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 fi أکثر
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.‎
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One of the considerable discussions for solving the nonlinear equations is to find the optimal iteration, and to use a proper termination criterion which is able to obtain a high accuracy for the numerical solution. In this paper, for a certain class of the family of op أکثر
One of the considerable discussions for solving the nonlinear equations is to find the optimal iteration, and to use a proper termination criterion which is able to obtain a high accuracy for the numerical solution. In this paper, for a certain class of the family of optimal two-point methods, we propose a new scheme based on the stochastic arithmetic to find the optimal number of iterations in the given iterative solution and obtain the optimal solution with its accuracy. For this purpose, a theorem is proved to illustrate the accuracy of the iterative method and the CESTAC$^1$\footnote{$^1$Controle et Estimation Stochastique des Arrondis de Calculs} method and CADNA$^2$\footnote{$^2$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, which is able to stop the process as soon as a satisfactory informatical solution is obtained. Some numerical examples are given to validate the results and show the efficiency and importance of using the stochastic arithmetic in place of the floating-point ‎arithmetic.‎
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In this paper, we apply the Newton’s and He’s iteration formulas in order to solve the nonlinear algebraic equations. In this case, we use the stochastic arithmetic and the CESTAC method to validate the results. We show that the He’s iteration formula أکثر
In this paper, we apply the Newton’s and He’s iteration formulas in order to solve the nonlinear algebraic equations. In this case, we use the stochastic arithmetic and the CESTAC method to validate the results. We show that the He’s iteration formula is more reliable than the Newton’s iteration formula by using the CADNA library.
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