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numerical integration - a NIntegrate in the NIntegrate



Here is our problem:
enter image description here
Suppose g[y,z] and f[x,z] is complicated,we have to solve it numerically.



Writing this in Mathematica:


Clear["`*"]
g[y_, w_] := 1/(w - y + I*0.1)
f[x_, w_] := (x*x)/(w - x + I*0.1)
NIntegrate[(g[k2, w]/(1 - NIntegrate[f[k1, w], {k1, 0, 30}])), {k2, 0,
   30}, {w, -1, 2}]

the error message is


    NIntegrate::inumr: The integrand x^2/((2. +0.1 I)-x) has evaluated to non-numerical values 
for all sampling points in the region with boundaries {{0,30}}.


Answer



This might be a duplicate of one of the examples in User-defined functions, numerical approximation, and NumericQ, but there isn't one with nested NIntegrate.


ClearAll[f, g, h];
g[y_, w_] := 1/(w - y + I*0.1)             (* note use of patterns y_, w_ *)
f[x_?NumericQ, w_?NumericQ] := (x*x)/(w - x + I*0.1)
h[w_?NumericQ] := NIntegrate[f[k1, w], {k1, 0, 30}];
NIntegrate[(g[k2, w]/(1 - h[w])), {k2, 0, 30}, {w, -1, 2},
 PrecisionGoal -> 2, AccuracyGoal -> 8]    (* for speed over accuracy; adjust as desired *)
(*  -0.0249226 - 0.0125844 I  *)


The trouble is that NIntegrate evaluates the integrand once symbolically. When the outer one evaluates the inner one, the symbol w does not have a value, so the inner NIntegrate complains. Note this error does not prevent the integral from being evaluated; but it is annoying and the messages slow things down.


For the use of patterns in defining functions, see the tutorial Defining Functions and its related tutorials.


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