An improved convergence criterion in the solution of nonlinear algebraic equations

Abstract

In most existing programs for the iterative solution of systems of nonlinear algebraic equations, convergence towards the solution is checked by requiring that either the sum of squares or the weighted sum of squares of the residuals decreases after each iteration. While this is often inadequate, we have previously presented a more suitable convergence test involving a modified merit function which equals exactly the squared distance between the current values of the unknowns and the solution vector in the case of linear systems of equations (Buzzi-Ferraris and Tronconi, Computers chem. Engng 10, 129–141, 1986). In its original formulation, evaluation of the merit function via an orthogonalization procedure called for additional O( N 3) operations at each iteration. We show in this note that the same convergence test can be evaluated via much simpler algebraic manipulations. The resulting criterion is computationally inexpensive and equivalent to one proposed by Deuflhard ( Numer. Math. 22, 289, 1974). Our geometric interpretation shows that it introduces natural weighing factors for the residuals in the objective function, which makes it recommended for implementation in general-purpose equation solvers.