JACOBI_POLYNOMIAL
Jacobi Polynomials

JACOBI_POLYNOMIAL is a MATLAB library which evaluates the Jacobi polynomial.

For a given choice of the parameters a and b, both greater than -1, the Jacobi polynomials are a set of polynomials which are pairwise orthogonal with respect to the integral:

        integral (-1<=x<=+1) J(i,a,b,x) J(j,a,b,x) (1-x)^a (1+x)^b dx
      

The standard Jacobi polynomials can be defined by a three term recurrence formula that is a bit too ugly to quote here.

It is worth noting that the definition of the Jacobi polynomials is general enough that it includes some familiar families as special cases:

• if a = b = 0, we have the Legendre polynomials, P(n,x);
• if a = b = -1/2, we have the Chebyshev polynomials of the first kind, T(n,x);
• if a = b = 1/2, we have the Chebyshev polynomials of the second kind, U(n,x);
• if a = b, we have the Gegenbauer polynomials;

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

JACOBI_POLYNOMIAL is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CHEBYSHEV_POLYNOMIAL, a MATLAB library which evaluates the Chebyshev polynomial and associated functions.

HERMITE_POLYNOMIAL, a MATLAB library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

JACOBI_RULE, a MATLAB program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_POLYNOMIAL, a MATLAB library which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a MATLAB library which evaluates the Legendre polynomial and associated functions.

POLPAK, a MATLAB library which evaluates a variety of mathematical functions.

TEST_VALUES, a MATLAB library which supplies test values of various mathematical functions.

Reference:

1. Theodore Chihara,
   An Introduction to Orthogonal Polynomials,
   Gordon and Breach, 1978,
   ISBN: 0677041500,
   LC: QA404.5 C44.
2. Walter Gautschi,
   Orthogonal Polynomials: Computation and Approximation,
   Oxford, 2004,
   ISBN: 0-19-850672-4,
   LC: QA404.5 G3555.
3. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
   NIST Handbook of Mathematical Functions,
   Cambridge University Press, 2010,
   ISBN: 978-0521192255,
   LC: QA331.N57.
4. Gabor Szego,
   Orthogonal Polynomials,
   American Mathematical Society, 1992,
   ISBN: 0821810235,
   LC: QA3.A5.v23.

Source Code:

• imtqlx.m, diagonalizes a symmetric tridiagonal matrix;
• j_double_product_integral.m, integral of J(i,x)*J(j,x)*(1-x)^a*(1+x)^b.
• j_integral.m, evaluates a monomial Jacobi integral for J(n,a,b,x).
• j_polynomial.m, evaluates the Jacobi polynomial J(n,a,b,x).
• j_polynomial_plot.m, plots one or more Jacobi polynomials J(n,a,b,x) over [-1,+1].
• j_polynomial_values.m, a few tabulated values of the Jacobi polynomial J(n,a,b,x).
• j_polynomial_zeros.m, returns zeros of the Jacobi polynomial J(n,a,b,x).
• j_quadrature_rule.m, returns quadrature rules associated with the Jacobi polynomial J(n,a,b,x).
• r8_factorial.m, computes the factorial function;
• r8_sign.m, returns the sign of an R8.
• r8mat_print.m, prints an R8MAT;
• r8mat_print_some.m, prints some of an R8MAT;
• r8vec_print.m, prints an R8VEC;
• r8vec2_print.m, prints a pair of R8VEC's;
• timestamp.m, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• jacobi_polynomial_test.m, calls all the tests;
• jacobi_polynomial_test_output.txt, the output file.
• jacobi_polynomial_test01.m, tests j_polynomial;
• jacobi_polynomial_test02.m, tests j_polynomial_zeros;
• jacobi_polynomial_test03.m, tests j_quadrature_rule;
• jacobi_polynomial_test04.m, tests j_double_product_integral;

You can go up one level to the MATLAB source codes.