Consider the problem of finding a zero :9I of a smooth function f: :9I. Such problems are commonly solved by some variant of Newton’s method, combined with a continuation procedure if a sufficiently accurate initial approximation to :9I is not available. In a continuation procedure, the original root-finding problem is embedded in a family of similar problems through a smooth homotopy h: :9I. Under appropriate conditions, there is a smooth path :9I. The idea is to follow this path from t_O to t₁; there are a variety of ways of accomplishing this (see [1]). The homotopy parameter t may arise naturally in the context of the original problem so that the entire path is of interest, or may be introduced artificially; for example, by defining) h(x,t) = f(x) + (t-1)f(x_{O: 0E), for O :9I.}

Among the aforementioned conditions is the assertion that :9I is nonsingular for t_O:9I. If we reparameterize in terms of arclength :9I, the analogous condition for the existence of a smooth path is that

:9I have maximal rank, namely n. A point :9I is singular bu :9I has rank n is called a turning point; a point :9I at which :9I is rank deficient is called a bifurcation point. A root t¯ for which f’(x:A) is singular is called a singular root. In all three cases, if the rank in question is n-A-1 and certain conditions involving second derivatives are satisfied, the turning point, bifurcation point, or singular root is termed simple.

In the aforementioned root-finding context, a turning point encountered during the path-following process is merely a nuisance; it reflects the fact that t does not increase monotonically from t_O to t₁ along the path. In other contexts, the identification of turning points is an end in itself. Algorithms have been developed for this purpose which recast a simple turning-point problem as an enlarged nonsingular root-finding problem; for a review, see [2]. This paper is concerned with using these algorithms to solve simple singular root-finding problems, and thence simple bifurcation problems. By introducing a homotopy analogous to the artificial one noted above, the simple singular root-finding problem is recast as a simple turning-point problem. When recast as an enlarged root-finding problem, a simple bifurcation point becomes a simple singular root.

The presentation is uneven and somewhat obscure at times; it assumes familiarity with [3], and preferably also [2]. Theoretical justification for the application to these two problems, along the lines sketched above, of the method developed in [3] is presented. Alternate approaches are discussed briefly, and two examples are provided.