On error estimation in the conjugate gradient method: Normwise backward error - Tichý, Petr (Proceedings of the Conference Algoritmy, 2016)
Using an idea of Duff and Vömel [BIT, 42 (2002), pp.~300--322 ] we suggest a simple algorithm that incrementally estimates the $2$-norm of Jacobi matrices that are available during the conjugate gradient (CG) computations. The estimate can be used, e.g., in stopping criteria based on the normwise backward error. Numerical experiments predict that the estimate approximates the $2$-norm of $A$ with a sufficient accuracy.