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Roots of a Jacobi polynomial
*To*: mathgroup at smc.vnet.net
*Subject*: [mg120503] Roots of a Jacobi polynomial
*From*: Jacopo Bertolotti <J.Bertolotti at utwente.nl>
*Date*: Tue, 26 Jul 2011 07:07:46 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
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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