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

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

$$ \begin{array}{*{20}{c}} {{\text{f[x,y]}}} & { = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\text{f}}({\text{x}}) - {\text{f}}({\text{y)}}}}{{{\text{x}} - {\text{y}}}}} & {{\text{ifx}} \ne {\text{y}}} \\ {{\text{f'(x}})} & {{\text{if x = y}}} \\ \end{array} ,} \right.} \\ {{\text{f[x,y,z]}}} & { = \left\{ {\begin{array}{*{20}{c}} {\frac{{{\text{f}}({\text{x,z}}) - {\text{f}}({\text{y,z)}}}}{{{\text{x}} - {\text{y}}}}} & {{\text{ifx}} \ne {\text{y}}} \\ {\frac{{{\text{f''(x}})}}{2}} & {{\text{if x = y}}} \\ \end{array} .} \right.} \\ \end{array} $$