Abstract
In this note we develop a simple finite differencing device to calculate approximations of derivativesx′(0),x″(0),x (3)(0), … of regular solution curvesx: ℜ ∋s →x(s) ∈ ℜn of nonlinear systems of equationsg(x)=0,g∈C k (ℜn + 1, ℜn) without having to compute points on the solution arcx(s).
The derivative vectorsx′(0),x″(0),x (3)(0),… can be used in the numerical approximation of the solution setg −1(0) in two ways. On one hand they can be applied to construct higher order predictors to be used in predictor-corrector branch following procedures. On the other they serve as order determining basis functions in the Reduced Basis Method.
The performance of the differencing method is demonstrated by some numerical examples.
Zusammenfassung
Hergeleitet werden einfache Differenzenprozesse zur Berechnung der Ableitungenx′(0),x″(0),x (3)(0),… regulärer Lösungskurvenx: ℜ ∋s →x(s) ∈ ℜn eines nichtlinearen Gleichungssystemesg(x)=0,g∈C k (ℜn + 1, ℜn). Abweichend von üblichen Vorgangsweisen werden dafür nicht mehrere Punkte der Lösungskurve benötigt.
Die Ableitungen finden etwa Verwendung bei der Konstruktion von Prädiktoren höherer Ordnung zur Verfolgung vong −1(0) sowie in der Methode der reduzierten Basis.
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References
Allgower, E. L., Georg, K.: Predictor-corrector and simplicial methods for approximating fixed points and zero points of nonlinear mappings, in: Mathematical Programming — The State of Art (Bachem, A., Grötschel, M., Korte, B., eds.), pp. 15–56. Heidelberg: Springer 1983.
Allgower, E. L., Georg, K.: Simplicial and continuation methods for approximating fixed points and solutions of systems of equations. Siam Rev.22, 28–85 (1980).
Deuflhard, P.: A stepsize control for continuation methods and its special application to multiple shooting techniques. Numer. Math.33, 115–146 (1979).
Fink, J. P., Rheinboldt, W. C.: On the error behavior of the reduced-basis technique for nonlinear finite-element approximations. ZAMM63, 21–28 (1983).
Gill, Ph. E., Murray, W., Wright, M. H.: Practical Optimization. London-New York-Toronto-Sydney-San Francisco: Academic Press 1981.
Mackens, W.: Kondensation großer nichtlinearer Gleichungssysteme mit der Methode der reduzierten Basis. Report, Institut für Geometrie und Praktische Mathematik, April 1988.
Menzel, R., Schwetlick, H.: Über einen Ordnungsbegriff bei Einbettungsalgorithmen zur Lösung nichtlinearer Gleichungen. Computing16, 187–199 (1976).
Porsching, T. A.: Estimation of the error in the reduced basis method solution of nonlinear equations. Math. Comp.45, 487–496 (1985).
Rheinboldt, W. C.: Solution fields of nonlinear equations and continuation methods. SIAM J. Numer. Anal.17, 221–237 (1980).
Schwetlick, H.: Lecture, given at the Institut für Angewandte Mathematik der Universität Hamburg, FRG Summer of 1984.
Schwetlick, H., Cleve, J.: Higher order predictors and adaptive steplength control in pathfollowing algorithms. SIAM J. Numer. Anal.24, 1382–1393 (1987).
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. New York-Heidelberg-Berlin: Springer 1980.
Wacker, H.-P. (ed.): Continuation Methods. New York: Academic Press 1978.
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Mackens, W. Numerical differentiation of implicitly defined space curves. Computing 41, 237–260 (1989). https://doi.org/10.1007/BF02259095
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DOI: https://doi.org/10.1007/BF02259095