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This paper gives a method for finding sharp a posteriori error bounds for Newton's method under the assumptions of Kantorovich's theorem. On the basis of this method, new error bounds are derived, and comparison is made among the known bounds of Dennis [2], Döring [4], Gragg-Tapia [5], Kantorovich [6, 7], Kornstaedt [9], Lancaster [10], Miel [11–13], Moret [14], Ostrowski [17, 18], Potra [19], and Potra-Pták [20].

1. Bartle, R.G.: Newton's method in Banach spaces. Proc. Amer. Math. Soc.6, 827–831 (1955) Google Scholar . 2. Dennis, J.E.: On the Kantorovich hypothesis for ...

16.09.2012 · Hi, PF Sometimes it is not easy to find roots of functions. Newton gave a nice clue: the Newton's Method formula: ##x_{n+1}=x_n-\\dfrac{f(x_n}{f'(x_n)}##. My concern is, now that I have understood and practiced it, comprehend what I've sketched in the summary. This is all taken from "Calculus...

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To obtain the last line we expand the denominator using the binomial expansion and then neglect all terms that have a higher power of than the leading term. Thus, we …

Best possible upper and lower bounds for the error in Newton's method are established under the hypotheses of the Kantorovich theorem.

Best possible upper and lower bounds for the error in Newton's method are established under the hypotheses of the Kantorovich theorem. Let X and Y be Banach ...

[1] J. E. Dennis, On the convergence of Newton-like methods. Numerical Methods for Nonlinear Algebraic Equations (ed. P. Rabinowitz), Gordon and Breach, New York, 1970, 163–181.

14.07.2006 · (2013) Estimating upper bounds on the limit points of majorizing sequences for Newton’s method. Numerical Algorithms 62 :1, 115-132. (2012) Secant-type methods and nondiscrete induction.

Sharp error bounds for Newton-like methods under weak smoothness assumptions - Volume 45 Issue 3

Jul 14, 2006 · Error Bounds for Newton’s Method Under the Kantorovich Assumptions. The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics, 197-208. (1986) Error bounds for Newton's iterates derived from the Kantorovich theorem.

In this paper, it is shown that the upper and lower bounds of the errors in the Newton iterates recently obtained by Potra-Pták [11] and Miel [7], with the use of nondiscrete induction and majorizing sequence, respectively, follow immediately from the Kantorovich theorem and the Kantorovich recurrence relations. It is also shown that the upper and lower bounds of Miel are …

The Newton–Fourier method is Joseph Fourier's extension of Newton's method to provide bounds on the absolute error of the root approximation, ...

Show activity on this post. I just have a very brief question regarding the formula for error bounds in Newton's method. Depending on where you look, this will either be written as: e n + 1 ≈ f ′ ′ ( r) 2 f ′ ( r) e n 2. or: e n + 1 ≈ − f ′ ′ ( r) 2 f ′ ( r) e n 2.

J. E. Dennis, Jr., “On the Kantorovich hypothesis for Newton’s method,” SIAM J. Numer.Anal., 6 (1969), 493–507. MathSciNet zbMATH CrossRef Google Scholar

Let α be the true root. We can define the error en+1 in the estimate xn+1 in three different ways. Way 1: en+1=xn+1−α. If that is the definition, ...

In this paper, it is shown that the upper and lower bounds of the errors in the Newton iterates recently obtained by Potra-Pták [11] and Miel [7], with the use of nondiscrete induction and majorizing sequence, respectively, follow immediately from the Kantorovich theorem and the Kantorovich recurrence relations. It is also shown that the upper and lower bounds of Miel are finer than those of ...

Given that we are using Newton's method to approximate a root of the function f(x). Suppose we have an approximation of the root xn which has an error of (r ...

Jan 18, 2022 · Hi, PF Sometimes it is not easy to find roots of functions. Newton gave a nice clue: the Newton's Method formula: ##x_{n+1}=x_n-\\dfrac{f(x_n}{f'(x_n)}##. My concern is, now that I have understood and practiced it, comprehend what I've sketched in the summary. This is all taken from "Calculus...

MAT 4310: 1. Newton's Method Error Bound. Theorem (Newton1–Raphson Method2 (1711)). Suppose f has 2 continuous derivatives on a neighborhood B of a root r.

Error bounds for the modified Newton's method - Volume 14 Issue 3. ... of the Kantorovich convergence theorem for the modified Newton's method is proved.

Affiliations. Department of Secretarial Science, Matsuyama Shinonome Junior College, 790, Matsuyama, Japan. Toshio Konishi. Department of Mathematics, Faculty of ...

Finally, results on the Newton method in partially ordered space are surveyed. A method is also described for estimating the componentwise errors for ...