Questions about ParallelTable for Transpose in numerical calculation

Question

Question details:

I used the GaussianQuadratureWeights to calculate the Integration of a simple expression. But when I used the ParallelTable, it turned out to be errors about the GaussianQuadratureWeights and Transpose.

My Mathematica kernels are 2.

---

Here are the original code for this example.

In[1]:= << NumericalDifferentialEquationAnalysis`

In[2]:= GLQ1D[func_, {a_, b_}, {Xdegree_}] := 
  Block[{Xnodes, x, Xweights, i},
   x = GaussianQuadratureWeights[Xdegree, a, b]; 
   Xnodes = Transpose[x][[1]]; Xweights = Transpose[x][[2]];
   Total[Flatten[
     Table[Xweights[[i]]*func[ Xnodes[[i]]], {i, 1, Xdegree}]]]];

In[3]:= f[x_] := x^2;

In[4]:= a = 1.;

In[5]:= b = 3.;

In[6]:= TrueValue = Integrate[x^2, {x, a, b}]

Out[6]= 8.66667

In[7]:= x1 = Table[i, {i, 1, 3 - 0.1, 0.1}];

In[8]:= x2 = Table[i, {i, 1 + 0.1, 3, 0.1}];

In[9]:= value1 = 
 Table[Total[
    Flatten[Table[
      GLQ1D[f, {x1[[i]], x2[[i]]}, {k}], {i, 1, Length[x1]}]]], {k, 1,
     5}] // AbsoluteTiming

Out[9]= {0.009066, {8.665, 8.66667, 8.66667, 8.66667, 8.66667}}

In[10]:= ListLinePlot[Abs[value1[[2]] - TrueValue]]

---

It was OK with Table for GaussianQuadratureWeights, but when it came for ParallelTable, it had errors like this:

In[11]:= value2 = 
 Table[Total[
    Flatten[ParallelTable[
      GLQ1D[f, {x1[[i]], x2[[i]]}, {k}], {i, 1, Length[x1]}]]], {k, 1,
     5}] // AbsoluteTiming

(kernel 2) Part::partw :  Part 2 of Transpose[NumericalDifferentialEquationAnalysis`GaussianQuadratureWeights[1,1.,1.1]] does not exist.


Out[11]= {4.62609, {42., 989.6, 2105.52, 3945.13, 6745.78}}

In[12]:= value3 = 
 ParallelTable[
   Total[Flatten[
     Table[GLQ1D[f, {x1[[i]], x2[[i]]}, {k}], {i, 1, 
       Length[x1]}]]], {k, 1, 5}] // AbsoluteTiming



(kernel 2) General::stop :  Further output of Part::partw will be suppressed during this calculation.

(kernel 1) General::stop :  Further output of Part::partw will be suppressed during this calculation.

(kernel 2) Part::partw :  Part 2 of Transpose[NumericalDifferentialEquationAnalysis`GaussianQuadratureWeights[4,1.,1.1]] does not exist.

(kernel 1) General::stop :  Further output of Part::partw will be suppressed during this calculation.

Out[12]= {0.343389, {{{40.}, {2.}}, {{323.091, 327.709}, {169.4, 
    169.4}}, {{520.948, 527.919, 534.891}, {172.919, 175.919, 
    172.919}}, {{796.147, 804.485, 815.365, 823.703}, {173.923, 
    178.792, 178.792, 173.923}}, {{1148.52, 1157.71, 1171.18, 1184.64
    ,1193.83}, {173.903, 179.946, 182.202, 179.946, 173.903}}}}

Answer 1

In order to use the package NumericalDifferentialEquationAnalysis on all kernels you need all kernels to load it. You can achieve this by replacing your first line

<< NumericalDifferentialEquationAnalysis`

with

ParallelNeeds["NumericalDifferentialEquationAnalysis`"];

See documentation for more information!