Weak Singularities and the Continuous Newton Method

Abstract

Some issues related to the determination of the singular roots of a nonlinear vector function f : ‫ޒ‬ n ª ‫ޒ‬ n are addressed in this paper. It is usually assumed that Newton-like fields are not defined at singularities; thus a particular treatment for these points is necessary. Nevertheless, in dimension 1 and in several higher dimensional instances it is possible to make a smooth extension of the field to singular points; when this is the case for a singular root, it can be treated in a way similar to that of regular ones. Necessary and sufficient conditions for this extension to be possible are given, under some structural assumptions, through the concept of weak singularity. The actual setting for this result is a general class of quotient functions which includes, in particular, the Newton field. For the specific case of the continuous-time Newton method, we enlarge some previous results concerning the relation between singular roots and equilibrium points of the extended field, as well as their asymptotic stability. Finally, a computational tool obtained from the extended continuous Newton method by means of the cell mapping technique is shown to be well behaved for the location of these singular roots. ᮊ

Stability of Singular Equilibria in Quasilinear Implicit Differential Equations

“…This corollary shows that domains of convergence of Newton method are not necessarily included in the domain of invertibility of the Jacobian matrix. Previous results in this direction can be seen in [46]…”

“…This is due to the fact that, at algebraic singularities x^, it is lim x Ä x &h(x)&= and, therefore, these points must be excluded from any domain of smooth definition of h. The converse is also true in the noncritical case: it may be proved that h is defined as a C m&1 vector field on a neighborhood of x* if this is a noncritical weak singularity [46]. It follows that weak singular points encompass situations where the domain of smooth definition of the field is larger than the domain of invertibility of A, against some common assumptions in the context of singular root-finding problems [20,21].…”

“…The continuous-time setting has been used to model analogues of root-finding algorithms [1,3,9,18,36,46,50,54], and also to formulate new methods for this kind of problems [4,25,47]. These schemes have been further developed in the context of homotopy techniques [17,26] and, more specifically, as trajectory methods (see [16] for a survey).…”

A Simple Method for Impasse Points Detection in Nonlinear Electrical Circuits

“…In the analysis of the existence of impasse points in (2), we shall use the classification of singularities given in [17]. If ℎ( ) = det ( ) and ( ) = (adj ( )) ( ), the classification corresponds to the analysis of the vector field ( )/ℎ( ) and is related to the existence of a continuous extension of such vector field.…”

A Numerical Study of Basins of Attraction of Zero-Finding Neural Nets Designed Using Control Theory

“…In particular, Zufiria and Guttalu [19], and Riaza and Zufiria [17] as well as Gomulka [11] define a taxonomy of singularities of the Newton vector field and analyze most of the types that could occur. Rather than detail these results here, we point out that the analysis is based on the fact the Newton vector fieldẋ = −D −1 f f(x) can be written as:…”