Polak-Ribiere formula

An alternative formula for P, is the Polak-Ribiere formula see Leach, p. 225.) The idea of the conjugate-gradient method (which really should be called the conjugate-direction method) is to choose each new step in a direction that is conjugate to the directions used in the previous steps (where the word conjugate has a certain technical... 

Different recipes exist to determine the coefficient jin (Jensen 2007) among which the Polak-Ribiere formula usually performs best for non-quadratic functions,... 

For quadratic functions this is identical to the Fletcher-Reeves formula but there is some evidence that the Polak-Ribiere may be somewhat superior to the Fletcher-Reeves procedure for non-quadratic functions. It is not reset to the steepest descent direction unless the energy has risen between cycles. 

Scales (1986) recommends the Polak Ribiere version because it has slightly better convergence properties. Scales also gives an algorithm which is used for both methods that differ only in the formula for the updating of the search vector. 

Three of the best known settings for (3 are titled the Fletcher-Reeves (FR), Polak-Ribiere (PR), and Hestenes-Stiefel (HS) formulas.66-71 77 78 They are given by the formulas... 

Powell (1977) explained why this last alternative performed better if no significant step from the operating point is obtained during the one-dimensional search, it results in gj gj j. In such a case, approaches 0 in Polak-Ribiere s formula and the new direction is oriented as the gradient in x . However, would approach 1 in Fletcher-Reeves s formula and the new direction might be ill-determined. 

A significant modification, that might at first have seemed trivial, was later introduced by Polak and Ribiere (1969) (see Polak (1971)). Using the feet that gkgk+i = 0 for quadratic functions, an equivalent formula for in step 3 above (at least in the quadratic case) is the following ... 

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