On the global convergence of Broyden's method

Authors:
J. J. Moré and J. A. Trangenstein

Journal:
Math. Comp. **30** (1976), 523-540

MSC:
Primary 65H10

DOI:
https://doi.org/10.1090/S0025-5718-1976-0418451-2

MathSciNet review:
0418451

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Abstract: We consider Broyden's 1965 method for solving nonlinear equations. If the mapping is linear, then a simple modification of this method guarantees global and *Q*-superlinear convergence. For nonlinear mappings it is shown that the hybrid strategy for nonlinear equations due to Powell leads to *R*-superlinear convergence provided the search directions form a uniformly linearly independent sequence. We then explore this last concept and its connection with Broyden's method. Finally, we point out how the above results extend to Powell's symmetric version of Broyden's method.

**[1]**C. G. Broyden,*A class of methods for solving nonlinear simultaneous equations*, Math. Comp.**19**(1965), 577–593. MR**0198670**, https://doi.org/10.1090/S0025-5718-1965-0198670-6**[2]**C. G. Broyden,*The convergence of single-rank quasi-Newton methods*, Math. Comp.**24**(1970), 365–382. MR**0279993**, https://doi.org/10.1090/S0025-5718-1970-0279993-0**[3]**C. G. Broyden, J. E. Dennis Jr., and Jorge J. Moré,*On the local and superlinear convergence of quasi-Newton methods*, J. Inst. Math. Appl.**12**(1973), 223–245. MR**0341853****[4]**J. E. Dennis Jr. and Jorge J. Moré,*A characterization of superlinear convergence and its application to quasi-Newton methods*, Math. Comp.**28**(1974), 549–560. MR**0343581**, https://doi.org/10.1090/S0025-5718-1974-0343581-1**[5]**J. E. DENNIS & J. J. MORÉ,*Quasi-Newton Methods, Motivation and Theory*, Cornell University Computer Science Technical Report TR 74-217, 1974.**[6]**P. E. Gill and W. Murray,*Quasi-Newton methods for unconstrained optimization*, J. Inst. Math. Appl.**9**(1972), 91–108. MR**0300410****[7]**J. M. Ortega and W. C. Rheinboldt,*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810****[8]**M. J. D. Powell,*A hybrid method for nonlinear equations*, Numerical methods for nonlinear algebraic equations (Proc. Conf., Univ. Essex, Colchester, 1969) Gordon and Breach, London, 1970, pp. 87–114. MR**0343589****[9]**M. J. D. Powell,*A Fortran subroutine for solving systems of nonlinear algebraic equations*, Numerical methods for nonlinear algebraic equations (Proc. Conf., Univ. Essex, Colchester, 1969) Gordon and Breach, London, 1970, pp. 115–161. MR**0343590****[10]**M. J. D. Powell,*A new algorithm for unconstrained optimization*, Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1970, pp. 31–65. MR**0272162****[11]**M. J. D. POWELL,*A FORTRAN Subroutine for Unconstrained Minimization, Requiring First Derivatives of the Objective Function*, Atomic Energy Research Establishment, Harwell, R.6469, 1970.**[12]**M. J. D. Powell,*Convergence properties of a class of minimization algorithms*, Nonlinear programming, 2 (Proc. Sympos. Special Interest Group on Math. Programming, Univ. Wisconsin, Madison, Wis., 1974) Academic Press, New York, 1974, pp. 1–27. MR**0386270****[13]**H. Schwetlick,*Über die Realisierung und Konvergenz von Mehrschrittverfahren zur iterativen Lösung nichtlinearer Gleichungen*, Z. Angew. Math. Mech.**54**(1974), 479–493 (German, with English and Russian summaries). MR**0378397**, https://doi.org/10.1002/zamm.19740540704**[14]**S. M. THOMAS,*Sequential Estimation Techniques for Quasi-Newton Algorithms*, Cornell University Ph.D. Thesis, 1974.

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DOI:
https://doi.org/10.1090/S0025-5718-1976-0418451-2

Article copyright:
© Copyright 1976
American Mathematical Society