Re: Calculus and InterpolatingFunction

*To*: mathgroup at smc.vnet.net*Subject*: [mg121718] Re: Calculus and InterpolatingFunction*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Mon, 26 Sep 2011 20:07:40 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201109221125.HAA26698@smc.vnet.net> <201109260813.EAA08552@smc.vnet.net>*Reply-to*: murray at math.umass.edu

Re your 2): exactly my complaint! There should be some way of converting an InterpolatingFunction into an "ordinary" (by formula -- presumably piecewise) function. Perhaps Normal could be extended to cover such things. On 9/26/11 4:13 AM, DrMajorBob wrote: > 1) Integrate handles symbolic functions like Sin and Cos. NIntegrate > handles numeric functions that cannot, even in principle, be integrated > symbolically. > > 2) Interpolation does NOT give a function "given by some formula". > > Bobby > > On Sun, 25 Sep 2011 10:36:32 -0500, Murray Eisenberg > <murray at math.umass.edu> wrote: > >> But why must NIntegrate be called by Integrate here? Is it merely the >> fact that the interpolating function has floating-point numbers in it? >> >> If I have what I would reasonably call an "explicit" function f, given >> by some formula, and if it's possible to find a (piecewise) >> antiderivative exactly, then I would expect Integrate to work directly >> with that function -- even in finding an indefinite integral >> Integrate[f[x],x]. >> >> On 9/24/11 10:33 PM, DrMajorBob wrote: >>> Interpolation does give "an explicit function" in any sense of >>> "explicit" >>> that I can think of. The problem you ran into is (IMHO) a "bug" or >>> "feature lack" in Integrate. It should call NIntegrate when necessary, >>> but >>> it did not, in the OP's example. >>> >>> Bobby >>> >>> On Fri, 23 Sep 2011 02:45:03 -0500, Murray Eisenberg >>> <murray at math.umass.edu> wrote: >>> >>>> Two remaining problems: >>>> >>>> (1) The Documentation Center page for Interpolation says, >>>> "Interpolation >>>> returns an InterpolatingFunction object, which can be used like any >>>> other pure function." >>>> >>>> Manifestly that is not the case. Thus the following, for a pure >>>> function, _does_ work: >>>> >>>> f = #^2& >>>> Integrate[f[x] + 1, {x, 1, 10}] >>>> >>>> (2) While the solutions you proposed both work, the latter using Map >>>> would be problematic for integrands involving the InterpolatingFunction >>>> in more complicated ways, e.g.: >>>> >>>> f = Interpolation[data]; >>>> Integrate[#, {x, 1, 10}]& /@ (Sin[f[x]]) >>>> 0.576208 >>>> NIntegrate[Sin[f[x]], {x, 1, 10}] >>>> 0.607007 >>>> >>>> Is there some way to obtain an explicit function from an >>>> InterpolatingFunction object? >>>> >>>> >>>> On 9/22/11 7:25 AM, Bob Hanlon wrote: >>>>> data = RandomReal[#]*2& /@ Range[1, 10]; >>>>> >>>>> f = Interpolation[data]; >>>>> >>>>> Integrate[f[x], {x, 1, 10}] >>>>> >>>>> 52.9041 >>>>> >>>>> Use NIntegrate >>>>> >>>>> NIntegrate[f[x] + 1, {x, 1, 10}] >>>>> >>>>> 61.9041 >>>>> >>>>> Or Map over the expression >>>>> >>>>> Integrate[#, {x, 1, 10}]& /@ (f[x] + 1) >>>>> >>>>> 61.9041 >>>>> >>>>> >>>>> Bob Hanlon >>>>> >>>>> ---- Just A Stranger<forpeopleidontknow at gmail.com> wrote: >>>>> >>>>> ============= >>>>> I'm trying to get a definite integral for an InterpolatingFunction. It >>>>> works >>>>> if it is the function by itself, but not for some reason >>>>> arithmetically >>>>> combining the InterpolatingFunction with another function makes it not >>>>> return a value. e.g. >>>>> >>>>> >>>>> In[1]:= >>>>> data = RandomReal[#]*2& /@ Range[1, 10]; >>>>> f = Interpolation[data]; >>>>> >>>>>> Integrate[f[x], {x, 1, 10}] >>>>> Out[1]:=40.098 >>>>> >>>>> So far so good. But just a little bit of arithmetic in the integral >>>>> and >>>>> it >>>>> doesn't work anymore: >>>>> In[2]:= >>>>> Integrate[f[x]+1, {x, 1, 10}] >>>>> Out[2]:= >>>>> Integrate[Plus[1, InterpolatingFunction[][x]], List[x, 1, 10]] >>>>> >>>>> (That last answer was actually the output with //FullForm applied) >>>>> Why won't it give me a numerical evaluation? Is there anyway to make a >>>>> continuous function from data that will seemlessly work with >>>>> Integrate? >>>>> I'm >>>>> thinking of constructing a piecwise function using Fit, Piecwise, and >>>>> a >>>>> Table for the arguments to Piecewise. But I would think Interpolation >>>>> might >>>>> have worked and been easier. I want to figure out if I am I doing >>>>> something >>>>> wrong with Interpolation before I start trying to tackle a slightly >>>>> more >>>>> complicated piecewise defined function ? >>>>> >>>>> >>>> >>> >>> >> > > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

**References**:**Re: Calculus and InterpolatingFunction***From:*Bob Hanlon <hanlonr@cox.net>

**Re: Calculus and InterpolatingFunction***From:*DrMajorBob <btreat1@austin.rr.com>

**Re: Bug with Sequence and Assignment by Part...**

**Re: A fast way to compare two vectors**

**Re: Calculus and InterpolatingFunction**

**Re: Calculus and InterpolatingFunction**