Estimation of errors in iterative solutions of a non-symmetric linear system

TL;DR

This paper introduces a new error estimation method for iterative solutions of general indefinite linear systems, enabling more accurate stopping criteria in large-scale problems where matrix properties are unknown.

Contribution

It presents a novel, efficient algorithm (BiCGQL) for estimating errors in nonsymmetric linear systems, extending existing methods to more general cases.

Findings

BiCGQL provides at least k/10 times more accuracy than residue-based criteria.

The method requires only O(1) time per iteration for quadratic form approximation.

Numerical results demonstrate improved error estimation in practical applications.

Abstract

Estimation of actual errors from the residue in iterative solutions is necessary for efficient solution of large problems when their condition number is much larger than one. Such estimators for conjugate gradient algorithms used to solve symmetric positive definite linear systems exist. This work presents error estimation for iterative solutions of general indefinite linear systems to provide accurate stopping and restarting criteria. In many realistic applications no properties of the matrices are known a priori; thus requiring such a general algorithm. Our method for approximating the required quadratic form of the residue (square of the A-norm of the error vector) when solving nonsymmetric linear systems with Bi-Conjugate Gradient (BiCG) algorithm, needs only O(1) time (per BiCG iteration). We also extend this estimate to approximate l2 norm of error vector using the relations of Hestenes and Stiefel. Using the heuristics of numerical results we observe that the developed algorithm (BiCGQL) is at least k/10 times more accurate than residue vector based stopping criteria (where k is the condition number of the system).