Numeric Integration of Tabulated Integrand Function: Part I
- To: mathgroup at smc.vnet.net
- Subject: [mg38406] Numeric Integration of Tabulated Integrand Function: Part I
- From: tasande at MIT.EDU
- Date: Fri, 13 Dec 2002 04:19:15 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
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