Fletcher-Reeves method

Results for Example 6.2 using the Fletcher-Reeves method... 

For problems jvith hundreds or thousands of variables, storing and manipulating the matrices H or V2/(x ) requires much time and computer memory, making conjugate gradient methods more attractive. These compute sk using formulas involving no matrices. The Fletcher-Reeves method uses... 

The one-step BFGS formula is usually more efficient than the Fletcher-Reeves method. It uses somewhat more complex formulas ... 

It is widely believed that, generally speaking, methods such as the Davidon-Fletcher-Powell method are superior to the Fletcher-Reeves method and, indeed, Fletcher suggests (see p. 82 of ref. 8) that typically the Fletcher-Reeves method will take about twice as many iterations as the Davidon-Fletcher-Powell method. 

As far as the Fletcher-Reeves method is concerned, it must clearly be the method of choice in linear coefficient optimization as it involves only the storage of gradient and direction vectors between iterations. It has been used by a number of authors (Sleeman,29 Fletcher,5 Kari and Sutcliffe,32 Claxton and Smith,51 and Weinstein and Pauncz52). It [is unfortunately possible, however, to sum up the experience so far gained of the method in quantum chemistry as disappointing, in the sense that in SCF caclulations the authors have found that the calculations proceed significantly more slowly than the conventional iterative procedure, when the conventional procedure converges at all. 

M. Al-Baali, Inst. Math. Appl. J. Numer. Anal., 5,121 (1985). Descent Property and Global Convergence of the Fletcher-Reeves Method with Inexact Linear Search. 

This chapter deals with the problem of finding the unconstrained minimum of a function P x) that involves the variables x e R with Wv 1. Section 3.4 showed that conjugate direction methods are useful in solving large-scale unconstrained minimization problems. The version that uses the Pollack-Ribiere and Fletcher-Reeves methods sequentially is often particularly effective. 

---