Pairwise Pivoting Research Articles

Growth factors of pivoting strategies associated with Neville elimination

Neville elimination is a direct method for solving linear systems. Several pivoting strategies for Neville elimination, including pairwise pivoting, are analyzed. Bounds for two different kinds of growth factors are provided. Finally, an approximation of the average normalized growth factor associated with several pivoting strategies is computed and analyzed using random matrices.

Communication-efficient parallel generic pairwise elimination

The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. In this paper, we consider the parallel complexity of generic pairwise elimination, special cases of which include Gaussian elimination with pairwise pivoting, Gaussian elimination over a finite field, generic Neville elimination and Givens reduction. We develop a new block-recursive, communication-efficient BSP algorithm for generic pairwise elimination.

A fault tolerant systolic mesh for linear system solution

A systolic N × (N + 1) mesh to solve a system of TV linear equations, using a variant of Gaussian elimination algorithm (GE), called successive GE (SGE), was presented in [2]. To ensure numerical stability, the authors have incorporated pairwise pivoting in the algorithm. However, they do not address the important issue of handling oversized problems, i.e., their design is not suitable for solving linear systems of arbitrary size. In this paper, in order to add flexibility to their design, we present two problem partitioning techniques so that problems of any size can be solved on the mesh. Further, in order to add robustness to their design, we suggest two fault tolerant schemes for handling processor failures. The first one is based on time redundancy and has 100% time overhead. It can tolerate upto 50% processor failures at no extra hardware cost. The second scheme is based on algorithm based fault tolerance and uses IN extra processors to tolerate upto 2N processor failures with very little time overhead.

On multiple error detection in matrix triangularizations using checksum methods

We introduced a unified checksum method for multiple transient error detection in three different matrix triangularizations: the LU decomposition, Gaussian elimination with pairwise pivoting, and the QR decomposition. We first develop the theoretical back-ground for multiple error detection in matrix triangularizations, and summarize the results by providing that we can detect all the transient errors that occur in a maximum of t different columns by introducing t checksum vectors. A floating-point error analysis, to determine the effects of the rounding errors in using the checksum method for multiple error detection and correction, is also presented.

An analysis of algorithm-based fault tolerance techniques

We introduce a unified checksum scheme for the LU decomposition, Gaussian elimination with pairwise pivoting, and the QR decomposition. The purpose is to detect and locate a transient error during a systolic array computation. We show how to represent the error as a rank-one perturbation to the original data, so that we need not worry when the error occurred. Finally, we perform a floating point error analysis to determine the effects of rounding errors on the checksum scheme.

Fault-tolerant matrix triangularizations on systolic arrays

Examines the checksum methods of Abraham et al. for LU decomposition on multiprocessor arrays. Their methods are efficient for detecting a transient error, but expensive for correcting it due to the need for a computation rollback. The authors show how to avoid the rollback by using matrix updating techniques, and they introduce new checksum methods for Gaussian elimination with pairwise pivoting and for QR decomposition on systolic arrays.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Analysis of Pairwise Pivoting in Gaussian Elimination

The method of Gaussian elimination using triangularization by elementary stabilized matrices constructed by pairwise pivoting is analyzed. It is shown that a variant of this scheme which is suitable for implementation on a paralle computer is numerically stable although the bound is larger than the one for the standard partial pivoting algorithm.