On an iterative method for solving absolute value equations

We suggest an iterative method for solving absolute value equation Ax ? |x| = b, where \({A\in R^{n\times n}}\) is symmetric matrix and \({b\in R^{n}}\), coupled with the minimization technique. We also discuss the convergence of the proposed method. Some examples are given to illustrate the implementation and efficiency of the method.     

An iterative method for solving semismooth equations

In this paper, we combine trust region technique with line search technique to develop an iterative method for solving semismooth equations. At each iteration, a trust region subproblem is solved. The solution of the trust region subproblem provides a descent direction for the norm of a smoothing function. By using a backtracking line search, a steplength is determined. The proposed method shares advantages of trust region methods and line search methods. Under appropriate conditions, the proposed method is proved to be globally and superlinearly convergent. In particular, we show that after finitely many iterations, the unit step is always accepted and the method reduces to a smoothing Newton method.     

An iterative method for solving nonlinear equations

An iterative method for finding a solution of the equation f(x)=0$f(x)=0$ is presented. The method is based on some specially derived quadrature rules. It is shown that the method can give better results than the Newton method.     

An iterative method for solving nonlinear functional equations

An iterative method for solving nonlinear functional equations, viz. nonlinear Volterra integral equations, algebraic equations and systems of ordinary differential equation, nonlinear algebraic equations and fractional differential equations has been discussed.     

求解对称张量绝对值方程问题的非光滑牛顿法

提出一类对称张量绝对值方程问题,给出了求解此类问题的一类非光滑牛顿法,并且在一般的假设条件下,给出了算法的局部收敛性.最后给出相关的数值实验表明了算法的有效性.     

On unique solvability of the absolute value equation

It is proved that the singular value condition σ _max(|B|) < σ _min(A) implies unique solvability of the absolute value equation Ax + B|x| = b for each right-hand side b. This is a generalization of an earlier result by Mangasarian and Meyer proved for the special case of B = −I. Supported by the Czech Republic Grant Agency under grants 201/09/1957 and 201/08/J020, and by the Institutional Research Plan AV0Z10300504.     

An iterative method for solving systems of linear algebraic equations

An iterative solution process for systems of linear algebraic equations is proposed. It converges starting from any initial approximation and theoretically does not require preliminary transformation of the input data.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 64–68, 1985.     

Two CSCS-based iteration methods for solving absolute value equations

Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration method for solving AVEs. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iteration solver for AVEs. The advantage of these methods is that they do not require the storage of coefficient matrices at all, and the sub-system of linear equations can be solved efficiently via the fast Fourier transforms (FFTs). Therefore, computational cost and storage can be saved in practical implementations. Numerical examples including numerical solutions of nonlinear fractional diffusion equations are reported to show the effectiveness of the proposed methods in comparison with some existing methods.     

A fifth-order iterative method for solving nonlinear equations

The object of this paper is to construct a new efficient iterative method for solving nonlinear equations. This method is mainly based on Javidi paper [1] by using a new scheme of a modified homotopy perturbation method. This new method is of the fifth order of convergence, and it is compared with the second-, third-, fifth-, and sixth-ordermethods. Some numerical test problems are given to show the accuracy and fast convergence of the method proposed.     

An iterative aggregation-disaggregation algorithm for solving linear equations

A general procedure is given for solving large sets of linear equations by first rewriting them in a form suitable for aggregation of both the variables and equations, followed by disaggregation. A computational algorithm which iteratively aggregates and disaggregates is shown to converge geometrically to the exact solution. Provided the original problem has a structure suitable for such aggregation, the algorithm exhibits fast computation times, small main-memory requirements, and robustness to the starting point. A rigorous foundation for aggregation and disaggregation is provided by the equations employed by this algorithm.     

An inverse-free Newton-Jarratt-type iterative method for solving equations under the gamma condition

A local as well as a semilocal convergence analysis for Newton–Jarratt–type iterative method for solving equations in a Banach space setting is studied here using information only at a point via a gamma-type condition (Argyros in Approximate Solution of Operator Equations with Applications, [2005]; Wang in Chin. Sci. Bull. 42(7):552–555, [1997]). This method has already been examined by us in (Argyros et al. in J. Comput. Appl. Math. 51:103–106, [1994]; Argyros in Comment. Mat. XXIII:97–108, [1994]), where the order of convergence four was established using however information on the domain of the operator. In this study we also establish the same order of convergence under weaker conditions. Moreover we show that all though we use weaker conditions the results obtained here can be used to solve equations in cases where the results in (Argyros et al. in J. Comput. Appl. Math. 51:103–106, [1994]; Argyros in Comment. Mat. XXIII:97–108, [1994]) cannot apply. Our method is inverse free, and therefore cheaper at the second step in contrast with the corresponding two–step Newton methods. Numerical Examples are also provided.     

Sufficient conditions for the unsolvability and solvability of the absolute value equation

We give linear-inequalities-based sufficient conditions for the unsolvability and solvability of the NP-hard absolute value equation: \(Ax-|x|=b\), where A is an \(n\times n\) square matrix. The satisfaction of the linear inequalities is easily verified using a linear program.     

A new eighth-order iterative method for solving nonlinear equations

In this paper we present an improvement of the fourth-order Newton-type method for solving a nonlinear equation. The new Newton-type method is shown to converge of the order eight. Per iteration the new method requires three evaluations of the function and one evaluation of its first derivative and therefore the new method has the efficiency index of , which is better than the well known Newton-type methods of lower order. We shall examine the effectiveness of the new eighth-order Newton-type method by approximating the simple root of a given nonlinear equation. Numerical comparisons are made with several other existing methods to show the performance of the presented method.     

Finite iterative method for solving coupled Sylvester-transpose matrix equations

This paper is concerned with solutions to the so-called coupled Sylvester-transpose matrix equations, which include the generalized Sylvester matrix equation and Lyapunov matrix equation as special cases. By extending the idea of conjugate gradient method, an iterative algorithm is constructed to solve this kind of coupled matrix equations. When the considered matrix equations are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, a numerical example is given to illustrate that the algorithm is effective.     

A fourth order iterative method for solving nonlinear equations

The present paper illustrates an iterative numerical method to solve nonlinear equations of the form f(x) = 0, especially those containing the partial and non partial involvement of transcendental terms. Comparative analysis shows that the present method is faster than Newton-Raphson method, hybrid iteration method, new hybrid iteration method and others. Cost is also found to be minimum than these methods. The beauty in our method can be seen because of the optimization in important effecting factors, i.e. lesser number of iteration steps, lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods. This work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative.     

Necessary and sufficient conditions for the solvability of a class of boundary value problems

We obtain necessary and sufficient conditions for the solvability of boundary value problems with one- and two-point boundary conditions.