LEGENDRE_PRODUCT Weighted Integrals of Legendre Polynomial Products

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

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

Tij = Integral ( -1 <= X <= +1 ) X^E * L(i)(X) * L(j)(X) dx
for I and J between 0 and P.

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

Tij = Integral ( -1 <= X <= +1 ) exp(B*X) * L(i)(X) * L(j)(X) dx
for I and J between 0 and P.

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 Legendre polynomials, LEGENDRE_LINEAR_PRODUCT will return the identity matrix when E=0, and LEGENDRE_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)
LEGENDRE_LINEAR_PRODUCT will return a symmetric tridiagonal matrix (with zero diagonal) when E=1.

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.

LEGENDRE_RULE, a MATLAB program which computes a Gauss-Legendre quadrature rule.

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

pce_legendre, a MATLAB program which assembles the system matrix associated with a polynomal chaos expansion of a 2D stochastic PDE, using Legendre polynomials;

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.