The Jacobian Matrix is a dense matrix. Rather than derive and evaluate analytic expressions for the Jacobian elements, we form the elements of the Jacobian numerically, through finite difference perturbations. This approach is justified since the accuracy of analytic Jacobians is not required for the modified Newton method described above. This is demonstrated by the successful use of old (and therefore inaccurate) Jacobians. We evaluate the numerical Jacobian elements from a one-sided finite difference formula as follows:

(16–9)

where

(16–10)

We choose the relative and absolute perturbations, and , to be the square root of the computer’s unit round-off.