Perturbation Theory for Evaluation Algorithms of Arithmetic Expressions

Abstract

Abstract. The paper presents the theoretical foundation of a forward error analysis of numerical algorithms under data perturbations, rounding error in arithmetic floating-point operations, and approximations in 'built-in' functions. The error analysis is based on the linearization method that has been proposed by many authors in various forms. Fundamental tools of the forward error analysis are systems of linear absolute and relative a priori and a posteriori error equations and associated condition numbers constituting optimal bounds of possible accumulated or total errors. Derivations, representations, and properties of these condition numbers are studied in detail. The condition numbers enable simple general, quantitative definitions of numerical stability, backward analysis, well-and ill-conditioning of a problem and an algorithm. The well-known illustration of algorithms and their linear error equations by graphs is extended to a method of deriving condition numbers and associated bounds. For many algorithms the associated condition numbers can be determined analytically a priori and be computed numerically a posteriori. The theoretical results of the paper have been applied to a series of concrete algorithms, including Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.

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

“…The term pi r/is, save for terms of second order in r/, a least upper bound for I P x~l with respect to all perturbations in the distribution (1),(2), which shall be proved now. The term pi r/is, save for terms of second order in r/, a least upper bound for I P x~l with respect to all perturbations in the distribution (1),(2), which shall be proved now.…”

“…with r/R = 5-10-7 and the condition numbers z R = z ~ + z~ a, fir = if0 + ffl i = 1 .... ,4, from _<(fif+4.527-02)t/R, I(C ~-z)il <-rf nR, (2) for t/R = 5.10 4, ~, r~ in (1), and i= 1,..., 4.…”

A stochastic roundoff error analysis for the convolution

“…The Xks, homogeneous linear functions of the local errors e = (ex, ... , ek), are known as the accompanying linear forms of the algorithm [10,11]. The firstorder absolute errors for the intermediate and final results of the algorithms to be considered are completely described by these forms.…”

Optimized Cramer’s Rule in WZ Factorization and Applications

“…However, Dunham [18] gave a counter example of 2 × 2 linear system to show that Cramer's rule is sufficient. Linear systems, especially 2 × 2 linear equations, solved by Cramer's rule can be forward stable or backward stable depending on the conditioning of the system [31,40]. Cramer's rule and Gaussian elimination requires about the same amount of arithmetic operations for finding the solution of 2 × 2 linear systems, but Cramer's rule yields a highly accuracy and stability than Gaussian elimination even with pivoting [16,29].…”