Matrices, moments and quadrature II: how to compute the norm of the error in iterative methods

Abstract. In this paper, we study the numerical computation of the errors in linear systems when using iterative methods. This is done by using methods to obtain bounds or approximations of quadratic forms uT A−1u where A is a symmetric positive definite matrix and u is a given vector. Numerical examples are given for the Gauss–Seidel algorithm. Moreover, we show that using a formula for the A–norm of the error from [2], very good bounds of the error can be computed almost for free during the iterations of the conjugate gradient method leading to a reliable stopping criteria. Key words. Iterative methods, Error computation, Conjugate gradient. 1. Introduction.. Let A be a large, sparse symmetric positive definite matrix of order n and suppose an iterative method is used to compute an approximate solution x ̃ of the linear system Ax = b,(1) where b is a given vector. The residual r is defined as