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functions - Double application of NIntegrate produces unexpected error messages




In the course of trying to answer 69601, I encountered unexpected error messages. In highly simplified form,


NIntegrate[NIntegrate[z y, {y, 0, 10}], {z, 0, 10}]

produces repeated messages,


NIntegrate::inumr: The integrand y z has evaluated to non-numerical values for all sampling points in the region with boundaries (0  10). >>

before returning the correct answer, 2500. Thus, the error messages seem to be harmless, apart from consuming time. Nonetheless, I would like understand their source and eliminate them by means other than Quiet. The option Method -> {Automatic, "SymbolicProcessing" -> False} reduces the number of messages to one, which is a step in the right direction. Suggestions?



Answer



The issue is that the interior integral tries to evaluate first and has a symbolic z. This fails and returns unevaluated before being passed to the outer integral.



The two solutions that come to mind would be to call NIntegrate as a double integral.


NIntegrate[z*y, {y,0,10},{z,0,10}]
(* 2500. *)

or to define the interior integral as a function that requires numeric inputs.


f[z_?NumericQ]:= NIntegrate[z * y, {y, 0, 10}]
NIntegrate[f[z], {z, 0, 10}]
(* 2500. *)

There may be some way to accomplish this with some option setting or by some non-standard evaluation that I'm not aware of but this seems cleanest.



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