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NYC 9/11 Public Portal Document

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derivatives.

The second technique is a Newton-Raphson procedure utilized in the computational solution of CATLAS and developed by Anas. This technique ignores the off-diagonal elements of the Jacobian in determining the step size at each iteration and evaluates the diagonal elements of the Jacobian matrix using analytical derivatives. It is easy to establish using either technique that the system of simultaneous equations at hand has a unique solution. This is done by starting from many highly different starting points and finding that convergence occurs to the same equilibrium point to within a very tightly defined tolerance.

MINPACK is potentially more accurate in determining the step size at each iteration, as long as the numerical approximation of the derivatives is not too crude. Hence, MINPACK may generally require a relatively small number of iterations. The CATUkS technique is potentially less accurate because it does not utilize the full Jacobian matrix. Hence, it may take a significantly larger number of iterations to find the equilibrium point.

However, it is not clear which technique is more accurate since it is possible that the numerical approximation of the derivatives (in MINPACK) is cruder than the approximation which results from neglecting the off-diagonal elements but accurately evaluating the diagonal elements (in CATLAS). Moreover, the cost of the CATLAS technique at each iteration is lower because the full LIk Jacobian is not evaluated. When the number of simultaneous equations is sufficiently large, the cost saving from not having to k evaluate the full Jacobian generally outweighs the cost saving from a potentially smaller number of iterations. As a result of this, the CATLAS technique is more appropriate for larger problems.

The For purposes of NYREG, the CATLAS procedure has been adopted. housing market part of the model has 847 equations which b

k The Jacobian matrix is evaluated by perturbing each element of the rent vector one at a time and evaluating the Jacobian matrix at each perturbation. For example, if there are n equations (hence, n rents), the first rent is perturbed by a small amount from its initial value and the effect of this rent perturbation on the elements of the Jacobian are evaluated (recall that the Jacobian is n X n) . Following this, the second rent is perturbed etc. Thus, the needed computations are of the order of n-cubed. Por example, if there are 400 I equations, 64,000,000 computations are needed. With 1800 equations, 5,832,000,000 computations are needed. To avoid excessive computations, MINPACK does not reevaluate the Jacobian matrix at every iteration. This, together with the fact that the Jacobian evaluation is not analytical, contributes to the inaccuracy of the MINPACK search direction.

-Arm.

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