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Abstract: The purpose of this paper is to propose a modified block iterative algorithm for find a common element of the set of common fixed points of an infinite family of quasi-ϕ-asymptotically nonexpansive mappings and the set of an equilibrium problem. Under suitable conditions, some strong convergence theorems are established in a uniformly smooth and strictly convex Banach space with the Kadec–Klee property. As an application, at the end of the paper a numerical example is given. The results presented in the paper improve and extend the corresponding results in Qin et al. [Convergence theorems of common elements for equilibrium problems and fixed point problem in Banach spaces, J. Comput. Appl. Math., 225, 2009, 20–30], Zhou et al. [Convergence theorems of a modified hybrid algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings, J. Appl. Math. Compt., 17 March, 2009, doi:10.1007/s12190-009-0263-4], Takahashi and Zembayshi [Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal., 70, 2009, 45–57], Wattanawitoon and Kumam [Strong convergence theorems by a new hybrid projection algorithm for fixed point problem and equilibrium problems of two relatively quasi-nonexpansive mappings, Nonlinear Anal. Hybrid Syst., 3, 2009, 11–20] and Matsushita and Takahashi [A strong convergence theorem for relatively nonexpansive mappings in Banach spaces, J. Approx. Theory, 134, 2005, 257–266] and others. [Copyright &y& Elsevier]

A trust-region algorithm in association with line search filter technique for solving nonlinear equality constrained programming is proposed in this paper. In the current iteration, the trial step providing sufficient descent is generated by solving a corresponding trust-region subproblem. Then, the step size is decided by backtracking line search together with filter technique to obtain the next iteration point. The advantage of this method is that resolving trust-region subproblem many times to determine a new iteration point in traditional trust-region method can be avoided and hence the expensive computation can be lessened. And the difficult decisions in regard to the choice of penalty parameters in the merit functions can be avoided by using filter technique. Second order correction steps are introduced in the proposed algorithm to overcome Maratos effect. Convergence analysis shows that fast local convergence can be achieved under some mild assumptions. The preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss the convergence of the double-shift and multi-shift QR algorithms for symmetric tridiagonal matrices. We analyze how to choose multi-shifts by comparing the relationships between the number of iterations, CPU time and the number of multi-shifts. Numerical tests and figures are performed. [Copyright &y& Elsevier]

Abstract: In this paper, we consider numerical simulation to a class of constrained dynamic problems where the overall dynamics are determined by the interactions between two sub-systems. We present an iterative algorithm that naturally decouples the computation of the two sub-systems and that ensures an accurate and efficient solution procedure. We conduct rigorous error analysis for the convergence of the iterative algorithm, and verify the analytical results through careful numerical tests. [Copyright &y& Elsevier]

Abstract: In this paper a modified gradient based algorithm for solving Sylvester equations is presented. Different from the gradient based method introduced by Ding and Chen and the relaxed gradient based algorithm proposed by Niu et al. , the information generated in the first half-iterative step is fully exploited and used to construct the approximate solution. Theoretical analysis shows that the new method converges under certain assumptions. Numerical results are given to verify the efficiency of the new method. [Copyright &y& Elsevier]

Abstract: This paper presents a hierarchical least squares iterative algorithm to estimate the parameters of multivariable Box–Jenkins-like systems by combining the hierarchical identification principle and the auxiliary model identification idea. The key is to decompose a multivariable systems into two subsystems by using the hierarchical identification principle. As there exist the unmeasurable noise-free outputs and noise terms in the information vector, the solution is using the auxiliary model identification idea to replace the unmeasurable variables with the outputs of the auxiliary model and the estimated residuals. A numerical example is given to show the performance of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, two iterative algorithms are proposed to solve the linear matrix equations . When the matrix equations are consistent, by the first algorithm, a solution can be obtained within finite iterative steps in the absence of roundoff-error for any initial value, furthermore, the minimum-norm solution can be got by choosing a special kind of initial matrix. Additionally, the unique optimal approximation solution to a given matrix can be derived by finding the minimum-norm solution of a new matrix equations . When the matrix equations are inconsistent, we present the second algorithm to find the least-squares solution with the minimum-norm. Finally, two numerical examples are tested by MATLAB, the results show that these iterative algorithms are efficient. [Copyright &y& Elsevier]

Abstract: In this paper we propose an iterative algorithm to solve large size linear inverse ill posed problems. The regularization problem is formulated as a constrained optimization problem. The dual Lagrangian problem is iteratively solved to compute an approximate solution. Before starting the iterations, the algorithm computes the necessary smoothing parameters and the error tolerances from the data. The numerical experiments performed on test problems show that the algorithm gives good results both in terms of precision and computational efficiency. [Copyright &y& Elsevier]

Abstract: This paper deals with fixed points methods related to the general class of demicontractive mappings (including the well-known classes of nonexpansive and quasi-nonexpansive mappings) in Hilbert spaces. Specifically, we point out some historical aspects concerning the concept of demicontactivity and we investigate a regularized variant of the Krasnoselski-Mann iteration that can be alternatively regarded as a simplified form of the inertial iteration (P-E. Maingé, J. Math. Anal. Appl. 344 (2008) 876–887) with non-constant relaxation factors. These two methods ensure the strong convergence of the generated sequence towards the least norm element of the set of fixed-points of demicontractive mappings. However, for convergence, our method does not require anymore the knowledge of some constant related to the involved demicontractive operator. A new and simpler proof is also proposed for its convergence even when involving non-constant relaxation factors. We point out the simplicity of this algorithm (at least from computational point of view) in comparison with other existing methods. We also present some numerical experiments concerning a convex feasibility problem, experiments that emphasize the characteristics of the considered algorithm comparing with a classical cyclic projection-type iteration. [Copyright &y& Elsevier]

Abstract: This paper is concerned with iterative solutions to a class of complex matrix equations, which include some previously investigated matrix equations as special cases. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of matrix equations. A sufficient condition is presented to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix with a real representation of a complex matrix as tools. By using some properties of the real representation, a convergence condition that is easier to compute is also given in terms of original coefficient matrices. A numerical example is employed to illustrate the effectiveness of the proposed methods. [Copyright &y& Elsevier]

Abstract: In this paper, the issue of multi-degree reduction of Bézier curves with C 1 and G 2-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to demonstrate the efficiency of this method. [Copyright &y& Elsevier]

Abstract: Chaotic catfish particle swarm optimization (C-CatfishPSO) is a novel optimization algorithm proposed in this paper. C-CatfishPSO introduces chaotic maps into catfish particle swarm optimization (CatfishPSO), which increase the search capability of CatfishPSO via the chaos approach. Simple CatfishPSO relies on the incorporation of catfish particles into particle swarm optimization (PSO). The introduced catfish particles improve the performance of PSO considerably. Unlike other ordinary particles, the catfish particles initialize a new search from extreme points of the search space when the gbest fitness value (global optimum at each iteration) has not changed for a certain number of consecutive iterations. This results in further opportunities of finding better solutions for the swarm by guiding the entire swarm to promising new regions of the search space and accelerating the search. The introduced chaotic maps strengthen the solution quality of PSO and CatfishPSO significantly. The resulting improved PSO and CatfishPSO are called chaotic PSO (C-PSO) and chaotic CatfishPSO (C-CatfishPSO), respectively. PSO, C-PSO, CatfishPSO, C-CatfishPSO, as well as other advanced PSO procedures from the literature were extensively compared on several benchmark test functions. Statistical analysis of the experimental results indicate that the performance of C-CatfishPSO is better than the performance of PSO, C-PSO, CatfishPSO and that C-CatfishPSO is also superior to advanced PSO methods from the literature. [Copyright &y& Elsevier]

Abstract: This paper is concerned with iterative solutions to a class of complex matrix equations. By applying the hierarchical identification principle, an iterative algorithm is constructed to solve this class of complex matrix equations. The range of the convergence factor is given to guarantee that the proposed algorithm is convergent for arbitrary initial matrix by applying a real representation of a complex matrix as a tool. By using some properties of the real representation, a sufficient convergence condition that is easier to compute is also given by original coefficient matrices. Two numerical examples are given to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a generalized global conjugate gradient squared method for solving nonsymmetric linear systems with multiple right-hand sides is presented. The method can be derived by using products of two nearby global BiCG polynomials and formal orthogonal polynomials, of which global CGS and global BiCGSTAB are just particular cases. We also show to apply the method for solving the Sylvester matrix equation. Finally, numerical examples are given to illustrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: In , we proposed some numerical verification methods for automatic proof of the existence of solution for obstacle problems. In this paper we propose a new iterative algorithm to automatically prove the existence of solutions for some generalized obstacle problems. [Copyright &y& Elsevier]

Abstract: In the paper, a new alternating-direction iterative method is proposed based on matrix splittings for solving saddle point problems. The convergence analysis for the new method is given. When the better values of parameters are employed, the proposed method has faster convergence rate and less time cost than the Uzawa algorithm with the optimal parameter and the Hermitian and skew-Hermitian splitting iterative method. Numerical examples further show the effectiveness of the method. [Copyright &y& Elsevier]

Abstract: The paper is concerned with the higher order nonlinear neutral delay differential equation Using the Banach fixed-point theorem, we establish five existence results of uncountably many bounded nonoscillatory solutions for the above equation, construct five Mann iterative sequences with mixed errors for approximating these nonoscillatory solutions and discuss five error estimates between the approximate solutions and these nonoscillatory solutions. To dwell the importance and applications of our results, five nontrivial examples are constructed. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the monotone affine variational inequality problem (AVIP for short). Based on a smooth reformulation of the AVIP, we propose a Newton-type method to solve the monotone AVIP, where a testing procedure is embedded into our algorithm. Under mild assumptions, we show that the proposed algorithm may find a maximally complementary solution to the monotone AVIP in a finite number of iterations. Preliminary numerical results are reported. [Copyright &y& Elsevier]

Abstract: In this paper, we study the convergence of the generalized accelerated overrelaxation (GAOR) iterative method. That is an extension of the classical convergence result of the generalized successive overrelaxation (GSOR) iterative method. We proposed some theorems, which they obtain better results than similar works. By some numerical examples we show the goodness of our results. [Copyright &y& Elsevier]

Abstract: In this paper two algorithms for computing the principal square root of a centrosymmetric H-matrix with positive diagonal entries are proposed. It is showed that our algorithms ensure significant savings in computational costs, as compared to the case of an arbitrary H-matrix A with positive diagonal entries. [Copyright &y& Elsevier]

Abstract: The ill-conditioned subproblems appeared in constrained optimization can often impede the preformance of optimization algorithms. In this paper we introduce a class of efficient iterative semi-refinement technique to solve the ill-conditioned trust-region subproblems. The nonmonotone technique is also used to increase the efficiency of the algorithm. The numerical results show that the presented method is competitive. [Copyright &y& Elsevier]

In this paper we establish several algorithms of parallel chaotic generalized AOR iterative methods for solving large nonsingular systems based on some given models. Under some different assumptions of coefficient matrix A and its multisplittings we obtain corresponding sufficient conditions of convergence for some relaxed parameters. [Copyright &y& Elsevier]