LAGUERRE_POLYNOMIAL\
Laguerre Polynomials {#laguerre_polynomial-laguerre-polynomials align=”center”}
=====================
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.
Licensing: {#licensing align=”center”}
The computer code and data files described and made available on this
web page are distributed under the GNU LGPL
license.
Languages: {#languages align=”center”}
LAGUERRE_POLYNOMIAL is available in a C
version and
a C++
version and
a FORTRAN77
version and
a FORTRAN90
version and
a MATLAB
version.
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.
GEN_LAGUERRE_RULE,
a C++ program which can compute and print a generalized Gauss-Laguerre
quadrature rule.
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_RULE, a C++
program which can compute and print a Gauss-Laguerre quadrature rule.
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: {#reference align=”center”}
- Theodore Chihara,\
An Introduction to Orthogonal Polynomials,\
Gordon and Breach, 1978,\
ISBN: 0677041500,\
LC: QA404.5 C44.
- Walter Gautschi,\
Orthogonal Polynomials: Computation and Approximation,\
Oxford, 2004,\
ISBN: 0-19-850672-4,\
LC: QA404.5 G3555.
- Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,\
NIST Handbook of Mathematical Functions,\
Cambridge University Press, 2010,\
ISBN: 978-0521192255,\
LC: QA331.N57.
- Gabor Szego,\
Orthogonal Polynomials,\
American Mathematical Society, 1992,\
ISBN: 0821810235,\
LC: QA3.A5.v23.
Source Code: {#source-code align=”center”}
Examples and Tests: {#examples-and-tests align=”center”}
List of Routines: {#list-of-routines align=”center”}
- 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.
You can go up one level to the C++ source codes.
Last revised on 11 March 2012.