Jacobi poly

jacobi_poly
Computes the coefficients of Jacobi polynomials
Other toolboxes required	none
Function category	Helper functions

jacobi_poly is a function that returns a vector containing the coefficients of the specified Jacobi polynomial.

Syntax

• JP = jacobi_poly(A,B,N)

Argument descriptions

• A: A real parameter (sometimes called alpha) of the Jacobi polynomials.
• B: A real parameter (sometimes called beta) of the Jacobi polynomials.
• N: The degree of the Jacobi polynomial (a non-negative integer).

Examples

The Jacobi polynomials are typically denoted by the notation \(P^{(\alpha,\beta)}_n\). In the \(\alpha = \beta = 1, n = 3\) case, we have \(P^{(1,1)}_3(z) = 7z^3 - 3z\), which we can see via the following code:

>> jacobi_poly(1,1,3)
 
ans =
 
     7     0    -3     0

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  JACOBI_POLY  Computes the coefficients of Jacobi polynomials

2. %   This function has three required arguments:

3. %     A: a real parameter (sometimes called alpha) of the Jacobi polynomials

4. %     B: a real parameter (sometimes called beta) of the Jacobi polynomials

5. %     N: the degree of the Jacobi polynomial (a non-negative integer)

6. %

7. %   JP = jacobi_poly(A,B,N) is an (N+1)-vector containing the coefficients

8. %   (starting with the coefficient of the highest-order term) of the Jacobi

9. %   polynomial P_N^{A,B}.

10. %

11. %   URL: http://www.qetlab.com/jacobi_poly

13. %   requires: nothing

14. %   author: Nathaniel Johnston (nathaniel@njohnston.ca)

15. %   package: QETLAB

16. %   last updated: March 6, 2013

18. function jp = jacobi_poly(a,b,n)

20.     if(n <= 0)

21.         jp = 1;

22.     elseif(n <= 1)

23.         jp = [a+b+2, a-b]/2;

24.     else

25.         prev_jp = jacobi_poly(a,b,n-1);

26.         jp = ((2*n+a+b-1)*(a^2-b^2)*[0,prev_jp] + (2*n+a+b-1)*(2*n+a+b)*(2*n+a+b-2)*[prev_jp,0] - 2*(n+a-1)*(n+b-1)*(2*n+a+b)*[0,0,jacobi_poly(a,b,n-2)]) / (2*n*(n+a+b)*(2*n+a+b-2));

27.     end

28. end