The idea is to reduce the number of derivative evaluations and the number of LU decompositions without hurting the convergence of the method.
The modified Newton formula is
The modified Newton step can be equivalently written as
This means that only one Jacobian () computation is required, and only one LU decomposition is performed, regardless of the number of iterations involved. All the linear systems () share the same ``coefficient'' matrix , and therefore we can re-use the same LU decomposition.
With modified Newton, we start with computing the matrix and its LU decomposition (with pivoting!). The algorithm does not work if is singular; if this happens we need to terminate the computations.
One step of the Modified Newton algorithm follows.
If after, say, iterations the speed of convergence
decreases, we need to update the derivative; set
From the third iteration on we will monitor the progress
of the iterations. For a good starting point,
if the iterations proceed all right, we have
If for 3 or 4 consecutive steps the iteration is likely to diverge, and a better starting point should be chosen; print the proper message and exit gracefully.