Bennett’s algorithm

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. 

The use of Bennett s algorithm is illustrated by the following numerical example ... 

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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See also in sourсe #XX -- [ , , , ]

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