LU implementation of the modified minimal polynomial extrapolation method for solving linear and nonlinear systems

We give an efficient implementation of the modified minimalpolynomial extrapolation (MMPE) method for solving linear andnonlinear systems. We will show how to choose the auxiliaryvectors in the MMPE method such that the resulting approximationsare always defined. This new implementation, which is basedon an LU factorization with a pivoting strategy, is inexpensiveboth in time and storage as compared with other extrapolationmethods.     

Application of homotopy perturbation methods for solving systems of linear equations

In this paper, homotopy perturbation methods (HPMs) are applied to obtain the solution of linear systems, and conditions are deduced to check the convergence of the homotopy series. Moreover, we have adapted the Richardson method, the Jacobi method, and the Gauss-Seidel method to choose the splitting matrix. The numerical results indicate that the homotopy series converges much more rapidly than the direct methods for large sparse linear systems with a small spectrum radius.     

A generalized iterative method and comparison results using projection techniques for solving linear systems

In [Y.-F. Jing, T.-Z. Huang, On a new iterative method for solving linear systems and comparison results, J. Comput. Appl. Math. 220 (2008) 74–84], Jing and Huang obtained a new iterative method for solving linear systems. This method can be considered as a projection method which uses a two-dimensional space at each step. In this paper, we generalize this method to a three-dimensional projection process. And a different approach is established, which is both theoretically and numerically proven to be better than (or at least the same as) [Jing and Huang’s (2008)].     

Parallel algorithms for solving large linear systems

The solution of linear systems continues to play an important role in scientific computing. The problems to be solved often are of very large size, so that solving them requires large computer resources. To solve these problems, at least supercomputers with large shared memory or massive parallel computer systems with distributed memory are needed.

This paper gives a survey of research on parallel implementation of various direct methods to solve dense linear systems. In particular are considered: Gaussian elimination, Gauss-Jordan elimination and a variant due to Huard (1979), and an algorithm due to Enright (1978), designed in relation to solving (stiff) ODEs, such that stepsize and other method parameters can easily be varied.

Some theoretical results are mentioned, including a new result on error analysis of Huard's algorithm. Moreover, practical considerations and results of experiments on supercomputers and on a distributed-memory computer system are presented.     

An optimal regularized projection method for solving ill-posed problems via dynamical systems method

A new regularized projection method was developed for numerically solving ill-posed equations of the first kind. This method consists of combining the dynamical systems method with an adaptive projection discretization scheme. Optimality of the proposed method was proved on wide classes of ill-posed problems.     

A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations

An iterative method based on Lanczos bidiagonalization is developed for computing regularized solutions of large and sparse linear systems, which arise from discretizations of ill-posed problems in partial differential or integral equations. Determination of the regularization parameter and termination criteria are discussed. Comments are given on the computational implementation of the algorithm.Dedicated to Peter Naur on the occasion of his 60th birthday     

A procedure for solving some second-order linear ordinary differential equations

The purpose of this work is to introduce the concept of pseudo-exactness for second-order linear ordinary differential equations (ODEs), and then to try to solve some specific ODEs.     

A primal-dual projection method for solving systems of linear inequalities

An algorithm is described for finding a feasible point for a system of linear inequalities. If the solution set has nonempty interior, termination occurs after a finite number of iterations. The algorithm is a projection-type method, similar to the relaxation methods of Agmon, Motzkin, and Schoenberg. It differs from the previous methods in that it solves for a certain “dual” solution in addition to a primal solution.     

A genetic algorithm for solving linear fractional bilevel problems

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. In this paper a genetic algorithm is proposed for the class of bilevel problems in which both level objective functions are linear fractional and the common constraint region is a bounded polyhedron. The algorithm associates chromosomes with extreme points of the polyhedron and searches for a feasible solution close to the optimal solution by proposing efficient crossover and mutation procedures. The computational study shows a good performance of the algorithm, both in terms of solution quality and computational time.     

A QMR-based interior-point algorithm for solving linear programs

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2×2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3×3-block systems to symmetric 2×2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefinite preconditioners are discussed. Finally, we report results of a few preliminary numerical experiments to illustrate the features of the new approach.     

A minimum residual algorithm for solving linear systems

A minimum residual algorithm for solving a large linear system (I+S)x=b, with b∈ℂ ⁿ and S∈ℂ ^n×n being readily invertible, is proposed. For this purpose Krylov subspaces are generated by applying S and S ^-1 cyclically. The iteration can be executed with every linear system once the coefficient matrix has been split into the sum of two readily invertible matrices. In case S is a translation and a rotation of a Hermitian matrix, a five term recurrence is devised. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65F10     

A new look at the Lanczos algorithm for solving symmetric systems of linear equations

Simple versions of the conjugate gradient algorithm and the Lanczos method are discussed, and some merits of the latter are described. A variant of Lanczos is proposed which maintains robust linear independence of the Lanczos vectors by keeping them in secondary storage and occasionally making use of them. The main applications are to problems in which (1) the cost of the matrix-vector product dominates other costs, (2) there is a sequence of right hand sides to be processed, and (3) the eigenvalue distribution of A is not too favorable.     

A parallel algorithm for solving Toeplitz linear systems

Numerical methods of solution are considered for systems which are Toeplitz and symmetric. In our case, the coefficient matrix is essentially tridiagonal and sparse. There are two distinct approaches to be considered each of which is efficient in its own way. Here we will combine the two approaches which will allow application of the cyclic reduction method to coefficient matrices of more general forms. The convergence of the approximations to the exact solution will also be examined. Solving linear systems by the adapted cyclic reduction method can be parallel processed.     

A fast algorithm for solving systems of linear equations with two variables per equation

We present a fast algorithm for solving m X n systems of linear equations Ax = c with at most two variables per equation. The algorithm makes use of a linear-time algorithm for constructing a spanning forest of an undirected graph, and it requires 5m + 2n – 2 arithmetic operations in the worst case.     

Preconditioning projection methods for solving algebraic linear systems

Numerical experiments have shown that projection methods are robust for solving the problem of finding a point satisfying a linear system of n variables and m equations; however, their qualities of convergence depend on certain parameters: an n × n symmetric positive definite matrix M, and a vector u with m components. We are concerned here with the choice of M. Through a link with Conjugate Gradient methods we determine an expedient M. Preliminary numerical results on a hard 3D partial differential equation are highly promising. We solve a discretized system that could not be solved by conventional methods. We also give hints on how to adapt our findings to the solution of a linear system of inequalities. This is the first stage of a forthcoming research. This revised version was published online in June 2006 with corrections to the Cover Date.     

A parallel algorithm for solving the linear complementarity problem

The concept of multitasking mathematical programs is discussed, and an application of multitasking to the multiple-cost-row linear programming problem is considered. Based on this, an algorithm for solving the Linear Complementarity Problem (LCP) in parallel is presented. A variety of computational results are presented using this multitasking approach on the CRAY X-MP/48. These results were obtained for randomly generated LCP's where thenxn dense matrixM has no special properties (hence, the problem is NP-hard). based on these results, an average time performance ofO(n ⁴) is observed.     

Computational acceleration of projection algorithms for the linear best approximation problem

This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra’s algorithm and the Halpern-Lions-Wittmann-Bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the Euclidean space. These algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes. We compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms. For the simultaneous versions we present a new component-averaging variant that substantially accelerates their initial behavior for sparse systems.     

A projection method and Kronecker product preconditioner for solving Sylvester tensor equations

The preconditioned iterative solvers for solving Sylvester tensor equations are considered in this paper.By fully exploiting the structure of the tensor equation,we propose a projection method based on the tensor format,which needs less flops and storage than the standard projection method.The structure of the coefficient matrices of the tensor equation is used to design the nearest Kronecker product(NKP) preconditioner,which is easy to construct and is able to accelerate the convergence of the iterative solver.Numerical experiments are presented to show good performance of the approaches.     

Optimization of the quasi-Monte Carlo algorithm for solving systems of linear algebraic equations

An interesting conclusion about error reduction of the modified quasi-Monte Carlo method for solving systems of linear algebraic equations is suggested. The Monte Carlo method is compared with the quasi-Monte Carlo method and its modification. The optimal choice of the parameters of the Markov chain for the modified Monte Carlo method applied to solving systems of linear equations is substantiated.