Inversion of jacobian matrices in metal-ligand-pH studies

Many non-linear regression programs which optimize the stability constants of chemical equilibria make use of Jacobian matrices for both the simulation of speciation by Newton-Raphson iteration and the optimization of parameters by Gauss-Newton iteration. An extended mathematical treatment is described here which shows that the full Jacobian matrix is partitioned into quadrants and that only one of these quadrants has been described in previous studies. This more complete treatment also corrects an error in the sign of the equation given in earlier work for the partial derivatives partial differential log h/ partial differential log beta (or partial differential pX/ partial differential log beta).