A stability analysis for perturbed nonlinear iterative methods
Authors:
Paul T. Boggs and J. E. Dennis
Journal:
Math. Comp. 30 (1976), 199-215
MSC:
Primary 65H10
DOI:
https://doi.org/10.1090/S0025-5718-1976-0395209-4
MathSciNet review:
0395209
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Abstract: This paper applies the asymptotic stability theory for ordinary differential equations to Gavurin’s continuous analogue of several well-known nonlinear iterative methods. In particular, a general theory is developed which extends the Ortega-Rheinboldt concept of consistency to include the widely used finite-difference approximations to the gradient as well as the finite-difference approximations to the Jacobian in Newton’s method. The theory is also shown to be applicable to the Levenberg-Marquardt and finite-difference Levenberg-Marquardt methods.
- Paul T. Boggs, The solution of nonlinear systems of equations by $A$-stable integration techniques, SIAM J. Numer. Anal. 8 (1971), 767–785. MR 297121, DOI https://doi.org/10.1137/0708071
- Paul T. Boggs, The convergence of the Ben-Israel iteration for nonlinear least squares problems, Math. Comp. 30 (1976), no. 135, 512–522. MR 416018, DOI https://doi.org/10.1090/S0025-5718-1976-0416018-3 W. E. BOSARGE, JR. (1968) Infinite Dimensional Iterative Methods and Applications, IBM Publications 230-2347, Houston.
- Kenneth M. Brown and J. E. Dennis Jr., Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation, Numer. Math. 18 (1971/72), 289–297. MR 303723, DOI https://doi.org/10.1007/BF01404679
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338 J. E. DENNIS, JR. (1971) "Algorithms for nonlinear problems which use discrete approximations to derivatives," Proc. ACM 1971 Nat’l. Conference, Chicago.
- Philip Rabinowitz (ed.), Numerical methods for nonlinear algebraic equations, Gordon and Breach Science Publishers, London-New York-Paris, 1970. MR 0331759
- M. K. Gavurin, Nonlinear functional equations and continuous analogues of iteration methods, Izv. Vysš. Učebn. Zaved. Mattmatika 1958 (1958), no. 5 (6), 18–31 (Russian). MR 0137932
- Allen A. Goldstein, Constructive real analysis, Harper & Row, Publishers, New York-London, 1967. MR 0217616
- G. H. Golub and V. Pereyra, The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate, SIAM J. Numer. Anal. 10 (1973), 413–432. MR 336980, DOI https://doi.org/10.1137/0710036
- Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
- James Hurt, Some stability theorems for ordinary difference equations, SIAM J. Numer. Anal. 4 (1967), 582–596. MR 221787, DOI https://doi.org/10.1137/0704053
- Kenneth Levenberg, A method for the solution of certain non-linear problems in least squares, Quart. Appl. Math. 2 (1944), 164–168. MR 10666, DOI https://doi.org/10.1090/S0033-569X-1944-10666-0
- Donald W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Indust. Appl. Math. 11 (1963), 431–441. MR 153071
- Gunter H. Meyer, On solving nonlinear equations with a one-parameter operator imbedding, SIAM J. Numer. Anal. 5 (1968), 739–752. MR 242366, DOI https://doi.org/10.1137/0705057
- James M. Ortega, Stability of difference equations and convergence of iterative processes, SIAM J. Numer. Anal. 10 (1973), 268–282. MR 339523, DOI https://doi.org/10.1137/0710026
- J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
- A. M. Ostrowski, Solution of equations and systems of equations, 2nd ed., Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London, 1966. MR 0216746
- C. Radhakrishna Rao and Sujit Kumar Mitra, Generalized inverse of matrices and its applications, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0338013
- R. A. Tapia, The differentiation and integration of nonlinear operators, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 45–101. MR 0285943
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Additional Information
Keywords:
Nonlinear iterative methods,
stability analysis,
consistent approximations,
steepest descent,
Newton’s method,
nonlinear least squares methods
Article copyright:
© Copyright 1976
American Mathematical Society