Sharp estimates for the convergence rate of Orthomin(k) for a class of linear systems
In this work we show that the convergence rate of Orthomin($k$) applied to systems of the form $(I+ρU) x = b$, where $U$ is a unitary operator and $0<ρ<1$, is less than or equal to $ρ$. Moreover, we give examples of operators $U$ and $ρ>0$ for which the asymptotic convergence rate of Orthomin($k$) is exactly $ρ$, thus showing that the estimate is sharp. While the systems under scrutiny may not be of great interest in themselves, their existence shows that, in general, Orthomin($k$) does not converge faster than Orthomin(1). Furthermore, we give examples of systems for which Orthomin($k$) has the same asymptotic convergence rate as Orthomin(2) for $k\ge 2$, but smaller than that of Orthomin(1). The latter systems are related to the numerical solution of certain partial differential equations.