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problem with integrating an interpolated list

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102841] problem with integrating an interpolated list
  • From: Tobias Baumann <ttobsen at hotmail.com>
  • Date: Sat, 29 Aug 2009 06:31:19 -0400 (EDT)

Hi

I've an problem with integrating an interpolated list. I use Mathematica 
7 and I'm a pretty newbie.

I generate a list of lists like this

List2D = {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 5,
     1, 0, 0}, {0, 0, 1, 4, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0,
     0}};

With MatrixForm it looks like

000000
000000
005100
001400
000000
000000

Now i make a ListInterpolation first order


MyFunction = ListInterpolation[List2D, {{0, 6}, {0, 6}}, 
InterpolationOrder -> 1];


After this I norm the new Function to value 1. So I integrate


NormFactor = Integrate[MyFunction[x, y], {x, 0, 6}, {y, 0, 6}]


This works very well, the result is 396/25. For the next step, I prepare 
a new function, to make shure that outside of the interpolaten range the 
function is 0. In this step I also normalize "MyFunction":


NewFunction[xy_List] := 0 /; xy[[1]] < 2
NewFunction[xy_List] := 0 /; xy[[2]] < 2
NewFunction[xy_List] := 0 /; xy[[1]] > MaxResolution - 2
NewFunction[xy_List] := 0 /; xy[[2]] > MaxResolution - 2
NewFunction[xy_List] := MyFunction [xy[[1]], xy[[2]]]/NormFactor


Now, if I make an integration of "NewFunction" ist works really good if 
I try


Output = Integrate[NewFunction[{x, y}], {x, -10, 10}, {y, -10, 10}];
Output

Out = 1


But I can't integrate something like this


Output = Integrate[NewFunction[{x+2, y+2}], {x, -10, 10}, {y, -10, 10}];
Output

Out = 25/396 Integrate[InterpolatingFunction[{{0,6}, {0,6}}, <>][2 + x, 
2 + y],{x,-10, 10}, {y,-10,10}]


In this case the Output is no numerical result. I allways get the 
inputstring back (in a bit more graphical way). The problem is that I 
need to transform the x and y coordinate.

Numerical integration is also no option. I get the right solution, but 
with the message

"NIntegrate::slwcon: Numerical integration converging too slowly; 
suspect one of the following: singularity, value of the integration is 
0, highly oscillatory integrand, or WorkingPrecision too small. >>"

It's also too slow for my project.

I hope someone has any ideas.


Thanks a lot,
Tobias

Complete Code for trying at home

List2D = {{0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 5, 1, 0,
     0}, {0, 0, 1, 4, 0, 0}, {0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0}};

List2D // MatrixForm

MaxResolution = Length[List2D];

MyFunction =
   ListInterpolation[List2D, {{0, MaxResolution}, {0, MaxResolution}},
    InterpolationOrder -> 1];

NormFactor = Integrate[MyFunction[x, y], {x, 0, 6}, {y, 0, 6}]

NewFunction[xy_List] := 0 /; xy[[1]] < 2
NewFunction[xy_List] := 0 /; xy[[2]] < 2
NewFunction[xy_List] := 0 /; xy[[1]] > MaxResolution - 2
NewFunction[xy_List] := 0 /; xy[[2]] > MaxResolution - 2
NewFunction[xy_List] := MyFunction [xy[[1]], xy[[2]]]/NormFactor

Output = Integrate[
    NewFunction[{x + 2, y + 2}], {x, -10, 10}, {y, -10, 10}];
Output


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