On a new iterative method for solving linear systems and comparison results

AbstractIn Ujević [A new iterative method for solving linear systems, Appl. Math. Comput. 179 (2006) 725–730], the author obtained a new iterative method for solving linear systems, which can be considered as a modification of the Gauss–Seidel method. In this paper, we show that this is a special case from a point of view of projection techniques. And a different approach is established, which is both theoretically and numerically proven to be better than (at least the same as) Ujević's. As the presented numerical examples show, in most cases, the convergence rate is more than one and a half that of Ujević

A Modification of Minimal Residual Iterative Method to Solve Linear Systems

We give a modification of minimal residual iteration (MR), which is 1V-DSMR to solve the linear system Ax=b. By analyzing, we find the modifiable iteration to be a projection technique; moreover, the modification of which gives a better (at least the same) reduction of the residual error than MR. In the end, a numerical example is given to demonstrate the reduction of the residual error between the 1V-DSMR and MR

A Novel Partitioning Method for Accelerating the Block Cimmino Algorithm

We propose a novel block-row partitioning method in order to improve the convergence rate of the block Cimmino algorithm for solving general sparse linear systems of equations. The convergence rate of the block Cimmino algorithm depends on the orthogonality among the block rows obtained by the partitioning method. The proposed method takes numerical orthogonality among block rows into account by proposing a row inner-product graph model of the coefficient matrix. In the graph partitioning formulation defined on this graph model, the partitioning objective of minimizing the cutsize directly corresponds to minimizing the sum of inter-block inner products between block rows thus leading to an improvement in the eigenvalue spectrum of the iteration matrix. This in turn leads to a significant reduction in the number of iterations required for convergence. Extensive experiments conducted on a large set of matrices confirm the validity of the proposed method against a state-of-the-art method

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also