Iterative refinement implies numerical stability for Gaussian elimination

Skeel, Robert D.

doi:10.1090/S0025-5718-1980-0572859-4

Iterative refinement implies numerical stability for Gaussian elimination
HTML articles powered by AMS MathViewer

by Robert D. Skeel PDF

Math. Comp. 35 (1980), 817-832 Request permission

Abstract:

Because of scaling problems, Gaussian elimination with pivoting is not always as accurate as one might reasonably expect. It is shown that even a single iteration of iterative refinement in single precision is enough to make Gaussian elimination stable in a very strong sense. Also, it is shown that without iterative refinement row pivoting is inferior to column pivoting in situations where the norm of the residual is important.

References

F. L. Bauer, Optimally scaled matrices, Numer. Math. 5 (1963), 73–87. MR 159412, DOI 10.1007/BF01385880
F. L. Bauer, J. Stoer, and C. Witzgall, Absolute and monotonic norms, Numer. Math. 3 (1961), 257–264. MR 130104, DOI 10.1007/BF01386026
George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler, Computer methods for mathematical computations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. MR 0458783
George E. Forsythe and Cleve B. Moler, Computer solution of linear algebraic systems, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219223

Numerical Errors in Sparse Linear Equations

Introduction to Applied Numerical Analysis

M. Jankowski and H. Woźniakowski, Iterative refinement implies numerical stability, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), no. 3, 303–311. MR 464560, DOI 10.1007/bf01932150
Webb Miller, On the stability of finite numerical procedures, Numer. Math. 19 (1972), 425–432. MR 351066, DOI 10.1007/BF01404925
Webb Miller and Celia Wrathall, Software for roundoff analysis of matrix algorithms, Computer Science and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 595503
W. Oettli and W. Prager, Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numer. Math. 6 (1964), 405–409. MR 168106, DOI 10.1007/BF01386090

Gaussian Elimination and Numerical Instability

Robert D. Skeel, Scaling for numerical stability in Gaussian elimination, J. Assoc. Comput. Mach. 26 (1979), no. 3, 494–526. MR 535268, DOI 10.1145/322139.322148
G. W. Stewart, Introduction to matrix computations, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0458818
J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0161456