Iterative methods for nearly singular linear systems

Authors
Citation
Ww. Hager, Iterative methods for nearly singular linear systems, SIAM J SC C, 22(2), 2000, pp. 747-766
Citations number
52
Categorie Soggetti
Mathematics
Journal title
SIAM JOURNAL ON SCIENTIFIC COMPUTING
ISSN journal
10648275 → ACNP
Volume
22
Issue
2
Year of publication
2000
Pages
747 - 766
Database
ISI
SICI code
1064-8275(20000831)22:2<747:IMFNSL>2.0.ZU;2-0
Abstract
Iterative methods are developed and studied for near-singular linear system s Cx = b. Our approach, called the transformed minimal residual algorithm ( TMRES), is derived from any convergent iterative scheme Sx(k+1) = Tx(k) + b associated with a splitting C = S-T. In each step of TMRES, the transforme d residual S-1 (b-Cx) is minimized over a Krylov space generated by S-1T. T he original iterative scheme typically converges slowly when C is nearly si ngular, while a Krylov space generated by S-1T often contains a much better approximation to a solution. TMRES is algebraically equivalent to the gene ralized minimal residual algorithm (GMRES) preconditioned by S-1, although there are numerical differences since a different matrix S-1C is used to ge nerate the Krylov space in preconditioned GMRES. Special attention is given to sparsity and convergence issues related to linear systems of the form ( AA(T) +sigma I) x = b, where sigma greater than or equal to 0.