On IOM(q): The Incomplete Orthogonalization Method for Large Unsymmetric Linear Systems

Abstract

The incomplete orthogonalization method (IOM(q)), a truncated version of the full orthogonalization method (FOM) proposed by Saad, has been used for solving large unsymmetric linear systems. However, no convergence analysis has been given. In this paper, IOM(q) is analysed in detail from a theoretical point of view. A number of important results are derived showing how the departure of the matrix A from symmetric affects the basis vectors generated by IOM(q), and some relationships between the residuals for IOM(q) and FOM are established. The results show that IOM(q) behaves much like FOM once the basis vectors generated by it are well conditioned. However, it is proved that IOM(q) may generate an ill‐conditioned basis for a general unsymmetric matrix such that IOM(q) may fail to converge or at least cannot behave like FOM. Owing to the mathematical equivalence between IOM(q) and the truncated ORTHORES(q) developed by Young and Jea, insights are given into the convergence of the latter. A possible strategy is proposed for choosing the parameter q involved in IOM(q). Numerical experiments are reported to show convergence behaviour of IOM(q) and of its restarted version.

Recent computational developments in Krylov subspace methods for linear systems

“…The advantages of these procedures over their restarted counterparts are not always apparent for general nonsymmetric matrices; see e.g., [273, sections 6.4.2, 6.5.6]. However, a recent analysis in [300] shows that if the original full (non-truncated) method converges smoothly and reasonably fast, then the truncated scheme only experiences a small convergence delay; see also Jia [192] for a similar discussion specific for truncated FOM. A natural question when discussing truncation strategies is whether it is necessarily the wisest thing to keep the latest basis vectors; in general, some of the discarded vectors might have been more significant than the ones that are kept.…”

Approximating the large sparse matrix exponential using incomplete orthogonalization and Krylov subspaces of variable dimension

“…It shows how these MGS sweeps can sometimes be much more time‐consuming than the matrix‐vector products in the case of the large biological problems that motivate our study. A gain in performance can therefore be achieved by instead using an incomplete orthogonalization process/method (IOP/IOM) as we shall do in this work, although in theory, this is potentially less numerically robust because orthonormality is traded . The IOP strategy of truncating the recurrences is well known and has been applied to linear systems and eigenvalue problems, but not so much to the matrix exponential until recently .…”

“…For the type of matrix exponential evaluations we are interested in, comparisons show that IOP can be a strong competitor to FOP. This contrasts with general nonsymmetric linear systems and eigenvalue problems, where IOP‐based methods have not become popular, considering they may fail to converge even though their FOP counterparts succeed there …”

“…A gain in performance can therefore be achieved by instead using an incomplete orthogonalization process/method (IOP/IOM) as we shall do in this work, although in theory, this is potentially less numerically robust because orthonormality is traded . The IOP strategy of truncating the recurrences is well known and has been applied to linear systems and eigenvalue problems, but not so much to the matrix exponential until recently . Our present study incorporates this strategy into the step‐by‐step integration scheme of Expokit, which we also extend to employ Krylov subspaces of variable dimension as done by Niesen and Wright .…”