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

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