Fast Linear System Research Articles

Fast Linear-System Solver for Shielding Current Analysis in High-Temperature Superconducting Thin Film without Crack: Development of H-Matrix Based Preconditioned Conjugate Gradient Method

Fast Linear-System Solver for Shielding Current Analysis in High-Temperature Superconducting Thin Film without Crack: Development of H-Matrix Based Preconditioned Conjugate Gradient Method

A fast algorithm for computing the Smith normal form with multipliers for a nonsingular integer matrix

A Las Vegas randomized algorithm is given to compute the Smith multipliers for a nonsingular integer matrix A , that is, unimodular matrices U and V such that A V = U S , with S the Smith normal form of A . The expected running time of the algorithm is about the same as required to multiply together two matrices of the same dimension and size of entries as A . Explicit bounds are given for the size of the entries in both unimodular multipliers. The main tool used by the algorithm is the Smith massager, a relaxed version of V , the unimodular matrix specifying the column operations of the Smith computation. From the perspective of efficiency, the main tools used are fast linear system solving and partial linearization of integer matrices. As an application of the Smith with multipliers algorithm, a fast algorithm is given to find the fractional part of the inverse of the input matrix.

Block pivoting implementation of a symmetric Toeplitz solver

Toeplitz matrices are characterized by a special structure that can be exploited in order to obtain fast linear system solvers. These solvers are difficult to parallelize due to their low computational cost and their closely coupled data operations. We propose to transform the Toeplitz system matrix into a Cauchy-like matrix since the latter can be divided into two independent matrices of half the size of the system matrix and each one of these smaller arising matrices can be factorized efficiently in multicore computers. We use OpenMP and store data in memory by blocks in consecutive positions yielding a simple and efficient algorithm. In addition, by exploiting the fact that diagonal pivoting does not destroy the special structure of Cauchy-like matrices, we introduce a local diagonal pivoting technique which improves the accuracy of the solution and the stability of the algorithm. • We improve the solution of symmetric Toeplitz linear systems in multicore systems. • We transform the Toeplitz matrix into a Cauchy-like one to obtain some benefits. • The problem is partitioned into two half-sized independent problems. • We use partial local pivoting to improve the accuracy of the solution. • We propose a special scheme to store data in memory that accelerates the algorithm.

Numerical Algorithms Based on Characteristic Domain Decomposition for Obstacle Problems

A new numerical solution algorithm for obstacle problems is proposed, where the characteristic domain decomposition into active and inactive subdomains separated by the free boundary is approximated by a Schwarz method. Such an approach gives an opportunity to apply fast linear system solvers to genuinely non-linear obstacle problems. Other solution algorithms, like projected relaxation methods and active set strategies, are compared to the new solution algorithm. Numerical experiments related to the elastoplastic torsion problem are included showing the efficiency of the new approach. © 1997 John Wiley & Sons, Ltd.

Fast linear system solution by neural networks

We present a highly parallel model for solving the m × n linear system Ax = b , based on the connectionist paradigm of neural networks. Using m + n linear, perceptron-like neurons, the neutral model converges fast to a problem solution, when one exists, or to a least-squares approximation, when the problem is inconsistent. The velocity of convergence of an analog realization of the network does not depend on m or n , but rather on the eigenvalues of A T A . Simulation results in linear system solution and suggections for applications are given.