BIT Numerical Mathematics

, Volume 24, Issue 1, pp 102–112 | Cite as

Backward error analysis for linear systems associated with inverses ofH-matrices

  • M. Neumann
  • R. J. Plemmons
Part II Numerical Mathematics


In this paper, bounds on the growth factors resulting from Gaussian elimination applied to inverses ofH-matrices are developed and investigated. These bounds are then used in the error analysis for solving linear systemsAx =b whose coefficient matricesA are of this type. For each such system our results show that the Gaussian elimination without pivoting can proceed safely provided that the elements of the inverse of a certainM-matrix (associated with the coefficient matrixA) are not excessively large. We exhibit a particularly satisfactory situation for the special case whenA itself is an inverse of anM-matrix. Part of the first section of this paper is devoted to a discussion on some results of de Boor and Pinkus for the stability of triangular factorizations of systemsAx =b, whereA is a nonsingular totally nonnegative matrix, and to the explanation of why the analysis of de Boor and Pinkus is not applicable to the case when the coefficient matrixA is an inverse of anM-matrix.


Growth Factor Linear System Computational Mathematic Error Analysis Coefficient matrixA 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Anderssen and P. Bloomfield,A time series approach to numerical differentiation, Technometrics 16 (1974), 69–75.Google Scholar
  2. 2.
    A. Berman and R. J. Plemmons,Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.Google Scholar
  3. 3.
    C. de Boor and A. Pinkus,Backward error analysis for totally positive linear systems, Numer. Math. 27 (1977), 485–490.CrossRefGoogle Scholar
  4. 4.
    M. Fiedler and V. Pták,Diagonally dominant matrices, Czech. Math. J. 17 (1967), 420–433.Google Scholar
  5. 5.
    R. E. Funderlic, M. Neumann and R. J. Plemmons,LU factorizations of generalized diagonally dominant matrices, Numer. Math. 40 (1982), 57–69.CrossRefGoogle Scholar
  6. 6.
    F. R. Gantmacher,The Theory of Matrices, Vol. I, Chelsea Publ., New York, 1959.Google Scholar
  7. 7.
    H. S. Leff,Correlation inequalities for coupled oscillators, J. Math. Phys. 12 (1971), 569–578.CrossRefGoogle Scholar
  8. 8.
    M. Lewin,Totally nonnegative, M-, and Jacobi matrices, SIAM J. Alg. Dis. Meth. 1 (1980), 419–421.Google Scholar
  9. 9.
    T. L. Markham,On oscillatory matrices, Lin. Alg. Appl. 3 (1970), 143–156.CrossRefGoogle Scholar
  10. 10.
    T. L. Markham,Factorizations of nonnegative matrices, Proc. AMS 32 (1972), 45–47.Google Scholar
  11. 11.
    A. Ostrowski,Determinanten mit überwiegender Hauptdiagonale und die absolute Konvergenz von Iterationprozessen, Comment. Math. Helv. 30 (1956), 175–210.Google Scholar
  12. 12.
    J. K. Reid,A note on the stability of Gaussian elimination, J. Inst. Math. Appl. 8 (1971), 374–375.Google Scholar
  13. 13.
    G. W. Stewart,Introduction to Matrix Computations, Academic Press, New York, 1973.Google Scholar
  14. 14.
    R. S. Varga,On recurring theorems on diagonal dominance, Lin. Alg. Appl. 13 (1976), 1–9.CrossRefGoogle Scholar
  15. 15.
    L. J. Watford,The Schur complement of a generalized M-matrix, Lin. Alg. Appl. 5 (1972), 247–255.CrossRefGoogle Scholar
  16. 16.
    H. Wielandt,Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642–648.Google Scholar
  17. 17.
    J. H. Wilkinson,Error analysis of direct methods of matrix inversion, J. Assoc. Comp. Mach. 8 (1961), 281–330.Google Scholar
  18. 18.
    R. A. Willoughby,The inverse M-matrix problem, Lin. Alg. Appl. 18 (1977), 75–94.CrossRefGoogle Scholar

Copyright information

© BIT Foundations 1984

Authors and Affiliations

  • M. Neumann
    • 1
    • 2
  • R. J. Plemmons
    • 1
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of South CarolinaColumbiaUSA
  2. 2.Departments of Mathematics and Computer ScienceNorth Carolina State UniversityRaleighUSA

Personalised recommendations