jacobiP

Jacobi polynomials

Syntax

jacobiP(n,a,b,x)

Description

jacobiP(n,a,b,x) returns the nth degree Jacobi polynomial with parameters a and b at x.

Examples

Find Jacobi Polynomials for Numeric and Symbolic Inputs

Find the Jacobi polynomial of degree 2 for numeric inputs.

jacobiP(2,0.5,-3,6)

ans =
    7.3438

Find the Jacobi polynomial for symbolic inputs.

syms n a b x
jacobiP(n,a,b,x)

ans =
jacobiP(n, a, b, x)

If the degree of the Jacobi polynomial is not specified, jacobiP cannot find the polynomial and returns the function call.

Specify the degree of the Jacobi polynomial as 1 to return the form of the polynomial.

J = jacobiP(1,a,b,x)

J =
a/2 - b/2 + x*(a/2 + b/2 + 1)

To find the numeric value of a Jacobi polynomial, call jacobiP with the numeric values directly. Do not substitute into the symbolic polynomial because the result can be inaccurate due to round-off. Test this by using subs to substitute into the symbolic polynomial, and compare the result with a numeric call.

J = jacobiP(300, -1/2, -1/2, x);
subs(J,x,vpa(1/2))
jacobiP(300, -1/2, -1/2, vpa(1/2))

ans =
101573673381249394050.64541318209
ans =
0.032559931334979678350422392588404

When subs is used to substitute into the symbolic polynomial, the numeric result is subject to round-off error. The direct numerical call to jacobiP is accurate.

Find Jacobi Polynomial with Vector and Matrix Inputs

Find the Jacobi polynomials of degrees 1 and 2 by setting n = [1 2] for a = 3 and b = 1.

syms x
jacobiP([1 2],3,1,x)

ans =
[ 3*x + 1, 7*x^2 + (7*x)/2 - 1/2]

jacobiP acts on n element-wise to return a vector with two entries.

If multiple inputs are specified as a vector, matrix, or multidimensional array, these inputs must be the same size. Find the Jacobi polynomials for a = [1 2;3 1], b = [2 2;1 3], n = 1 and x.

a = [1 2;3 1];
b = [2 2;1 3];
J = jacobiP(1,a,b,x)

J =
[ (5*x)/2 - 1/2,     3*x]
[       3*x + 1, 3*x - 1]

jacobiP acts element-wise on a and b to return a matrix of the same size as a and b.

Visualize Zeros of Jacobi Polynomials

Plot Jacobi polynomials of degree 1, 2, and 3 for a = 3, b = 3, and -1<x<1. To better view the plot, set axis limits by using axis.

syms x
fplot(jacobiP(1:3,3,3,x))
axis([-1 1 -2 2])
grid on

ylabel('P_n^{(\alpha,\beta)}(x)')
title('Zeros of Jacobi polynomials of degree=1,2,3 with a=3 and b=3');
legend('1','2','3','Location','best')

Prove Orthogonality of Jacobi Polynomials with Respect to Weight Function

The Jacobi polynomials P(n,a,b,x) are orthogonal with respect to the weight function $${\left(1-x\right)}^a{\left(1-x\right)}^b$$ on the interval [-1,1].

Prove P(3,a,b,x) and P(5,a,b,x) are orthogonal with respect to the weight function $${\left(1-x\right)}^a{\left(1-x\right)}^b$$ by integrating their product over the interval [-1,1], where a = 3.5 and b = 7.2.

syms x
a = 3.5;
b = 7.2;
P3 = jacobiP(3, a, b, x);
P5 = jacobiP(5, a, b, x);
w = (1-x)^a*(1+x)^b;
int(P3*P5*w, x, -1, 1)

ans =
0

Input Arguments

n — Degree of Jacobi polynomial

nonnegative integer | vector of nonnegative integers | matrix of nonnegative integers | multidimensional array of nonnegative integers | symbolic nonnegative integer | symbolic variable | symbolic vector | symbolic matrix | symbolic function | symbolic expression | symbolic multidimensional array

Degree of Jacobi polynomial, specified as a nonnegative integer, or a vector, matrix, or multidimensional array of nonnegative integers, or a symbolic nonnegative integer, variable, vector, matrix, function, expression, or multidimensional array.

a — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic expression | symbolic multidimensional array

Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, expression, or multidimensional array.

b — Input

number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic expression | symbolic multidimensional array

Input, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, expression, or multidimensional array.

x — Evaluation point

number | vector | matrix | multidimensional array | symbolic number | symbolic vector | symbolic matrix | symbolic function | symbolic expression | symbolic multidimensional array

Evaluation point, specified as a number, vector, matrix, multidimensional array, or a symbolic number, vector, matrix, function, expression, or multidimensional array.

More About

Jacobi Polynomials

• The Jacobi polynomials are given by the recursion formula

  $$\begin{array}{l}2n{c}_n{c}_{2n-2}P\left(n,a,b,x\right)=c_{2n-1}\left(c_{2n-2}{c}_{2n}x+a^2-b^2\right)P\left(n-1,a,b,x\right)\\ -2\left(n-1+a\right)\left(n-1+b\right)c_{2n}P\left(n-2,a,b,x\right),\\ \text{where}\\ c_n=n+a+b\\ P\left(0,a,b,x\right)=1\\ P\left(1,a,b,x\right)=\frac{a-b}{2}+\left(1+\frac{a+b}{2}\right)x.\end{array}$$

• For fixed real a > -1 and b > -1, the Jacobi polynomials are orthogonal on the interval [-1,1] with respect to the weight function $$w\left(x\right)={\left(1-x\right)}^a{\left(1+x\right)}^b$$.

  $${\displaystyle \underset{-1}{\overset{1}{\int }}P(n,a,b,x)P(m,a,b,x){\left(1-x\right)}^a{\left(1+x\right)}^{b}dx}=\{\begin{array}{cc}0& \text{if }n\ne m\\ \frac{2^{a+b+1}}{2n+a+b+1}\frac{\Gamma \left(n+a+1\right)\Gamma \left(n+b+1\right)}{\Gamma \left(n+a+b+1\right)n!}& \text{if }n=m.\end{array}$$

• For a = 0 and b = 0, the Jacobi polynomials P(n,0,0,x) reduce to the Legendre polynomials P(n, x).

• The relation between Jacobi polynomials P(n,a,b,x) and Chebyshev polynomials of the first kind T(n,x) is

  $$T\left(n,x\right)=\frac{2^{2n}{\left(n!\right)}^2}{\left(2n\right)!}P\left(n,-\frac{1}{2},-\frac{1}{2},x\right).$$

• The relation between Jacobi polynomials P(n,a,b,x) and Chebyshev polynomials of the second kind U(n,x) is

  $$U\left(n,x\right)=\frac{2^{2n}n!\left(n+1\right)!}{\left(2n+1\right)!}P\left(n,\frac{1}{2},\frac{1}{2},x\right).$$

• The relation between Jacobi polynomials P(n,a,b,x) and Gegenbauer polynomials G(n,a,x) is

  $$G\left(n,a,x\right)=\frac{\Gamma \left(a+\frac{1}{2}\right)\Gamma \left(n+2a\right)}{\Gamma \left(2a\right)\Gamma \left(n+a+\frac{1}{2}\right)}P\left(n,a-\frac{1}{2},a-\frac{1}{2},x\right).$$

Version History

Introduced in R2014b