Abstract
In this paper the well-known modified (underrelaxed, damped) Newton method is extended in such a way as to apply to the solution of ill-conditioned systems of nonlinear equations, i.e. systems having a “nearly singular” Jacobian at some iterate. A special technique also derived herein may be useful, if only bad initial guesses of the solution point are available. Difficulties that arose previously in the numerical solution of nonlinear two-point boundary value problems by multiple shooting techniques can be removed by means of the results presented below.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bard, Y.: Comparison of gradient methods for the solution of nonlinear parameter estimation problems. SIAM J. Numer. Anal.7, 157–186 (1970)
Bauer, F. L.: Optimally scaled matrices. Num. Math.5, 73–87 (1963)
Ben-Israel, Adi: A Newton-Raphson method for the solution of systems of equations. J. Math. Anal. Appl.15, 243–252 (1966)
Brown, K. M., Dennis, J. E., Jr.: Derivative-free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation. Num. Math.18, 289–297 (1972)
Broyden, C. G.: A class of methods for solving nonlinear simultaneous equations. Math. Comp.19, 577–583 (1965)
Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Vortrag im Lehrgang „Flugbahnoptimierung” der Carl-Cranz-Gesellschaft e. V., Okt. 1971
Bulirsch, R., Stoer, J., Deuflhard, P.: Numerical solution of nonlinear two-point boundary value problems I. To be published in Num. Math., Handbook Series Approximation
Businger, P., Golub, G. H.: Linear least squares solutions by Householder transformations. Num. Math.7, 269–276 (1965)
Cauchy, A.: Méthode générale pour la résolution des systèmes d'équations simultanées. C. R. Acad. Sci. Paris25, 536–538 (1847)
Deuflhard, P.: Ein Newton-Verfahren bei fastsingulärer Funktionalmatrix zur Lösung von nichtlinearen Randwertaufgaben mit der Mehrzielmethode. Universität zu Köln, Mathematisches Institut: Dissertation, 1972
Dickmanns, E. D.: Optimale dreidimensionale Gleifflugbahnen beim Eintritt in Planetenatmosphäre. Raumfahrtforschung14, Heft 3 (1970)
Dickmanns, E. D.: Optimal control for synenergetic plane change. Proc. XXth Int. Astronautical Congress 597-631 (1969)
Fletcher, R.: Generalized inverse methods for the best least squares solution of systems of nonlinear equations. Comp. J.10, 392–399 (1968)
Goldstein, A. A.: Cauchy's Methode der Minimierung. Num. Math.4, 146–150 (1962)
Golub, G. H., Reinsch, C.: Singular value decomposition and least squares solutions. Num. Math.14, 403–420 (1970)
Greenstadt, J.: On the relative efficiencies of gradient methods. Math. Comp.21, 360–367 (1967)
Householder, A. S.: Principles of numerical analysis. New York: McGraw-Hill 1953
Kantorovič, L., Akilow, G.: Functional analysis in normed spaces. Moscow: Fizmatgiz 1959. German Translation: Berlin: Akademie-Verlag 1964
Keller, H. B.: Numerical methods for two-point boundary-value problems. London: Blaisdell 1968
Kowalik, J., Osborne, M. R.: Methods for unconstrained optimization problems. New York: American Elsevier Publ. Comp., Inc. 1968
Levenberg, K. A.: A method for the solution of certain nonlinear problems in least squares. Quart. Appl. Math.2, 164–168 (1944)
Marquardt, D. W.: An algorithm for least-squares-estimation of nonlinear parameters. SIAM J. Appl. Math.11, 431–441 (1963)
Meyer, G. H.: On solving nonlinear equations with a one-parameter operator imbedding. University of Maryland, Computer Science Center: Techn. Rep. 67-50 (Sept. 1967)
Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York-London: Academic Press 1970
Osborne, M. R.: On shooting methods for boundary value problems. J. Math. Anal. Appl.27, 417–433 (1969)
Penrose, R.: A generalized inverse for matrices. Proc. Cambridge Philos. Soc.51, 406–413 (1955)
Pesch, H.-J.: Numerische Berechnung optimaler Steuerungen mit Hilfe der Mehrzielmethode dokumentiert am Problem der Rückführung eines Raumgleiters unter Berücksichtigung von Aufheizungsbegrenzungen. Universität Köln, Mathematisches Institut: Diplomarbeit, 1973
Powell, M. J. D.:A Fortran subroutine for solving systems of non-linear algebraic equations. A.E.R.E. Rep. 5947, Harwell (1968)
Rheinboldt, W. C.: Local mapping relations and global implicit function theorems. University of Maryland, Computer Science Center: Techn. Rep. 68-52 (Febr. 1968)
Rosenbloom, P. C.: The method of steepest descent. Proc. Symp. Appl. Math. AMS6, 127–176 (1956)
Stoer, J.: Einführung in die Numerische Mathematik I. Heidelberger Taschenbuch 105. Berlin-Heidelberg-New York: Springer 1972
Stoer, J., Bulirsch, R.: Einführung in die Numerische Mathematik II. Heidelberger Taschenbuch 114. Berlin-Heidelberg-New York: Springer 1973
Wilkinson, J. H.: Rounding errors in algebraic processes. London: Her Majesty's Stationary Office 1963 (German translation: Rundungsfehler. Heidelbeger Taschenbuch 44. Berlin-Heidelberg-New York: Springer 1969)
Zimmermann, U.: Numerische Berechnung optimaler Steuerungen unter Verwendung der Mehrzielmethode bei mehrfacher Beschränkung im Phasenraum mit Beispielen aus der Flugbahnoptimierung. Universität Köln, Mathematisches Institut: Diplomarbeit, 1973
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Deuflhard, P. A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting. Numer. Math. 22, 289–315 (1974). https://doi.org/10.1007/BF01406969
Issue Date:
DOI: https://doi.org/10.1007/BF01406969