• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.

Verlag der Österreichischen Akademie der Wissenschaften
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ETNA - Electronic Transactions on Numerical Analysis



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Verlag der Österreichischen Akademie der Wissenschaften
Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2,
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400
https://verlag.oeaw.ac.at, e-mail: bestellung.verlag@oeaw.ac.at
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Preconditioning without a preconditioner using randomized block Krylov subspace methods

    Tyler Chen, Caroline Huber, Ethan Lin, Hajar Zaid

ETNA - Electronic Transactions on Numerical Analysis, pp. 63-92, 2026/02/03

doi: 10.1553/etna_vol65s63

doi: 10.1553/etna_vol65s63


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doi:10.1553/etna_vol65s63



doi:10.1553/etna_vol65s63

Abstract

We describe a randomized variant of the block conjugate gradient method for solving a single positive definite linear system of equations. This method provably outperforms the preconditioned conjugate gradient method with a broad class of Nyström-based preconditioners, without ever explicitly constructing a preconditioner. In analyzing our algorithm, we derive theoretical guarantees for new variants of the Nyström-preconditioned conjugate gradient method, which may be of separate interest. We also describe how our approach yields fast algorithms for key data-science tasks such as computing the entire ridge regression regularization path and generating multiple independent samples from a high-dimensional Gaussian distribution.

Keywords: preconditioning, randomized, conjugate gradient, block Krylov subspace methods