Re: Implicit integrals?

"kirpekar@xxxxxxxxx" <kirpekar@xxxxxxxxx> wrote in message <120f0ecd-
c79e-42d5-be15-b46052d26f72@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>...
Roger, Thanks for the info. I do understand that I am dealing with an
non-homogeneous Fredholm equation and there are many ways (e.g. series
solutions) to solve it. I've been using the classical text by Dettman
( http://www.amazon.com/Mathematical-Methods-Physics-Engineering-
Dettman/dp/0486656497/ref=sr_1_3?
ie=UTF8&s=books&qid=1207762892&sr=1-3
) for the details.

I have been able to setup MATLAB for Riemann sums / Simpson's
quadrature, but the final solution is a strong function of the number
of points, for less than about a 1000 points. The solution converges
between 2000 to 10000 points, but this is computationally very
expensive. It would be great if quad could handle this, but it seems
quad can handle only functions, not vectors.

FYI: In my case I-G is nonsingular.

Regards
----------
When you say, "the final solution is a strong function of the number of
points, for less than about a 1000 points," do you mean 1000 points in the
discrete steps in f or 1000 points in discrete values of g? If you mean 1000
points in f, that surprises me somewhat! For 1000 points to be necessary to
accurately integrate with respect to y along a single dimension, the product g
(y,x)*f(y), would require that g be a very irregular-looking function, it seems
to me. For a reasonably smooth integrand, the Riemann sum I mentioned
should yield an accurate integral with many fewer points. Of course you
could always make the subinterval spacings non-uniform, as in Gaussian
integration, or perhaps use some kind of higher-order integration, and
thereby hopefully achieve a higher order accuracy, and this would still permit
the kind of matrix solution I mentioned.

When you say, "I have been able to setup MATLAB for Riemann sums /
Simpson's quadrature", do you mean along the lines I have described, or in
some kind of iterative procedure? The reason I ask is that you originally
stated that, "I understand that I'll have to take an initial guess and perform
fixed point iterations." There is no iterative or convergence aspect to the
method I mentioned. It gets its answer in one (massive) step. Whether that
step is accurate or not depends of course on the accuracy of the integration
approximation and therefore on the number of points used.

The problem with using a routine like 'quad' is that you have no control over
the placement of points at which it may choose to evaluate its integrands, so
in theory you would have had to evaluate f(x) at infinitely many points to
allow for all possibilities. Of course in practice you could evaluate a discrete
set of f values and then set up some function which generates other f values
by interpolation, but that seems like an act born of desperation.

I believe what you need is a single meshwork of closely-spaced points in the
x and y directions at which g is to be evaluated, one time only. They need
not be uniformly-spaced as long as the spacings are the same in both
directions. Then you need some method of integration which uses only these
points in accurately achieving the integrals in your integral equations.
Whether this is done by a one-step approach like the one I recommended or
by some iterative technique which uses successively more accurate f values
(hopefully,) depends on which yields the best results (or any result at all) in a
reasonable amount of time. However it is done, there is no getting around
the need for computing n different integrations in order to achieve n different
values of f, and the consequent restriction that each integration cannot make
use of more than these n values of f. That pretty well lays the law down as to
what sort of quadrature algorithm you can make use of.

Roger Stafford

.

Relevant Pages

  • Re: Positive/Negative after taking the square root
    ... logical reason for this? ... the confusion: ... You cannot make a substitution u for x which has ... the interval of integration, i.e. a and b are ...
    (sci.math)
  • Re: Error using quad...
    ... >> that solution with FZERO for whatever reason, ... affected how I view symbolic integration and when I recommend it. ... > explicit formulas such as the one available in this case, ...
    (comp.soft-sys.matlab)
  • Re: Self-teaching of integration methods
    ... > He apparently didn't realize that the reason that I'm not a ... and I think that teaching them will go OK. ... > However, when I hit the part in my book about integration techniques, I ... > integration by parts well beyond the scope of a typical high school ...
    (sci.math)
Loading