Re: Calculus and InterpolatingFunction

• To: mathgroup at smc.vnet.net
• Subject: [mg121684] Re: Calculus and InterpolatingFunction
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Mon, 26 Sep 2011 04:13:00 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201109221125.HAA26698@smc.vnet.net>

```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 ?
>>>>
>>>>
>>>
>>
>>
>

--
DrMajorBob at yahoo.com

```

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