Forward error analysis of gaussian elimination

StummelFriedrich,

doi:10.1007/bf01389492

Cited by 7 publications

(5 citation statements)

References 10 publications

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“…Where: Cond (A) = ‖ ‖ ‖ ‖ condition number. [4], Cond ‖ ‖ = denote the spectral norm of A (The norm of a matrix is in some sense a measure of the magnitude of the matrix). ‖ ‖ | ( )| ⁄ The spectral norm.…”

Section: Remarkmentioning

confidence: 99%

A Note on the Perturbation of arithmetic expressions

Journal¹

2016

Baghdad Sci.J

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In this paper we present the theoretical foundation of forward error analysis of numerical algorithms under;• Approximations in "built-in" functions.• Rounding errors in arithmetic floating-point operations.• Perturbations of data.The error analysis is based on linearization method. The fundamental tools of the forward error analysis are system of linear absolute and relative a prior and a posteriori error equations and associated condition numbers constituting optimal of possible cumulative round – off errors. The condition numbers enable simple general, quantitative bounds definitions of numerical stability. The theoretical results have been applied a Gaussian elimination, and have proved to be very effective means of both a priori and a posteriori error analysis.

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Section: Remarkmentioning

confidence: 99%

A Note on the Perturbation of arithmetic expressions

Journal¹

2016

Baghdad Sci.J

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“…By substituting the representations (5) for the absolute rounding errors in (3), (4), error and residual representations are obtained whose a priori first order approximations have been determined already in [5], 2. (23), (24), (25).…”

Section: A2-y= Aa X = -(A a 1 + F) ~C + Lfmentioning

confidence: 99%

“…Further, the solution of a five-point difference approximation of a Dirichlet boundary-value problem for the Poisson equation has been studied in [5]. Further, the solution of a five-point difference approximation of a Dirichlet boundary-value problem for the Poisson equation has been studied in [5].…”

Section: Examplesmentioning

confidence: 99%

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

Stummel

1986

Computing

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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.

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“…The accompanying linear forms that we use to describe the first-order effects of the local relative errors on the intermediate results, and therefore on the output, were introduced by Stummel and Heiner [20] (see also Stummel [19]). For fixed-point arithmetic this technique was used earlier by Henrici [11].…”

Section: Literaturementioning

confidence: 99%

A Stochastic Roundoff Error Analysis for the Fast Fourier Transform

Calvetti¹

1991

Mathematics of Computation

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Abstract. We study the accuracy of the output of the Fast Fourier Transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication. We compare the results with the corresponding ones for the direct algorithm for the Discrete Fourier Transform, and we give indications of the relative performances when different rounding schemes are used. We also present the results of numerical experiments run to test the theoretical bounds and discuss their significance.

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Forward error analysis of gaussian elimination

Cited by 7 publications

References 10 publications

A Note on the Perturbation of arithmetic expressions

A Note on the Perturbation of arithmetic expressions

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

A Stochastic Roundoff Error Analysis for the Fast Fourier Transform

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