BIT Numerical Mathematics

, Volume 41, Issue 1, pp 53–70 | Cite as

A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories

  • Zhong-Zhi Bai
  • Iain S. Duff
  • Andrew J. Wathen
Article

Abstract

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.

Preconditioning linear systems sparse least squares modified Gram-Schmidt orthogonalization Givens rotations incomplete orthogonal factorizations 

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Copyright information

© Swets & Zeitlinger 2001

Authors and Affiliations

  • Zhong-Zhi Bai
    • 1
  • Iain S. Duff
    • 2
  • Andrew J. Wathen
    • 3
  1. 1.State Key Laboratory of Scientific/Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering ComputingChinese Academy of SciencesBeijingP. R.China
  2. 2.Atlas CentreRutherford Appleton LaboratoryOxonEngland, UK
  3. 3.Oxford University Computing LaboratoryOxfordUK

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