Broyden-Bennett algorithm

Another significantly faster procedure for solving the Newton-Raphson equations, called the Broyden-Bennett method, was proposed by Hess et al.10 The Broyden-Bennett algorithm may be used to solve the equations for an absorber-type column accompanied by a chemical reaction in the following manner. 

Step 1. For the first trial, assume X0, and use the Broyden-Bennett algorithm (see Chap. 4) to find an improved set of rjJs which satisfy Eqs. (8-8) and (8-11), and which make R0 = 0. 

This problem may be solved by use of any of the three formulations of the IN Newton-Raphson method which were presented in Chap. 4. The classification of the components and the selection of the appropriate stages between the pinches are performed in precisely the same manner as described in Sec. 11-1. It should be noted, however, that if analytical expressions for the partial derivatives are used, one must not overlook the fact that in the case of the separated lights, the vector will have derivatives since it contains Au. Because of the slight differences in the equations for the distributed and separated components, Broyden s method, or the Broyden-Bennett algorithm is recommended. 

Instead of applying Householder s formula, the calculation of an inverse of the jacobian may be avoided altogether by use of the algorithm proposed by Bennett for updating the LU factors of the jacobian matrix. Example 4-9 will show that fewer numerical operations are required to compute the LU factors than are required to compute the inverse of a matrix. Bennett s algorithm is applied to the Broyden equations as follows. 

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

Less time is consumed by procedure 3 than by procedure 1. Calculation of the LU factors of the matrix J in step 2 of procedure 3 requires approximately n3/3 operations, whereas the calculation of the inverse of J in step 2 of procedure 2 requires approximately n3 operations, where the matrix J is a square matrix of order n. To update the LU factors in step 6 of procedure 3 by use of Bennett s algorithm requires approximately In2 operation, whereas approximately 3n2 operations are required to update the inverse of J by use of Householder s formula as proposed by Broyden in step 6 of procedure 2. 

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