laguerre_polynomial

laguerre_polynomial, an Octave code 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 information on this web page is distributed under the MIT license.

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 and an Octave version and a Python version.

Related Data and Programs:

laguerre_polynomial_test

companion_matrix, an Octave code which computes the companion matrix for a polynomial. The polynomial may be represented in the standard monomial basis, or as a sum of Chebyshev, Gegenbauer, Hermite, Laguerre, or Lagrange basis polynomials. All the roots of the polynomial can be determined as the eigenvalues of the corresponding companion matrix.

gen_laguerre_rule, an Octave code which computes and prints a generalized Gauss-Laguerre quadrature rule.

octave_polynomial, an Octave code which analyzes a variety of polynomial families, returning the polynomial values, coefficients, derivatives, integrals, roots, or other information.

polpak, an Octave code which evaluates a variety of mathematical functions.

polynomial_conversion, an Octave code which converts representations of a polynomial between monomial, Bernstein, Chebyshev, Hermite, Lagrange, Laguerre and other forms.

test_values, an Octave code 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:

• imtqlx.m, diagonalizes a symmetric tridiagonal matrix.
• l_exponential_product.m, exponential product table for L(n,x).
• l_integral.m, evaluates a monomial integral associated with L(n,x).
• l_polynomial.m, evaluates the Laguerre polynomial L(n,x).
• l_polynomial_coefficients.m, coefficients of the Laguerre polynomial L(n,x).
• l_polynomial_values.m, some values of the Laguerre polynomial L(n,x).
• l_polynomial_zeros.m, zeros of the Laguerre polynomial L(n,x).
• l_power_product.m, power product table for L(n,x).
• l_quadrature_rule.m, Gauss-Laguerre quadrature based on L(n,x).
• lf_integral.m, evaluates a monomial integral associated with Lf(n,alpha,x).
• lf_function.m, evaluates the Laguerre function Lf(n,alpha,x).
• lf_function_values.m, returns values of the Laguerre function Lf(n,alpha,x).
• lf_function_zeros.m, returns the zeros of Lf(n,alpha,x).
• lf_quadrature_rule.m, Gauss-Laguerre quadrature rule for Lf(n,alpha,x);
• lm_integral.m, evaluates a monomial integral associated with Lm(n,m,x).
• lm_polynomial.m, evaluates Laguerre polynomials Lm(n,m,x).
• lm_polynomial_coefficients.m, coefficients of Laguerre polynomial Lm(n,m,x).
• lm_polynomial_values.m, returns values of Laguerre polynomials Lm(n,m,x).
• lm_polynomial_zeros.m, returns the zeros for Lm(n,m,x).
• lm_quadrature_rule.m, Gauss-Laguerre quadrature rule for Lm(n,m,x);
• r8_sign.m, returns the sign of an R8.
• r8mat_print.m, prints an R8MAT.
• r8mat_print_some.m, prints some of an R8MAT.
• r8vec_print.m, prints an R8VEC.
• r8vec2_print.m, prints a pair of R8VEC's.