2-norm error bounds and estimates for Lanczos approximations to linear systems and rational matrix functions (Q2866223)

The focus of the paper is on bounding the 2-norm error when applying an iterative Lanczos procedure to compute \(f(A)b\) with \(A\) a positive definite matrix, \(b\) a vector and \(f\) a function defined on the positive real axis. In particular, if \(f\) is a rational function with poles outside the spectrum of \(A\), a multishift version of Lanczos gives easy access to the 2-norm of the error that can be represented as an integral of a known rational function that can be approximated by a Gauss-type (Gauss, Gauss-Radau or Gauss-Lobatto) quadrature with \(k\) nodes. The necessary quantities for the quadrature can be obtained from a restarted Lanczos, or, as is proved here, can be obtained from the trailing part of size \(2k+1\) of the tridiagonal matrix generated by the Lanczos process. This gives a cheap way of estimating (upper and lower bound) the 2-norm of the error, especially useful as a stopping criterion. A comparison with alternative methods and numerical experiments are included.