PAPER Imperfect Singular Solutions of Nonlinear Equations and a Numerical Method of Proving

1.Байду номын сангаас
Introduction
In this paper, we are concerned with the problem of proving the existence of singular solutions numerically for the following system of nonlinear equations: f (x) = 0, f : Rn → Rn , (1)
where f is sufficiently smooth. Various methods such as Krawczyk’s method have been proposed for calculating the regular solutions of Eq. (1) with guaranteed accuracy [3]. A way of calculating singular solutions is to resolve the singularity. The bordering methods have been proposed in this way. In these methods, extended systems are proposed such that singular solutions of the original systems become regular ones for the extended systems. Thus it is natural to consider that it may be possible to prove the existence of the singular solutions of Eq. (1) by applying Krawczyk’s method to the extended systems. However, in the extended systems additional variables are necessary in order to resolve the singularities. A regular solution of the extended system becomes a singular solutions of Eq. (1) when these additional variables are equal to zero. Usually, it is numerically undecidable whether such variables are equal to zero or not. In this paper, based on this consideration the concept of an imperfect singular solution of Eq. (1) is proposed as an exact solution of the extended system of Eq. (1) whose additional variables have norms of small values. Imperfect singular solution always becomes a true singular solution if additional variables are added to the original equation Eq. (1) as perturbations. This is the motivation why we have introduced a new concept.
Yuchi KANZAWA† and Shin’ichi OISHI† , Members
SUMMARY A new concept of “an imperfect singular solution” is defined as an approximate solution which becomes a singular solution by adding a suitable small perturbation to the original equations. A numerical method is presented for proving the existence of imperfect singular solutions of nonlinear equations with guaranteed accuracy. A few numerical examples are also presented for illustration. key words: numerical computation with guaranteed accuracy, Krawczyk-based interval validation method, imperfect singular solutions extended systems
Manuscript received May 7, 1998. Manuscript revised November 12, 1998. † The authors are with School of Science and Engineering, Waseda University, Tokyo, 169–8555 Japan.
Remark that proving the existence of imperfect singular solutions is not proving one of true singular solutions. In this point of view, it may be no sense proving the existence of imperfect singular solutions. Since we think there never exists methods to prove the existence of singular solutions numerically, we regard the method proposed in this paper as the limit of numerical approach to prove the existence of singular solutions. In the field of engineering, we need proving the existence of singular solutions to avoid them. For example, electrical circuits must be designed to have no singular operating points. We have the case that a real electrical circuit has singular operating points while the equation describing this electrical circuit has no singular solutions because of modeling error. Since there is no way to estimate this modeling error completely, it is safe to avoid the points which become singular operating points by some noise. In this point, we think it is valuable to prove the existence of imperfect singular solutions. The plan of this paper is as follows. In the 2nd section, simple isolated singular solution is defined. Some notations and definitions for interval analysis are introduced. An algorithm is also introduced to enclose the regular solution of Eq. (1). In the 3rd section, an extended system for simple isolated singular solution of Eq. (1) is introduced. We shall define the imperfect isolated simple singular solution of Eq. (1) and shall propose an algorithm to enclose it. We shall also indicate a theorem indicating that the proposed algorithm succeeds for sufficiently good approximate solution of Eq. (1). In the 4th section, we shall treat imperfect singular solutions with higher singularity. Finally, numerical examples are also presented for illustration. 2. Preliminaries
In this section, we shall briefly explain notations and definitions which will be used in this paper. 2.1 Singular Point Regularity of functions is defined as follows: Let D be a bounded open subset of Rn . Let f : D → Rn be C 1 . f is regular at x if the Jacobian matrix fx (x) is regular, otherwise f is singular at x. such a point x is called singular point. y ∈ Rn is