Question

Somebody have idea how to use all cores for calculating integration? I need to use parallelize or parallel table but how?

 f[r_] := Sum[(((-1)^n*(2*r - 2*n - 7)!!)/(2^n*n!*(r - 2*n - 1)!))*
 x^(r - 2*n - 1), {n, 0, r/2}]; 


 Nw := Transpose[Table[f[j], {i, 1}, {j, 5, 200, 1}]]; 

 X1 = Integrate[Nw . Transpose[Nw], {x, -1, 1}]; 

 Y1 = Integrate[D[Nw, {x, 2}] . Transpose[D[Nw, {x, 2}]], {x, -1, 1}]; 

 X1//MatrixForm
 Y1//MatrixForm

Answer 1

If one helps Integrate a bit by expanding the matrix elements first,
things are doable with a little bit of effort.

On a quad-core laptop with Windows and Mathematica 8.0.4 the following code below runs
for the asked DIM=200 in about 13 minutes,
for DIM=50 the code runs in 6 second.

---

$starttime = AbsoluteTime[]; Quiet[LaunchKernels[]]; 
DIM = 200; 
Print["$Version = ", $Version, "  |||  ", "Number of Kernels : ", Length[Kernels[]]]; 
f[r_] := f[r] = Sum[(((-1)^n*(-(2*n) + 2*r - 7)!!)*x^(-(2*n) + r - 1))/(2^n*n!*(-(2*n) + r - 1)!), {n, 0, r/2}]; 
Nw = Transpose[Table[f[j], {i, 1}, {j, 5, DIM, 1}]]; 
nw2 = Nw . Transpose[Nw]; 
Print["Seconds for expanding Nw.Transpose[Nm] ", Round[First[AbsoluteTiming[nw3 = ParallelMap[Expand, nw2]; ]]]]; 
Print["do the integral once: ", Integrate[x^n, {x, -1, 1}, Assumptions -> n > -1]]; 
Print["the integration can be written as a simple rule: ", intrule = (pol_Plus)?(PolynomialQ[#1, x] & ) :> 
     (Select[pol,  !FreeQ[#1, x] & ] /. x^(n_.) /; n > -1 :> ((-1)^n + 1)/(n + 1)) + 2*(pol /. x -> 0)]; 
Print["Seconds for integrating Nw.Transpose[Nw] : ", Round[First[AbsoluteTiming[X1 = ParallelTable[row /. intrule, {row, nw3}]; ]]]]; 
Print["expanding: ", Round[First[AbsoluteTiming[preY1 = ParallelMap[Expand, D[Nw, {x, 2}] . Transpose[D[Nw, {x, 2}]]]; ]]]]; 
Print["Seconds for integrating : ", Round[First[AbsoluteTiming[Y1 = ParallelTable[py /. intrule, {py, preY1}]; ]]]]; 
Print["X1 = ", (Shallow[#1, {4, 4}] & )[X1]]; 
Print["Y1 = ", (Shallow[#1, {4, 4}] & )[Y1]]; 
Print["seq Y1 : ", Simplify[FindSequenceFunction[Diagonal[Y1], n]]]; 
Print["seq X1 0 : ",Simplify[FindSequenceFunction[Diagonal[X1, 0], n]]]; 
Print["seq X1 2: ",Simplify[FindSequenceFunction[Diagonal[X1, 2], n]]]; 
Print["seq X1 4: ",Simplify[FindSequenceFunction[Diagonal[X1, 4], n]]]; 
Print["overall time needed in seconds: ", Round[AbsoluteTime[] - $starttime]];