Some theoretical aspects of generalised quadrature methods
Given a weight function in the class Cn[a, b], a generalised quadrature method can be found using Lagrange’s identity. The process involves forcing the integral of Lagrange’s identity to be exact for a set of basis functions, so obtaining a set of linear algebraic equations. In some earlier papers, this process was found to be very effective on a range of test examples with irregular oscillations, and the accuracy of the resulting quadrature rule appeared to remain high despite the condition factor for the underlying linear equations suggesting otherwise. In this paper the two obvious implementations are shown to be equivalent and under fairly general conditions the accuracy of the quadrature rule remains machine precision ε as long as the condition factor κ of the underlying matrix remains less than 1/ε. These effects are demonstrated using various weights including J100(130 cos x), and ln x on an infinite range. This work provides some theoretical/foundational basis that underpins earlier works on generalised quadrature methods.
Citation : Evans, G.A. and Chung, K.C. (2003) Some theoretical aspects of generalised quadrature methods. Journal of Complexity, 19(3), pp. 272-285.
ISSN : 1090-2708
Research Group : Scientific Computation