Extending Elman's bound for GMRES

If the numerical range of a matrix is contained in the open right half of the complex plane, the GMRES algorithm for solving linear systems will reduce the norm of the residual at every iteration. In his Ph.D. dissertation, Howard Elman derived a bound that guarantees convergence. When the numerical range contains the origin, GMRES need not make progress at every step and Elman's bound does not apply, even if all the eigenvalues have positive real part. By solving a Lyapunov equation, one can construct an inner product in which the numerical range is contained in the open right half-plane. One can then bound GMRES (run in the standard Euclidean norm) by applying Elman's bound (or most other GMRES bounds) in this new inner product, at the cost of a multiplicative constant that characterizes the distortion caused by the change of inner product. Using Lyapunov inverse iteration, one can build a family of such inner products, trading off the location of the numerical range with the size of constant. This approach complements techniques that Greenbaum and colleagues have recently proposed for excising the origin from the numerical range to gain greater insight into the convergence of GMRES for nonnormal matrices.