Broyden-method

Supercomputers, such as the CRAY X-MP, CRAY Y-MP, and CRAY-2, are partially available and used for flow-sheet and optimization studies (7—10). Optimization modules usiag linear and nonlinear programming (LINPRO and UNLPl, based on a revised simplex, and Davidson-Eletcher-PoweU and Broyden methods, respectively) are available ia MicroMENTOR (11). 

EX253 2.5.3 Liquid-liquid equilibrium by Broyden method M32... 

The Broyden method is one of the simplest quasi-Newton method. The aim of quasi Newton methods is to achieve convergence properties comparable to those... 

The (k+l)-th iteration of the Broyden method consists of updating the inverse according to (2.38) and then performing a correction by (2.35). 

To prove this relationship, show that B-1B = I. Solve the system Ax - b = 0 with a square, nonsingular A by the Broyden method and test whether or not the final matrix H will satisfy the equality... 

In the following main program equations (2.61-2.64) are solved using the Broyden method. The distribution coefficients are computed from equations (2.58 - 2.60) written for C = 3 components. The starting values used are very simple, for the extract phase we start from poor Furfural and for the raffinate phase from the original n-Heptane - Cyclohexane mixture. Negative mole numbers and mole fractions are not allowed. 

Wegstein) was used. Switching to the Broyden method successfully converged to flowsheet. Figure 6.97 gives the temperature and composition profiles in the two reactors. 

Just as LU factorization is more efficient and more accurate than calculating the inverse, the use of the Bennett method is less time consuming than the Broyden method. 

Tomich recommends filling and inverting the Jacobian only once and using the Broyden method (Sec. 4.2.6) to update the inverse. This reduces computer time per trial but increases the number of column trials or passes through the procedure and for 6ome columns may also decrease reliability of the method. It is the author 6 experience that for many columns, solution is easier to reach when the Jacobian is filled and inverted in each trial and Brcyden s method is not used. 

Adjust U to make sme the incompressibihty constraint

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Remark 1 Step 5 must still be specified. The matrix has been introduced to indicate that the matrix k is modified by a (low-rank) correction matrix. When step 5 is reset by a specific update formula, the quasi-Newton method is named after that update (for instance BFGS-method, DFP-method, Broyden-method,. ..). 

A direct recycle configuration was then added (Figure 2.4). Depending on the type of pretreatment, the recycle stream vapor can contain water or a mixture of water and chemicals. From the convergence methods available in Aspen, the Broyden method was selected to solve the direct recycle flowsheet. Several simulations were run in order to analyze the operation cost (energy cost + chemical cost) as a function of the vapor fraction sent to the recycle the vapor fraction with the minimum cost was then identified. 

To start, we typically use a crude approximation ofthe Jacobian such as 5l l = /.In general, convergence is slower with Broyden s method than with Newton s method. Note, however, that the quadratic convergence of Newton s method is only obtained near the solution, and that this update formula does not require the N additional function evaluations of the finite difference approximation. This reduction in workload per Newton iteration makes the Broyden method a popular choice when we do not supply a routine to evaluate the Jacobian matrix analytically. For more information on qnasi-Newton methods and. Jacobian estimation, consult Nocedal Wright (1999). 

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