Interval iteration for zeros of systems of equations

Neumaier, Arnold

doi:10.1007/bf01935003

Cited by 43 publications

(11 citation statements)

References 11 publications

(25 reference statements)

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“…Interval quantities are written in square brackets, matrices in capital letters, vectors and scalars in small letters. The following formulae (which are proved, e.g., in [4], [14], [15]) will be used repeatedly. Let …”

Section: Notation and Basic Factsmentioning

confidence: 99%

Efficient methods for enclosing solutions of systems of nonlinear equations

Frommer

Mayer

1990

Computing

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--ZusammenfassungEfficient Methods for Enclosing Solutions of Systems of Nonlinear Equations. We consider modifications of the interval Newton method which combine two ideas: Reusing the same evaluation of the Jacobian several (say s) times and approximately solving the Newton equation by some 'linear' iterative process. We show in particular that the R-order of these methods may become s + 1. We illustrate our results by a numerical example. AMS Subject Classifications (1985): 65G10, 65H10Key words: Newton's method, Newton-Gauss-Seidel method, nonlinear equations, R-order, interval analysis Effiziente Verfahren zur Einsehlieflung yon Liisungen nichtlinearer Gleichungssysteme. Wir betrachten Modifikationen des Intervall-Newton-Veffahrens, welche zwei Ansgtze miteinander verbinden: Mehrfache (z.B. s-fache) Verwendung derselben Auswertung der Jacobi-Matrix und n~iherungsweise L6sung der Newton-Gleichung mit einem ,,linearen" IterationsprozeB. Insbesondere zeigen wir, dab die ROrdnung dieser Verfahren s + 1 werden kann. Wir illustrieren unsere Ergebnisse an einem numerischen Beispiel.

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Section: Notation and Basic Factsmentioning

confidence: 99%

Efficient methods for enclosing solutions of systems of nonlinear equations

Frommer

Mayer

1990

Computing

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“…Conversely, if (t.4) Xk ~ Xk = 0 then there are no solutions of the system in (1.1) in X k. (cf. [16], theorems 5.1.7, 5.1.8, etc. for specifics.…”

Section: Introduction Notation and Goalsmentioning

confidence: 98%

Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems

Kearfott

1991

Computing

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Abstract--Zusammenfassung Decomposition of Arithmetic Expressions to Improve the Behavior of Interval Iteration for NonlinearSystems. Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration.In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example. AMS Subject Classifications: Primary: 65H10; secondary: 65G10Key words: Nonlinear algebraic systems, interval arithmetic, automatic differentiation. Verbesserung der Intervalliteration fiir niehtlineare Gleichungssysteme durch Zerlegung arithmetischerAusdriicke. Intervalliterationen k6nnen in Verbindung mit anderen Verfahren verwendet werden, um alle Ltsungen eines nichlinearen Gleichungsystems in einem gegebenen Gebiet mit Sicherheit abzuschgtzen, und auch um Approximationen der Ltsungen solcher Systeme zu veriflzieren. Die Abschiitzungen in den Veffahren sind jedoch manchmal nieht hinreichend genau, da (Jbersehfitzungen in der Berechnung uud in dem Gebraueh der Invervall-Jacobi Matrix auflreten.In der vorliegenden Arbeit wcrden Intervalliterationen auf einem erweiterten Gleichungssystem behandelt. In diesem System gibt es keine Oberschiitzungen der Einzelkomponenten der Intervall-Jacobi Matrix, und fiir die Nichtlinearitfiten ktnnen Absch~tzungen angegeben werden. Anhand eines Beispiels wird die Wirkungsweise der behandelten Algorithmen demonstriert.

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“…IA was originally introduced for the computation of the rounding errors when using numerical resolution strategies [5,6]. It was then extended to deal with nonlinear equations [7] as well as optimization problems [8,9]. Recently, this method has been successfully applied to compute the bounds of the radiation pattern when array excitation errors exist [4,[10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Tolerance Analysis of Antenna Array Pattern and Array Synthesis in the Presence of Excitation Errors

Zhang

Zhao

Wang

et al. 2017

International Journal of Antennas and Propagation

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This paper analyzes array pattern tolerance against excitation errors. The nonprobabilistic interval analysis algorithm is used for tolerance analysis of the nonideal uniform linear array in this work. Toward this purpose, corresponding interval models of the power pattern functions are established, respectively, with the consideration of the amplitude errors, phase errors, or both simultaneously, in antenna arrays. The tolerance for the amplitude-phase error of the main function parameters including the beamwidth, sidelobe level, and the directivity is simulated by computer according to the indicators and the actual requirements. Accordingly, the worst admissible performance of an array can be evaluated, which may provide theoretical reference for optimal antenna array design. As for the problem of array synthesis in the presence of various array errors, interval analysis-convex programming (IA-CP) is presented. Simulation results show that the proposed IA-CP based synthesis technique is robust for the amplitude and phase errors.

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Interval iteration for zeros of systems of equations

Cited by 43 publications

References 11 publications

Efficient methods for enclosing solutions of systems of nonlinear equations

Efficient methods for enclosing solutions of systems of nonlinear equations

Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems

Tolerance Analysis of Antenna Array Pattern and Array Synthesis in the Presence of Excitation Errors

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