in the interval
(0, \infty), if it is a principal value singularity you might deform
the contour into the complex plane and take the real part. I do this
in the above-mentioned article, as well as in a forthcoming column,
"Feeling no pain in the Argand plane" which you can also get at the
above URL. The example I did there was inspired by a query to this
newsgroup! 

-- 
Julian V. Noble
Professor Emeritus of Physics
jvn@lessspamformother.virginia.edu
    ^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/
   "For there was never yet philosopher that could endure the 
    toothache patiently."  
        -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.

Relevant Pages

  • Re: Integrable singularity
    ... >> I wish to numerically integrate a real function with an integrable ... >> What are the recommended methods for accomplishing the above? ... > integration region. ... > remove the singularity. ...
    (comp.lang.fortran)
  • Re: Integrable singularity
    ... >> I wish to numerically integrate a real function with an integrable ... >> What are the recommended methods for accomplishing the above? ... > integration region. ... > remove the singularity. ...
    (sci.math.num-analysis)
  • Re: Integrable singularity
    ... > I wish to numerically integrate a real function with an integrable ... > What are the recommended methods for accomplishing the above? ... integration region. ... remove the singularity. ...
    (sci.math.num-analysis)
  • Re: Integrable singularity
    ... > I wish to numerically integrate a real function with an integrable ... > What are the recommended methods for accomplishing the above? ... integration region. ... remove the singularity. ...
    (comp.lang.fortran)
  • Re: help with the code
    ... Since i use "dblquad" to perform the double integration, d and phi have to be made vectors since in the dblquad routine, x is a vector and d and phi are calculated variables for different values of x. ... I have chosen the numerical integration approach since i am ... *snip to show just a few lines in the integrand function* ... which is often a sign that there's a singularity in your function at that ...
    (comp.soft-sys.matlab)
Loading