Bounding the zeros of an interval equation

doi:10.1016/j.amc.2009.03.041

Applied Mathematics and Computation

Volume 213, Issue 2, 15 July 2009, Pages 466-478

https://doi.org/10.1016/j.amc.2009.03.041 Get rights and content

Abstract

In this paper, we consider the problem of finding reliably and with certainty all zeros of an interval equation within a given search interval for continuously differentiable functions over real numbers. We propose a new formality of interval arithmetic which is treated in a theoretical manner to develop and prove a new method, lying on the context of interval Newton methods. Some important theoretical aspects of the new method are stated and proved. Finally, an algorithmic realization of our method is proposed to be applied on a set of test functions, where the promising theoretical results are verified.

Section snippets

Introductory notions on interval equations

In this work we deal with the often occurring problem of finding reliably and with certainty all zeros of the equation $f (x) = 0$ for a continuously differentiable function $f : R \to R$ , where $R$ denotes the set of all real numbers.

In many fields of science (e.g. chemical engineering: [2], computer graphics: [3], robotics: [12], control theory: [16], etc) the determination of all zeros of (1.1) remains a crucial and well-timed issue. Interval methods have shown a great performance in solving it, resulting

On intervals

In the following, we give some basic definitions on intervals and related terms. A more thoroughly study can be found in [1], [6], [10].

Classic interval Newton method and drawback issues

The derivation of classic interval Newton arise from the application of Mean Value Theorem in combination with the Fundamental Theorem of Interval Arithmetic [8], resulting in the following relation: $x^{*} \in m - \frac{F (m)}{F^{'} ([x])} = : N ([x]),$ with $x^{*} \in [x]$ , being a zero of $f (x) = 0$ , $m \in [x]$ , F and $F^{'}$ interval extensions of f and $f^{'}$ , respectively, and $N ([x])$ denoting the interval Newton operator over interval $[x]$ . The corresponding iterative scheme of (3.1) is given by $\begin{matrix} m_{k} = m ([x]_{k}), \\ N ([x]_{k}) = m_{k} - \frac{F (m_{k})}{F^{'} ([x]_{k})}, \\ [x]_{k + 1} = [x]_{k} \cap N ([x] \end{matrix}$

The proposed method

We consider the interval equation $f (x) = 0$ and we want to find all interval zeros over a given search interval $[x]$ . If we write $F (m) = [\underset{̲}{f_{m}}, \bar{f_{m}}]$ and $F^{'} ([x]) = [\underset{̲}{d}, \bar{d}]$ , where $m \in [x]$ , the classic interval Newton method becomes $N ([x]) : = m - \frac{[\underset{̲}{f_{m}}, \bar{f_{m}}]}{[\underset{̲}{d}, \bar{d}]} .$ The constant m is usually set to be the midpoint of search interval $[x]$ . If we make use of the proposed hull interval arithmetic (2.4), the relation (4.1) takes the following form: $N ([x]) : = m - (\frac{\underset{̲}{f_{m}}}{[\underset{̲}{d}, \bar{d}]} ⊔ \frac{\bar{f_{m}}}{[\underset{̲}{d}, \bar{d}]}) = (m - \frac{\underset{̲}{f_{m}}}{[\underset{̲}{d}, \bar{d}]}) ⊔ (m - \frac{\bar{f_{m}}}{[\underset{̲}{d}, \bar{d}]}) .$ Thus, we can restate interval

Theoretical results

In this section, we state and prove some important properties of the proposed method. Firstly we prove the existence of an interval zero in the interval derived by the proposed operator.

Lemma 5.1

Let $a, b$ be real numbers such that $a, b \in [x] \in IR$ . If $a ⩽ b$ then $[a, b] \subseteq [\underset{̲}{x}, \bar{x}]$ .

Proof

The proof of the above lemma is trivial and it is omitted. □

Theorem 5.2

Let f be a continuously differentiable interval function as defined in Definition 1.1 and $[x] \in IR$ the search interval. Moreover, let $f_{L}$ and $f_{U}$ be boundary functions of f, and $N_{L}$ , $N_{U}$

Algorithmic formulation

In this section, we will present an algorithmic formulation of the method, described in the previous sections. The following algorithm, called HIN (Hull Interval Newton) takes as input the interval natural extension of interval function f, the searching interval $[x]_{0}$ , the tolerance $ε$ and returns a result list $L$ containing the found interval zeros. At the following lines a brief description of the proposed algorithm is given. Firstly, in Step 2 we initialize the result list $L$ to be the empty

Numerical results

Through the literature, according to our knowledge, there exist plenty of test problems concerning the problem of solving a non-interval equation, but a very small sample of interval equations either in the form of examples or in application problems. Thus, we put out a set of interval functions to cover the most important cases of the proposed method, as well as interval equations occurring in applications. In Table 1 we present a set of twelve interval functions with their corresponding

Conclusions, results and future work

In this work, we state the weaknesses of the existing methods and algorithms in solving, generally, the problem of finding reliably and with certainty all zeros of an interval equation. Even though the proposed method is, essentially, based on interval Newton method for non-interval equations, achieves to produce efficient solutions to the problem of solving interval equations.

The most significant result of our work is focused in generating an interval $[r]$ which consists a guaranteed interior

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Bounding the zeros of an interval equation

Abstract

Section snippets

Introductory notions on interval equations

On intervals

Classic interval Newton method and drawback issues

The proposed method

Theoretical results

Algorithmic formulation

Numerical results

Conclusions, results and future work

Automatica

Introduction to Interval Computations

Application of interval newton’s method to chemical engineering problems

Reliable Computing

Robust and efficient ray intersection of implicit surfaces

Reliable Computing

C++ Toolbox for Verified Computing I

A globally convergent interval method for computing and bounding real roots

BIT

Global Optimization using Interval Analysis