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I have to (numerically) integrate this function over all p from to get a function g(x):

f = 0.102354 Exp[(11876.9 p^2 - 154.386 p x + (0.000556163 + 0.501713 x) x)] AiryAi[0.0275599 + 89099. p^2 + 0.00335015 x - 1158.19 p x + 3.7638 x^2]

I tried this:

tabx = Table[{x1, NIntegrate[f /. x -> x1, {p, -Infinity, Infinity}, MinRecursion -> 10, MaxRecursion -> 20]}, {x1, -20000, 1000, 100}];

However, it takes a long time, I get a lot of error messages and the solution doesn't look like it should, especially for large negative values of x. Is there a clever way to play with methods (or any other way), to make this more accurate and faster?

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  • $\begingroup$ Try Table[{x1, NIntegrate[f /. x -> x1, {p, -Infinity, Infinity}, Method -> "ExtrapolatingOscillatory", AccuracyGoal -> 9, PrecisionGoal -> 9, MinRecursion -> 10, MaxRecursion -> 20]}, {x1, -2000, 1000, 100}].This results in . $\endgroup$ Commented Jul 15, 2021 at 17:47
  • $\begingroup$ {{... , {-1600, -2.34539*10^-46}, {-1500, 7.44701*10^-163}, {-1400, 0.}, {-1300, 5.80957*10^-222}, {-1200, -2.76047*10^-148}, {-1100, \ -2.64119*10^-70}, {-1000, -1.98239*10^-104}, {-900, 3.56913*10^-77}, {-800, 0.000755784}, {-700, 1.38624*10^-20}, {-600, 0.000679051}, {-500, 0.000800901}, {-400, 0.000715841}, {-300, 0.000555767}, {-200, 0.000407082}, {-100, 0.000296536}, {0, 0.000222761}, {100, 0.000177177}, {200, 0.000152183}, {300, 0.000143298}, {400, 0.00014962}, {500, 0.000174703}, {600, 0.000229498}, {700, 6.98312*10^-27}, {800, 0.000571349}, ....} $\endgroup$ Commented Jul 15, 2021 at 17:48
  • $\begingroup$ thx but that's even worse ;) $\endgroup$ Commented Jul 15, 2021 at 19:39
  • $\begingroup$ Luke (@does not work.) : Sorry, don't understand you. Can you present the results to compare? TIA and regard, $\endgroup$ Commented Jul 15, 2021 at 20:01
  • $\begingroup$ I guess the issue is that I evaluated from -20000 to 1000, you somehow lost a 0 $\endgroup$ Commented Jul 15, 2021 at 21:07

1 Answer 1

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  1. You should try plotting your functions before assuming they're oscillatory. Some are not. Some have numerics issues.
  2. You're asking a lot if you want people to help you with 211 slow integrals. A MWE should be minimal.

Here's an idea to help with a few of the 211 integrals: Figure out where the support of the integral lies, and make sure it's sampled.

rat = Rationalize[Rationalize@#, 0] &; integrand = Rationalize[rat@f, 0]; min = p /. First@ Solve[ D[0.0275599` + 89099.` p^2 + 0.00335015` x - 1158.19` p x + 3.7638` x^2 // rat, p] == 0, p]; dp = 1/D[ 0.0275599` + 89099.` p^2 + 0.00335015` x - 1158.19` p x + 3.7638` x^2 // rat, p, p]; roots = p /. NSolve[ SetPrecision[ 0.0275599` + 89099.` p^2 + 0.00335015` x - 1158.19` p x + 3.7638` x^2 == 500, 32], p, WorkingPrecision -> 32]; g[x0_ /; -8.180990019212768` < x0 < 1479.8684784944994`] := Block[{x = x0}, NIntegrate[ integrand, {p, -Infinity, min - 10 dp, min, min + 10 dp, Infinity}, Method -> "GaussKronrodRule", WorkingPrecision -> 16] ]; g[x0_] := Block[{x = x0}, With[{central = NIntegrate[integrand, {p, roots[[1]], Mean@roots, roots[[2]]}, Method -> "LevinRule", WorkingPrecision -> 32, PrecisionGoal -> 6]}, central + NIntegrate[integrand, {p, -Infinity, roots[[1]]}, Method -> "GaussKronrodRule", WorkingPrecision -> 32, PrecisionGoal -> 6, AccuracyGoal -> 16 - Min[Log10@central, 0]] + NIntegrate[integrand, {p, roots[[2]], Infinity}, Method -> "GaussKronrodRule", WorkingPrecision -> 16, AccuracyGoal -> 16 - Min[Log10@central, 0]] ]];

Examples:

g[-20000] (* 1.0286007852287060785130193502038*10^572 *) g[-2000] (* 37.208582874780115313397802617944 *) g[200] (* 0.0001521828356334408 *)
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