Sherman-Morrison formula

The fact that may be proved using the Sherman-Morrison formula. 

In general, the solution of system [50a] requires 0(n3) operations, and the multiplication in [50b] demands 0(n2) work however, the updating context of these problems can be used to formulate recursive lower-cost solutions in terms of previously computed quantities. Such formulas may update the factors of the previously decomposed matrix, or use an iteration process based on the Sherman-Morrison-Woodbury formula6 for (B + uvT)-1 in combination with a recursive process suggested by Matthies and Strang.124 The second procedure [50b] is generally preferred in large-scale applications, as it is computationally more economical. 

A proof of Eq. (47) was given by Allen (1971). Another (perhaps more accessible) path to Eq. (47) starts with the basic deletion formulas known as the Sherman-Morrison-Woodbury theorem, illustrated by Rao (1973) in an exercise. 

Since a quasi-Newton procedure requires the inverse of the matrices (see procedure I below), a matrix computed by one of the update formulae given above cannot directly be employed in such a procedure. But the Sherman-Morrison-Woodbury formula (see lemma 1) enables to invert an updated matrix in a simple way if the matrix mJ is known. 

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