Re: Roots of a Jacobi polynomial

*To*: mathgroup at smc.vnet.net*Subject*: [mg120543] Re: Roots of a Jacobi polynomial*From*: Emu <samuel.thomas.blake at gmail.com>*Date*: Thu, 28 Jul 2011 07:54:31 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <j0ool2$kr9$1@smc.vnet.net>

On Jul 27, 8:18 pm, Bob Hanlon <hanl... at cox.net> wrote: > Don't use machine precision > > eqn1 = JacobiP[25, -1/2, -1/2, x] == 0; > > soln1 = NSolve[eqn1, x, WorkingPrecision -> 15]; > > Length[soln1] > > 25 > > And @@ (eqn1 /. soln1) > > True > > And @@ (-1 < # < 1 & /@ (x /. soln1)) > > True > > eqn2 = JacobiP[25, -0.5`20, -0.5`20, x] == 0; > > soln2 = NSolve[eqn2, x, WorkingPrecision -> 15]; > > soln1 == soln2 > > True > > Bob Hanlon > > ---- Jacopo Bertolotti <J.Bertolo... at utwente.nl> wrote: > > ============= > Dear MathGroup, > Mathematica implements Jacobi polynomials as JacobiP[n,a,b,x] where n is > the order of the polynomial. As it can be checked plotting it a Jacobi > polynomial has n real roots in the interval [-1,1] and it goes rapidly > to infinity outside this interval (at least when both a and b are >-1). > The problem arise when you try to find the roots of such a polynomial > for a relatively high value of n. As an example the command > NSolve[JacobiP[20, -0.5, -0.5, x] == 0, x] correctly returns > > {{x -> -0.99702}, {x -> -0.972111}, {x -> -0.92418}, {x -> -0.852398}, > {x -> -0.760555}, {x -> -0.649375}, {x -> -0.522529}, {x -> -0.382672}, > {x -> -0.233449}, {x -> -0.0784582}, {x -> 0.0784591}, {x -> 0.233445}, > {x -> 0.382684}, {x -> 0.522504}, {x -> 0.649423}, {x -> 0.760466}, {x > -> 0.852539}, {x -> 0.924002}, {x -> 0.972267}, {x -> 0.996958}} > > while NSolve[JacobiP[25, -0.5, -0.5, x] == 0, x] gives > > {{x -> -1.01869}, {x -> -0.979859 - 0.0479527 I}, {x -> -0.979859 + > 0.0479527 I}, {x -> -0.870962 - 0.070991 I}, {x -> -0.870962 + 0.070991 > I}, {x -> -0.71378 - 0.0505783 I}, {x -> -0.71378 + 0.0505783 I}, {x -> > -0.571283}, {x -> -0.486396}, {x -> -0.367829}, {x -> -0.248377}, {x -> > -0.125513}, {x -> -0.0000434329}, {x -> 0.125442}, {x -> 0.2489}, {x -> > 0.365644}, {x -> 0.496977}, {x -> 0.555743}, {x -> 0.717741- 0.0573399 > I}, {x -> 0.717741+ 0.0573399 I}, {x -> 0.87423- 0.0652273 I}, {x -> > 0.87423+ 0.0652273 I}, {x -> 0.977876- 0.0422422 I}, {x -> 0.977876+ > 0.0422422 I}, {x -> 1.01494}} > > i.e. both complex roots and roots outside the [-1,1] interval. > Substituting any of these values back into the polynomial easily show > that these values are not roots at all. Also notice that using the > command Root (e.g. Sort@Table[Root[JacobiP[25, -0.5, -0.5, x], i], {i, > 1, 25}]) gives different but still wrong results. > On a related note: NIntegrate sometimes gives wrong results when > integrating function of the form (1-x)^a (1+x)^b f[x] and I feel this > might related to the Gauss-Jacobi quadrature failing to retrieve the > correct roots of a Jacobi polynomial. > > Do anyone have a solution for that? > > Thank you > > Jacopo Here's one way to get them in Alpha. http://www.wolframalpha.com/input/?i=numerical+solutions+to+JacobiP%5B25%2C+-1%2F2%2C+-1%2F2%2C+x%5D+%3D%3D+0 Sam