A symbolic computation procedure for the generation of Gauss quadrature rules with a user-defined weight function
✍ Scribed by Maucher, Roland
- Publisher
- John Wiley and Sons
- Year
- 1996
- Tongue
- English
- Weight
- 222 KB
- Volume
- 12
- Category
- Article
- ISSN
- 1069-8299
No coin nor oath required. For personal study only.
✦ Synopsis
The advance of powerful software for symbolic and numerical computations such as Mathernatica sheds a new light on a paper by Golub and Welsch from 1969. Based on this paper the author describes a Mathernatica procedure for determining the weights and abscissae of a Gauss quadrature rule with a userdefined weight function. After a brief description of the algorithm and its implementation examples demonstrate the usefulness of the procedure. The procedure is extremely useful if one has to compute many integrals with the same, possibly weakly singular, weight function. This might happen, for example, in the boundary element method. KEY WORDS numerical integration; Gauss quadrature rules
GOLUB AND WELSCH'S ALGORITHM
We want to compute the value of an integral by a weighted sum of N function values:
The weights wi and abscissae xi are chosen in such a way that the quadrature rule (1) is exact if f(x) is a polynomial of degree less or equal to 2N -1. A construction of a Gauss quadrature rule is only possible if w ( x ) 2 0 and w ( x ) x k dx < 00, k = 0, 1, 2, . . . . For some standard weight function w(x) the weights w , and the abscissae xi are known explicitly and tabled, but unfortunately from time to time there arises the need for a quadrature rule with a non-standard weight function w ( x ) . w ( x ) x k dx ~j (XI dx = d i j All the roots of p N are distinct and lie inside [a, b ] . They are the abscissae for the N-point Gauss rule. The w-orthonormal polynomials p i ( x ) satisfy a three-term recurrence relationship: pj(x) = (ajx + bj)pj-, (x) -~, p ~-~( x ) , j = 1,2, ... where po = 0, p , = 1 (2)
where the numbers aj, bj and cj are in general unknown.'