×

Ralstons Quadratur. (German) Zbl 0574.65010

A. Ralston [J. Assoc. Comput. Mach. 6, 384-394 (1959; Zbl 0117.114)] introduced and analyzed a class of quadratures \[ Q_ nf:=a_{on}[f(-1)-f(1)]+\sum^{n}_{i=1}a_{in}f(t_{in}),\quad n\in {\mathbb{N}},\quad f\in C[-1,1], \] which are exact for polynomials of degree 2n or less. Obviously, these formulas are particularly useful when the quadrature is to be done by subdividing the interval [-1,1] into a number of subintervals because the ordinates at the end of each subinterval drop out except for those at the end of the complete interval. Ralston derived estimates for the remainder term and computed the weights \(a_{on}\), \(a_{in}\) and knots \(t_{in}\), \(i=1(1)n\), for \(n=1\) and \(n=2\). Now a more suitable form of the remainder is derived showing that the quadrature is definite. Furthermore, explicit computable expressions for the weights are given, and the polynomial with the roots \(t_{in}\) is characterized in terms of Gauss-Jacobi polynomials. Details of the proofs and the underlying general concept to analyze quadratures may be found in the author’s Habilitationsschrift.

MSC:

65D32 Numerical quadrature and cubature formulas
41A55 Approximate quadratures

Citations:

Zbl 0117.114
PDFBibTeX XMLCite