JACOBI_POLYNOMIAL
Jacobi Polynomials

JACOBI_POLYNOMIAL is a C++ 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 C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

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

CHEBYSHEV_POLYNOMIAL, a C++ 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.

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

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

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

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

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

POLPAK, a C++ library which evaluates a variety of mathematical functions.

TEST_VALUES, a C++ 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:

• jacobi_polynomial.cpp, the source code.
• jacobi_polynomial.hpp, the source code.
• jacobi_polynomial.sh, BASH commands to compile the source code.

Examples and Tests:

• jacobi_polynomial_prb.cpp, a sample calling program.
• jacobi_polynomial_prb.sh, BASH commands to compile and run the sample program.
• jacobi_polynomial_prb_output.txt, the output file.

List of Routines:

• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• J_DOUBLE_PRODUCT_INTEGRAL: integral of J(i,x)*J(j,x)*(1-x)^a*(1+x)^b.
• J_INTEGRAL evaluates a monomial integral associated with J(n,a,b,x).
• J_POLYNOMIAL evaluates the Jacobi polynomials J(n,a,b,x).
• J_POLYNOMIAL_VALUES returns some values of the Jacobi polynomial.
• J_POLYNOMIAL_ZEROS: zeros of Jacobi polynomial J(n,a,b,x).
• J_QUADRATURE_RULE: Gauss-Jacobi quadrature based on J(n,a,b,x).
• R8_FACTORIAL computes the factorial of N.
• R8_SIGN returns the sign of an R8.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8VEC_PRINT prints an R8VEC.
• R8VEC2_PRINT prints an R8VEC2.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.