Re: Numerical convolution

*To*: mathgroup at smc.vnet.net*Subject*: [mg116434] Re: Numerical convolution*From*: Bob Hanlon <hanlonr at cox.net>*Date*: Tue, 15 Feb 2011 06:33:56 -0500 (EST)*Reply-to*: hanlonr at cox.net

f1[x_] = PDF[NormalDistribution[0, 1], x]; Prior to version 8 the PDF for LogNormalDistribution was defined as f2[x_] = 1/(E^((1/2)*Log[x]^2)*(Sqrt[2*Pi]*x)); This required the user to avoid negative arguments (which I did not). However, with version 8 the PDF for one-sided distributions uses Piecewise. f3[x_] = PDF[LogNormalDistribution[0, 1], x] Piecewise[{{1/(E^((1/2)*Log[x]^2)*(Sqrt[2*Pi]*x)), x > 0}}, 0] convolve[f_, g_, t_?NumericQ] := NIntegrate[f[u]*g[t - u], {u, 0, t}]; NIntegrate[convolve[f1, f2, t], {t, -Infinity, 0}] // Re 0.5 NIntegrate[convolve[f1, f3, t], {t, -Infinity, 0}] // Re NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option. >> 0. NIntegrate[convolve[f1, f2, t], {t, -Infinity, Infinity}] // Re 1. NIntegrate[convolve[f1, f3, t], {t, -Infinity, Infinity}] // Re 0.5 Without Piecewise in its definition the PDF for LogNormalDistribution acts as if two-sided and gives the different result. Alternatively, the definition of convolve could be changed for one-sided distributions. convolve2[f_, g_, t_?NumericQ] := NIntegrate[f[u]*g[t - u], {u, 0, Max[0, t]}]; NIntegrate[convolve2[f1, f2, t], {t, -Infinity, 0}] // Re NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option. >> 0. NIntegrate[convolve2[f1, f2, t], {t, -Infinity, Infinity}] // Re 0.5 Bob Hanlon ---- "Rob Y. H. Chai" <yhchai at ucdavis.edu> wrote: ============= Dear Bob, I found your suggested solution in MathGroup from 2007 for numerical integration of a convolution integral. I tried it in Mathematica 8.0 and was surprised to see that the results are now different from yours earlier. More specifically NIntegrate[convolve[f1,f2,t],{t,-Infinity,Infinity}]//Re gives 0.5 as opposed to 1.0 Any idea what might have happened? Thanks and sincerely Rob Chai, UC Davis ============================================================== Re: Numerical Integrate * To: mathgroup at smc.vnet.net * Subject: [mg72883] Re: [mg72864] Numerical Integrate * From: Bob Hanlon <hanlonr at cox.net> * Date: Wed, 24 Jan 2007 05:30:04 -0500 (EST) * Reply-to: hanlonr at cox.net _____ Needs["Statistics`"]; f1[x_]=PDF[NormalDistribution[0, 1],x]; f2[x_]=PDF[LogNormalDistribution[0, 1],x]; convolve[f_,g_, t_?NumericQ]:= NIntegrate[f[u]*g[t-u],{u,0,t}]; NIntegrate[convolve[f1,f2,t],{t,-Infinity,0}]//Re 0.5 NIntegrate[convolve[f1,f2,t],{t,-Infinity,Infinity}]//Re 1. Bob Hanlon ---- "mailcwc at gmail.com" <mailcwc at gmail.com> wrote: > I need to perform the following integration: > > Integrate[ Convolution[G(t), X(t)], {t, -inf, 0}], > > where G is Normal distribution, and X is Log-normal distribution. > > As I know, there is no analytical form for the convolution. > I tried NIntegrate to perform numerical calculation, but Mathematica > can't accept it. > Does anyone know how to deal such problem?