General Iterative Method Research Articles

The general modulus-based Jacobi iteration method for linear complementarity problems

For the large sparse linear complementarity problem, by reformulating it as an implicit fixed point equation problem, Bai propose a class of modulus-based matrix splitting iteration methods in [12]. In this paper, we discuss one form of these methods-the general modulus-based Jacobi iteration method, proved the convergence, and derive the domain and the optimum value of the parameter for one special situation. Numerical results show that this method is superior to some modulus-related methods in computing efficiency and feasible aspects in some situations.

On the Numerical Behavior of Matrix Splitting Iteration Methods for Solving Linear Systems

We study the numerical behavior of stationary one-step or two-step matrix splitting iteration methods for solving large sparse systems of linear equations. We show that inexact solutions of inner linear systems associated with the matrix splittings may considerably influence the accuracy of the approximate solutions computed in finite precision arithmetic. For a general stationary matrix splitting iteration method, we analyze two mathematically equivalent implementations and discuss the conditions when they are componentwise or normwise forward or backward stable. We distinguish two different forms of matrix splitting iteration methods and prove that one of them is significantly more accurate than the other when employing inexact inner solves. The theoretical results are illustrated by numerical experiments with an inexact one-step and an inexact two-step splitting iteration method.

A general composite iterative method for strictly pseudocontractive mappings in Hilbert spaces

In this paper, we introduce a new general composite iterative method for finding a fixed point of a strictly pseudocontractive mapping in Hilbert spaces. We establish the strong convergence of the sequence generated by the proposed iterative method to a fixed point of the mapping, which is the unique solution of a certain variational inequality. In particular, we utilize weaker control conditions than previous ones in order to show strong convergence. Our results complement, develop, and improve upon the corresponding ones given by some authors recently in this area.

A modified general modulus-based matrix splitting method for linear complementarity problems of H-matrices

In this paper we establish a modified general modulus-based matrix splitting iteration method for solving the large sparse linear complementarity problems, and present the convergence analysis when the system matrix is an H+-matrix. Our result always improves some existing ones. In addition, numerical examples are given to show the effectiveness of the new method.

Generalized Schultz iterative methods for the computation of outer inverses

We consider a general matrix iterative method of the type Xk+1=Xkp(AXk) for computing an outer inverse AR(G),N(G)(2), for given matrices A∈Cm×n and G∈Cn×m such that AR(G)⊕N(G)=Cm. Here p(x) is an arbitrary polynomial of degree d. The convergence of the method is proven under certain necessary conditions and the characterization of all methods having order r is given. The obtained results provide a direct generalization of all known iterative methods of the same type. Moreover, we introduce one new method and show a procedure how to improve the convergence order of existing methods. This procedure is demonstrated on one concrete method and the improvement is confirmed by numerical examples.

Five-axis trajectory generation based on kinematic constraints and optimisation

This paper proposed a five-axis trajectory generation method based on kinematic constraints and optimisation. The optimisation algorithms based on double solutions filtering and singularity determination are designed during inverse kinematics, which can choose the appropriate solution set within the movement ranges of rotary axes. Trajectory generation with kinematic constraints for each axis is significant in high speed machining, however, the component acceleration for each drive does not only depend on the value of tangential velocity (feed), but also on its changing rate. It is inefficient to deal with this problem by using general iterative methods. This paper proposes an approximate constraint method which keeps constant feed for each small segment by eliminating the effect from feed changing, and then connects the constant feed stages by trapezoidal acceleration transition. Simulation for a five-axis tool path in impeller machining is implemented. The results show that the solution profile which overruns the angle movement range is filtered, and both the velocity and acceleration on each axis are confined.

An Algebraic Solution for the Candecomp/PARAFAC Decomposition with Circulant Factors

The Candecomp/PARAFAC decomposition (CPD) is an important mathematical tool used in several fields of application. Yet, its computation is usually performed with iterative methods which are subject to reaching local minima and to exhibiting slow convergence. In some practical contexts, the data tensors of interest admit decompositions constituted by matrix factors with particular structure. Often, such structure can be exploited for devising specialized algorithms with superior properties in comparison with general iterative methods. In this paper, we propose a novel approach for computing a circulant-constrained CPD, i.e., a CPD of a hypercubic tensor whose factors are all circulant (and possibly tall). To this end, we exploit the algebraic structure of such tensor, showing that the elements of its frequency-domain counterpart satisfy homogeneous monomial equations in the eigenvalues of square circulant matrices associated with its factors, which we can therefore estimate by solving these equations. Then, we characterize the sets of solutions admitted by such equations under Kruskal's uniqueness condition. Simulation results are presented, validating our approach and showing that it can help avoiding typical disadvantages of iterative methods.

Common Fixed Point Iterations of Generalized Asymptotically Quasi-Nonexpansive Mappings in Hyperbolic Spaces

We introduce a general iterative method for a finite family of generalized asymptotically quasi- nonexpansive mappings in a hyperbolic space and study its strong convergence. The new iterative method includes multi-step iterative method of Khan et al. [1] as a special case. Our results are new in hyperbolic spaces and generalize many known results in Banach spaces and CAT(0) spaces, simultaneously.

Convergence of Viscosity Iteration Process for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

We introduce a general iteration method for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in the paper extend and improve some recent results in the works by Shahzad and Udomene (2006); L. Qihou (2001); Khan et al. (2008).

Strong Convergence of a Unified General Iteration fork-Strictly Pseudononspreading Mapping in Hilbert Spaces

We introduce a unified general iterative method to approximate a fixed point ofk-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of ak-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.

A general iterative method for two maximal monotone operators and 2-generalized hybrid mappings in Hilbert spaces

Let C be a closed and convex subset of a real Hilbert space H. Let T be a 2-generalized hybrid mapping of C into itself, let A be an α-inverse strongly-monotone mapping of C into H, and let B and F be maximal monotone operators on and respectively. The purpose of this paper is to introduce a general iterative scheme for finding a point of which is a unique solution of a hierarchical variational inequality, where is the set of fixed points of T, and are the sets of zero points of and F, respectively. A strong convergence theorem is established under appropriate conditions imposed on the parameters. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to mixed equilibrium problems and the set of fixed points of a 2-generalized hybrid mapping in a real Hilbert space.

A General Iterative Method for Solving Constrained Convex Minimization Problems

It is well known that the gradient-projection algorithm plays an important role in solving minimization problems. In this paper, we will use the idea of regularization to establish a general method so that the sequence generated by the general method can be strongly convergent to a minimizer of constrained convex minimization problems, which solves a variational inequality under suitable conditions.

Research on Iterative Method in Solving Linear Equations on the Hadoop Platform

Solving linear equations is ubiquitous in many engineering problems, and iterative method is an efficient way to solve this question. In this paper, we propose a general iteration method for solving linear equations. Our general iteration method doesnt contain denominators in its iterative formula, and this relaxes the limits that traditional iteration methods require the coefficient aii to be non-zero. Moreover, as there is no division operation, this method is more efficient. We implement this method on the Hadoop platform, and compare it with the Jacobi iteration, the Guass-Seidel iteration and the SOR iteration. Experiments show that our proposed general iteration method is not only more efficient, but also has a good scalability.

Right-Hand Side Decomposition for Variational Inequalities

We consider a general class of variational inequality problems in a finite-dimensional space setting. The cost mapping need not be the gradient of any function. By using a right-hand side allocation technique, we transform such a problem into a collection of small-dimensional variational inequalities. The master problem is a set-valued variational inequality. We suggest a general iterative method for the problem obtained, which is convergent under monotonicity assumptions. We also show that regularization of partial problems enables us to create single-valued approximations for the cost mapping of the master problem and to propose simpler solution methods.

The modified general iterative methods for asymptotically nonexpansive semigroups in Banach spaces

In this paper, we introduce the modified general iterative methods for finding a common fixed point of asymptotically nonexpansive semigroups, which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Banach space which has a uniformly Gâteaux differentiable norm and admits the duality mapping and uniform normal structure. The main result extends various results existing in the current literature.

Iterative methods for constrained convex minimization problem in Hilbert spaces

In this paper, based on Yamada's hybrid steepest descent method, a general iterative method is proposed for solving constrained convex minimization problem. It is proved that the sequences generated by proposed implicit and explicit schemes converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality. MSC: 58E35; 47H09; 65J15

On summability, integrability and impulsive differential equations in Banach spaces

To study summability of families indexed by well-ordered sets of in normed spaces. To derive integrability criteria for step mappings and for right regulated mappings from an interval of to a Banach space. To study solvability of impulsive differential equations. A generalized iteration method presented, e.g., (Heikkila and Lakshmikantham in Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, 1994). Summability of families in normed spaces indexed with well-ordered subsets of . Necessary and sufficient conditions for global and local HK, HL, Bochner and Riemann integrability of step mappings and right regulated mappings defined on an interval of , and having values in a Banach space. Applications to impulsive differential equations are also presented. Families indexed with well-ordered subsets of are used to represent impulsive parts of considered equations and to approximate their solutions. MSC: 26A06, 26A18, 26A39, 26B12, 26E20, 34A36, 34A37, 34G20, 40A05, 40D05, 40F05, 47H07, 47H10.

The general iterative methods for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces

In this paper, the researcher introduces the general iterative scheme for finding a common element of the set of equilibrium problems and fixed point problems of a countable family of nonexpansive mappings in Hilbert spaces. The results presented in this paper improve and extend the corresponding results announced by many others. MSC: 47H10; 47H09

An Iterative Method for Approximating the Common Solutions of Nonexpansive Semigroups, a System of Variational Inequalities, Variational Inequalities and Equilibrium Problems

In this work, we consider an general iterative method for finding a common element of the set of common fixed points of a one-parameter nonexpansive semigroup problem, a system of variational inequality problems, a variational inequality problem and the set of solutions of a suitable equilibrium problem in a real Hilbert space. Further we establish a strong convergence theorem based on this method. Our results improve and extend corresponding ones announced by many others.

Constrained SART algorithm for inverse problems in image reconstruction

In this paper we integrate the SART (Simultaneous Algebraic Reconstruction Technique) algorithm into a general iterative method, introduced in [8]. This general method offers us the possibility of achieving a new convergence proof of the SART method and prove the convergence of the constrained version of SART. Systematic numerical experiments, comparing SART and Kaczmarz-like algorithms, are made on two phantoms widely used in image reconstruction literature.