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By comparing the first two runs, it is seen that increasing the number of work zones from one to three does not change the number of iterations but increases computational time by a factor of 2.5 because of the larger number of terms involved in each equation. With only one workplace, it takes 19 seconds to evaluate the Jacobian matrix. When two more workplaces are added, the time it takes to evaluate the Jacobian matrix goes up to 52 seconds. When the number of equations goes from 400 to 1800, the time needed to evaluate the much bigger Jacobian goes to 377 seconds. Netting out the time it takes to evaluate the Jacobian, we find that one iteration takes a bit more than 5 one hundredths of a second. We also see that the problem with 1800 equations is difficult to solve by means of MINPACK when the vacancy rate is low: the CPU time limit of 900 seconds was exceeded. This experience suggested that MINPACK would be too slow for large realistic problems. The slowness is due to the time it takes to evaluate the Jacobian matrix and due to the fact that the direction-finding is inaccurate because the Jacobian is not reevalauated at each iteration. Hence, it is imperative to ask whether alternative techniques which do not rely on a full Jacobian evaluation would be faster.

Table 5.2 Comparison of MINPACK and CATLAS on the CRAY-YMP (Dimensions: i=3, j=1800, k=l, m=2, 1=6, n=2)

Vacancy Equilibrium 0.7 X (Equil. Low Random Rate Method Rents Rents) Rents (1) n 1 6 13

45% MINPACK CPU 378.51 385.08 389.621

n 1 4 5

CATLAS CPU 0.978 1.394 1.520

n 1 40

4.5% MINPACK CPU 378.51 781.13 > 900 (2)

n 1 10 14

CATLAS CPU 0.978 2.078 2.537

(1) The starting rent of each zone is a random number generated from the range ($0 - $1,000). (2) 900 second CPU time was exceeded. n : the number of iterations needed to converge. CPU : the computational time includes input-output time.

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