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


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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  1. 1.
    O. Axelsson, Iterative Solution Methods, Cambridge University Press, New York, 1994.Google Scholar
  2. 2.
    O. Axelsson and V. A. Barker, Finite Element Solutions of Boundary Value Problems. Theory and Computation, Academic Press, New York, 1985.Google Scholar
  3. 3.
    P. Concus, G. H. Golub, and G. Meurant, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6 (1985), pp. 220–252.Google Scholar
  4. 4.
    J. Daniel, W. B. Gragg, L. Kaufman, and G. W. Stewart, Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization, Math. Comp., 30 (1976), pp. 772–95.Google Scholar
  5. 5.
    J. M. Donato and T. F. Chan, Fourier analysis of incomplete factorization preconditioners for three dimensional anisotropic problems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 319–338.Google Scholar
  6. 6.
    H. C. Elman, A stability analysis of incomplete LU factorizations, Math. Comp., 47 (1986), pp. 191–217.Google Scholar
  7. 7.
    H. C. Elman, Relaxed and stabilized incomplete factorizations for non-self-adjoint linear systems, BIT, 29 (1989), pp. 890–915.Google Scholar
  8. 8.
    A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Alg. Appl., 34 (1980), pp. 69–83.Google Scholar
  9. 9.
    A. George and E. Ng, On the complexity of sparse QR and LU factorisation of finite element matrices, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 849–861.Google Scholar
  10. 10.
    J. R Gilbert, E. Ng, and W. Peyton, Separators and structure prediction in sparse orthogonal factorisation, Linear Alg. Appl., 262 (1997), pp. 83–97.Google Scholar
  11. 11.
    I. Gustafsson, A class of first order factorizations, BIT, 18 (1978), pp. 142–156.Google Scholar
  12. 12.
    A. Jennings and M. A. Ajiz, Incomplete methods for solving A′Ax = b, SIAM J. Sci. Statist. Comput., 5:4 (1984), pp. 978–987.Google Scholar
  13. 13.
    T. A. Manteuffel, An incomplete factorization technique for positive definite linear systems, Math. Comp., 34 (1980), pp. 473–497.Google Scholar
  14. 14.
    J. A. Meijerink and H. A. van der Vorst, An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31 (1977), pp. 148–162.Google Scholar
  15. 15.
    Y. Saad, Preconditioning techniques for nonsymmetric and indefinite linear systems, J. Comput. Appl. Math., 24 (1988), pp. 89–105.Google Scholar
  16. 16.
    Y. Saad, ILUT: A dual threshold incomplete ILU factorization, Numer. Linear Algebra Appl., 1 (1994), pp. 387–402.Google Scholar
  17. 17.
    Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, MA, 1996.Google Scholar
  18. 18.
    X. Wang, K. A. Gallivan, and R. Bramley, CIMGS: A incomplete orthogonalization preconditioner, SIAM J. Sci. Comput., 18 (1997), pp. 516–536.Google Scholar
  19. 19.
    Z. Zlatev, Comparison of two pivotal strategies in sparse plane rotations, Internat. J. Comput. Appl., 8 (1982), pp. 119–135.Google Scholar

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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