Re: Integrating Interpolation functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg89029] Re: Integrating Interpolation functions*From*: oshaughn <oshaughn at northwestern.edu>*Date*: Sat, 24 May 2008 03:52:55 -0400 (EDT)*References*: <g15qk7$p9t$1@smc.vnet.net>

On May 23, 3:09 am, Hugh Goyder <h.g.d.goy... at cranfield.ac.uk> wrote: > Below I given an illustrative example where I have a two-dimensional > interpolation function, f1[x,t], and I integrate over x and obtain a > one-dimensional expression g1 which is an interpolation function > depending on t. The interpolation function g1 has a built-in argument > of t ie it ends in [t]. I would prefer it to be a pure function so > that I could use any variable instead of t, like the original > interpolation function, f1. I give a work-around which defines a new > function independent of t. However, I feel that there should be > > 1. A better way of getting the interpolation function from Integrate > without the built in t > > 2.A method of using NIntegrate rather than Integrate which should be > able to use the information that the function is interpolated > > 3. A method that would work on a product of interpolating functions > > Thanks > > Hugh > > data1 = Table[{x, t, Exp[(-x)*t]}, {x, 0, 1, 0.1}, {t, 0, 1, 0.1}]; > > data2 = Table[{x, t, Sin[x*t]}, {x, 0, 1, 0.1}, {t, 0, 1, 0.1}]; > > f1 = Interpolation[Flatten[data1, 1]] > > f2 = Interpolation[Flatten[data2, 1]] > > g1 = Integrate[f1[x, t], {x, 0, 1}] (1) Use g2new = Head[g1]. This strips off the 't' and can be used as a pure function. (2,3) If you aren't integrating a pure InterpolatingFunction (e.g., if you even multiply it by a polynomial) this method doesn't work. For a quick black box, try FunctionInterpolation g3new = FunctionInterpolation[NIntegrate[f1[x, t]*f2[x, t], {x, 0, 1}], {t, 0, 1}] Because of all the different ways one can imagine constructing expressions involving interpolating functions (F[D[f2,{x,5}], Exp[Sinh[f1[x-4]], ....]) I would not look too hard for a completely general code. But if there's a limited set of operations you need to perform involving simply-weighted products of low-order interpolations, you could hardcode them in using the explicit form of the interpolating polynomials. That too is more trouble than it is worth. My two cents.