Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The Lanczos algorithm with partial reorthogonalization


Author: Horst D. Simon
Journal: Math. Comp. 42 (1984), 115-142
MSC: Primary 65F10; Secondary 65F25
MathSciNet review: 725988
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Lanczos algorithm is becoming accepted as a powerful tool for finding the eigenvalues and for solving linear systems of equations. Any practical implementation of the algorithm suffers however from roundoff errors, which usually cause the Lanczos vectors to lose their mutual orthogonality. In order to maintain some level of orthogonality, full reorthogonalization (FRO) and selective orthogonalization (SO) have been used in the past as a remedy. Here partial reorthogonalization (PRO) is proposed as a new method for maintaining semiorthogonality among the Lanczos vectors. PRO is based on a simple recurrence, which allows us to monitor the loss of orthogonality among the Lanczos vectors directly without computing the inner products. Based on the information from the recurrence, reorthogonalizations occur only when necessary. Thus substantial savings are made as compared to FRO. In some numerical examples we apply the Lanczos algorithm with PRO to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the Lanczos vectors. The results obtained compare favorably with the conjugate gradient method.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65F10, 65F25

Retrieve articles in all journals with MSC: 65F10, 65F25


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1984-0725988-X
PII: S 0025-5718(1984)0725988-X
Article copyright: © Copyright 1984 American Mathematical Society



Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia