Re: Suming InterpolatingFunction Objects

On Thu, 14 Jul 2011, DrMajorBob wrote:

>> sum = FunctionInterpolation[
>>   Evaluate[Table[
>>     D[Evaluate[x[t] + y[t] /. s], {{t}, k}], {k, 0, 2}]], {t, 0, 20},
>>   InterpolationPoints -> 30]
>> (* then you get away with less InterpolationPoints *)
>
> I have two questions about that:
>
> (1) The second Evaluate has no effect -- correct? -- since
> FunctionInterpolation holds ENTIRE arguments, not parts of them.

Yes.

>
> (2) Help gives an example something like this, but among the basic function
> forms at the top of the page, there are no templates involving a Table as the
> first argument. What is Help trying NOT to explain?
>
> Bobby
>

Bobby, unfortunately I do not understand the question. I just copied stuff
from the help page.

Oliver

> On Thu, 14 Jul 2011 06:14:09 -0500, Oliver Ruebenkoenig
> <ruebenko at wolfram.com> wrote:
>
>> On Thu, 14 Jul 2011, Gabriel Landi wrote:
>>
>>> Hello everyone.
>>>
>>> I encountered the following problem, which I am not sure is a bug or
>>> simply
>>> a annoying detail.
>>> I will use as an example a piece of code from the mathematica tutorials.
>>>
>>> Say I solve a system of ODEs
>>>
>>> s = NDSolve[{x'[t] == -y[t] - x[t]^2, y'[t] == 2 x[t] - y[t]^3,
>>>  x[0] == y[0] == 1}, {x, y}, {t, 20}]
>>>
>>> My quantity of interest could be x[t] + y[t]. I can plot it:
>>>
>>> Plot[x[t] + y[t] /. s, {t, 0, 20}]
>>>
>>> But say I wish to store it as a single object.
>>> So I do (this is the kinky part):
>>>
>>> sum = FunctionInterpolation[x[t] + y[t] /. s, {t, 0, 20}]
>>>
>>> Now, when I plot the result, it comes out completely different. It looks
>>> like something changed with the interpolation order or something.
>>>
>>> Plot[sum[t], {t, 0, 20}]
>>>
>>> What do you guys think?
>>>
>>> Best regards,
>>>
>>> Gabriel
>>>
>>
>> Gabriel,
>>
>> here are two alternatives:
>>
>> You could sample at more points via
>>
>> sum = FunctionInterpolation[Evaluate[x[t] + y[t] /. s], {t, 0, 20},
>>   InterpolationPoints -> 100]
>>
>>
>> additionally specifying the derivatives may be a good idea:
>>
>> sum = FunctionInterpolation[
>>   Evaluate[Table[
>>     D[Evaluate[x[t] + y[t] /. s], {{t}, k}], {k, 0, 2}]], {t, 0, 20},
>>   InterpolationPoints -> 30]
>> (* then you get away with less InterpolationPoints *)
>>
>> or do it manually as follows:
>>
>> The good thing here is that the NDSolve solutions are sampled at the same
>> points.
>>
>> tutorial/NDSolvePackages#120436095
>>
>> s = NDSolve[{x'[t] == -y[t] - x[t]^2, y'[t] == 2 x[t] - y[t]^3,
>>    x[0] == y[0] == 1}, {x, y}, {t, 20}]
>>
>> Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"]
>>
>> domain = InterpolatingFunctionDomain[if1]
>>
>> {coords1} = InterpolatingFunctionCoordinates[if1];
>> {coords2} = InterpolatingFunctionCoordinates[if2];
>>
>> Norm[coords1 - coords2, 1]
>>
>> vals1 = InterpolatingFunctionValuesOnGrid[if1];
>> vals2 = InterpolatingFunctionValuesOnGrid[if2];
>>
>> if3 = Interpolation[Transpose[{coords1, vals1 + vals2}]]
>>
>> Plot[if3[x], {x, 0, 20}]
>>
>> Hope this helps.
>>
>> Oliver
>>
>
>
>

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