Solving nonlinear systems of equations with only one nonlinear variable

摘要

We describe a method for solving systems of N + 1 nonlinear equations in N + 1 unknowns y \te; R and z \te; RN of the form A(y)z + b(y) = 0, where the (N + 1) × N matrix A(y) and vector b(y) are functions of y alone. Such equations arise in minimax approximation. We reduce the problem to one equation in y only. An efficient quadratically convergent numerical technique based on Newton's method in one variable is used to solve this equation. Computational details and results are provided, and two generalizations are discussed.