# LAGUERRE_POLYNOMIAL Laguerre Polynomials

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

The Laguerre polynomial L(n,x) can be defined by:

```        L(n,x) = exp(x)/n! * d^n/dx^n ( exp(-x) * x^n )
```
where n is a nonnegative integer.

The generalized Laguerre polynomial Lm(n,m,x) can be defined by:

```        Lm(n,m,x) = exp(x)/(x^m*n!) * d^n/dx^n ( exp(-x) * x^(m+n) )
```
where n and m are nonnegative integers.

The Laguerre function can be defined by:

```        Lf(n,alpha,x) = exp(x)/(x^alpha*n!) * d^n/dx^n ( exp(-x) * x^(alpha+n) )
```
where n is a nonnegative integer and -1.0 < alpha is a real number.

### Languages:

LAGUERRE_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 evaluates the Chebyshev polynomial and associated functions.

GEGENBAUER_POLYNOMIAL, a C library which evaluates the Gegenbauer polynomial and associated functions.

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

JACOBI_POLYNOMIAL, a C library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_EXACTNESS, a C program which tests the polynomial exactness of Gauss-Laguerre quadrature rules for integration over [0,+oo) with density function exp(-x).

LAGUERRE_RULE, a C program which can compute and print a Gauss-Laguerre quadrature rule for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

LAGUERRE_TEST_INT, a C library which defines test integrands for integration over [A,+oo).

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

LEGENDRE_SHIFTED_POLYNOMIAL, a C library which evaluates the shifted Legendre polynomial, with domain [0,1].

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.

### List of Routines:

• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the minimum of two I4's.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• L_EXPONENTIAL_PRODUCT: exponential product table for L(n,x).
• L_INTEGRAL evaluates a monomial integral associated with L(n,x).
• L_POLYNOMIAL evaluates the Laguerre polynomials L(n,x).
• L_POLYNOMIAL_COEFFICIENTS: coeffs for Laguerre polynomial L(n,x).
• L_POLYNOMIAL_VALUES returns some values of the Laguerre polynomial L(n,x).
• L_POLYNOMIAL_ZEROS: zeros of the Laguerre polynomial L(n,x).
• L_POWER_PRODUCT: power product table for L(n,x).
• LF_FUNCTION evaluates the Laguerre function Lf(n,alpha,x).
• LF_FUNCTION_VALUES: some values of the Laguerre function Lf(n,alpha,x).
• LF_FUNCTION_ZEROS returns the zeros of Lf(n,alpha,x).
• LF_INTEGRAL evaluates a monomial integral associated with Lf(n,alpha,x).
• LM_INTEGRAL evaluates a monomial integral associated with Lm(n,m,x).
• LM_POLYNOMIAL evaluates Laguerre polynomials Lm(n,m,x).
• LM_POLYNOMIAL_COEFFICIENTS: coefficients of Laguerre polynomial Lm(n,m,x).
• LM_POLYNOMIAL_VALUES: some values of the Laguerre polynomial Lm(n,m,x).
• LM_POLYNOMIAL_ZEROS returns the zeros for Lm(n,m,x).
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_FACTORIAL computes the factorial of N.
• R8_GAMMA evaluates Gamma(X) for an R8.
• R8_SIGN returns the sign of an R8.
• R8MAT_PRINT prints an R8MAT.
• R8MAT_PRINT_SOME prints some of an R8MAT.
• R8VEC_DOT_PRODUCT computes the dot product of a pair of R8VEC's.
• 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.

Last revised on 11 August 2013.