New zero-finders for trust-region computations

作者：Alkilayh Maged; Reichel Lothar^*; Yuan Jin Yun

来源：Numerical Algorithms, 2017, 76(2): 361-375.

DOI：10.1007/s11075-016-0260-2

• Zero-finder
• Trust-region method
• Cubic convergence

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 psi(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 ''(x) can be evaluated inexpensively when the first derivative psi'(x) is known. Computed examples illustrate the performance of the zero-finder proposed.

• 出版日期2017-10