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Improving the efficiency index in enclosing a root of an equation

Повыщение индекса эффективности нахождения оценки корня уравнения

  • Mathematical Research
  • Published:
Reliable Computing

Abstract

Recently several algorithms have been developed which achieve high efficiency index in enclosing a root of the equationf(x)=0 in an interval [a, b] over whichf(x) is continuous andf(a)f(b)<0. The highest efficiency index, 1.6686 ..., was achieved in [4] using the inverse cubic interpolation. This paper studies the possibility of improving efficiency index by using high order inverse interpolations. A class of algorithms are presented and the optimal one of the class has achieved the efficiency index 1.7282 ... With a user-given accurary ɛ and starting with the initial interval [a 1,b 1]=[a, b], these algorithms guarantee to find in finitely many iterations an enclosing interval [a n ,b n ] that contains a root of the equation and whose lengthb n –a n is smaller than ɛ. Numerical experiments indicate that the new algorithm performs very well in practice.

Abstract

В последнее время было разработано несколько алгоритмов, нозволяящнх достичь высокого индекса эффективности при нахождении оценки корня рравненияf(x)=0 в интервале [a, b], на котором фрнкцияf(x) непрерывна иf(a)f(b)<0. Наилрчий инлекс эффективности, равный 1,6686 ..., был достигнрт в аботе [4] с иснользованием обратной кубической интерцоляции. В настоящей работе исследуется возможность дальнейего улугщения индекса эффективности с иснользованием обратной интерноляции более высокого норядка. Представлен класс алгоритмов, наилуящий представитель которого достигает индекса эффектнвностн 1.7282 ... Начав с нсходного интервала [a 1 ,b 1 ]=[a, b], эти алгоритмы гарантнрованно находят эа конеяное число итераций включающий интервал [a n ,b n ], содержащий корень уравнения, чья иринаb n –a n не нревынает заданной пользователем величины погрещности ε. Численные эксперименты ноказывают, что производительность данного алгоритма на практических задачах достачно велика.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Bloomsburg University of Pennsylvania, 17815, Bloomsburg, PA, USA

    Yixun Shi

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  1. Yixun Shi
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© Y. Shi, 1996

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Shi, Y. Improving the efficiency index in enclosing a root of an equation. Reliable Comput 2, 357–372 (1996). https://doi.org/10.1007/BF02389896

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  • DOI: https://doi.org/10.1007/BF02389896

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