LegendreP Evaluation Mystery

*To*: mathgroup at smc.vnet.net*Subject*: [mg75055] LegendreP Evaluation Mystery*From*: "Antonio" <aneves at gmail.com>*Date*: Mon, 16 Apr 2007 04:05:23 -0400 (EDT)

Dear all, I have found that Mathematica v5.2 evaluates BesselJ and LegendreP differently. Why is this? I have written this test below to illustrate the timing. In[11]:= k=10000; n=51; m=Random[Integer,{0,n}]; x=Random[Real,{0,1}]; P[y_]=LegendreP[n,m,y]; J[y_]=BesselJ[n,y]; Timing[Do[LegendreP[n,m,x],{k}]] Timing[Do[P[x],{k}]] Timing[Do[BesselJ[n,x],{k}]] Timing[Do[J[x],{k}]] Out[17]= {4.531 Second,Null} Out[18]= {0.484 Second,Null} Out[19]= {0.61 Second,Null} Out[20]= {0.657 Second,Null} The emphasis here is with the Associated Legendre Function, since it is a bottel neck of my current calculations for big n's (n>50), used inside NIntegrate. The form LegendreP[n,m,x] takes longer to evaluate than in extended form (P[y])? If I try to use the extended form to do some numerical integration, it results in a wrong result even though it is faster. When plotting the functions, it seems to show some numerical instabilities (for low m): n = 51; m = 3; Plot[LegendreP[n, m, x], {x, 0, 1}, PlotRange -> All, PlotPoints -> 100]; Plot[P[x], {x, 0, 1}, PlotRange -> All, PlotPoints -> 100]; And it is worse for m=0. Does Mathematica evaluate LegendreP differently for high n's, why does it take so long? Is there any way that I build an array of extended Associated Legendre functions, so as to speed up calculations and wouldn't fail numerically as above? Antonio