Two algorithms for the construction of product formulas (Q1207051)

A familiar problem in numerical quadrature is that of integration over a finite interval, with a specified (usually classical) weight function. The author here deals with a variant problem, which turns out to be quite different. In this application in integral equations, the weight function \(K(x,y)\) depends on a parameter \(y\); a family (one-parameter system) of quadrature formulas are required in which the abscissas are fixed, but the weights depend on the particular kernel, i.e., depend on \(y\). This leads to a family of interpolatory rules. The author discusses this problem and gives an outline of two algorithms. The first employs as abscissas the zeros of a Jacobi polynomial. The second involves a Kronrod type extension. She demonstrates, by example, how successful this approach can be and how much it costs. The FORTRAN programs are available from the author.