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15 Orthogonal Polynomials: General Aspects
15.1 Function Overview
polymeth The SPECFUNPHYS superclass polymeth comes with some methods
common for several orthogonal polynomials. The class orthpoly creates
from an arbitrary orthogonal polynomial an object, e.g., to use the methods of
polymeth.
Jacobi polynomial The SPECFUNPHYS class jacobipoly computes the polynomial coefficients of Jacobi polynomials and the class jacobix evaluates
directly without detours about polynomial coefficients the polynomial values.
The SPECFUNPHYS function jacobizero computes the zeros of a Jacobi
polynomial.
Gegenbauer polynomial The SPECFUNPHYS class gegenbauerpoly computes the polynomial coefficients of the Gegenbauer or ultraspherical polynomials. The class gegenbauerx directly evaluates the polynomial values.
The SPECFUNPHYS function gegenbauerzero computes the roots of the
Gegenbauer polynomial.
15.2 General Definitions
Orthogonal polynomials are playing a crucial rôle in physics under analytical and
computational aspects, with many applications, e.g., in quantum- and electrodynamics. The polynomial structure uncovers rather analytical aspects, 1 but are not
sufficiently computational robust for higher orders. Therefore we will compute both,
the polynomial structure and directly the polynomial values.
Orthogonal polynomials can be produced by starting with 1, x, x 2 , · · · and
employing the Gram–Schmidt orthogonalization process. However, although this
is quite general, we took a more elegant approach that simultaneously uncovers
aspects of interest to physicists. The following definitions, lemmas, and theorems
can be found in [1, 4], applications with respect to quantum dynamics, e.g., in [3].
Also the following discussion is restricted to real variables the polynomials can
be continued into the complex domain. In this section we will denote by [a, b] an
interval of the real axis, by w(x) a weight function for [a, b], and by φ i (x) an
orthogonal polynomial of degree i. A function w(x) is called a weight function
for [a, b] if and only if
w(x) ≥ 0 on [a, b]
and
b
a
x
n w(x)dx = μ n
exists and remains finite for each n = 0, 1, 2, . . . and μ 0 > 0 . The numbers
μ n defined above are called moments of w(x). The polynomials {φ i (x), i =
1 To uncover the polynomial structure in the corresponding arrays use the MATLAB format rat via
>> format rat.
15 Orthogonal Polynomials: General Aspects
15.1 Function Overview
polymeth The SPECFUNPHYS superclass polymeth comes with some methods
common for several orthogonal polynomials. The class orthpoly creates
from an arbitrary orthogonal polynomial an object, e.g., to use the methods of
polymeth.
Jacobi polynomial The SPECFUNPHYS class jacobipoly computes the polynomial coefficients of Jacobi polynomials and the class jacobix evaluates
directly without detours about polynomial coefficients the polynomial values.
The SPECFUNPHYS function jacobizero computes the zeros of a Jacobi
polynomial.
Gegenbauer polynomial The SPECFUNPHYS class gegenbauerpoly computes the polynomial coefficients of the Gegenbauer or ultraspherical polynomials. The class gegenbauerx directly evaluates the polynomial values.
The SPECFUNPHYS function gegenbauerzero computes the roots of the
Gegenbauer polynomial.
15.2 General Definitions
Orthogonal polynomials are playing a crucial rôle in physics under analytical and
computational aspects, with many applications, e.g., in quantum- and electrodynamics. The polynomial structure uncovers rather analytical aspects, 1 but are not
sufficiently computational robust for higher orders. Therefore we will compute both,
the polynomial structure and directly the polynomial values.
Orthogonal polynomials can be produced by starting with 1, x, x 2 , · · · and
employing the Gram–Schmidt orthogonalization process. However, although this
is quite general, we took a more elegant approach that simultaneously uncovers
aspects of interest to physicists. The following definitions, lemmas, and theorems
can be found in [1, 4], applications with respect to quantum dynamics, e.g., in [3].
Also the following discussion is restricted to real variables the polynomials can
be continued into the complex domain. In this section we will denote by [a, b] an
interval of the real axis, by w(x) a weight function for [a, b], and by φ i (x) an
orthogonal polynomial of degree i. A function w(x) is called a weight function
for [a, b] if and only if
w(x) ≥ 0 on [a, b]
and
b
a
x
n w(x)dx = μ n
exists and remains finite for each n = 0, 1, 2, . . . and μ 0 > 0 . The numbers
μ n defined above are called moments of w(x). The polynomials {φ i (x), i =
1 To uncover the polynomial structure in the corresponding arrays use the MATLAB format rat via
>> format rat.