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, $Ω(κ\log(1/\varepsilon))$ matrix--vector products are necessary to solve a linear system with condition number $κ$ 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.

翻译：矩阵-向量算法，特别是 Krylov 子空间方法，被广泛视为求解大规模线性方程组最有效的算法。本文建立了此类算法为近似求解一般线性系统所需矩阵-向量乘积数量在最坏情况下的下界。第一个主要结果是：对于任何允许使用随机化且能执行矩阵及其转置乘积的矩阵-向量算法，求解条件数为 $κ$、精度为 $\varepsilon$ 的线性系统至少需要 $Ω(κ\log(1/\varepsilon))$ 次矩阵-向量乘积，这与法方程上共轭梯度法的上界相匹配。第二个主要结果是：单侧算法（即无法访问转置的算法）必须使用 $n$ 次矩阵-向量乘积才能求解 $n \times n$ 线性系统，即使该问题是完全良态的。两个主要结果均包含与已知上界至多相差四倍因子的显式常数。这些结果严格证明了矩阵-向量算法的局限性，并证实了广泛使用的 Krylov 子空间算法的最优性。