orthpoly::jacobi

The Jacobi polynomials

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

orthpoly::jacobi(n, a, b, x)

Description

orthpoly::jacobi(n,a,b,x) computes the value of the n-th degree Jacobi polynomial with parameters a and b 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::jacobi(2, a, b, x)

orthpoly::jacobi(3, 4, 5, x[1])

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

orthpoly::jacobi(2, 4, b, 6*x)

orthpoly::jacobi(2, 0, I, x[1] + 2)

"Arithmetical expressions" include numbers:

orthpoly::jacobi(2, 1/2, -1/2, sqrt(2)),
orthpoly::jacobi(3, 2, 5, 8 + I),
orthpoly::jacobi(1000, 1, 2, 0.3);

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

orthpoly::jacobi(n, a, b, x)

Example 2

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

orthpoly::jacobi(100, 1/2, 3/2, 0.9)

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:

P100 := orthpoly::jacobi(100, 1/2, 3/2, x):
evalp(P100, x = 0.9)

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

DIGITS := 30: evalp(P100, x = 0.9)

delete P100, DIGITS:

Parameters

n

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

a, b

Arithmetical expressions.

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 Jacobi polynomial at this point is returned as an arithmetical expression. If n is an arithmetical expression, then orthpoly::jacobi returns itself symbolically.

Algorithms

The Jacobi polynomials are given by the recursion formula

with ci = i + a + b and

.

For fixed real a > - 1, b > - 1 the Jacobi polynomials are orthogonal on the interval [- 1, 1] with respect to the weight function w(x) = (1 - x)a (1 + x)b.

For special values of the parameters a, b the Jacobi polynomials are related to the Legendre polynomials

,

to the Chebyshev polynomials of the first kind

,

to the Chebyshev polynomials of the second kind

,

and to the Gegenbauer polynomials, respectively:

.

Was this topic helpful?