Summary: The need to inexpensively determine upper and lower bounds for matrix functionals of the form \(F(A):= u^Tf(A)u\) arises in a large number of applications. Here \(A\) denotes a large symmetric matrix and \(u\) is a vector. G. Golub and collaborators have described how such bounds can be computed by using Gauss and Gauss-Radau quadrature rules when the derivatives \({d^d\over dt^j}f(t)\), \(j= 1,2,\dots\), are of constant sign in an interval that contains the spectrum of \(A\). However, many matrix functionals of interest in applications are defined by functions \(f\) whose derivatives do not have constant sign on the spectrum of \(A\).
We describe a new method for inexpensively computing candidates for upper and lower bounds of \(F(A)\) based on the application of pairs of Gauss and anti-Gauss quadrature rules. This method does not require the sign of the derivatives of \(f\) to be constant on an interval that contains the spectrum of \(A\). Anti-Gauss rules are modifications of Gauss rules recently introduced by D. P. Laurie [Math. Comput. 65, No. 214, 739-747 (1996; Zbl 0843.41020)]. We also discuss applications to matrix functionals with nonsymmetric matrices.
For the entire collection see [Zbl 0919.00065].