Any positive residual history is possible for the Arnoldi method for Lyapunov matrix equations

Abstract

In this paper we consider the Lyapunov equation AX+XA^T+bb^T = 0, where A is negative definite n by n matrix and b in R^n. The Arnoldi method is an iterative algorithm which can be used to compute an approximate solution. However, the convergence can be very slow and in this paper we show how to explicitly construct a Lyapunov equation with a given residual curve. The matrix A can be chosen as symmetric negative definite and it is possible to arbitrarily specify the elements on the diagonal of the Cholesky factor of -A. If the symmetry is dropped, then it is possible to arbitrarily specify A+A^T, while retaining the residual curve