Applications of Anti-Gauss Quadrature Rules in Linear Algebra
Daniela Calvetti, Lothar Reichel, Fiorella Sgallari · Birkhäuser Basel eBooks · 1999
The need to inexpensively determine upper and lower bounds for matrix functionals of the form F(A):= u T f (A) u arises in a large number of applications. Here A denotes a large symmetric matrix and u is a vector. Golub and collaborators have described how such bounds can be computed by using Gauss and Gauss-Radau quadrature rules when the derivatives $$ \frac{{{d^j}}}{{d{t^j}}}f\left( t \right),$$ j = 1, 2,…, 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 derivates 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 Laurie. We also discuss applications to matrix functionals with nonsymmetric matrices.