G. Buzziferraris et E. Tronconi, AN IMPROVED CONVERGENCE CRITERION IN THE SOLUTION OF NONLINEAR ALGEBRAIC EQUATIONS, Computers & chemical engineering, 17(10), 1993, pp. 1053-1056
Citations number
4
Categorie Soggetti
Computer Application, Chemistry & Engineering","Computer Applications & Cybernetics","Engineering, Chemical
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.