Strict optimal a posteriori error and residual bounds for Gaussian elimination in floating-point arithmetic

Abstract

Zusammenfassung Strict Optimal A Posteriori Error and Residual Bounds for Ganssian Elimination in Floating-PointArithmetic. Exact representations of errors and residuals of approximate solutions of linear algebraic systems under data perturbations and rounding errors of a floating-point arithmetic are established from which strict optimal a posteriori error and residual bounds are obtained. These bounds are formulated by means of a posteriori error and residual condition numbers. Condition numbers, error and residual bounds can be computed completely in the range of nonnegative numbers using the arithmetic operations +, x, / only. It is shown that computations in this range are numerically very stable. The general results are applied to a series of numerical examples. AMS-MOS Suhfect Class(fications: 65F. 65G.Key words." Gaussian elimination, rounding error analysis, strict optimal error and residual estimates. Strikte optimale a posteriori Fehler-and Residuenschranken fiir die Ganfl-Elimination in Gleitpunkt-arithmetik. Exakte Darstellungen der Fehler und Residuen yon N~iherungsl6sungen linearer algebraischer Gleichungssysteme unter Datenst6rungen und Rundungsfehlern einer Gleitpunktarithmetik werden hergeleitet, aus denen strikte, optimale, a posteriori Fehler-und Residuenschranken gewonnen werden. Diese Schranken verwenden a posteriori Fehler-und Residuenkonditionszahlen. Die Konditionszahlen, Fehler-und Residuenschranken k6nnen ganz im Bereich nichtnegativer Zahlen nur mit den arithmetischen Operatoren +, x, / berechnet werden. Es wird gezeigt, dab numerische Rechnungen dieser Art sehr stabil sind. Die allgemeinen Ergebnisse werden auf numerische Beispiele angewandt.

A mathematical view of automatic differentiation

“…In particular, it led Seppo Linnainmaa to the first publication of the reverse mode in English (Linnainmaa 1983). The factor l i=1 |v i v i | has been extensively used by Stummel (1981) and others as a condition estimate for the function evaluation procedure. Braconnier and Langlois (2001) used the adjointsv i to compensate evaluation errors.…”

ZAMM ‐ Journal of Applied Mathematics and Mechanics

“…With the computable zh>lzh,O we assume, see where the errors are given for the first components of z. This high accuracy is obtained in essentially the same computation time as the final grid in the KREISSstrategy and is better by many orders of magnitude than in [6]. and, see…”

“…(8) -IIAt; -A / l n l -k , s 5 1~'~~k , J,. (t,) = A , , tt 2 t , , the discrete Newton iteration ( 6 ) i s welldefined for 7c = 1, ... , m and the iterates sati.yfy the error estimntes IIy(k) -x I I , , -~,~ (= ch2L , k = 1, ... , m .…”

Forward error analysis of gaussian elimination

“…For instance, if the coefficients and right-hand sides of the linear systems are rounded to floating-point numbers, the associated data errors are bounded by (1) with r/o=~/R and the above weights (8). (2) The bounds a~ qR,'Cj fiR in (10) are optimal with respect to the rounding error distribution (2) in the sense described above.…”