The convergence of the Ben-Israel iteration for nonlinear least squares problems

Author:
Paul T. Boggs

Journal:
Math. Comp. **30** (1976), 512-522

MSC:
Primary 65K05; Secondary 34D20

DOI:
https://doi.org/10.1090/S0025-5718-1976-0416018-3

MathSciNet review:
0416018

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Abstract | References | Similar Articles | Additional Information

Abstract: Ben-Israel [1] proposed a method for the solution of the nonlinear least squares problem where . This procedure takes the form where denotes the Moore-Penrose generalized inverse of the Fréchet derivative of *F*. We give a general convergence theorem for the method based on Lyapunov stability theory for ordinary difference equations. In the case where there is a connected set of solution points, it is often of interest to determine the minimum norm least squares solution. We show that the Ben-Israel iteration has no predisposition toward the minimum norm solution, but that any limit point of the sequence generated by the Ben-Israel iteration is a least squares solution.

**[1]**Adi Ben-Israel,*A Newton-Raphson method for the solution of systems of equations*, J. Math. Anal. Appl.**15**(1966), 243–252. MR**0205445**, https://doi.org/10.1016/0022-247X(66)90115-6**[2]**P. T. BOGGS, "On the use of Lyapunov theory for the analysis of nonlinear iterative methods,"*Proc. of*1975*Conference on Information Sciences and Systems*, April 2-4, 1975.**[3]**Paul T. Boggs and J. E. Dennis Jr.,*A stability analysis for perturbed nonlinear iterative methods*, Math. Comp.**30**(1976), 199–215. MR**0395209**, https://doi.org/10.1090/S0025-5718-1976-0395209-4**[4]**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**0303723**, https://doi.org/10.1007/BF01404679**[5]**P. Deuflhard,*A modified Newton method for the solution of ill-conditioned systems of nonlinear equations with application to multiple shooting*, Numer. Math.**22**(1974), 289–315. MR**0351093**, https://doi.org/10.1007/BF01406969**[6]**R. Fletcher,*Generalized inverse methods for the best least squares solution of systems of non-linear equations*, Comput. J.**10**(1967/1968), 392–399. MR**0221748**, https://doi.org/10.1093/comjnl/10.4.392**[7]**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****[8]**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. Collection of articles dedicated to the memory of George E. Forsythe. MR**0336980**, https://doi.org/10.1137/0710036**[9]**Wolfgang Hahn,*Über die Anwendung der Methode von Ljapunov auf Differenzengleichungen*, Math. Ann.**136**(1958), 430–441 (German). MR**0099543**, https://doi.org/10.1007/BF01347793**[10]**James Hurt,*Some stability theorems for ordinary difference equations*, SIAM J. Numer. Anal.**4**(1967), 582–596. MR**0221787**, https://doi.org/10.1137/0704053**[11]**Kenneth Levenberg,*A method for the solution of certain non-linear problems in least squares*, Quart. Appl. Math.**2**(1944), 164–168. MR**0010666**, https://doi.org/10.1090/qam/10666**[12]**A. M. LYAPUNOV, "Problème générale de la stabilité du mouvement," Kharkov, 1892; French transl.,*Ann. Fac. Sci. Univ. Toulouse*, v. (2) 9, 1907, pp. 203-474; reprint, Ann. of Math. Studies, no. 17, Princeton Univ. Press, Princeton, N. J., 1949. MR**9**, 34.**[13]**Donald W. Marquardt,*An algorithm for least-squares estimation of nonlinear parameters*, J. Soc. Indust. Appl. Math.**11**(1963), 431–441. MR**0153071****[14]**James M. Ortega,*Stability of difference equations and convergence of iterative processes*, SIAM J. Numer. Anal.**10**(1973), 268–282. Collection of articles dedicated to the memory of George E. Forsythe. MR**0339523**, https://doi.org/10.1137/0710026**[15]**J. M. Ortega and W. C. Rheinboldt,*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810****[16]**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****[17]**David A. Sánchez,*Ordinary differential equations and stability theory: An introduction*, W. H. Freeman and Co., San Francisco, Calif.-London, 1968. MR**0227492****[18]**Taro Yoshizawa,*Stability theory by Liapunov’s second method*, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966. MR**0208086**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1976-0416018-3

Keywords:
Ben-Israel iteration,
generalized inverses,
nonlinear least squares,
Lyapunov stability for difference equations

Article copyright:
© Copyright 1976
American Mathematical Society