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Abstract
In this paper we study convergence rates using quotient convergence factors and root convergence factors as described by Ortega and Rheinboldt for Hestenes' Gram-Schmidt conjugate direction method without derivatives. We do study this computationally and analytically by comparing this conjugate direction method for minimizing a non quadratic function f with Newton's method for solving ∇f = 0. This method of Hestenes is different from that of Smith, Powell, and Zangwill in its mathematical development. All of these ideas have already been developed and based upon 1952 paper of Hestenes and Siefel, the 1980 book Conjugate Direction Methods in Optimization by Hestenes and the 1975 paper by Dennemeyer and Mookini. The primary purpose of this work is to provide details analytically and computationally of what has been done before.
