AN IMPROVED CONVERGENCE CRITERION IN THE SOLUTION OF NONLINEAR ALGEBRAIC EQUATIONS

In most existing programs for the iterative solution of systems of non linear algebraic equations, convergence towards the solution is checke d by requiring that either the sum of squares or the weighted sum of s quares of the residuals decreases after each iteration. While this is often inadequate, we have previously presented a more suitable converg ence test involving a modified merit function which equals exactly the squared distance between the current values of the unknowns and the s olution vector in the case of linear systems of equations (Buzzi-Ferra ris and Tronconi, Computers chem. Engng 10, 129-141, 1986). In its ori ginal formulation, evaluation of the merit function via an orthogonali zation procedure called for additional O(N3) operations at each iterat ion. We show in this note that the same convergence test can be evalua ted via much simpler algebraic manipulations. The resulting criterion is computationally inexpensive and equivalent to one proposed by Deufl hard (Numer. Math. 22, 289, 1974). Our geometric interpretation shows that it introduces natural weighing factors for the residuals in the o bjective function, which makes it recommended for implementation in ge neral-purpose equation solvers.