NDLI: Improved Gram–Schmidt Type Downdating Methods

Content Provider	Springer Nature Link
Author	Barlow, Jesse L. Smoktuwicz, Alicja Erbay, Hasan
Copyright Year	2005
Abstract	The problem of deleting a row from a Q–R factorization (called downdating) using Gram–Schmidt orthogonalization is intimately connected to using classical iterative methods to solve a least squares problem with the orthogonal factor as the coefficient matrix. Past approaches to downdating have focused upon accurate computation of the residual of that least squares problem, then finding a unit vector in the direction of the residual that becomes a new column for the orthogonal factor. It is also important to compute the solution vector of the related least squares problem accurately, as that vector must be used in the downdating process to maintain good backward error in the new factorization. Using this observation, new algorithms are proposed.One of the new algorithms proposed is a modification of one due to Yoo and Park [BIT, 36:161–181, 1996]. That algorithm is shown to be a Gram–Schmidt procedure.Also presented are new results that bound the loss of orthogonality after downdating. An error analysis shows that the proposed algorithms’ behavior in floating point arithmetic is close to their behavior in exact arithmetic. Experiments show that the changes proposed in this paper can have a dramatic impact upon the accuracy of the downdated Q–R decomposition.
Starting Page	259
Ending Page	285
Page Count	27
File Format	PDF
ISSN	00063835
Journal	BIT Numerical Mathematics
Volume Number	45
Issue Number	2
e-ISSN	15729125
Language	English
Publisher	Springer-Verlag
Publisher Date	2005-01-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	orthogonalization Q–R factorization matrix splittings residual Numeric Computing Mathematics Computational Mathematics and Numerical Analysis
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computer Networks and Communications Computational Mathematics Software

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