sk AXfcr AXk

Since the jacobian matrix J may be stated in terms of its factors L and U as demonstrated in procedure 1, the above expression for Jfc+ j may be restated in the following form

Bennett proposed the algorithm presented in Fig. 4-4 for updating the matrices Lk and Ufc to obtain the updated matrices Lfc+1 and Ufc+1. When Bennett's algorithm is used to make the Broyden correction, the following calcu-lational procedure is used.

Step 1. Same as step 1 of procedure 2.

Step 2. The partial derivatives of J0 are found in the same manner as shown in step 2 of procedure 2. Then find the factors Lc and U0 of J0 such that

as described by Hess et al.;14 see also Conte and de Boor.9

Step 3. On the basis of Lfc, Uk, and ffc (the most recent values of L, U, and f) compute AXfc as follows

LfcUk AXfc = -ffc Step 4. Same as step 4 of procedure 2.

Figure 4-4 The Bennet algorithm for updating an LU factorization.1

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