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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A characterization of superlinear convergence and its application to quasi-Newton methods
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by J. E. Dennis and Jorge J. Moré PDF
Math. Comp. 28 (1974), 549-560 Request permission


Let F be a mapping from real n-dimensional Euclidean space into itself. Most practical algorithms for finding a zero of F are of the form \[ {x_{k + 1}} = {x_k} - B_k^{ - 1}F{x_k},\] where $\{ {B_k}\}$ is a sequence of nonsingular matrices. The main result of this paper is a characterization theorem for the superlinear convergence to a zero of F of sequences of the above form. This result is then used to give a unified treatment of the results on the superlinear convergence of the Davidon-Fletcher-Powell method obtained by Powell for the case in which exact line searches are used, and by Broyden, Dennis, and Moré for the case without line searches. As a by-product, several results on the asymptotic behavior of the sequence $\{ {B_k}\}$ are obtained. An interesting aspect of these results is that superlinear convergence is obtained without any consistency conditions; i.e., without requiring that the sequence $\{ {B_k}\}$ converge to the Jacobian matrix of F at the zero. In fact, a modification of an example due to Powell shows that most of the known quasi-Newton methods are not, in general, consistent. Finally, it is pointed out that the above-mentioned characterization theorem applies to other single and double rank quasi-Newton methods, and that the results of this paper can be used to obtain their superlinear convergence.
    K. M. Brown & J. E. Dennis, Jr., "A new algorithm for nonlinear least-squares curve fitting," in Mathematical Software, John R. Rice, Editor, Academic Press, New York, 1971, pp. 391-396. C. G. Broyden, J. E. Dennis & J. J. Moré, On the Local and Superlinear Convergence of Quasi-Newton Methods, Cornell Computer Science Technical Report 72-137, 1972; J. Inst. Math. Appl. (To appear.) R. Fletcher, "A new approach to variable metric algorithms," Comput. J., v. 13, 1970, pp. 317-322.
  • A. A. Goldstein and J. F. Price, An effective algorithm for minimization, Numer. Math. 10 (1967), 184–189. MR 217995, DOI 10.1007/BF02162162
  • H. Y. Huang, Unified approach to quadratically convergent algorithms for function minimization, J. Optim. Theory Appl. 5 (1970), 405–423. MR 288939, DOI 10.1007/BF00927440
  • Garth P. McCormick and Klaus Ritter, Methods of conjugate directions versus quasi-Newton methods, Math. Programming 3 (1972), 101–116. MR 302204, DOI 10.1007/BF01584978
  • J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York-London, 1970. MR 0273810
  • M. J. D. Powell, On the convergence of the variable metric algorithm, J. Inst. Math. Appl. 7 (1971), 21–36. MR 279977
  • M. J. D. Powell, Private communication, 1972. K. Ritter, "Superlinearly convergent methods for unconstrained minimization problems," Proc. ACM, Boston, 1972, pp. 1137-1145. R. Voigt, Rates of Convergence for Iterative Methods for Nonlinear Systems of Equations, Ph.D. Thesis, University of Maryland, College Park, Md., 1969.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Math. Comp. 28 (1974), 549-560
  • MSC: Primary 65H05
  • DOI:
  • MathSciNet review: 0343581