A unified derivation of several error bounds for Newton's process

摘要

This paper gives a unified derivation of the upper and lower bounds of the errors in Newton's process, obtained by Dennis [4], Tapia [27], Gragg-Tapia [6] and recently by Potra-Pták [20] and Miel [15], with the use of different techniques. Our argument is based upon the Kantorovich theorem and the Kantorovich recurrence relations. From the proof, it is concluded that the recent upper and lower bounds of Miel are finer than those of Potra-Pták, but do not improve the basic error bounds which are directly obtained from the Kantorovich theorem. Further, it is shown that the error bounds obtained with the use of the recurrence relations are the same as those obtained by the majorant principle. Finally, results on the Newton method in partially ordered space are surveyed. A method is also described for estimating the componentwise errors for computed solutions.