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
$$ \begin{array}{*{20}{c}} {{\text{f'(x)}} \in {\text{f'(X),}}} & {{\text{x}} \in {\text{X}} \subseteq {{{\text{X}}}^{{\text{O}}}}} \\ {{\text{f'(x)}} \subseteq {\text{f'(Y),}}} & {{\text{X}} \subseteq {\text{Y}} \subseteq {{{\text{X}}}^{{\text{O}}}}} \\ {{\text{d(f'(X))}} \leqslant {\text{cd(X),}}} & {{\text{X}} \subseteq {{{\text{X}}}^{{\text{O}}}}} \\ \end{array} $$
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
$$ \begin{array}{*{20}{c}} {{\text{f[x,y]}}} & { = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\text{f}}({\text{x}}) - {\text{f}}({\text{y)}}}}{{{\text{x}} - {\text{y}}}}} & {{\text{ifx}} \ne {\text{y}}} \\ {{\text{f'(x}})} & {{\text{if x = y}}} \\ \end{array} ,} \right.} \\ {{\text{f[x,y,z]}}} & { = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\text{f}}({\text{x,z}}) - {\text{f}}({\text{y,z)}}}}{{{\text{x}} - {\text{y}}}}} & {{\text{ifx}} \ne {\text{y}}} \\ {\frac{{{\text{f''(x}})}}{2}} & {{\text{if x = y}}} \\ \end{array} .} \right.} \\ \end{array} $$


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  1. [1]
    Alefeld, G., Potra, F.: A new class of interval methods with higher order of convergence. Computing 42, 69–80 (1989).CrossRefGoogle Scholar
  2. [2]
    Illg, B.: Über einige Verfahren höherer Ordnung zur iterativen Einschließung bei nicht linearen Gleichungssystemen. Diplomarbeit. Universität Karlsruhe 1989. (Not available).Google Scholar
  3. [3]
    Schmidt, J. W.: Eine Übertragung der Regula falsi auf Gleichungen in Banach-räumen II. ZAMM 43, 97–110 (1963).CrossRefGoogle Scholar

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© Springer Basel AG 1991

Authors and Affiliations

  • G. Alefeld
    • 1
  • B. Illg
    • 1
  • F. Potra
    • 2
  1. 1.Institut für Angewandte MathematikUniversität KarlsruheKarlsruheDeutschland
  2. 2.Department of MathematicsUniversity of IowaIowa CityUSA

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