Mixed block elimination for linear systems with wider borders

The paper is about the stable solution of possibly ill-conditionedbordered linear systems. Given stable solvers for matrix A andfor A^T, we prove that the Govaerts Mixed Block Elimination (BEM)method constitutes a stable solver for the matrix consistingof A or A^T with a border of width 1, and hence by recursionfor a border of any width. We express the algorithm in an efficient,iterative, form. We analyse its operation count, and verifythe theory by extensive numerical experiments. ^*Senior Research Associate of the Belgian National Fund of ScientificResearch NFWO.     

On the Convergence of Iterated Remeshing

In the context of constructing a one-dimensional space-meshby equidistributing a weight function (monitor function) asintroduced by de Boor, we prove that under not too unrealistican approximation to the process as carried out in practice,iterating the remeshing process gives a convergent sequenceof meshes. The limit mesh has useful asymptotic smoothness properties.We suggest ways in which this could be exploited for improvederror control in BVPs. Numerical experiments show the convergencebehaviour, and give preliminary support to the error-controlproposal.     

Error Control of Phase-Function Shooting Methods for Sturm-Liouville Problems

A popular method for computing eigenvalues and eigenfunctionsof Sturm-Liouville systems is to convert to phase-amplitudevariables (, r) in the (y', y)- plane (a Prüer transformation)and solve the resulting equations by a shooting method usingan ODE initial-value code. We study the error control of suchmethods with specific reference to two published codes basedon this technique, and make comparisons and recommendations.Improved error estimates are derived and evidence for theirefficacy presented.     

Multiplicative Error Analysis of Matrix Transformation Algorithms

The author's recently introduced relative error measure forvectors is applied to the error analysis of algorithms whichproceed by successive transformation of a matrix. Instead ofmodelling the roundoff errors at each stage by A: = T(A)+E onemodels them by A: =e^E T(A) where E is a small linear transformation.This can simplify analyses considerably. Applications to theparallel Jacobi method for eigenvalues, and to Gaussian elimination,are given.