Broyden s method

Sparse Matrix Methods. In order to get around the limitations of the sequential modular architecture for use in design and optimization, alternate approaches to solving flowsheeting problems have been investigated. Attempts to solve all or many of the nonlinear equations simultaneously has led to considerable interest in sparse matrix methods generally as a result of using the Newton-Raphson method or Broyden s method (22, 23, 24 ). ... 

Distillation Calculations, Work done with flash calculations and sparse matrix methods was extended to distillation calculations. Holland and Gallun (51) explored the use of Broyden s method coupled with sparse updating procedures to distillation calculations with highly non-ideal solutions. Shah and Boston (52), and Ross and Seider (53j discuss the case of multiple liquids phases on a tray. 

Broyden s method converges rapidly in the linear region (near the solution) to a tight tolerance. 

This would suggest that initial approximations to J (X) using Wegstein s method should be followed by using Broyden s method. 

Holland, C. D. Gallun, S. E., "Modifications of Broyden s Method for the Solution of Distillation Problems Involving Highly Non-Ideal Solutions",... 

The Newton method uses an estimate of the gradient at each step to calculate the next iteration, as described in Section 1.9.6. Quasi-Newton methods such as Broyden s method use linearized secants rather than gradients. This approach reduces the number of calculations per iteration, although the number of iterations may be increased. 

Two extremes are encountered in flowsheeting software. At one extreme, the entire set of equations (and inequalities) representing the process is employed. This representation is known as the equation-oriented method of flowsheeting. The equations can be solved in a sequential fashion analogous to the modular representation described below or simultaneously by Newton s method, Broyden s method, or by employing sparse matrix techniques to reduce the extent of matrix manipulations. Refer to the review by Evans and Chapter 5. ... 

Broyden s method and Wegstein s method are two examples of this approach that use different approximations for In the former, which is based... 

This chapter listed many of the possible units in the Model Library of Aspen Plus. The ammonia process illustrated the procedures (and computer windows) you used to set the process conditions and examine the results. The thermodynamics choices can be verified by comparison with data reported in the literature. Sometimes the calculations do not converge, and then the Wegstein method, or Broyden s method, are useful for accelerating convergence. 

Tomich25 was the first to apply Broyden s method (developed in chap. 15) to the solution of distillation problems. Broyden s method is based on the use of numerical approximations of the partial derivatives appearing in the jacobian matrix. The approach proposed by Broyden permits the inverse of the jacobian matrix to be updated each trial after the first through the use of Householder s formula.6 Thus, it is necessary to invert the jacobian matrix only once. Since approximate values for the partial derivatives are used, procedure 2 generally requires more trials than does procedure 1. However, since the evaluation of the partial derivatives and the inversion of the jacobian matrix are not generally required after the first trial of procedure 2, it requires less computer time per trial than does procedure 1. 

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... 

In this algorithm, Broyden s method is applied by updating the jacobian matrices by use of Householder s formula.13 Let J0 be the initial approximation of the jacobian matrix with which the iterative procedure is started. Then... 

First, try s0 = 1 (see procedure 2, Broyden s method in Chap. 4). Thus... 

Thus, the inequality given of Broyden s method (see procedure 2, Chap. 4) is satisfied, that is,... 

Schubert proposed a modification of Broyden s method which takes advantage of the fact that in the case of sparse jacobian matrices, most of the elements... 

Example 5-5 was used by Gallun and Holland9 to compare Broyden s method implemented with the new algorithm to the Newton-Raphson method and to Schubert s21 modification of Broyden s method. The statement of this example is given in Tabta5-8 and the solution is presented in Table 5-9. 

This algorithm is based on the use of newly defined energy and volatility parameters as the primary successive approximation variables, and Broyden s method is used to iterate on these parameters. A brief review of the Boston-Sullivan Algorithm follows. 

Equation (3) and the definition of a provide the basis for an algorithm to efficiently solve the series of linear equations that arise during the implementation of Broyden s method. 

The solution may be effected by use of analytical expressions for the partial derivatives of the R/s or by use of Broyden s method, which makes use of numerical approximations of the partial derivatives 5 see Chaps. 4 and 15. 

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. 

Newton-Raphson with Broyden s method (one trial for each choice of plates) 11-4 20 184.93 WATFIV... 

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