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, $\Omega(\kappa \log(1/\varepsilon))$ 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.