Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2007
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2007

[Date Index] [Thread Index] [Author Index]

Search the Archive

Re: Evaluating a convolution integral in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg80269] Re: Evaluating a convolution integral in Mathematica
  • From: Randall Beer <rdbeer at indiana.edu>
  • Date: Thu, 16 Aug 2007 04:46:06 -0400 (EDT)
  • References: <f9ud4q$a5q$1@smc.vnet.net> <46C32C19.3090203@gmail.com>

Thanks for your response.  I defined B as 1/2 (-Sign[-max + x] + Sign 
[-min + x]). Yes, I did provide Mathematica with the assumptions I  
stated in my message.  However, the result Mathematica gives seems to  
produce incorrect answers when specific values of n, m, and the mins  
and maxs are substituted in.

I will take a look at the MathSource notebook that you suggested.

Thanks,
Randy Beer


On Aug 15, 2007, at 12:38 PM, Jean-Marc Gulliet wrote:

> rdbeer at indiana.edu wrote:
>> I am having trouble evaluating a convolution integral in Mathematica.
>> Define B[x,{min,max}] to be a function that takes on the value 1 when
>> x is between min and max and 0 otherwise.  I need to find the
>> convolution of x^n B[x, {min1, max1}] with x^m B[x, {min2, max2}],
>> where x is Real, n and m are nonnegative integers, and the mins and
>> maxs are Real with the constraints that min1<=max1 and min2<=max2.
>> The resulting convolution integeral is Integrate[t^n B[t, {min1,
>> max1}] (x-t)^m B[x-t, {min2, max2}], {t, -Infinity, Infinity}].
>> Mathematica 6.0.1 has no problem evaluating this integral when
>> constant values are substituted for n, m, and the mins and maxs.
>> However, I need the general value of this integral.  Mathematica also
>> claims to be able to evaluate this general integral, returning a
>> complicated peicewise expression involving gamma functions and
>> hypergeometric functions.  However, when specific values for n, m and
>> the mins and maxs are then substituted into this general expression,
>> it always returns either 0 or Indeterminate.
>> Any help with evaluating this integral in general would be greatly
>> appreciated.
>
> It is not clear, at least for me, how you coded your function B?  
> Did you use Piecewise, If, Which, Boole, ... ? Here is a simple  
> example with Boole,
>
> B[x_, {min_, max_}] := Boole[min <= x <= max]
>
> Also, it is not clear whether you have tried to set up your  
> assumptions in Integrate itself. For instance,
>
> Integrate[
>  t^n B[t, {min1, max1}] (x - t)^m B[
>    x - t, {min2, max2}], {t, -Infinity, Infinity},
>  Assumptions -> {{x, min1, mmax1, min2, max2} \[Element]
>     Reals, {n, m} \[Element] Integers, n >= 0, m >= 0, min1 <= max1,
>    min2 <= max2}]
>
> returns a long piecewise function with some hypergeometric  
> functions (I  did not try to simplify nor to check the validity of  
> the expression since I did not know what the function B really was).
>
> Also, it may be worthwhile to try Maxim Rytin's "Integration of  
> Piecewise Functions with Applications" package available at
>
> http://library.wolfram.com/infocenter/MathSource/5117/
>
> Especially, you should check the function *PiecewiseIntegrate*.  
> From the notebook piecewise.nb, we can read,
>
> "PiecewiseIntegrate[f,{x,xmin,xmax},{y,ymin,ymax},\[Ellipsis]]  
> gives the definite integral of function f. It is intended for  
> integrating piecewise continuous functions, and also generalized  
> functions. It handles integrands and integration bounds involving  
> the following expressions:
>
> . UnitStep, Sign, Abs, Min, Max
> . Floor, Ceiling, Round, IntegerPart, FractionalPart, Quotient, Mod
> . DiracDelta and its derivatives, DiscreteDelta, KroneckerDelta
> . If, Which, Element, NotElement
> . Piecewise, Boole, Clip"
>
> For instance,
>
> PiecewiseIntegrate[
>  t^n B[t, {min1, max1}] (x - t)^m B[
>    x - t, {min2, max2}], {t, -Infinity, Infinity},
>  Assumptions -> {{x, min1, mmax1, min2, max2} \[Element]
>     Reals, {n, m} \[Element] Integers, n >= 0, m >= 0, min1 <= max1,
>    min2 <= max2}]
>
> returns a long answer made of nested If constructs that return some  
> simple expressions such as
>
> If[m == 0 && min1 < max1 && min2 == -max1 + max2 + min1 &&
>   x == max1 + min2,
>  If[min1 > 0, (-min1^(1 + n) + max1^n (-min2 + x))/(1 + n), [...]
>
> (Again, I have not try to simplify nor to check the validity of the  
> expression.)
>
> HTH
> -- 
> Jean-Marc
>



  • Prev by Date: Re: Complexity explosion in linear solve
  • Next by Date: Re: Nested Dialogs
  • Previous by thread: Re: Evaluating a convolution integral in Mathematica
  • Next by thread: Re: Evaluating a convolution integral in Mathematica