orthpoly::gegenbauer

The Gegenbauer (ultraspherical) polynomials

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

orthpoly::gegenbauer(n, a, x)

Description

orthpoly::gegenbauer(n,a,x) computes the value of the n-th degree Gegenbauer polynomial with parameter a at the point x.

Evaluation for real floating-point values x from the interval [- 1.0, 1.0] is numerically stable. See Example 2.

Examples

Example 1

Polynomials of domain type DOM_POLY are returned, if identifiers or indexed identifiers are specified:

orthpoly::gegenbauer(2, a, x)

orthpoly::gegenbauer(3, 2, x[1])

However, using arithmetical expressions as input the "values" of these polynomials are returned:

orthpoly::gegenbauer(2, a, 6*x)

orthpoly::gegenbauer(3, 2, x[1] + 2)

"Arithmetical expressions" include numbers:

orthpoly::gegenbauer(2, a, sqrt(2)), 
orthpoly::gegenbauer(3, 0.4, 8 + I),
orthpoly::gegenbauer(1000, -1/3, 0.3)

If the degree of the polynomial is a variable or expression, then orthpoly::gegenbauer returns itself symbolically:

orthpoly::gegenbauer(n, a, x)

Example 2

If a floating-point value is desired, then a direct call such as

orthpoly::gegenbauer(200, 4, 0.3)

is appropriate and yields a correct result. One should not evaluate the symbolic polynomial at a floating-point value, because this may be numerically unstable:

G200 := orthpoly::gegenbauer(200, 4, x):
DIGITS := 10: evalp(G200, x = 0.3)

This result is caused by numerical round-off. Also with increased DIGITS only a few leading digits are correct:

DIGITS := 20: evalp(G200, x = 0.3)

delete DIGITS, G200:

Parameters

n

A nonnegative integer or an arithmetical expression representing a nonnegative integer: the degree of the polynomial.

a

An arithmetical expression.

x

An indeterminate or an arithmetical expression. An indeterminate is either an identifier (of domain type DOM_IDENT) or an indexed identifier (of type "_index").

Return Values

If x is an indeterminate, then a polynomial of domain type DOM_POLY is returned. If x is an arithmetical expression, then the value of the Gegenbauer polynomial at this point is returned as an arithmetical expression. If n is an arithmetical expression, then orthpoly::gegenbauer returns itself symbolically.

Algorithms

The Gegenbauer polynomials are given by the recursion formula

with G(0, a, x) = 1, G(1, a, x) = 2 ax.

For fixed real these polynomials are orthogonal on the interval [- 1, 1] with respect to the weight function .

coincides with the Legendre polynomial P(n, x).

G(n, 1, x) coincides with the Chebyshev polynomial U(n, x) of the second kind.

The polynomials G(n, 0, x) are trivial.

See Also

MuPAD Functions

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