Sharp estimates for the convergence rate of Orthomin(k) for a class of linear systems

Details Sharp estimates for the convergence rate of Orthomin(k) for a class of linear systems Sharp estimates for the convergence rate of Orthomin(k) for a class of linear systems

2011-09-13

arxiv.org/abs/1109.2669v1

Mathematics

Numerical Analysis

20 pages, 6 figures

Scientific paper

In this work we show that the convergence rate of Orthomin($k$) applied to systems of the form $(I+\rho U) x = b$, where $U$ is a unitary operator and $0<\rho<1$, is less than or equal to $\rho$. Moreover, we give examples of operators $U$ and $\rho>0$ for which the asymptotic convergence rate of Orthomin($k$) is exactly $\rho$, 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.

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