The matrix-vector complexity of $Ax=b$

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, $\hat{I}\text{c◯}(\hat{I}º\log (1/\epsilon ))$ matrix--vector products are necessary to solve a linear system with condition number $\hat{I}º$ to accuracy $\epsilon$, 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.