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Three steps modified Levenberg–Marquardt method for nonlinear equations was introduced by Yang (2013). This method uses the addition of the Levenberg–Marquardt (LM) step and two approximate LM steps as the trial step at every iteration. Using trust region technique, the global and biquadratic convergence of the method is proved by Yang. The main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique. Since the search direction of Yang method may be not a descent direction, standard line searches cannot be used directly. In this paper we propose a new nonmonotone third order Armijo type line search technique which guarantees the global convergence of this method while we use an adaptive LM parameter. It is proved that the convergence order of the new method is biquadratic. Numerical results show the new algorithm is efficient and promising. [ABSTRACT FROM AUTHOR]

The paper studies the convergence of some parallel multisplitting block iterative methods for the solution of linear systems arising in the numerical solution of Euler equations. Some sufficient conditions for convergence are proposed. As special cases the convergence of the parallel block generalized AOR (BGAOR), the parallel block AOR (BAOR), the parallel block generalized SOR (BGSOR), the parallel block SOR (BSOR), the extrapolated parallel BAOR and the extrapolated parallel BSOR methods are presented. Furthermore, the convergence of the parallel block iterative methods for linear systems with special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, building on the previous work by Greif and Schötzau [Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numer. Linear Algebra Appl. 14 (2007) 281–297] and Benzi and Olshanskii [An augmented lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28 (2006) 2095–2113], we present the improved preconditioning techniques for the iterative solution of the saddle point linear systems, which arise from the finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The modified block diagonal and triangular preconditioners considered are based on augmentation with using the symmetric nonsingular weighted matrix. We discuss the spectral properties of the preconditioned matrix in detail and generalize the results of the above-mentioned paper by Greif and Schötzau. Numerical experiments are given to demonstrate the efficiency of the presented preconditioners. [Copyright &y& Elsevier]

Abstract: In this paper, we use a projected gradient algorithm to solve a nonlinear operator equation with -norm constraint. Gradient iterations with -norm constraints have been studied recently both in the context of inverse problem and of compressed sensing. In this paper, the constrained gradient iteration is implemented via a projected operator. We establish the -norm convergence of sequence constructed by the constrained gradient iteration when . The performance of the method is testified by a numerical example. [Copyright &y& Elsevier]

In this paper we propose an iterative numerical method for approximating solutions of fractional integral equations of the second kind, based on Picard iteration and a suitable quadrature formula. Under certain conditions, we prove the existence and uniqueness of the solution and give error estimates of the approximations. Numerical examples are given, which illustrate the good approximations. [ABSTRACT FROM AUTHOR]

Abstract: The problems of calculating a dominant eigenvector or a dominant pair of singular vectors, arise in several large scale matrix computations. In this paper we propose a minimum norm approach for solving these problems. Given a matrix, , the new method computes a rank-one matrix that is nearest to , regarding the Frobenius matrix norm. This formulation paves the way for effective minimization techniques. The methods proposed in this paper illustrate the usefulness of this idea. The basic iteration is similar to that of the power method, but the rate of convergence is considerably faster. Numerical experiments are included. [Copyright &y& Elsevier]

Abstract: Schröder’s methods of the first and second kind for solving a nonlinear equation , originally derived in 1870, are of great importance in the theory and practice of iteration processes. They were rediscovered several times and expressed in different forms during the last 130 years. It was proved in the paper of Petković and Herceg (1999)  that even seven families of iteration methods for solving nonlinear equations are mutually equivalent. In this paper we show that these families are also equivalent to another four families of iteration methods and find that all of them have the origin in Schröder’s generalized method (of the second kind) presented in 1870. In the continuation we consider Smale’s open problem from 1994 about possible link between Schröder’s methods of the first and second kind and state the link in a simple way. [Copyright &y& Elsevier]

Abstract: The paper studies the convergence of some block iterative methods for the solution of linear systems when the coefficient matrices are generalized -matrices. A truth is found that the class of conjugate generalized -matrices is a subclass of the class of generalized -matrices and the convergence results of R. Nabben [R. Nabben, On a class of matrices which arises in the numerical solution of Euler equations, Numer. Math. 63 (1992) 411–431] are then extended to the class of generalized -matrices. Furthermore, the convergence of the block AOR iterative method for linear systems with generalized -matrices is established and some properties of special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a system of nonlinear variational inclusions involving -accretive operators studied by Huang and Fang in -uniformly smooth Banach spaces. Using resolvent operator technique, we suggest an iterative algorithm for finding an approximate solution to the system of variational inclusions. Further, we discuss convergence criteria for the approximate solution of the system of variational inclusions. The theorems presented in this paper improve and unify many known results of variational inclusions. [Copyright &y& Elsevier]

It is known that the concept of optimality is not defined for multidimensional iterative methods for solving nonlinear systems of equations. However, usually optimal fourth-order schemes (extended to the case of several variables) are employed as starting steps in order to design higher order methods for this kind of problems. In this paper, we use a non-optimal (in scalar case) iterative procedure that is specially efficient for solving nonlinear systems, as the initial steps of an eighth-order scheme that improves the computational efficiency indices of the existing methods, as far as the authors know. Moreover, the method can be modified by adding similar steps, increasing the order of convergence three times per step added. This kind of procedures can be used for solving big-sized problems, such as those obtained by applying finite differences for approximating the solution of diffusion problem, heat conduction equations, etc. Numerical comparisons are made with the same existing methods, on standard nonlinear systems and Fisher’s equation by transforming it in a nonlinear system by using finite differences. From these numerical examples, we confirm the theoretical results and show the performance of the proposed schemes. [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]

In this paper, we propose two new approaches for model order reduction of large-scale multi-input multi-output (MIMO) linear time invariant dynamical systems (LTI). These methods are based on a generalization of the global Arnoldi algorithm which is used to generate projection subspaces. An adaptive procedure for the selection of shift parameters is proposed and some simple Arnoldi relations are established in order to compute error bounds on the transfer function. Numerical experiments showing the effectiveness of these approaches are provided. [ABSTRACT FROM AUTHOR]

In this paper we propose some modifications in the schemes for the iterative splitting techniques defined in Geiser (2009) for partial differential equations and introduce the parallel version of these modified algorithms. Theoretical results related to the order of the iterative splitting for these schemes are obtained. In the numerical experiments we compare the obtained results by applying iterative methods to approximate the solutions of the nonlinear systems obtained from the discretization of the splitting techniques to the mixed convection–diffusion Burgers’ equation and a momentum equation that models a viscous flow. The differential equations in each splitting interval are solved by the back-Euler–Newton algorithm using sparse matrices. [ABSTRACT FROM AUTHOR]

Linear systems with multiple right-hand sides arise in many applications. To solve such systems efficiently, a new deflated block GCROT( m , k ) method is explored in this paper by exploiting a modified block Arnoldi deflation. Incorporating this modified block Arnoldi deflation, the new algorithm can address the possible linear dependence at each iteration during the block Arnoldi procedure and reduces expensive computational operations. Moreover, as a block version of GCROT( m , k ), the new method inherits the property of easy operability. Finally, some numerical examples also illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The GMRES method is one of the most popular iterative schemes for the solution of large linear systems of equations with a square nonsingular matrix. GMRES-type methods also have been applied to the solution of linear discrete ill-posed problems. Computational experience indicates that for the latter problems variants of the standard GMRES method, that require the solution to live in the range of a positive power of the matrix of the linear system of equations to be solved, generally yield more accurate approximations of the desired solution than standard GMRES. This paper investigates properties of these variants of GMRES. [ABSTRACT FROM AUTHOR]

In this paper, we present the two preconditioners I + S ˆ and I + S ˆ + R for solving M -matrix linear systems and discuss the convergence of the two preconditioned iterative methods. Meanwhile, we obtain comparison theorems between the two preconditioned iterative methods and consider the solution of M -matrix linear systems by preconditioned Krylov subspace methods. Numerical experiments are given to validate the performance of the preconditioners. [ABSTRACT FROM AUTHOR]

In this paper, a method with very high order of convergence is constructed and analyzed. The method is used to compute generalized inverses. The efficiency index has been employed to show its superiority. Numerical experiments re-verify that the proposed iterative expression is more effective than the existing methods of the same type. [ABSTRACT FROM AUTHOR]

Many physics problems can only be studied by coupling various numerical codes, each modeling a subaspect of the physics problem that is addressed. In most cases, the “brute force” technique of running the codes one after the other in a loop until convergence is reached requires excessive CPU time. The present paper illustrates that re-writing the coupling as a root-finding problem, to which a quasi-Newton method–here the (Inverse) Column Updating Method–can be applied, is useful to push down the computation time, at the expense of a very modest amount of supplementary programming. A simplified version of the set of codes commonly used to describe plasma heating by radio frequency waves in a tokamak plasma is adopted for illustrating the potential of the speed-up method. It consists of a wave equation as well as a Fokker–Planck velocity space diffusion and a radial energy diffusion model. It is shown that with this approach a substantial reduction in CPU time needed for convergence can be obtained. [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]

For the augmented system of linear equations, by considering a new splitting of the coefficient matrix, a modified SOR-like (MSOR-like) method is presented in this paper. The convergence of this method is discussed under suitable restrictions on iteration parameters, the optimal parameters and the corresponding optimal convergence factor are determined. Numerical examples are reported to confirm the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we first present a shift-splitting preconditioner for saddle point problems. The preconditioner is based on a shift-splitting of the saddle point matrix, resulting in an unconditional convergent fixed-point iteration. Based on the idea of the splitting, we further propose a local shift-splitting preconditioner. Some properties of the local shift-splitting preconditioned matrix are studied. These preconditioners extend those studied by Bai, Yin and Su for solving non-Hermitian positive definite linear systems (Bai et al., 2006). Finally, numerical experiments of a model Stokes problem are presented to show the effectiveness of the proposed preconditioners. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, based on GCRS algorithm in Zhang and Zhao (2010) and the ideas in Gu et al. (2007), we present an improved generalized conjugate residual squared (IGCRS) algorithm that is designed for distributed parallel environments. The new improved algorithm reduces two global synchronization points to one by changing the computation sequence in the GCRS algorithm in such a way that all inner products per iteration are independent so that communication time required for inner products can be overlapped with useful computation. Theoretical analysis and numerical comparison of isoefficiency analysis show that the IGCRS method has better parallelism and scalability than the GCRS method, and the parallel performance can be improved by a factor of about 2. Finally, some numerical experiments clearly show that the IGCRS method can achieve better parallel performance with a higher scalability than the GCRS method and the improvement percentage of communication is up to 52.19% averagely, which meets our theoretical analysis. [Copyright &y& Elsevier]

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]

Abstract: In this paper, we propose a reweighted nuclear norm minimization algorithm based on the weighted fixed point method (RNNM–WFP algorithm) to recover a low rank matrix, which iteratively solves an unconstrained minimization problem introduced as a nonconvex smooth approximation of the low rank matrix minimization problem. We prove that any accumulation point of the sequence generated by the RNNM–WFP algorithm is a stationary point of the minimization problem. Numerical experiments on randomly generated matrix completion problems indicate that the proposed algorithm has better recoverability compared to existing iteratively reweighted algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, a class of nonlinear matrix equations is considered. An equivalent form of this equation is derived, and the properties of the solutions are studied. Then Newton’s iterative method is applied to solve this equivalent one. Finally, two numerical examples are given to illustrate the efficiency of Newton’s method. [Copyright &y& Elsevier]

Abstract: Recently, Zheng, Bai and Yang studied the parameterized Uzawa method for solving singular saddle point problems (B. Zheng, Z.-Z. Bai, X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl. 431 (2009) 808–817). In this paper, we discuss the inexact Uzawa method, which covers the Uzawa method, the preconditioned Uzawa method, and the parameterized Uzawa method to solve the singular saddle point problems. We prove the semi-convergence result under restrictions by verifying two necessary and sufficient conditions, that is, all elementary divisors associated with the eigenvalue of its iterative matrix are linear, and the pseudo-spectral radius of the iterative matrix is less than 1. Sufficient conditions for the semi-convergence of several Uzawa-type methods are also provided. In addition, numerical examples are given to demonstrate the semi-convergence of Uzawa-type methods. [Copyright &y& Elsevier]

Abstract: In this paper, based on the damped Newton method, we present an inversion-free iterative method with a stepsize parameter for solving the nonlinear matrix equation , where is a Hermitian positive definite matrix and . The proposed method is an extension of Algorithm 2.2 in [Z. Peng, S.M. El-Sayed, X. Zhang, Iterative methods for the extremal positive definite solution of the matrix equation , J. Comput. Appl. Math. 200 (2007) 520–527]. Preliminary numerical results show that the proposed method can improve numerical effect. [Copyright &y& Elsevier]

Abstract: We consider the nonnegative inverse eigenvalue problem with partial eigendata, which aims to find a nonnegative matrix such that it is nearest to a pre-estimated nonnegative matrix and satisfies the prescribed eigendata. In this paper, we propose several iterative schemes based on the alternating direction method of multipliers for solving the nonnegative inverse problem. We also extend our schemes to the symmetric case and the cases of prescribed lower bounds and of prescribed entries. Numerical tests (including a practical engineering application in vibrations) show the efficiency of the proposed iterative schemes. [Copyright &y& Elsevier]

Abstract: Alternating methods for image deblurring and denoising have recently received considerable attention. The simplest of these methods are two-way methods that restore contaminated images by alternating between deblurring and denoising. This paper describes Krylov subspace-based two-way alternating iterative methods that allow the application of regularization operators different from the identity in both the deblurring and the denoising steps. Numerical examples show that this can improve the quality of the computed restorations. The methods are particularly attractive when matrix–vector products with a discrete blurring operator and its transpose can be evaluated rapidly, but the structure of these operators does not allow inexpensive diagonalization. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a two stage strategy for waveform relaxation (WR) iterations, applied to initial value problems for differential-algebraic equations (DAEs) in the form . Outer iterations of TSWR are defined by , where , , and each iteration is computed using an inner iterative process, based on another splitting . Meanwhile, by the means of the Theta method, the discretized TSWR of DAEs is realized. Furthermore, when is an Hermitian positive semi-definite matrix with -regular splittings, the convergence and the comparison theorems of TSWR are analyzed. Finally, the numerical experiments are presented. [Copyright &y& Elsevier]

Abstract: Radial basis function interpolation involves two stages. The first is fitting, solving a linear system corresponding to the interpolation conditions. The second is evaluation. The systems occurring in fitting problems are often very ill-conditioned. Changing the basis in which the radial basis function space is expressed can greatly improve the conditioning of these systems resulting in improved accuracy, and in the case of iterative methods, improved speed, of solution. The change of basis can also improve the accuracy of evaluation by reducing loss of significance errors. In this paper new bases for the relevant space of approximants, and associated preconditioning schemes are developed which are based on Floater’s mean value coordinates. Positivity results and scale independence results are shown for schemes of a general type. Numerical results show that the given preconditioning scheme usually improves conditioning of polyharmonic spline and multiquadric interpolation problems in and by several orders of magnitude. The theory indicates that using the new basis elements (evaluated indirectly) for both fitting and evaluation will reduce loss of significance errors on evaluation. Numerical experiments confirm this showing that such an approach can improve overall accuracy by several significant figures. [Copyright &y& Elsevier]

Abstract: This paper introduces an error propagation formula of a certain class of multi-level iterative aggregation–disaggregation (IAD) methods for numerical solutions of stationary probability vectors of discrete finite Markov chains. The formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific Markov chains. Numerical experiments indicate that the same type of the formula could be used for a wider class of the multi-level IAD methods. Using the formula we show that for given data there is no relation between convergence of two-level and of multi-level IAD methods. [Copyright &y& Elsevier]

Abstract: Fluid–structure interaction problems arise in many fields of application such as flows around elastic structures and blood flow in arteries. The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov–Poincaré operators. This interface equation is solved by a Newton iteration, for which directional derivatives involving shape derivatives with respect to the interface perturbation have to be evaluated appropriately. One step of the Newton iteration requires the solution of several decoupled linear sub-problems in the structure and the fluid domains. These sub-problems are spatially discretized by a finite element method on hybrid meshes. For the time discretization, implicit first-order methods are used for both sub-problems. The discretized equations are solved by algebraic multigrid methods. [Copyright &y& Elsevier]

Abstract: A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA–GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of -error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we present a finite difference scheme for the solution of an initial-boundary value problem of the Schrödinger–Boussinesq equation. The scheme is fully implicit and conserves two invariable quantities of the system. We investigate the existence of the solution for the scheme, give computational process for the numerical solution and prove convergence of iteration method by which a nonlinear algebra system for unknown is solved. On the basis of a priori estimates for a numerical solution, the uniqueness, convergence and stability for the difference solution is discussed. Numerical experiments verify the accuracy of our method. [Copyright &y& Elsevier]

Abstract: In this paper, some semismooth methods are considered to solve a nonsmooth equation which can arise from a discrete version of the well-known Hamilton–Jacobi–Bellman equation. By using the slant differentiability introduced by Chen, Nashed and Qi in 2000, a semismooth Newton method is proposed. The method is proved to have monotone convergence by suitably choosing the initial iterative point and local superlinear convergence rate. Moreover, an inexact version of the proposed method is introduced, which reduces the cost of computations and still preserves nice convergence properties. Some numerical results are also reported. [Copyright &y& Elsevier]

Abstract: The variational iteration method and the homotopy analysis method, as alternative methods, have been widely used to handle linear and nonlinear models. The main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently. This paper deals with the numerical solutions of nonlinear fractional differential equations, where the fractional derivatives are considered in Caputo sense. The main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations. In these algorithms, Legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work. The proposed algorithms are capable of reducing the size of calculations, improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals. Numerical examples are examined to show the efficiency of the algorithms. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a trust region method for unconstrained optimization that can be regarded as a combination of conic model, nonmonotone and line search techniques. Unlike in traditional trust region methods, the subproblem of our algorithm is the conic minimization subproblem; moreover, our algorithm performs a nonmonotone line search to find the next iteration point when a trial step is not accepted, instead of resolving the subproblem. The global and superlinear convergence results for the algorithm are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems. [ABSTRACT FROM AUTHOR]

Abstract: The inexact GMRES algorithm is a variant of the GMRES algorithm where matrix–vector products are performed inexactly, either out of necessity or deliberately, as part of a trading of accuracy for speed. Recent studies have shown that relaxing matrix–vector products in this way can be justified theoretically and experimentally. Research, so far, has focused on decreasing the workload per iteration without significantly affecting the accuracy. But relaxing the accuracy per iteration is liable to increase the number of iterations, thereby increasing the overall runtime, which could potentially end up being greater than that of the exact GMRES if there were not enough savings in the matrix–vector products. In this paper, we assess the benefit of the inexact approach in terms of actual CPU time derived from realistic problems, and we provide cases that provide instructive insights into results affected by the build-up of the inexactness. Such information is of vital importance to practitioners who need to decide whether switching their workflow to the inexact approach is worth the effort and the risk that might come with it. Our assessment is drawn from extensive numerical experiments that gauge the effectiveness of the inexact scheme and its suitability for use in addressing certain problems, depending on how much inexactness is allowed in the matrix–vector products. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we introduce and analyze Uzawa algorithms for non-symmetric saddle point systems. Convergence for the algorithms is established based on new spectral results about Schur complements. A new Uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework. Numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, two new matrix iterative methods are presented to solve the matrix equation , the minimum residual problem and the matrix nearness problem , where is the set of constraint matrices, such as symmetric, symmetric -symmetric and -symmetric, and is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in Deng et al. (2006) , Huang et al. (2008) , Peng (2005)  and Lei and Liao (2007) . Paige’s algorithms are used as the frame method for deriving these matrix iterative methods. Numerical examples are used to illustrate the efficiency of these new methods. [ABSTRACT FROM AUTHOR]

Abstract: For the iterative solution of linear systems of equations arising from finite element discretization of elliptic problems there exist well-established techniques to construct numerically efficient and computationally optimal preconditioners. Among those, most often preferred choices are Multigrid methods (geometric or algebraic), Algebraic MultiLevel Iteration (AMLI) methods, Domain Decomposition techniques. In this work, the method in focus is AMLI. We extend its construction and the underlying theory over to systems arising from discretizations of parabolic problems, using non-conforming finite element methods (FEM). The AMLI method is based on an approximated block two-by-two factorization of the original system matrix. A key ingredient for the efficiency of the AMLI preconditioners is the quality of the utilized block two-by-two splitting, quantified by the so-called Cauchy–Bunyakowski–Schwarz (CBS) constant, which measures the abstract angle between the two subspaces, associated with the two-by-two block splitting of the matrix. The particular choice of space discretization for the parabolic equations, used in this paper, is Crouzeix–Raviart non-conforming elements on triangular meshes. We describe a suitable splitting of the so-arising matrices and derive estimates for the associated CBS constant. The estimates are uniform with respect to discretization parameters in space and time as well as with respect to coefficient and mesh anisotropy, thus providing robustness of the method. [ABSTRACT FROM AUTHOR]

Abstract: For large sparse saddle point problems, Chen and Jiang recently studied a class of generalized inexact parameterized iterative methods (see [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765–771]). In this paper, the methods are modified and some choices of preconditioning matrices are given. These preconditioning matrices have advantages in solving large sparse linear system. Numerical experiments of a model Stokes problem are presented. [Copyright &y& Elsevier]

Abstract: In this paper, on the basis of matrix splitting, two preconditioners are proposed and analyzed, for nonsymmetric saddle point problems. The spectral property of the preconditioned matrix is studied in detail. When the iteration parameter becomes small enough, the eigenvalues of the preconditioned matrices will gather into two clusters—one is near and the other is near —for the PPSS preconditioner no matter whether is Hermitian or non-Hermitian and for the PHSS preconditioner when is a Hermitian or real normal matrix. Numerical experiments are given, to illustrate the performances of the two preconditioners. [Copyright &y& Elsevier]

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]

Abstract: In this paper, we present a new iterative method (successive projection iterative method) to solve matrix equation , where is a symmetric positive definite (SPD) matrix. Based on this method an algorithm is proposed and proved to be convergent. In addition, analysis of the algorithm and numerical experiments are also given to show the efficiency of the method. [Copyright &y& Elsevier]

Abstract: In this paper, by using the method of lower and upper solutions coupled with the monotone iterative technique, we give conditions for existence of extreme solutions for first order functional differential equations with a new impulsive integral condition. Some comparison results are also formulated. [Copyright &y& Elsevier]

Abstract: In this paper, we first present a class of structure-oriented hybrid two-stage iteration methods for solving the large and sparse blocked system of linear equations, as well as the saddle point problem as a special case. And the new methods converge to the solution under suitable restrictions, for instance, when the coefficient matrix is positive stable matrix generally. Numerical experiments for a model generalized saddle point problem are given, and the results show that our new methods are feasible and efficient, and converge faster than the Classical Uzawa Method. [Copyright &y& Elsevier]