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2.5.3 Quasi Newton     continued...

For an update of rank one, written as , we obtain from (53) the condition that is a vector in the direction of . This produces the general rank one update formula as:

for . Broyden's QN method, for example, uses . While Broyden's update does not guarantee symmetry, it is useful for solving nonlinear equations and for deriving a more effective, rank two update. To restrict the general rank one update form of (53) further, we can impose the condition of symmetry. Symmetry will be ``inherited'' from to if for some scalar . Letting that be , we obtain the general symmetric rank 1 update (SR1) as follows:

Now, SR1 will only be positive-definite if . Thus, rank two updates (e.g., ) were thought until very recently to be more suitable for optimization.