The Feasibility of Continuation Methods for Nonlinear Equations

Let F:D ⊂ Rⁿ → Rⁿ be a given nonlinear mapping. The idea underlying all continuation methods for solving the equation $F(x) = 0$ is to consider a homotopy H:[ 0,1 ] × D → D with the property that there exists a continuous solution curve x:[0,1] → D of H(t,x) = 0,t ∈ [0,1], starting at a known point x₀ = x(0) and ending at a solution of $F(x) = 0$. In the basic numerical continuation process, a partition 0 < t₀ < t₁ < ⋯ < tN = 1 is introduced and, starting from an approximate solution of the kth problem H(tₖ ,x) = 0, a corresponding solution of the $(k + 1)$st problem is obtained by means of a locally convergent iterative process. A general theorem is proved which establishes the feasibility of the basic numerical continuation process under near minimal conditions. This result is then applied to several iterative methods. The problem of estimating t-parameter step length a priori is examined and several results are given. A generalized numerical continuation process is introduced which includes previous approaches such as the Davidenko method. The theory is applied to a nonlinear two-point boundary value problem and numerical results are given.