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In this paper we propose an alternative for the study of local convergence radius and the uniqueness radius for some third-order methods for multiple roots whose multiplicity is known. The main goal is to provide an alternative that avoids the use of sophisticated properties of divided differences that are used in already published papers about local convergence for multiple roots. We defined the local study by using a technique taking into consideration a bounding condition for the (m + i) th derivative of the function f (x) with i=1,2. In the case that the method uses first and second derivative in its iterative expression and i=1 in case the method only uses first derivative. Furthermore we implement a numerical analysis in the following sense. Since the radius of local convergence for high-order methods decreases with the order, we must take into account the analysis of ITS behaviour when we introduce a new iterative method. Finally, we have used these iterative methods for multiple roots for the case where the multiplicity m is unknown, so we estimate this factor by different strategies comparing the behaviour of the corresponding estimations and how this fact affect to the original method. [ABSTRACT FROM AUTHOR]

In this paper we present a new family of efficient iterative methods, in order to approximate the simple roots of various nonlinear equations. By some numerical examples we test the accuracy of our methods making a comparative study with other well known iterative methods. [ABSTRACT FROM AUTHOR]

Purpose – The weighted linear least squares (WLLS) problem has many different applications in sciences and engineering. The purpose of this paper is to introduce an iterative scheme for solving the WLLS problem. Design/methodology/approach – By considering the splitting techniques in conjunction with generalized accelerated over-relaxation (GAOR) method the authors design a new iterative method to solve the WLLS problem. Furthermore, within the computational framework, some models of iterative schemes candidates are investigated and evaluated. Findings – In this paper, the authors propose an efficient iterative scheme for solving the WLLS problem. The proposed scheme presented promising results from the aspects of both convergence behavior and performance. Moreover, comparative results for the proposed schemes are also presented. Research limitations/implications – Comparison between the new methods and other similar methods for the studied problem shows a remarkable agreement and reveals that the new model is much better in comparison with the existing methods in point of view rate of convergence and computing efficiency, as illustrated by the theoretical analysis and numerical results presented. Originality/value – For solving WLLS more attention has recently been paid on a special class of splitting techniques with the preconditioned GAOR method. In this paper, the authors use a different splitting for the GAOR method and present a promising class of methods. The convergence results of the iterative algorithm are also proposed. Several examples are given to show the efficiency of the presented methods. [ABSTRACT FROM AUTHOR]

In this paper we propose a generating function method for constructing new two- and three-point iterations with p  (3 ≤  p  ≤ 8) order of convergence. This approach allows us to derive a new family of the optimal order iterative methods that include well known methods as special cases. The necessary and sufficient conditions for p th order convergence of the proposed iterations are given in terms of parameters τ n and α n . We also propose some generating functions for τ n and α n . We give the extension of a class of optimal fourth-order Jarratt’s type iterations with a ≠ 2 3 . We develop a unified representation of all optimal eighth-order methods. Several numerical results are given to demonstrate the efficiency and the performance of the presented methods and compare them with some other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, a new iteration method is proposed for solving the complex symmetric linear systems. In theory, we show that the convergence factor or the upper bound of the spectral radius of the iteration matrix of the new method are smaller than that of the PMHSS method proposed by Bai et al. (2011). Moreover, we analyze and compare the parameter-free versions and the spectrum distributions of the preconditioned matrix of the new method and the PMHSS method. Finally, we present some numerical experiments on a few model problems to illustrate the theoretical results and show the effectiveness of our new method. [ABSTRACT FROM AUTHOR]

The behavior of an iterative method applied to nonlinear equations may be considerably sensitive to the starting points. Comparisons between iterative methods are supported by the study of the basins of attraction in the complex plane C . However, usually, nothing is said about the rate of convergence. In this paper, by making recourse to several examples of algebraic and transcendental equations, a numerical comparison is performed between three methods with the same structure, namely BSC, Halley's and Euler–Chebyshev's methods. The study takes into account both the basins of attraction and the rate of convergence which is measured as the number of iterations required to obtain an equation root with a given tolerance. [ABSTRACT FROM AUTHOR]

In this paper we prove that, among all one-point iterative processes without memory of order p , the most efficient processes are of order p = 3 . Moreover, the computational efficiency of one-point iterative processes without memory decreases to 1 as p increases, i.e., the efficiency index of higher order of convergence methods is low. We find the upper and lower bounds of the Ostrowski–Traub index of computational efficiency in a wider class of iterative methods with unit informational efficiency. [ABSTRACT FROM AUTHOR]

This paper concerns all acceleration method for fixed-point iterations that originated in work of D. G. Anderson [J. Assoc. Comput. Mach., 12 (1965), pp. 547 560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anderson mixing; however, it seems to have been untried or underexploited in many other important applications. Moreover, while other acceleration methods have been extensively studied by the mathematics and numerical analysis communities, this method has received relatively little attention from these communities over the years. A recent paper by H. Fang and Y. Saad [Numer. Linear Algebra Appl., 16 (2009), pp. 197-221] has clarified a remarkable relationship of Anderson acceleration to quasi-Newton (secant updating) methods and extended it to define a broader Anderson family of acceleration methods. In this paper, our goals are to shed additional light on Anderson acceleration and to draw further attention to its usefulness as a general tool. We first show that, on linear problems, Anderson acceleration without truncation is "essentially equivalent" in a certain sense to the generalized minimal residual (GMRES) method. We also show that the Type 1 variant in the Fang-Saad Anderson family is similarly essentially equiv- alent to the Arnoldi (full orthogonalization) method. We then discuss practical considerations for implementing Anderson acceleration and illustrate its performance through numerical experiments involving a variety of applications. [ABSTRACT FROM AUTHOR]

This paper deals with novel mathematical methods to locate the edges of heated planar objects employing simple photodiodes. Thereby, both the lateral edge position and the axial distance of the object can be determined. Furthermore, a method which allows the estimation of the rotation angle of the investigated object is discussed. In contrast to an infrared camera, photodiodes and the required electronic components are insensitive to mechanical vibrations commonly experienced in industrial fabrication processes. Nevertheless, the limited spatial resolution of such photodiode ensemble demands for a signal-processing-based resolution enhancement, which is presented in this paper. By means of simulation results, the efficiency of the novel methods is verified. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a new family of iterative methods for solving nonlinear equations with sixth and seventh order convergence. The new methods are obtained by composing known methods of third and fourth order with Newton’s method and using an adequate approximation for the last derivative, which provides high order of convergence and reduces the required number of functional evaluations per step. The new methods attain efficiency indices of 1.5651 and 1.6266, which makes them competitive. We introduce a new efficiency index involving the computational effort as well as the functional evaluations per iteration. We use this new index, in combination with the usual efficiency index, in order to compare the methods described in the paper with other known methods and present several numerical tests. [Copyright &y& Elsevier]

Abstract: In this paper, a finite-difference scheme for incompressible flow problems is treated. The scheme uses non-staggered grid for velocity approximation. A special stabilization is introduced to ensure the well-posedness and optimal approximation properties of the scheme. The stability estimate is proved in the form of a mesh-independent bound for the norm of discrete operator inverse. The finite-difference method is particularly suitable for problems in which velocity vector is involved in additional quantities that enter the system of flow equations as, for example, in the Bingham problem. We describe this application in the paper in some detail. Results of numerical experiments are included that confirm the applicability and optimality of the method. [Copyright &y& Elsevier]

Because the spectral radius is only an asymptotic measure of the rate of convergence of a linear iterative method, Golub and dePillis [Toward an effective two-parameter method, in Iterative Methods for Large Linear Systems, Academic Press, New York, 1990] have raised in a recent paper the question of determining, for each κ ≥ 1, a relaxation parameter ω ∈(0,2) and a pair of relaxation parameters ω1 and ω2 which minimize the Euclidean norm of the κth power of the SOR and MSOR iteration matrices, respectively, associated with a real symmetric positive definite matrix with ‘Property A.’ Here we use a reduction of these operators which they derived from the SVD of the associated block Jacobi matrix to obtain the minimizing relaxation parameters for the case κ = 1 for both operators. We conclude the paper with two brief sections in which we assess what our results imply. [ABSTRACT FROM AUTHOR]

In this paper, we propose new iterative schemes for the computation of outer inverse which reduce the total number of matrix multiplications per iteration. In particular, we consider how the hyper-power method of orders 5 and 9 can be accelerated such that they require 4 and 5 matrix multiplications per iteration, respectively. These improvements are tested against quadratically convergent Schultz’ method and fastest Horner scheme hyper-power method of order three. Numerical results show the superiority and practical applicability of the proposed methods. Finally, it is shown that a possibly more efficient method should have the order at least r ≥ 14 , making it useless for practical applications. [ABSTRACT FROM AUTHOR]

In recent years the growing popularity of supercomputers has fostered the development of algorithms able to take advantage of the massive parallelism offered by multiple processors. Direct methods, though robust and computationally efficient, hardly exploit high degrees of parallelism. By contrast, Krylov methods preconditioned by Factored Sparse Approximate Inverses (FSAI) provide, at least in principle, a perfectly parallel approach but are often thwarted by an excessive set-up cost. In this paper we extend the concept of supernode from sparse LU factorizations to approximate inverses, and use it to accelerate the computation of an FSAI-type preconditioner. The numerical experiments on real-world problems show that the overall FSAI efficiency can be significantly increased while preserving its intrinsic parallelism. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with some generalizations of the Hermitian and skew-Hermitian splitting (HSS) iteration for solving continuous Sylvester equations. The main contents we will introduce are the normal and skew-Hermitian splitting (NSS) iteration methods for the continuous Sylvester equations. It is shown that the new schemes can outperform the standard HSS method in some situations. Theoretical analysis shows that the NSS methods converge unconditionally to the exact solution of the continuous Sylvester equations. Moreover, we derive the upper bound of the contraction factor of the NSS iterations. Numerical experiments further show the effectiveness of our new methods. [Copyright &y& Elsevier]

In this paper, we present a family of optimal, in the sense of Kung-Traub's conjecture, iterative methods for solving nonlinear equations with eighth-order convergence. Our methods are based on Chun's fourth-order method. We use the Ostrowski's efficiency index and several numerical tests in order to compare the new methods with other known eighth-order ones. We also extend this comparison to the dynamical study of the different methods. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the order of convergence and the efficiency of four families of iterative methods using frozen divided differences. The first two families correspond to a generalization of the secant method and the implementation made by Schmidt and Schwetlick. The other two frozen schemes consist of a generalization of Kurchatov method and an improvement of this method applying the technique used by Schmidt and Schwetlick previously. An approximation of the local convergence order is generated by the examples, and it numerically confirms that the order of the methods is well deduced. Moreover, the computational efficiency indexes of the four algorithms are presented and computed in order to compare their efficiency. [ABSTRACT FROM AUTHOR]

Spectral preconditioners are based on the fact that the convergence rate of the Krylov subspace methods is improved if the eigenvalues of the smallest magnitude of the system matrix are ‘removed’. In this paper, two preconditioning strategies are studied to solve a set of linear systems associated with the numerical integration of the time-dependent neutron diffusion equation. Both strategies can be implemented using the matrix–vector product as the main operation and succeed at reducing the total number of iterations needed to solve the set of systems. [ABSTRACT FROM PUBLISHER]

Abstract: The goal of this paper is to present an accelerated iterative method for computing weighted Moore–Penrose inverse. Analysis of convergence is included to show that the proposed scheme has sixth-order convergence. Using a proper initial matrix, a sequence of iterates will be produced, which is convergent to the weighted Moore–Penrose inverse. Numerical experiments are reported to show the efficiency of the new method. [Copyright &y& Elsevier]

In this paper, we propose a Guass–Newton-like method for finding least-square solutions to inverse eigenvalue problems. We show that the proposed method converges under some mild conditions. In particular, if the method converges to the exact solution, the convergence rate is at least quadratic in the root sense. Numerical examples are given to justify the theoretical result. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper we introduce a new third-order iterative method for solving nonlinear equations. This method is converging on larger intervals than some other similar known methods. A comparison between the new method and other third-order methods is presented by using the basins of attraction for the real roots for some test problems. The numerical results are also presented. [Copyright &y& Elsevier]

In this paper, we study the convergence ofP-regular splitting iterative methods for singular and non-singular non-Hermitian positive semidefinite linear systems, which generalize the known results. Some numerical experiments are performed to illustrate the convergence results. [ABSTRACT FROM PUBLISHER]

In this paper, we propose to solve the Toeplitz linear systems Tnx = b by a recursive-based method. The method is based on repeatedly dividing the original problem into two subproblems that involve the solution of systems containing the Schur complement of the leading principal submatrix of the previous level. The idea is to solve the linear systems Smy = d, where Sm is the Schur complement of T2m (the principal submatrix of Tn), by using a self preconditioned iterative methods. The preconditioners, which are the approximate inverses of Sm, are constructed based on famous Gohberg-Semencul formula. All occurring matrices are represented by proper generating vectors of their displacement rank characterization. We show that, for well conditioned problems, the proposed method is efficient and robust. For ill-conditioned problems, by using some iterative refinement method, the new method would be efficient and robust. Numerical experiments are presented to show the effectiveness of our new method. [ABSTRACT FROM AUTHOR]

In this paper, a general form of single-step iterative methods for multiple roots of nonlinear equations is derived under a number of assumptions of optimization. Definition of multiplicational combinations and their properties are used upon the optimization procedure. Among all, we construct a family of iterative methods with nine parameters and simplest terms, and we obtain 23 simplest iterative methods within the family, those including all existing methods of single-step scheme. Numerical comparisons between the methods also present interesting and noteworthy results. [ABSTRACT FROM AUTHOR]

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]

In this paper, we develop new families of sixth-order methods for solving simple zeros of non-linear equations. These methods are constructed such that the convergence is of order six. Each member of the families requires two evaluations of the given function and two of its derivative per iteration. These methods have more advantages than Newton's method and other methods with the same convergence order, as shown in the illustration examples. [ABSTRACT FROM AUTHOR]

SUMMARY This paper is devoted to the analysis of the eigenvalue distribution of two classes of block preconditioners for the generalized saddle point problem. Most of the bounds developed improve those appeared in previously published works. Numerical results onto a realistic test problem give evidence of the effectiveness of the estimates on the spectrum of preconditioned matrices. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the modification of Chebyshevʼs iterative method constructed in Amat et al. (2008) is revisited. The behavior of this method when considering quadratic nonlinear operators is analyzed. In this case, the iterative method has a competitive behavior due to its computational efficiency. Moreover, a new result of semilocal convergence assuming only a pointwise condition is obtained, improving the result given in Amat et al. (2008) . The domain of uniqueness of the solution is also improved. The new technique used in the proof of these results allows us to achieve all these improvements. Finally, some theoretical and numerical applications for a quadratic system of equations are presented. [Copyright &y& Elsevier]

Regularization methods for the solution of ill-posed inverse problems can be successfully applied if a right estimation of the regularization parameter is known. In this paper, we consider the L1-regularized image deblurring problem and evaluate its solution using the iterative forward–backward splitting method. Based on this approach, we propose a new adaptive rule for the estimation of the regularization parameter that, at each iteration, dynamically updates the parameter value, following the evolution of the objective functional. The iterative algorithm automatically stops, without requiring any assumption about the perturbation process, when the parameter has reached a seemingly near optimal value. In spite of the fact that the optimality of this value has not yet been theoretically proved, a large number of numerical experiments confirm that the proposed rule yields restoration results competitive with those of the best state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we use the decomposition technique of Noor and Noor [M. A. Noor and K. I. Noor, Three-step iterative methods for nonlinear equations, preprint (2006)] to suggest and analyze a new iterative methods for solving the integral equations. Our method of developing this new method is very simple as compared with other methods. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results reveal that the proposed method is very effective and simple. Our method can be considered as an improvement of the existing methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive a new family of eighth-order methods for obtaining simple roots of nonlinear equations by using the weight function method. Each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which are optimal according to the Kung and Traub’s conjecture (1974) . Numerical comparisons are made to show the performance of the derived method, as is shown in the numerical section. [Copyright &y& Elsevier]

Purpose – The purpose of this paper is to find the efficient iterative methods for solving the general matrix equation A1X+ XA2+A3XH+XHA4=B (including Lyapunov and Sylvester matrix equations as special cases) with the unknown complex (reflexive) matrix X. Design/methodology/approach – By applying the principle of hierarchical identification and the Hermitian/skew-Hermitian splitting of the coefficient matrix quadruplet A1; A2; A3; A4 the authors propose a shift-splitting hierarchical identification (SSHI) method to solve the general linear matrix equation A1X+XA2+A3XH+XHA4=B. Also, the proposed algorithm is extended for finding the reflexive solution to this matrix equation. Findings – The authors propose two iterative methods for finding the solution and reflexive solution of the general linear matrix equation, respectively. The proposed algorithms have a simple, neat and elegant structure. The convergence analysis of the methods is also discussed. Some numerical results are given which illustrate the power and effectiveness of the proposed algorithms. Originality/value – So far, several methods have been presented and used for solving the matrix equations by using vec operator and Kronecker product, generalized inverse, generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD) of matrices. In several cases, it is difficult to find the solutions by using matrix decomposition and generalized inverse. Also vec operator and Kronecker product enlarge the size of the matrix greatly therefore the computations are very expensive in the process of finding solutions. To overcome these complications and drawbacks, by using the hierarchical identification principle and the Hermitian=skew-Hermitian splitting of the coefficient matrix quadruplet (A1; A2; A3; A4), the authors propose SSHI methods for solving the general matrix equation. [ABSTRACT FROM AUTHOR]

This paper presents two methods based on domain decomposition concepts for determining the diagonal of the inverse of specific matrices. The first uses a divide-and-conquer principle and the Sherman-Morrison-Woodbury formula and assumes that the matrix can be decomposed into a 2 x 2 block-diagonal matrix and a low-rank matrix. The second method is a standard domain decomposition approach in which local solves are combined with a global correction. Both methods can be successfully combined with iterative solvers and sparse approximation techniques. The efficiency of the methods usually depends on the specific implementation, which should be fine-tuned for different test problems. Preliminary results for some two-dimensional (2D) problems are reported to illustrate the proposed methods. [ABSTRACT FROM AUTHOR]

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: The rational Krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems. However, its success has been hindered by the lack of a parameter-free procedure, which would effectively generate the sequence of shifts used to build the space. In this paper we propose an adaptive computation of these shifts. The whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix, while further information is automatically generated during the process. The approach is a full generalization to the nonsymmetric case of the idea first proposed in Druskin et al. (2010) and it is used for two important problems in control: the approximation of the transfer function and the numerical solution of large Lyapunov equations. The procedure can be naturally extended to other related problems, such as the solution of the Sylvester equation, and parametric or higher order systems. Several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors. [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]

In this paper, we introduce and consider a new class of variational inequalities involving two operators, which is called the general nonconvex quasi variational inequality. Several special cases are discussed. We use the projection technique to establish the equivalence between the general nonconvex quasi variational inequalities and the fixed point problems. This alternative equivalent formulation is used to study the existence of a solution of the general nonconvex quasi variational inequalities. Using these equivalent formulations, we suggest and analyze a wide class of new extragradient methods for solving the general nonconvex quasi variational inequalities. Convergence criteria of these new iterative methods is considered under some suitable conditions. [ABSTRACT FROM AUTHOR]

Abstract: A multilevel preconditioner for the Helmholtz equation using two types of incremental unknowns (IU) is developed. The transition between the two occurs when the mesh size reaches a predetermined fraction of the wavelength – roughly one quarter wavelength. Conventional IUs based on bilinear interpolation are employed for fine meshes like in an earlier paper by the authors. For coarse mesh sizes, novel IUs are defined using a Helmholtz/wave equation-based interpolation. The interpolation coefficients for the coarse meshes with dimension higher than one are derived numerically for stencils resembling integral representations for interior points. In two dimensions, the IUs are located on crosses surrounded by square contours. Numerical experiments on 1D and 2D examples have demonstrated the efficacy of the proposed approach in reducing the condition numbers and accelerating convergence for coarse grids with mesh sizes exceeding the wavelength. [ABSTRACT FROM AUTHOR]

Abstract: Recently, spline approximations have been proposed for the reconstruction of piecewise smooth functions from Fourier data. That approach makes possible to retrieve the functions from their Fourier coefficients for any given degree of accuracy when the discontinuity points are known. In this paper we present iterative methods based on those spline approximations, for several degrees, to find locations and amplitudes of the jumps of a piecewise smooth function, given its Fourier coefficients. We also present numerical experiments comparing with different previous approaches. [Copyright &y& Elsevier]

Abstract: In this paper, we present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra–Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture Kung and Traub (1974) , that establishes for an iterative method based on evaluations an optimal order is fulfilled, getting the highest efficiency indices for orders and , which are 1.587 and 1.682. We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods. [Copyright &y& Elsevier]

The aim of this paper is to compare strong and weak coupling of the PEEC and MoM method applied to the modeling of magnetoharmonic problem, and to generalize the weak coupling approach. MoM and PEEC are dedicated to the modeling of specific parts of the overall device. As an example of validation, we have chosen the modeling of a simple transformer. Resolutions of several weak couplings are compared to the strong coupling approach. A new strategy of weak coupling is proposed to improve the time of resolution. [ABSTRACT FROM AUTHOR]

This paper proposes a framework of waveform relaxation methods to simulate electromagnetic fields coupled to electric networks. Within this framework, a guarantee for convergence and stability of Gauß-Seidel-type methods is found by partial differential algebraic equation (PDAE) analysis. It is shown that different time step sizes in different parts of the model can be automatically chosen according to the problem's dynamics. A finite-element model of a transformer coupled to a circuit illustrates the efficiency of multirate methods. [ABSTRACT FROM AUTHOR]

Solving systems of nonlinear equations is a relatively complicated problem for which a number of different approaches have been proposed. In this paper, we employ the Homotopy Analysis Method (HAM) to derive a family of iterative methods for solving systems of nonlinear algebraic equations. Our approach yields second and third order iterative methods which are more efficient than their classical counterparts such as Newton’s, Chebychev’s and Halley’s methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a class of Uzawa-SOR methods for saddle point problems, and prove the convergence of the proposed methods. We solve a lower triangular system per iteration in the proposed methods, instead of solving a linear equation . Actually, the new methods can be considered as an inexact iteration method with the Uzawa as the outer iteration and the SOR as the inner iteration. Although the proposed methods cannot achieve the same convergence rate as the GSOR methods proposed by Bai et al. [Z.-Z. Bai, B.N. Parlett, Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005) 1–38], but our proposed methods have less workloads per iteration step. Experimental results show that our proposed methods are feasible and effective. [Copyright &y& Elsevier]

Abstract: The Extended Krylov subspace has recently received considerable attention as a powerful tool for matrix function evaluations and other problems involving large matrices. In this paper we show that this space has a great potential within projection-type methods for effectively solving several other important large-scale algebraic problems: we focus on the solution of shifted systems and of more general parameter-dependent matrix equations, and on the approximation of the transfer function by projection. Numerical experiments stemming from real applications show the effectiveness of the approach. [Copyright &y& Elsevier]

In this paper, we construct some improvements of the Jarratt method for solving non-linear equations. A new sixth-order method are developed and numerical examples are given to support that the method obtained can compete with other sixth-order iterative methods. [ABSTRACT FROM AUTHOR]

In this paper, we present some new third-order iterative methods for finding a simple root α of nonlinear scalar equation f( x)=0 in R. A geometric approach based on the circle of curvature is used to construct the new methods. Analysis of convergence shows that the new methods have third-order convergence, that is, the sequence { x n} generated by each of the presented methods converges to α with the order of convergence three. The efficiency of the methods are tested on several numerical examples. It is observed that our methods can compete with Newton’s method and the classical third-order methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents reduced-rank linearly constrained minimum variance (LCMV) beamforming algorithms based on joint iterative optimization of filters. The proposed reduced-rank scheme is based on a constrained joint iterative optimization of filters according to the minimum variance criterion. The proposed optimization procedure adjusts the parameters of a projection matrix and an adaptive reduced-rank filter that operates at the output of the bank of filters. We describe LCMV expressions for the design of the projection matrix and the reduced-rank filter. We then describe stochastic gradient and develop recursive least-squares adaptive algorithms for their efficient implementation along with automatic rank selection techniques. An analysis of the stability and the convergence properties of the proposed algorithms is presented and semi-analytical expressions are derived for predicting their mean squared error (MSE) performance. Simulations for a beamforming application show that the proposed scheme and algorithms outperform in convergence and tracking the existing full-rank and reduced-rank algorithms while requiring comparable complexity. [Copyright &y& Elsevier]