Continuation Methods

Continuation methods are based on piecewise linear approximation of a variety (e.g. curve or surface) by means of numerical solution of an initial value problem [7]. In other words, they compute solution varieties of nonlinear systems usually expressed in terms of an equation

$$f({\text{p}}) = 0$$

((6.1))

with f:ℝ^n002B;d → ℝⁿ a real function. The solution of this equation is called zero set (i.e. a particular level set).

As studied in Chapter 1, a zero set is a variety that consists of regular pieces called manifolds, which are joined at singular solutions (which are also solution manifolds, but of a system with lower d). The regular pieces are manifold curves when d = 1, manifold surfaces when d = 2, and d-manifolds in general. These systems arise frequently in engineering and scientific problems, because these problems are often formulated in terms of the computation of a function that satisfies some set of equations, for example, the Navier-Stokes equations, Maxwell's equations, or Newton's law.