Rank-one and Rank-two Corrections to Positive Definite Matrices Expressed in Product Form

A New Derivation of Symmetric Positive Definite Secant Updates

“…All the algorithms of this section have already been suggested by Goldfarb [14] using the Brodlie, Gourlay, and Greenstadt [1] factored form of the BFGS and DFP updates and the orthogonal decompositon update ideas of Gill, Golub, Murray, and Saunders [12]. Our purpose is to point out that they follow very naturally from the preceding derivations.…”

Some practical procedures for the solution of nonlinear finite element equations

“…We can now calculate Sti) and yCi) from eqs. (25) and (26) and proceed with the evaluation of the matrix update that satisfies eq. (27).…”

“…Evaluate the correction to the coefficient matrix. In the BFGS method the updated matrix can be expressed in product form (see [25]):…”

Direct Secant Updates of Matrix Factorizations

“…This leaves the method for specification of {Bk} to be chosen. Many choices are available; most commonly Bk+X is obtained either by a Broyden update of Bk, (yk -P-ksk)sI (1)(2)(3)(4) Bk+x=Bk + T SkSk withyk = F(xk + sk) -F(xk) and sk = xk+x -xk, or from the exact Jacobian or a finite difference approximant, or by taking Bk+l = Bk [7]. The Broyden update is the unique solution to the Frobenius norm minimization problem:…”

“…The algorithm outlined here is representative of a general technique for producing inexpensive updates of matrix factorizations. This is not the same as techniques for obtaining matrix updates in factored form as in the case of rank 1 and rank 2 updates [1], [12]. Here, the update is defined implicitly by the updated factors [5].…”