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Re: Roots of a Jacobi polynomial

  • To: mathgroup at smc.vnet.net
  • Subject: [mg120521] Re: Roots of a Jacobi polynomial
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Wed, 27 Jul 2011 06:13:03 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • Reply-to: hanlonr at cox.net

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.Bertolotti 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





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