# An Enclosure Method with Higher Order of Convergence-Applications to the Algebraic Eigenvalue Problem

• G. Alefeld
• B. Illg
• F. Potra
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique book series (ISNM, volume 96)

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

## References

1. [1]
Alefeld, G., Potra, F.: A new class of interval methods with higher order of convergence. Computing 42, 69–80 (1989).
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).

## 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