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On equivalence of three-parameter iterative methods for singular symmetric saddle-point problem

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Abstract

There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of convergence for the parameters are found, while in three of them, optimal parameters are determined, and in one of the latter, many more cases, than in all the others, are distinguished, analyzed, and studied. It turns out that two of the optimal parameters coincide making the optimal three-parameter methods be equivalent to the optimal two-parameter known ones. Our aim in this work is manifold: (i) to show that the iterative methods we present are equivalent, (ii) to slightly change some statements in one of the main papers, (iii) to complete the analysis in another one, (iv) to explain how the transition from any of the methods to the others is made, (v) to extend the iterative method to cover the singular symmetric case, and (vi) to present a number of numerical examples in support of our theory. It would be an omission not to mention that the main material which all researchers in the area have inspired from and used is based on the one of the most cited papers by Bai et al. (Numer. Math. 102:1–38, 2005).

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Acknowledgments

The authors are most grateful to the three reviewers, especially for their patience and for their essential and constructive criticism as well as for their specific comments and suggestions which greatly improved the quality of this work.

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Authors and Affiliations

  1. Department of Electrical & Computer Engineering, University of Thessaly, GR-382 21, Volos, Greece

    Apostolos Hadjidimos

  2. Department of Mathematics, University of Ioannina, GR-451 10, Ioannina, Greece

    Michael Tzoumas

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  1. Apostolos Hadjidimos
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Correspondence to Apostolos Hadjidimos.

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Hadjidimos, A., Tzoumas, M. On equivalence of three-parameter iterative methods for singular symmetric saddle-point problem. Numer Algor 86, 1391–1419 (2021). https://doi.org/10.1007/s11075-020-00938-1

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  • DOI: https://doi.org/10.1007/s11075-020-00938-1

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