جستجوی مقالات مرتبط با کلیدواژه « stochastic arithmetic » در نشریات گروه « علوم پایه »

One of the schemes to find the optimal shape parameter and optimal number of points in the radial basis function (RBF) methods is to apply the stochastic arithmetic (SA) in place of the common floating-point arithmetic (FPA). The main purpose of this work is to introduce a reliable approach based on this new arithmetic to compute the local optimal shape parameter and number of points in multiquadric and Gaussian RBF-meshless methods for solving differential equations, in the iterative process. To this end, the CESTAC method is applied. Also, in order to implement the proposed algorithms, the CADNA library is performed. The examples illustrate the efficiency and importance of using this library to validate the results.

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.‎

The aim of this research is to apply the stochastic arithmetic (SA) for validating the Sinc-collocation method (S-CM) with single or double exponentially decay to find the numerical solution of second kind Fredholm integral equation (IE). To this end, the CESTAC(Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library are applied. Using this method, the optimal iteration of S-CM, the optimal approximation, the absolute error and the numerical instabilities can be determined. A theorem is proved which shows the accuracy of the S-CM by means of the concept of common significant digits. Some IEs are presented and the numerical results of comparison between the single exponentially decay (SE) and the double exponentially decay (DE) are demonstrated in the tables.

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 ýarithmetic.ý

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.