BIT Numerical Mathematics

, Volume 41, Issue 1, pp 53–70

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

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

DOI: 10.1023/A:1021913700691

Cite this article as:
Bai, ZZ., Duff, I.S. & Wathen, A.J. BIT Numerical Mathematics (2001) 41: 53. doi:10.1023/A:1021913700691


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.

Preconditioninglinear systemssparse least squaresmodified Gram-Schmidt orthogonalizationGivens rotationsincomplete orthogonal factorizations

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