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 )
      

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) )
      

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) )
      

Licensing:

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

Languages:

LAGUERRE_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 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.

Source Code:

• laguerre_polynomial.c, the source code.
• laguerre_polynomial.h, the include file.

Examples and Tests:

• laguerre_polynomial_test.c, a sample calling program.
• laguerre_polynomial_test.txt, the output file.

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).
• L_QUADRATURE_RULE: Gauss-Laguerre quadrature based on 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).
• LF_QUADRATURE_RULE: Gauss-Laguerre quadrature rule for 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).
• LM_QUADRATURE_RULE: Gauss-Laguerre quadrature rule for Lm(n,m,x);
• R8_ABS returns the absolute value of an R8.
• R8_ADD adds two R8's.
• 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.

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