arXiv subjects

Daniel B Szyld

Publications and source records attributed to Daniel B Szyld.

3 recordsLinked to original sources

arXiv2026

Do you precondition on the left or on the right?

This work is a follow-up to a poster that was presented at the DD29 conference. Participants were asked the question: ``Do you precondition on the left or on the right?''. Here we report on the results of this social experiment. We also provide context on left, right and split preconditioning, share our literature review on the topic, and analyze some of the finer points. Two examples illustrate that convergence bounds can sometimes lead to misleading conclusions.

math.NA↗

arXiv2025

Improved Polynomial Bounds and Acceleration of GMRES by Solving a min-max Problem on Rectangles, and by Deflating

Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered. The well-known polynomial bounds are generalized to the cases considered, and then reduced to solving a min-max problem on rectangles on the complex plane. Several approaches are considered and compared. The new bounds can be improved by using specific deflation spaces and preconditioners. This in turn accelerates the convergence of GMRES. Numerical examples illustrate the results obtained.

math.NA↗

arXiv2024

New Convergence Analysis of GMRES with Weighted Norms, Preconditioning and Deflation, Leading to a New Deflation Space

New convergence bounds are presented for weighted, preconditioned, and deflated GMRES for the solution of large, sparse, non-Hermitian linear systems. These bounds are given for the case when the Hermitian part of the coefficient matrix is positive definite, the preconditioner is Hermitian positive definite, and the weight is equal to the preconditioner. The new bounds are a novel contribution in and of themselves. In addition, they are sufficiently explicit to indicate how to choose the preconditioner and the deflation space to accelerate the convergence. One such choice of deflating space is presented, and numerical experiments illustrate the effectiveness of such space.

math.NA↗