where is the Jacobian matrix . If this method is applied to (9.3), we obtain

(9.5)

sufficient conditions for the convergence of Newton's method for a system are rather complicated. However, when (9.5) is applied to a stiff system, Convergence is usually obtained without a restriction on of comparable severity to that implied by (9.4), provided that we can supply a sufficiently accurate initial estimate, : a separate predictor can be used for this last purpose. Note, however, that (9.5) calls for the re-evaluation of the Jacobian and the consequent re-inversion of the matrix at each step of the iteration . This can be very expensive in terms of computing time, and a commonly used device is to hold the value of in (9.5) constant for a number of consecutive iteration steps. If the iteration converges when so modified, then it will converge to the theoretical solution of (9.3); if after a few step (typically three) it appears not to be converging, then the Jacobian is re-evaluated and the corresponding matrix re-inverted. For problems for which varies very slowly, it may even prove possible to hold the same value for for a number of consecutive integration steps.