JACOBI_POLYNOMIAL
Jacobi Polynomials

JACOBI_POLYNOMIAL, 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 made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a MATLAB library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a MATLAB library which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

GEGENBAUER_POLYNOMIAL, a MATLAB library which evaluates the Gegenbauer 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_polynomial_test

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.

LOBATTO_POLYNOMIAL, a MATLAB library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

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.