variable values listed in Tables 7-1 and 7-3, respectively. The index

ordering chosen was i., il, .2. This ordering allows a full precedence

ordering for both ii (because it can be incremented for some function

types and decremented for others) and i2. The solution procedure

converged to the solution in 14 iterations, taking 2.17 seconds of CPU

time on an IBM 370/165. The same problem, solved by Naphtali's method

(Naphtali and Sandholm, 1971), converged in five iterations and required

16.21 seconds of CPU time on an IBM 360/65. The implementation of

Naphtali's method employed a convergence accelerator, whereas the Gauss-

Seidel solution did not. Although a comparison of execution times from

different computers is speculative, the Gauss-Seidel solution appears

to be faster. This only means that for this particular type of problem

the Gauss-Seidel solution procedure may be better. As will be seen,

Nrphi'ali's method will solve problems with a wide range of numerical

characteristics, while the Gauss-Seidel method will not.

 The second example problem is a model of an absorber, with four

components exhibiting a wide range of boiling points. Fhe FVIM remains

as; in Fij. 7-1. The FVIM for output set assi nment, however, becomes

that in Fig. 7-3. Note that although it is known that tie absorber

will have 4 components, a blocking factor of 3 is stil is ed for the

analyses. The estimated variable values are listed in fable 7-4. Using

these values, the expanded Jacobian matrix is calculated. Then, using

the index output offsets, the FVIM weights in Fig. 7-3 are calculated.

The maximum product output set is again along the diagonal. For this

problem, however, the maximum eigenvalue of the convergence matrix has a

rmgnituide of 4.75x!00. This indicates that the Gauss--Seidel solution

procedure is a bad choice.