Solution of Linear Systems

• Published in: Handbook of Numerical Analysis Vol. 13; pp. 755 - 824
• Main Authors: Schenk, O., van der Vorst, H.A.
• Format: Book Chapter
• Language: English
• Published: The Netherlands Elsevier Science & Technology 2005
• Subjects: ELECTRONICS & COMMUNICATIONS ENGINEERING; Maths for engineers; Signal processing
• ISBN: 9780444513755; 0444513752
• ISSN: 1570-8659
• DOI: 10.1016/S1570-8659(04)13008-1

This chapter gives an overview over different sparse direct methods and the related literature. The chapter discusses state of art numerical methods for the solution of linear systems. These methods fall belong to the traditional two different classes: direct solution methods based on Gaussian elimination techniques and iterative methods. Developing an efficient parallel or even serial, direct solver for sparse systems of linear equations is a challenging task that has been a subject of research for the past four decades. Recent algorithmic improvements alone have reduced the time required for the sequential direct solution of unsymmetric sparse systems of linear equations by almost an order of magnitude. The main conclusions that can be drawn from this chapter are: (1) the multilevel nested dissection algorithm used to find a fill-in reducing ordering is substantially better than multiple minimum degrees for three-dimensional (3D) irregular matrices, and (2) the multiple minimum degree method performs better for most of the two-dimensional (2D) problems.