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Nonlinear least squares — the Levenberg algorithm revisited*

Published online by Cambridge University Press:  17 February 2009

M. R. Osborne
M. R. Osborne
Affiliation:
Computer Centre, Australian National University, Canberra, A. C. T. 2601, Australia.
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Abstract

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One of the most succesful algorithims for nonlinear least squares calculations is that associated with the names of Levenberg, Marquardt, and Morrison. This algorithim gives a method which depends nonlinearly on a parameter γ for computing the correction to the current point. In this paper an attempt is made to give a rule for choosing γ which (a) permits a satisfactory convergence theorem to be proved, and (b) is capable of satisfactory computer implementation. It is beleieved that the stated aims have been met with reasonable success. The convergence theorem is both simple and global in character, and a computer code is given which appears to be at least competitive with existing alternatives.

Type
Research Article
Information
The ANZIAM Journal , Volume 19 , Issue 3 , June 1976 , pp. 343 - 357
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Beale, E. M. L., ‘Numerical methods’ in Nonlinear Programming (ed. Abadie, J.), North Holland (1967) 135–205.Google Scholar
[2]Daniel, J. W., The approximate minimisation of functionals, Prentice Hall, (1971).Google Scholar
[3]Goldstein, A. A., ‘On steepest descent’, Siam J. Control 3 (1965), 147–151.Google Scholar
[4]Levenberg, K., ‘A method for the solution of certain nonlinear problems in least squares’, Quant. Appl. Math. 2 (1994), 164–168.Google Scholar
[5]Marquardt, D. W., ‘An algorathim for least squares estimation of nonlinear parameters’, Siam J. Appl. Math. 11 (1963), 431–441.CrossRefGoogle Scholar
[6]Morrison, D. D., ‘Methods for nonlinear least squares problems and convergence proofs’, JPL Seminar Proceedings. (1960)Google Scholar
[7]Osborne, M. R., ‘Some aspects of nonlinear least squares calculations in Numerical Methods for Nonlinear Optimization (ed. Lootsma, F. A.), Academic Press, (1972).Google Scholar
[8]Ostrowski, A. M., ‘Solutions of equations and systems of equations’, Academic Press, (1966).Google Scholar