Re: Numeric Integration of Tabulated Integrand Function: Part I
- To: mathgroup at smc.vnet.net
- Subject: [mg38443] Re: Numeric Integration of Tabulated Integrand Function: Part I
- From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
- Date: Mon, 16 Dec 2002 02:20:21 -0500 (EST)
- Organization: Universitaet Leipzig
- References: <atcau9$3m7$1@smc.vnet.net>
- Reply-to: kuska at informatik.uni-leipzig.de
- Sender: owner-wri-mathgroup at wolfram.com
Hi,
make an Interpolation[] from the data and use Integrate[],
or use the standard add on package
NumericalMath`ListIntegrate`
Mathematica don't use splines, it make pice wise polynomials
and integrate the polynomial. Since you have the source
code of ListIntegrate you can look by your self.
Regards
Jens
tasande at mit.edu wrote:
>
> Dear Mathemticons( * ):
>
> I have a very simple question. I have a function that is NOT specified
> as an analytic expression. Rather, the function is a previsouly
> tabulated list of the form:
>
> F( x ):=
>
> { ( x( 1 ),F( x( 1 ) ) ), ... , ( x( i ),F( x( i ) ) , ... ,
> ( x( N ),F( x( N ) ) ) }
>
> The list is comprised of N pairs of values, one for the independent varaible
> x and the other for the dependent function F( x ).
>
> The x values are NOT equally spaced. They are tabulated at unevenly spaced
> intervals.
>
> Also, I do NOT have the freedom to compute additional list values between the
> ones initially provided. In a sense, this list can be conceived of as DATA,
> although it is actually generated by a previous complex computation.
>
> GOAL:
>
> I wish to integrate this tabulated function from x( 1 ) to x( N )
>
> SIMPLE SOLUTION:
>
> Apply the trapaziodal rule.
>
> DESIRED SOLUTION:
>
> I wish to integrate this tabulated function with a higher degree of accuracy
> or "fidelity", however this is construed. The caveat, is, as previously
> mentioned, that I cannot direclty compute a more densely spaced list of
> function values.
>
> PROVISIONAL DESIRED SOLUTION ?
>
> Should I implement a Romberg Type integration scheme where I "synthetically"
> sample progressively more tightly spaced function values via interpolation until
> the integral convereges to its final value within a pre-specified precision?
>
> What interpolation scheme should I employ? Cubic spine? Spath Monotonicity
> Preserving Hermite? Cubic Spline with Tension? I only required vlaues of
> the function, and not its higher derivatives. I am afraid to use Cubic Splines
> in general since these often "bulge" and may produce an answer worse that
> the linear interpolation that the Trapazoidal Rule uses.
>
> What technique, if any, does Mathematica employ?
>
> I wish to do this both on Mathematica as well as in an F90 code.
>
> The technique should be fast, since I will be integrating 1000's of such
> functions sequentially.
>
> Any advice will be greatly appreciated!
>
> Sincerely,
>
> Theodore Sande
> MIT Department of Physics
>
> ( * ) Mathematicon: The fundamental quantum of the Mathematica User's Field