Elementary proof of constructive versions of the tangent direction theorem and the implicit function theorem
Given a vector-valued function \(F: \mathbb R^n\to\mathbb R^m\), \(F=(f^1,\dots,f^m)^T\), the authors seek an efficient way of solving the system \(F(x)=0\), i.e., finding at least one point of the set \(X=\{x: F(x)=0\}\). Let \(F_x^{\dagger}\) be the pseudoinverse of the Jacobian matrix \(F_x\), and \(N(x)=F_x^\dagger(x)F(x)\). The authors consider the iterative method \(x_{i+1}=x_i - N(x_i)-\frac{1}{2}F_x^\dagger (x_i)N^{T}(x_i)F_{xx}(x_i)N(x_i)\). Assuming that \(F\in C^2(\mathbb R^m)\) and there exists a point \(x_*\in X\) at which the rank of \(F_x\) is \(m\), they show the following (Theorem 2): There exists a neighborhood \(U\) of \(x_*\) in which the above iterative process converges at a cubic rate for any starting point \(x_1\in U\). Specifically, \(\mathrm{dist}\,(x_{i+1}(X))\leq c\, \mathrm{dist}\,(x_{i}(X))^3\), where \(c\) is a constant. The authors also give a quantitative version of the implicit function theorem for the implicit function \(y=y(x)\) determined by the equation \(F(x,y)=0\) with \(F\in C^2\). The paper concludes with extensions of the above results to Banach spaces.
scientific article; zbMATH DE number 6032794