The Gegenbauer (ultraspherical) polynomials
This functionality does not run in MATLAB.
orthpoly::gegenbauer(n, a, x)
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.
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)
However, using arithmetical expressions as input the "values" of these polynomials are returned:
orthpoly::gegenbauer(2, a, 6*x)
orthpoly::gegenbauer(3, 2, x + 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 no integer degree is specified, then orthpoly::gegenbauer returns itself symbolically:
orthpoly::gegenbauer(n, a, x), orthpoly::gegenbauer(1/2, 2, x)
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:
A nonnegative integer: the degree of the polynomial.
An arithmetical expression.
An indeterminate or an arithmetical expression. An indeterminate is either an identifier (of domain type DOM_IDENT) or an indexed identifier (of type "_index").
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 not a nonnegative integer, then orthpoly::gegenbauer returns itself symbolically.
The Gegenbauer polynomials are given by the recursion formula
with G(0, a, x) = 1, G(1, a, x) = 2 a x.
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.