A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories
 ZhongZhi Bai,
 Iain S. Duff,
 Andrew J. Wathen
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We present a class of incomplete orthogonal factorization methods based on Givens rotations for large sparse unsymmetric matrices. These methods include: Incomplete Givens Orthogonalization (IGOmethod) and its generalisation (GIGOmethod), which drop entries from the incomplete orthogonal and upper triangular factors by position; Threshold Incomplete Givens Orthogonalization (TIGO(τ)method), which drops entries dynamically by their magnitudes; and its generalisation (GTIGO(τ,p)method), which drops entries dynamically by both their magnitudes and positions. Theoretical analyses show that these methods can produce a nonsingular sparse incomplete upper triangular factor and either a complete orthogonal factor or a sparse nonsingular incomplete orthogonal factor for a general nonsingular matrix. Therefore, these methods can potentially generate efficient preconditioners for Krylov subspace methods for solving large sparse systems of linear equations. Moreover, the upper triangular factor is an incomplete Cholesky factorization preconditioner for the normal equations matrix from leastsquares problems.
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 Title
 A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories
 Journal

BIT Numerical Mathematics
Volume 41, Issue 1 , pp 5370
 Cover Date
 20010101
 DOI
 10.1023/A:1021913700691
 Print ISSN
 00063835
 Online ISSN
 15729125
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 Preconditioning
 linear systems
 sparse least squares
 modified GramSchmidt orthogonalization
 Givens rotations
 incomplete orthogonal factorizations
 Industry Sectors
 Authors

 ZhongZhi Bai ^{(1)}
 Iain S. Duff ^{(2)}
 Andrew J. Wathen ^{(3)}
 Author Affiliations

 1. State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, P.O. Box 2719, Beijing, 100080, P. R.China
 2. Atlas Centre, Rutherford Appleton Laboratory, Chilton, Oxon, OX11 0QD, England, UK
 3. Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK