Almost block diagonal linear systems: sequential and parallel solution techniques, and applications. (English) Zbl 1051.65018

Amodio, P.; Cash, J. R.; Roussos, G.; Wright, R. W.; Fairweather, G.; Gladwell, I.; Kraut, G. L.; Paprzycki, M.

Numer. Linear Algebra Appl. 7, No. 5, 275-317 (2000).

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently.
The authors survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise when boundary value problems are solved by means of finite differences, multiple shooting and orthogonal spline collocation. They outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, they examine parallel iterative methods for solving BABD systems. The survey character of the paper is strengthened by references numbering 149 entries.

Reviewer: Jan Chleboun (Praha)

MSC:

65F05	Direct numerical methods for linear systems and matrix inversion
65F10	Iterative numerical methods for linear systems
65Y05	Parallel numerical computation
65L10	Numerical solution of boundary value problems involving ordinary differential equations
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
65L12	Finite difference and finite volume methods for ordinary differential equations
65L60	Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05	Linear boundary value problems for ordinary differential equations

Keywords:

almost block diagonal systems; direct algorithms; iterative algorithms; parallel computing; boundary value problems; collocation methods; multiple shooting methods; numerical examples; finite differences; survey paper

Software:

PMIRKDC; NAG; BLAS; nag; COLSYS

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