# Mathematica taking forever to compute

I want to simulate the fresnel diffraction through circular aperture.But mathematica is taking forever to calculate.Here is my code.

  f[x_?NumericQ, y_?NumericQ, d_?NumericQ, p_?NumericQ, t_?NumericQ] :=
NIntegrate[(\[Zeta]*(Exp[
I*20000 ((d^2 + \[Zeta]^2)^(0.5) + ((p^2 + (x - \[Zeta]*
Cos[t])^2 + (y - \[Zeta]*
Sin[t])^2)^(0.5)))]))/(d^2 + \[Zeta]^2)^(0.5), {\
\[Zeta], 0, 0.008}]
w[x_?NumericQ, y_?NumericQ, d_?NumericQ, p_?NumericQ] :=
NIntegrate[f[x, y, d, p, t], {t, 0, 2*Pi}]
g[x_, y_, d_, p_] := (Abs[w[x, y, d, p]])^2
Plot3D[g[x, y, 0.004, 20], {x, -100, 100}, {y, -100, 100},
PlotRange -> Full]


What's the way out? *Specs:i5 7th gen intel processor,12 gb ram

Let's compile the integrand into a Listable CompiledFunction:

cintegrand = Block[{x, y, d, p, ζ, t},
With[{code =
N[(ζ (Exp[I 20000 ((d^2 + ζ^2)^(1/2) + ((p^2 + (x - ζ Cos[t])^2 + (y - ζ Sin[t])^2)^(1/2)))]))/(d^2 + ζ^2)^(1/2)]
},
Compile[{{x, _Real}, {y, _Real}, {d, _Real}, {p, _Real}, {ζ, _Real}, {t, _Real}},
code,
CompilationTarget -> "C",
RuntimeAttributes -> {Listable},
Parallelization -> True
]
]
];


Next, pick a Gauss quadrature rule for high order polynomials (the integrand is very smooth but quite oscillatory):

{pts, weights, errweights} = NIntegrateGaussRuleData[9, $MachinePrecision];  Divide the integration domain into$m \times n$rectangles and set up the quadrature points and weights: m = 20; n = 20; ζdata = Partition[Subdivide[0., 0.008, m], 2, 1]; tdata = Partition[Subdivide[0., 2 Pi, n], 2, 1]; {ζ, t} = Transpose[Tuples[{Flatten[ζdata.{1. - pts, pts}], Flatten[tdata.{1. - pts, pts}]}]]; ω = Flatten[KroneckerProduct[ Flatten[KroneckerProduct[Differences /@ ζdata, weights]], Flatten[KroneckerProduct[Differences /@ tdata, weights]] ]];  Define the mapping that parameterizes the graph of OP's function w: W = {x, y} \[Function] {x, y, Abs[cintegrand[x, y, 0.004, 20., ζ, t].ω]^2};  Compute the values of W on a$101 \times 101$grid: R = 2.; data = Outer[W, Subdivide[-R, R, 100], Subdivide[-R, R, 100]]; // AbsoluteTiming // First  28.2668 Plot the result ListPlot3D[Flatten[data, 1], PlotRange -> All, AxesLabel -> {"x", "y", "w"}]  • Could you please recomend a source for learning such techniques for reducing time of computation,like the use of block,listablecompiledfunction etc..? – user157588 May 24 '18 at 13:26 • This is a good start. The rest is basically experience, some basics in numerical Mathematics and computer science, reading a lot on StackExchange,... and experience of course. – Henrik Schumacher May 24 '18 at 13:32 • Oh, and I would not call Block a way of reducing computational time. It is just a scoping construct that works a bit more predictably in conjunction with Compile than Module. In general, I would advise Module. – Henrik Schumacher May 24 '18 at 13:34 TIP: In Mathematica use exact values,because have infinty precision. f[x_?NumericQ, y_?NumericQ, d_?NumericQ, p_?NumericQ] := NIntegrate[(ζ*(Exp[I*20000 ((d^2 + ζ^2)^(1/2) + ((p^2 + (x - ζ*Cos[t])^2 + (y - ζ*Sin[t])^2)^(1/2)))]))/(d^2 + ζ^2)^(1/2), {ζ, 0, 8/1000}, {t, 0, 2*Pi}, Method -> {"LocalAdaptive", "SymbolicProcessing" -> 0}]; g[x_, y_, d_, p_] := Abs[f[x, y, d, p]]^2; n = 1/6; (*The smaller the value , 3D plot is smoother,but CPU time will increase !!! *) ListPlot3D[Partition[Flatten[Table[{x, y, g[x, y, 4/1000, 20]}, {x, -2, 2, n}, {y, -2, 2, n}]], 3]] (*In Range: -2<x<2 and -2<y<2.If you increase range domain CPU time will increase !!! *)  On my a 100$old laptop calculation time is about 24` minutes.