A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories
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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 (IGO-method) and its generalisation (GIGO-method), 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 least-squares problems.
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- A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories
BIT Numerical Mathematics
Volume 41, Issue 1 , pp 53-70
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
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- linear systems
- sparse least squares
- modified Gram-Schmidt orthogonalization
- Givens rotations
- incomplete orthogonal factorizations
- Industry Sectors
- 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