Dynamical Control of Computations ‎U‎sing the Iterative Methods to Solve Absolute Value ‎Equation

Abstract

In this work, the absolute value equation (AVE) $ Ax-vert x vert= b$ is solved by the Gauss-Seidel and Jacobi iterative methods based on the stochastic arithmetic, where $A$ is an arbitrary square matrix whose singular values exceed one. An algorithm is proposed to find the optimal number of iterations in the given iterative scheme and obtain the optimal solution with its accuracy. To this aim, the CESTAC $^{1}$footnote{Controle et Estimation Stochastique des Arrondis de Calculs} method and the CADNA $^{2}$footnote{Control of Accuracy and Debugging for Numerical Application} library are applied which allows us to estimate the round-off error effect on any computed result. The classical criterion to terminate the iterative procedure is replaced by a criterion independent of the given accuracy $(epsilon)$ such that the best solution is evaluated numerically. Numerical examples are solved to validate the results and show the efficiency and importance of using the stochastic arithmetic in place of the floating-point arithmetic. Moreover, this method is applied to solve two-point boundary value ‎problem.‎