legendre_product

legendre_product, an Octave code 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

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

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)

Licensing:

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

Languages:

legendre_product is available in a MATLAB version and an Octave version.

Related Data and Programs:

legendre_product_test

hermite_polynomial, an Octave code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

legendre_rule, an Octave code which computes a Gauss-Legendre quadrature rule.

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

quad_rule, an Octave code 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:

• legendre_exponential_product.m, computes a table of exponential weighted Legendre polynomial products.
• legendre_linear_product.m, computes a table of linearly weighted Legendre polynomial products.
• legendre_polynomial.m, evaluates the Legendre polynomials at a point.
• legendre_quadrature_rule.m, computes a Legendre 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.