Re: LegendreP Evaluation Mystery

*To*: mathgroup at smc.vnet.net*Subject*: [mg75083] Re: LegendreP Evaluation Mystery*From*: "Norbert Marxer" <marxer at mec.li>*Date*: Mon, 16 Apr 2007 20:14:02 -0400 (EDT)*References*: <evvaqs$95r$1@smc.vnet.net>

On 16 Apr., 10:06, "Antonio" <ane... at gmail.com> wrote: > 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 Hello If you use SetDelayed (:=) instead of Set (=) in your defintions for m, x, P and J then all your mysteries and questions will (probably) evaporate. With your definition the random numbers are evaluated only once and therefore m is constant. Similarly your associated Legendre polynomial (P) is evaluated only once (for specific values of n and m) and set to a specific constant polynomial with the independent variable y. The evaluation of this polynomial takes obviously less time than the calculation and evaluation of the associated Legendre polynomial. Best Regards Norbert Marxer