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 formsu^{T}A^{−1}u whereA is a symmetric positive definite matrix andu is a given vector. Numerical examples are given for the Gauss-Seidel algorithm.

Moreover, we show that using a formula for theA-norm of the error from Dahlquist, Golub and Nash [1978] 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 criterion.