On the Optimal Solution of Large Linear Systems

Abstract

The information-based study of the optimal solution of large linear systems is initiated by studying the case of Krylov information. Among the algorithms that use Krylov information are minimal residual, conjugate gradient, Chebyshev, and successive approximation algorithms. A "sharp" lower bound on the number of matrix-vector multiplications required to compute an å-approximation is obtained for any orthogonally invariant class of matrices. Examples of such classes include many of practical interest such as symmetric matrices, symmetric positive definite matrices, and matrices with bounded condition number. It is shown that the minimal residual algorithm is within at most one matrix-vector multiplication of the lower bound. A similar result is obtained for the generalized minimal residual algorithm. The lower bound is computed for certain classes of orthogonally invariant matrices. How the lack of certam properties (symmetry, positive definiteness) increases the lower bound is shown. A conjecture and a number of open problems are stated.