The matrix-vector complexity of $Ax=b$

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, $\mathrm{\Omega }(\kappa \log (1\mathrm{/}\epsilon ))$ matrix–vector products are necessary to solve a linear system with condition number $\kappa$ 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.