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Abstract:Matrix--vector algorithms, particularly Krylov subspace methods, are widely viewed as the most effective algorithms for solving large systems of linear equations. This paper establishes lower bounds on the worst-case number of matrix--vector products needed by such an algorithm to approximately solve a general linear system. The first main result is that, for any matrix--vector algorithm which is allowed the use of randomization and can perform products with both a matrix and its transpose, $\Omega(\kappa \log(1/\varepsilon))$ matrix--vector products are necessary to solve a linear system with condition number $\kappa$ to accuracy $\varepsilon$, matching an upper bound for conjugate gradient on the normal equations. The second main result is that one-sided algorithms, which lack access to the transpose, must use $n$ matrix--vector products to solve an $n \times n$ linear system, even when the problem is perfectly conditioned. Both main results include explicit constants that match known upper bounds up to a factor of four. These results rigorously demonstrate the limitations of matrix--vector algorithms and confirm the optimality of widely used Krylov subspace algorithms.

Submission history

From: Raphael Meyer [view email]
[v1] Wed, 4 Feb 2026 18:32:52 UTC (44 KB)
[v2] Wed, 18 Feb 2026 01:58:02 UTC (45 KB)
[v3] Thu, 9 Jul 2026 22:42:19 UTC (45 KB)