A numerical approach for solving the high-order linear singular differential-difference equations

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential-difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential-difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).     

Gradient-based maximal convergence rate iterative method for solving linear matrix equations

This paper is concerned with numerical solutions to general linear matrix equations including the well-known Lyapunov matrix equation and Sylvester matrix equation as special cases. Gradient based iterative algorithm is proposed to approximate the exact solution. A necessary and sufficient condition guaranteeing the convergence of the algorithm is presented. A sufficient condition that is easy to compute is also given. The optimal convergence factor such that the convergence rate of the algorithm is maximized is established. The proposed approach not only gives a complete understanding on gradient based iterative algorithm for solving linear matrix equations, but can also be served as a bridge between linear system theory and numerical computing. Numerical example shows the effectiveness of the proposed approach.     

Parallel methods for solving equations

Two classes of algorithms for equation solving are presented and analyzed. These algorithms have been devised in recent years because of the computational facility of the multiprocessor. The first class consists of parallel search methods while the second class consists of asynchronous methods. The first class of methods are fail safe. That is they always provide an approximation to the root as well as the smallest possible interval (for the work done) guaranteed to contain the root. The second class frees the intrinsically interlocked nature of the more complicated forms of algorithms designed for multiprocessors by omitting the synchrony usually demanded in computation.     

Best convergence rates of linear multistep methods for Volterra first kind equations

In two papers Holyhead et al. (1975, 1976) analyzed the convergence of general linear multistep methods under minimum continuity assumptions. This paper is concerned with determining the maximum orders of convergence of these methods given that the truncation error has an asymptotic expansion with sufficiently many terms.     

Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations

Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper. Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter. It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.     

Subdivision methods for solving polynomial equations

This paper presents a new algorithm for solving a system of polynomials, in a domain of Rⁿ${\mathbb{R}}^n$. It can be seen as an improvement of the Interval Projected Polyhedron algorithm proposed by Sherbrooke and Patrikalakis [Sherbrooke, E.C., Patrikalakis, N.M., 1993. Computation of the solutions of nonlinear polynomial systems. Comput. Aided Geom. Design 10 (5), 379–405]. It uses a powerful reduction strategy based on univariate root finder using Bernstein basis representation and Descarte’s rule  . We analyse the behavior of the method, from a theoretical point of view, shows that for simple roots, it has a local quadratic convergence speed and gives new bounds for the complexity of approximating real roots in a box of Rⁿ${\mathbb{R}}^n$. The improvement of our approach, compared with classical subdivision methods, is illustrated on geometric modeling applications such as computing intersection points of implicit curves, self-intersection points of rational curves, and on the classical parallel robot benchmark problem.     

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.     

Semiconvergence of P-regular splittings for solving singular linear systems

We investigate necessary and sufficient conditions for semiconvergence of a splitting for solving singular linear systems, where the coefficient matrix A is a singular EP matrix. When A is a singular Hermitian matrix, necessary and sufficient conditions for semiconvergence of P-regular splittings are given, which generalize known results. As applications, the necessary and sufficient conditions for semiconvergence of block AOR and SSOR iterative methods are derived. A numerical example is given to illustrate the theoretical results. The work is supported by the National Natural Science Foundation of China under grant 10371056, the Foundation for the Authors of the National Excellent Doctoral Thesis Award of China under grant 200720 and the Natural Science Foundation of Jiangsu Province of China under grant BK2006725.     

On the convergence of subproper (multi)-splitting methods for solving rectangular linear systems

Abstract We give a convergence criterion for stationary iterative schemes based on subproper splittings for solving rectangular systems and show that, for special splittings, convergence and quotient convergence are equivalent. We also analyze the convergence of multisplitting algorithms for the solution of rectangular systems when the coefficient matrices have special properties and the linear systems are consistent. Keywords: Rectangular linear system, iterative method, proper splitting, subproper splitting, regularity, Hermitian positive semi-definite matrix, multi-splitting, quotient convergence AMS Subject Classification: 65F10, 65F15     

An efficiently implementable Gauss-Newton-like method for solving singular nonlinear equations

A Gauss-Newton-like method for solving singular nonlinear equations is presented. The local convergence analysis shows that this method converges quadratically. The algorithm requires second derivative information in the formF″ ab only, which makes it attractive from the viewpoint of computational effort.     

An iterative method for solving partitioned linear equations

A new iterative scheme, using two partitions of the coefficient matrix of a given linear and non-singular system of equationsAx=b, is shown to always converge to the solution. The concept of two vector spaces approaching orthogonality is quantified and used to show that the eigenvalues of the iteration matrix approach zero as the vector spaces defined by the two partitions ofA approach orthogonality.     

Pivot tightening for direct methods for solving symmetric positive definite systems of linear interval equations

The paper considers systems of linear interval equations, i.e., linear systems where the coefficients of the matrix and the right hand side vary between given bounds. We focus on symmetric matrices and consider direct methods for the enclosure of the solution set of such a system. One of these methods is the interval Cholesky method, which is obtained from the ordinary Cholesky decomposition by replacing the real numbers by the related intervals and the real operations by the respective interval operations. We present a method by which the diagonal entries of the interval Cholesky factor can be tightened for positive definite interval matrices, such that a breakdown of the algorithm can be prevented. In the case of positive definite symmetric Toeplitz matrices, a further tightening of the diagonal entries and also of other entries of the Cholesky factor is possible. Finally, we numerically compare the interval Cholesky method with interval variants of two methods which exploit the Toeplitz structure with respect to the computing time and the quality of the enclosure of the solution set.     

The AOR method for solving linear interval equations

This paper is motivated by the paper [7], where the SOR method for solving linear interval equations was considered. It is known that sometimes the AOR method for systems of linear (“point”) equations converges faster than the SOR method. We give some sufficient conditions for the convergence of the interval AOR method for the same class of interval matrices which are considered in [7].     

求解块三对角线性方程组的含参并行算法

提出了分布式环境下求解含有两个参数的矩阵分裂方式的一种交替方向迭代并行算法,通过引入两个参数并巧妙分解系数矩阵A得到新算法,从理论上给出了该算法收敛的两个充分条件,并讨论了参数的选择范围.基于局域网的MPI异构环境,在HP rx2600集群上进行了数值实验,并与多分裂方法比较.比较的结果表明,此算法是可行的,具有良好的并行效率.     

Fourth-order derivative-free methods for solving non-linear equations

Two new one-parameter families of methods for finding simple and real roots of non-linear equations without employing derivatives of any order are developed. Error analysis providing the fourth-order convergence is given. Each member of the families requires three evaluations of function per step, and therefore the method has an efficiency index of 1.587. Numerical examples are presented and the performance of the method presented here is compared with methods available in the literature.     

周期块三对角线性方程组的一种并行算法

该文提出了分布式环境下求解周期块三对角线性方程组的一种并行算法,该算法通过对系数矩阵进行一次预处理后,充分利用系数矩阵结构的特殊性,使算法只在相邻处理机间通信两次。并从理论上给出了算法收敛的一个充分条件。最后,在HPrx2600集群上进行了数值试验,结果表明,实算与理论是一致的,并行性也很好。     

Multipoint iterative parallel methods for solving equations

In this paper we show the results of some research carried out on parallel iterative methods to solve equations. In particular we study general classes of one point parallel methods and multipoint ones without memory, and we point out the convergence order of these methods and the conditions which are both necessary and sufficient for them to be optimal. In addition we prove that the convergence order for multipoint parallel procedures without memory cannot be more thenr(r+) ^m−1 , wherer indicates the number of the parallel processor used andm the number of the functions and eventual derivatives, calculated not simultaneously in every iteration.

---

Sommario In questo articolo presentiamo alcuni risultati concernenti i metodi iterativi paralleli per risolvere equazioni. In particolare analizziamo alcune classi generali di procedimenti ad un punto ed a più punti senza memoria, il loro ordine di convergenza e le condizioni necessarie e sufficienti per ottenere l'ottimalità. Inoltre dimostriamo che l'ordine di convergenza di un procedimento iterativo senza memoria non può eccedere:r(r+1) ^m−1 , dover indica il numero di processor in parallelo usati edm indica il numero di funzioni ed eventuali derivate calcolate non simultaneamente in ogni iterazione.

     

Global convergence of a filter-trust-region algorithm for solving nonsmooth equations

In this paper, we present a new algorithm for solving nonsmooth equations, where the function is locally Lipschitzian. The algorithm attempts to combine the efficiency of filter techniques and the robustness of trust-region method. Global convergence for this algorithm is established under reasonable assumptions.