Re: Why I got inumr??

*To*: mathgroup at smc.vnet.net*Subject*: [mg81020] Re: Why I got inumr??*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>*Date*: Sat, 8 Sep 2007 03:50:05 -0400 (EDT)*Organization*: The Open University, Milton Keynes, UK*References*: <fbqpld$6eu$1@smc.vnet.net>

hj.gegewu wrote: > I have a problem when I try to Nintegrate following code: > > a = 0; b = 2; > > P = (Max[0, (Min[b, #2*Cos[#3] + Sqrt[#1^2 - (#2*Sin[#3])^2]] - > Max[a, #2*Cos[#3] - Sqrt[#1^2 - (#2*Sin[#3])^2]])/(b - > a)]) &; > > ProbR1 = (NIntegrate[ > P[(-#1*Cos[z] + Sqrt[#2^2 - (#1 *Sin[z])^2])/#3, x, > y]/(4*Pi^2*(b - a)), {z, Pi - ArcSin[#2/#1], > Pi + ArcSin[#2/#1]}, {y, 0, 2*Pi}, {x, a, > Min[b, (-#1*Cos[z] + Sqrt[#2^2 - (#1 *Sin[z])^2])/(#3* > Abs[Sin[y]])]}, > Method -> {"GlobalAdaptive", > "SingularityHandler" -> "DuffyCoordinates"}]) &; > > ProbR2 = (NIntegrate[ > P[(-#1*Cos[z] - Sqrt[#2^2 - (#1 *Sin[z])^2])/#3, x, > y]/(4*Pi^2*(b - a)), {z, Pi - ArcSin[#2/#1], > Pi + ArcSin[#2/#1]}, {y, 0, 2*Pi}, {x, a, > Min[b, (-#1*Cos[z] + Sqrt[#2^2 - (#1 *Sin[z])^2])/(#3* > Abs[Sin[y]])]}, > Method -> {"GlobalAdaptive", > "SingularityHandler" -> "DuffyCoordinates"}]) &; > > ProbR1[130, 125, 50] > ProbR2[130, 125, 50] > > while the line of "ProbR1[130, 125, 50]" can be caculated correctly, the line of "ProbR2[130, 125, 50]" keeps giving me the warning of > > NIntegrate::inumr: "The integrand \ > Max[0,1/2\(-Max[0,Times[<<2>>]+Times[<<2>>]]+Min[2,x\Cos[<<1>>]+Sqrt[\ > Plus[<<2>>]]])]/(8\\[Pi]^2) has evaluated to non-numerical values for \ > all sampling points in the region with boundaries \ > {{0.5,0.75},{0.125,0.0625},{0.5,0.25}}." > > The only difference of these two sentence is the sign of "Sqrt[#2^2 - (#1 *Sin[z])^2])". Why this happened? What version of Mathematica are you using? I would not blindly trust the result for the first expression: Mathematica 6.0.1 returns some long warnings before outputting the "result". In[5]:= ProbR1[130, 125, 50] During evaluation of In[5]:= NIntegrate::slwcon: Numerical \ integration converging too slowly; suspect one of the following: \ singularity, value of the integration is 0, highly oscillatory \ integrand, or WorkingPrecision too small. >> During evaluation of In[5]:= NIntegrate::eincr: The global error of \ the strategy GlobalAdaptive has increased more than 2000 times. The \ global error is expected to decrease monotonically after a number of \ integrand evaluations. Suspect one of the following: the working \ precision is insufficient for the specified precision goal; the \ integrand is highly oscillatory or it is not a (piecewise) smooth \ function; or the true value of the integral is 0. Increasing the \ value of the GlobalAdaptive option MaxErrorIncreases might lead to a \ convergent numerical integration. NIntegrate obtained \ 0.37889873229233717` and 0.00026093286784722514` for the integral and \ error estimates. >> Out[5]= 0.378899 Regards, -- Jean-Marc