Backward error analysis for linear systems associated with inverses ofH-matrices
- 57 Downloads
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.
KeywordsGrowth Factor Linear System Computational Mathematic Error Analysis Coefficient matrixA
Unable to display preview. Download preview PDF.
- 1.R. Anderssen and P. Bloomfield,A time series approach to numerical differentiation, Technometrics 16 (1974), 69–75.Google Scholar
- 2.A. Berman and R. J. Plemmons,Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.Google Scholar
- 4.M. Fiedler and V. Pták,Diagonally dominant matrices, Czech. Math. J. 17 (1967), 420–433.Google Scholar
- 6.F. R. Gantmacher,The Theory of Matrices, Vol. I, Chelsea Publ., New York, 1959.Google Scholar
- 8.M. Lewin,Totally nonnegative, M-, and Jacobi matrices, SIAM J. Alg. Dis. Meth. 1 (1980), 419–421.Google Scholar
- 10.T. L. Markham,Factorizations of nonnegative matrices, Proc. AMS 32 (1972), 45–47.Google Scholar
- 11.A. Ostrowski,Determinanten mit überwiegender Hauptdiagonale und die absolute Konvergenz von Iterationprozessen, Comment. Math. Helv. 30 (1956), 175–210.Google Scholar
- 12.J. K. Reid,A note on the stability of Gaussian elimination, J. Inst. Math. Appl. 8 (1971), 374–375.Google Scholar
- 13.G. W. Stewart,Introduction to Matrix Computations, Academic Press, New York, 1973.Google Scholar
- 16.H. Wielandt,Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642–648.Google Scholar
- 17.J. H. Wilkinson,Error analysis of direct methods of matrix inversion, J. Assoc. Comp. Mach. 8 (1961), 281–330.Google Scholar