Modifications of Broydens Method

As originally proposed, the sparsity of the jacobian matrix is destroyed by Broyden s method. Two procedures (or modifications) which preserves the sparsity of the jacobian matrices are presented. The procedures are demonstrated by use of simple algebraic examples and applied to the solution of distillation problems whose jacobian matrices are sparse. 

Since all derivatives may be evaluated numerically in Broyden s method,5 the necessity for programming the expressions needed for the derivatives appearing in the Newton-Raphson equations is avoided by use of these methods. The wide variety of thermodynamic packages which are available make these approaches very attractive. 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

Two methods have been proposed for retaining the desirable characteristics of Broyden s method and eliminating the undesirable characteristic of the loss of sparsity of the jacobian matrix through the use of inverses. In both of these modifications of Broyden s method, the necessity for the development of analytical expressions for the partial derivations is eliminated. To initiate the calcula-tional procedure in each of these modified versions of Broyden s method, the partial derivatives appearing in the jacobian matrix are evaluated numerically, and the jacobian matrix is updated in subsequent trials through the use of functional evaluations. The first modified form of Broyden s method is the one proposed by Gallun and Holland,9 and the second modification is the one proposed by Schubert.21 

As shown in Chap. 4, Broyden proposed the following formula for updating the jacobian matrix Jfc to obtain Jfc+1 

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