Abstract
In [1] we have considered the nonlinear equation f(x) = 0 where f is a continuous differentiate real function of a real variable. We suppose that f is strictly monotone on an interval X0. Without loss of generality we may assume that f is strictly increasing on X0. We assume that by using interval arithmetic methods it is possible to compute two positive numbers ℓ1, ℓ2 such that 0 < ℓ1 < f′(x) < ℓ2 for all x ∈ X0. Let us denote by L the interval [ℓ1, ℓ2]. We suppose that the derivative f′ (x) ∈ IR, x ∈ X0, has an interval extension f′ (X),X ∈ X0, satisfying the following conditions
where c is a constant independent of X and where d denotes the diameter of an interval. Furthermore we assume that these three relations also hold for the second derivative of f. Together with f and its derivatives we consider its divided differences
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References
Alefeld, G., Potra, F.: A new class of interval methods with higher order of convergence. Computing 42, 69–80 (1989).
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Dedicated to L. Collatz on the occassion of his 80th birthday
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© 1991 Springer Basel AG
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Alefeld, G., Illg, B., Potra, F. (1991). An Enclosure Method with Higher Order of Convergence-Applications to the Algebraic Eigenvalue Problem. In: Albrecht, J., Collatz, L., Hagedorn, P., Velte, W. (eds) Numerical Treatment of Eigenvalue Problems Vol. 5 / Numerische Behandlung von Eigenwertaufgaben Band 5. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 96. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6332-2_2
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DOI: https://doi.org/10.1007/978-3-0348-6332-2_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-6334-6
Online ISBN: 978-3-0348-6332-2
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