Iterative update of the Hessian matrix

Quadratic expansion of a function Vk(q) about a local extremum takes the form 

The practical problem is to find a way to use the information obtained in each iterative step without making unjustified changes of the Hessian. This could be 

These formulas update the rank-m projection of F° or G°, using the nonhermitian projection operator Vm = Ap(Aqf Ap) 1Aqf, such that V nAq = Aq, VmAp = Ap, and VmVm = Vm. This operator projects onto the m-dimensional vector space spanned by the specified set of gradient vectors. The Rm update has the undesirable property of altering the complementary projection of the updated matrix. 

The BFGS (Broyden [42], Fletcher [124], Goldfarb [145], Shanno [379]) algorithm is an update procedure for the Hessian matrix that is widely used in iterative optimization [125]. The simpler Rm update takes the form 

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