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Automatic error bounds on real zeros of rational functions

Authors:

Robert H. Dargel,

Frank R. Loscalzo,

Thomas H. WittAuthors Info & Claims

Communications of the ACM, Volume 9, Issue 11

Pages 806 - 809

https://doi.org/10.1145/365876.365913

Published: 01 November 1966 Publication History

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Abstract

A procedure for implementing an interval arithmetic version of the Newton-Raphson method is proposed. The procedure requires only a starting interval over which the zeros of a given rational function are to be located. The method automatically provides bounds for roundoff error.

References

[1]

GORN, SAUL. The automatic analysis and control of computing errors. J. Soc. Indust. AppI. Math. 2 (June 1954), 69-81.

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MOORE, RAMON E. The automatic analysis and control of error in digital computation based on the use of interval numbers. Error in Digital Computation, Vol. I, L. B. Rall (ED.), John Wiley & Sons, Inc., New York, 1965, pp. 61-130.

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--. Interval Analysis. Prentice-Hall, Inc., Englewood Cliffs, N. J., 1966.

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REITER, ALLEN. Interval arithmetic package (INTERVAL). M.R.C. Program #2, COOP Organ. Code-WISC, Math. Res. Center, U. of Wisconsin, Madison, Wis.

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Cited By

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Moore R(2006)Bounding Sets in Function Spaces with Applications to Nonlinear Operator EquationsSIAM Review10.1137/102006820:3(492-512)Online publication date: 2-Aug-2006
https://dl.acm.org/doi/10.1137/1020068
Alefeld GHerzberger J(2006)Über Simultanverfahren zur Bestimmung reeller PolynomwurzelnZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik10.1002/zamm.1974054060654:6(413-420)Online publication date: 22-Nov-2006
https://doi.org/10.1002/zamm.19740540606
Herzberger J(2006)Bemerkungen zu einem Verfahren von R. E. MooreZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik10.1002/zamm.1973053051453:5(356-358)Online publication date: 22-Nov-2006
https://doi.org/10.1002/zamm.19730530514
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Automatic error bounds on real zeros of rational functions
1. Theory of computation

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Let f(z)=@!(t-z)^-^1d@m(t) be a Markov function, where @m is a positive measure with compact support in R. We assume that supp(@m)@?(-1, 1), and investigate the best rational approximants to f in the Hardy space H^0"2(V), where V@?{z@?C||z|>1} and H^0"2(...
Orthogonal rational functions and quadrature on an interval

Rational functions with real poles and poles in the complex lower half-plane, orthogonal on the real line, are well known. Quadrature formulas similar to the Gauss formulas for orthogonal polynomials have been studied. We generalize to the case of ...

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Published In

Communications of the ACM Volume 9, Issue 11

Nov. 1966

44 pages

ISSN:0001-0782

EISSN:1557-7317

DOI:10.1145/365876

Editor:
Gerard Salton
Cornell Univ., Ithaca, NY

Issue’s Table of Contents

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 November 1966

Published in CACM Volume 9, Issue 11

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Moore R(2006)Bounding Sets in Function Spaces with Applications to Nonlinear Operator EquationsSIAM Review10.1137/102006820:3(492-512)Online publication date: 2-Aug-2006
https://dl.acm.org/doi/10.1137/1020068
Alefeld GHerzberger J(2006)Über Simultanverfahren zur Bestimmung reeller PolynomwurzelnZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik10.1002/zamm.1974054060654:6(413-420)Online publication date: 22-Nov-2006
https://doi.org/10.1002/zamm.19740540606
Herzberger J(2006)Bemerkungen zu einem Verfahren von R. E. MooreZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik10.1002/zamm.1973053051453:5(356-358)Online publication date: 22-Nov-2006
https://doi.org/10.1002/zamm.19730530514
Hansen E(2005)A generalized interval arithmeticInterval Mathematics10.1007/3-540-07170-9_2(7-18)Online publication date: 23-May-2005
https://doi.org/10.1007/3-540-07170-9_2
(1983)BIBLIOGRAPHYIntroduction to Interval Computation10.1016/B978-0-08-091636-1.50032-0(309-328)Online publication date: 1983
https://doi.org/10.1016/B978-0-08-091636-1.50032-0
Mancini LWilde D(1980)Interval Arithmetic in Unidimensional Signomial ProgrammingAdvances in Geometric Programming10.1007/978-1-4615-8285-4_19(375-387)Online publication date: 1980
https://doi.org/10.1007/978-1-4615-8285-4_19
Mancini LWilde D(1978)Interval arithmetic in unidimensional signomial programmingJournal of Optimization Theory and Applications10.1007/BF0093340826:2(277-289)Online publication date: Oct-1978
https://doi.org/10.1007/BF00933408

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Orthogonal rational functions and quadrature on the real half line

Asymptotic Error Estimates for L2 Best Rational Approximants to Markov Functions

Orthogonal rational functions and quadrature on an interval

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