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.
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.
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:
-
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:
Examples and Tests:
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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the C++ source codes.
Last revised on 11 March 2012.