Bisection is optimal

Sikorski, K.

doi:10.1007/bf01459080

Cited by 92 publications

(34 citation statements)

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“…For some nonlinear problems it may happen that adaptive information operators are significantly better than nonadaptive, see [5], [9] and Chap. 8 of [10].…”

Section: Optimality Resultsmentioning

confidence: 98%

Optimal solution of nonlinear equations satisfying a Lipschitz condition

Sikorski

1984

Numer. Math.

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Summary.For a given nonnegative e we seek a point x* such that if(x*)[ l) satisfying a Lipschitz condition with the constant K and having a zero in B.The information operator on f consists of n values of arbitrary linear functionals which are computed adaptively. The point x* is constructed by means of an algorithm which is a mapping depending on the information operator. We find an optimal algorithm, i.e., algorithm with the smallest error, which uses n function evaluations computed adaptively. We also exhibit nearly optimal information operators, i.e., the linear functionals for which the error of an optimal algorithm that uses them is almost minimal. Nearly optimal information operators consists of n nonadaptive function evaluations at equispaced points xj in the cube B. This result exhibits the superiority of the T. Aird and J. Rice procedure ZSRCH (IMSL library [1]) over Sobol's approach [7] for solving nonlinear equations in our class of functions. We also prove that the simple search algorithm which yields a point x*=x k such that If(Xk)]= min If(xj)l is nearly optimal. The coml<=j<=n plexity, i.e., the minimal cost of solving our problem is roughly equal to

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“…For some nonlinear problems it may happen that adaptive information operators are significantly better than nonadaptive, see [5], [9] and Chap. 8 of [10].…”

Section: Optimality Resultsmentioning

confidence: 98%

Optimal solution of nonlinear equations satisfying a Lipschitz condition

Sikorski

1984

Numer. Math.

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“…it possesses asymptotically the best possible rate of convergence in the worst-case (Sikorski, 1982(Sikorski, , 2001). This means that it is guaranteed to converge within the predefined number of iterations and moreover, no other method has this property.…”

Section: The Jacobi-bisection Methodsmentioning

confidence: 99%

Sign-based learning schemes for pattern classification

Anastasiadis

Magoulas

Vrahatis

2005

Pattern Recognition Letters

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This paper introduces a new class of sign-based training algorithms for neural networks that combine the sign-based updates of the Rprop algorithm with the composite nonlinear Jacobi method. The theoretical foundations of the class are described and a heuristic Rprop-based Jacobi algorithm is empirically investigated through simulation experiments in benchmark pattern classification problems. Numerical evidence shows that this new modification of the Rprop algorithm exhibits improved learning speed in all cases tested, and compares favorably against the Rprop and a recently proposed modification, the improved Rprop.

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“…They permit adaptive (sequential) evaluations of arbitrary linear functionals and arbitrary transformations of this information as algorithms. This conjecture was established in [2]. That is for n fixed, the bisection information and algorithm are optimal in the worst case setting.…”

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confidence: 87%

“…The first lemma, 2.1, was proved in [2]. Namely, let I,, i= 1 .... , k, be closed intervals in [0, 1] and In the next lemma, we construct a family of functions from G needed in the proof of Theorem 1.1. with the convention dist(0, Y) = + oo.…”

Section: Auxiliary Lemmasmentioning

confidence: 97%

Asymptotic near optimality of the bisection method

Sikorski

Trojan

1990

Numer. Math.

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Summary. We seek an approximation to a zero of an infinitely differentiable function f: [0, 1]--.9~ such that f(0)<0 and f(1)>0. It is known that the error of the bisection method using n function evaluations is 2 -t"+ t~. If the information used are function values, then it is known that bisection information and the bisection algorithm are optimal. Traub and Wo~niak-owski conjectured in [5] that the bisection information and algorithm are optimal even if far more general information is permitted. They permit adaptive (sequential) evaluations of arbitrary linear functionals and arbitrary transformations of this information as algorithms. This conjecture was established in [2]. That is for n fixed, the bisection information and algorithm are optimal in the worst case setting. Thus nothing is lost by restricting oneself to function values.One may then ask whether bisection is nearly optimal in the asymptotic worst case sense, that is, possesses asymptotically nearly the best rate of convergence. Methods converging fast asymptotically, like Newton or secant type, are of course, widely used in scientific computation. We prove that the answer to this question is positive for the class F of functions having zeros of infinite multiplicity and information consisting of evaluations of continuous linear functionals. Assuming that every f in F has zeroes with bounded multiplicity, there are known hybrid methods which have at least quadratic rate of convergence as n tends to infinity, see e.g., Brent [1], Traub [4] and Sect. 1.

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Bisection is optimal

Cited by 92 publications

References 1 publication

Optimal solution of nonlinear equations satisfying a Lipschitz condition

Optimal solution of nonlinear equations satisfying a Lipschitz condition

Sign-based learning schemes for pattern classification

Asymptotic near optimality of the bisection method

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