Simplified anti-Gauss quadrature rules with applications in linear algebra
The authors develop inexpensive techniques for the efficient computation of upper and lower bounds for expressions of the form \(F(A) := w^{\mathrm T} f(A) v\) where \(A\) is a large square \(n\)-by-\(n\) matrix, \(f\) is a function, and \(v, w \in \mathbb R^n\). In contrast to earlier works, there are no highly restrictive assumptions such as the symmetry of \(A\), \(v=w\), or the requirement that certain derivatives of \(f\) do not change sign. Also, the methods developed here do not require any knowledge about the spectrum of \(A\). The approach is based on rewriting \(F(A)\) in the form of a Stieltjes integral and on using a Gaussian quadrature rule associated to its distribution function in combination with a second quadrature rule to obtain the desired upper and lower bounds. The second rule is essentially an anti-Gaussian rule, slightly modified in order to significantly reduce the computational cost of its construction. The authors provide a set of relatively mild conditions under which one can prove that one of the formulas yields a lower bound for the required value and the other one gives an upper bound.
scientific article; zbMATH DE number 6836929
scientific article