The Jacobian matrices given by both fixed point algorithms are the same:
The full Newton algorithm leads to the following derivative matrix of
the activity coefficients:
The following Jacobian matrix is also obtained:
For the scenario with a high ionic strength, the Jacobian matrices Z are
very different for the full Newton and fixed-point algorithms, leading
to very different Newton steps (). Moreover, we find an increase in the
condition number of matrix Z for the full Newton algorithm. The
condition number of matrix is 21.6, whereas it equals 1 for matrix .
The symmetry of the Jacobian matrix in equations and is specific to this
test case. As shown in equation , the Z matrix for the fixed-point
algorithm is symmetric, whereas equation proves that it is usually not
symmetric for the full Newton algorithm.