New zero-finders for trust-region computations

Abstract

Trust-region methods are among the most popular schemes for determining a local minimum of a nonlinear function in several variables. These methods approximate the nonlinear function by a quadratic polynomial, and a trust-region radius determines the size of the sphere in which the quadratic approximation of the nonlinear function is deemed to be accurate. The trust-region radius has to be computed repeatedly during the minimization process. Each trust-region radius is computed by determining a zero of a nonlinear function ψ(x). This is often done with Newton’s method or a variation thereof. These methods give quadratic convergence of the computed approximations of the trust-region radius. This paper describes a cubically convergent zero-finder that is based on the observation that the second derivative $\psi ^{\prime \prime }(x)$ can be evaluated inexpensively when the first derivative $\psi ^{\prime }(x)$ is known. Computed examples illustrate the performance of the zero-finder proposed.