Mathematica Asked by 123infinity on December 20, 2020

I’m trying out the code below and it takes quite a long time to compute and throws eincr and slwcon warnings. Is there a way to speed this up?

```
phase[u_, m_, [Theta]_, [Phi]_] :=
u^3 + 2 m [Theta] (3/(4 m))^(2/3) Cos[[Phi]] u^2 +
m [Theta]^2 (3/(4 m))^(1/3) u - m [Theta] Cos[[Phi]];
integrand[u_,
m_, [Theta]_, [Phi]_] := (-2 (3/(4 m))^(1/3)
u + [Theta] ) Exp[-I phase[u, m, [Theta], [Phi]]];
integral[[Theta]_, [Phi]_, m_, {ulo_, uhi_}] :=
NIntegrate[integrand[u, m, [Theta], [Phi]], {u, ulo, uhi},
MaxRecursion -> 20, WorkingPrecision -> 20] //
Timing
integraltab =
Monitor[Table[{[Theta], [Phi],
integral[[Theta], [Phi], 100, {-Infinity, Infinity}]}, {[Theta], 0,
Pi, Pi/32}, {[Phi], 0, 2 Pi,
2 Pi/63}], {[Theta], [Phi]}];
```

The integrand only depends on ` Cos[[Phi]`

, that's why you can decrease the integration range to `{[Phi], 0, Pi}`

(- 50% evaluation time!)!

Include the option `Method -> {Automatic, "SymbolicProcessing" -> 0}`

inside `NIntegrate`

:

```
integral[[Theta]_, [Phi]_, m_, {ulo_, uhi_}] :=
NIntegrate[integrand[u, m, [Theta], [Phi]], {u, ulo, uhi} ,MaxRecursion -> 20, WorkingPrecision -> 20
,Method -> {Automatic, "SymbolicProcessing" -> 0}]
```

the grid of 101x101 tablevalues is evaluated in nearly 75s!

```
integraltab = Monitor[Table[{[Theta], [Phi],integral[[Theta], [Phi],100, {-Infinity, Infinity}]}
, {[Theta], 0, Pi,Pi/10}, {[Phi], 0, Pi, Pi/10}], {[Theta], [Phi]}]; // Timing
(*{74.7813, Null}*)
```

Without these modifictions the evaluation of 101x101 grid lasts around 1700s ( **speedup factor 23** ) !

Hope it helps!

Answered by Ulrich Neumann on December 20, 2020

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