LAGUERRE_PRODUCT
Weighted Integrals of Laguerre Polynomial Products

LAGUERRE_PRODUCT, a MATLAB library which computes weighted integrals of products of Laguerre polynomials.

LAGUERRE_LINEAR_PRODUCT takes as input the maximum degree P and the (nonnegative integer) exponent E and computes

Tij = Integral ( 0 <= X < +oo ) X^E * L(i)(X) * L(j)(X) * exp(-X) dx

LAGUERRE_EXPONENTIAL_PRODUCT takes as input the maximum degree P and the coefficient B and computes

Tij = Integral ( 0 <= X < +oo ) exp(B*X) * L(i)(X) * L(j)(X) * exp(-X) dx

When the polynomial chaos expansion is used to study stochastic differential equations, it is a common task to have to form and compute integrals of the sort considered here.

Note that, because of the orthonormality of the Laguerre polynomials, LAGUERRE_LINEAR_PRODUCT will return the identity matrix when E=0, and LAGUERRE_EXPONENTIAL_PRODUCT will return the identity matrix when B=0.

What is more interesting is that, because of the recursion relationship

i*L(i+1)(X) = (2*i-1 - X) * L(i)(X) -(i-1)*L(i-1)(X)

Licensing:

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

Languages:

laguerre_product is available in a MATLAB version.

Related Data and Programs:

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

laguerre_product_test

LAGUERRE_RULE, a MATLAB program which computes a Gauss-Laguerre quadrature rule.

LEGENDRE_PRODUCT, a MATLAB program which computes weighted integrals of products of Legendre polynomials.

POLPAK, a MATLAB library which evaluates a variety of mathematical functions.

QUADRULE, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions.

Reference:

1. Milton Abramowitz, Irene Stegun,
   Handbook of Mathematical Functions,
   National Bureau of Standards, 1964,
   ISBN: 0-486-61272-4,
   LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
3. Daniel Zwillinger, editor,
   CRC Standard Mathematical Tables and Formulae,
   30th Edition,
   CRC Press, 1996,
   ISBN: 0-8493-2479-3,
   LC: QA47.M315.

Source Code:

• laguerre_exponential_product.m, computes a table of exponential weighted Laguerre polynomial products.
• laguerre_linear_product.m, computes a table of linearly weighted Laguerre polynomial products.
• laguerre_polynomial.m, evaluates the Laguerre polynomials at a point.
• laguerre_quadrature_rule.m, computes a Laguerre quadrature rule.
• r8mat_print.m, prints an R8MAT, with an optional title;
• r8mat_print_some.m, prints some of an R8MAT;
• s_len_trim.m, returns the length of a string to the last nonblank.
• timestamp.m, prints the current YMDHMS date as a time stamp.