# JACOBI_POLYNOMIAL Jacobi Polynomials

JACOBI_POLYNOMIAL is a FORTRAN77 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
```
That is, this integral is 0 unless i = j. J(i,a,b,x) indicates the Jacobi polynomial of degree i.

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;

### 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 FORTRAN77 library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a FORTRAN77 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 FORTRAN77 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

JACOBI_RULE, a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Laguerre polynomial, the generalized Laguerre polynomials, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a FORTRAN77 library which evaluates the Legendre polynomial and associated functions.

POLPAK, a FORTRAN77 library which evaluates a variety of mathematical functions.

TEST_VALUES, a FORTRAN77 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.

### 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).