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BROYDEN'S METHOD FOR OPERATORS WITH REGULARLY CONTINUOUS DIVIDED DIFFERENCES
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 Title & Authors
BROYDEN'S METHOD FOR OPERATORS WITH REGULARLY CONTINUOUS DIVIDED DIFFERENCES
Galperin, Anatoly M.;
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 Abstract
We present a new convergence analysis of popular Broyden's method in the Banach/Hilbert space setting which is applicable to non-smooth operators. Moreover, we do not assume a priori solvability of the equation under consideration. Nevertheless, without these simplifying assumptions our convergence theorem implies existence of a solution and superlinear convergence of Broyden's iterations. To demonstrate practical merits of Broyden's method, we use it for numerical solution of three nontrivial infinite-dimensional problems.
 Keywords
nonlinear operator equations;Broyden's method;convergence analysis;regular continuity;
 Language
English
 Cited by
 References
1.
M. Aigner, Discrete Mathematics, American Mathematical Society, Providence, RI, 2007.

2.
C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19 (1965), 577-593. crossref(new window)

3.
Y. Eidelman, V. Milman, and A. Tsolomitis, Functional Analysis, American Mathematical Society, Providence, RI, 2004.

4.
F. Facchinei and J.-S. Pang, Finite-dimensional Variational Inequalities and Complemetarity Problems, Springrer-Verlag, N.Y., 2003.

5.
A. Galperin, Secant method with regularly continuous divided differences, J. Comput. Appl. Math. 193 (2006), no. 2, 574-595. crossref(new window)

6.
A. Galperin, On a class of systems of difference equations and their invariants, J. Difference Equ. Appl. 13 (2007), no. 5, 357-381. crossref(new window)

7.
A. Galperin, Ulm's method without derivatives, Nonlinear Anal. 71 (2009), no. 5-6, 2094-2113. crossref(new window)

8.
A. Griewank, The local convergence of Broyden-like methods on Lipschitzian problems in Hilbert space, SIAM J. Numer. Anal. 24 (1987), no. 3, 684-705. crossref(new window)

9.
P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Program. 48 (1990), no. 2, 161-220. crossref(new window)

10.
D. M. Hwang and C. T. Kelley, Convergence of Broyden's method in Banach spaces, SIAM J. Optim. 2 (1992), no. 3, 505-532. crossref(new window)

11.
L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, Elmsford, 1982.

12.
C. T. Kelley, Iterative methods for linear and nonlinear equations, SIAM, Philadelphia, 1995.

13.
C. T. Kelley and E. W. Sachs, Broyden's method for approximate solution of nonlinear integral equation, J. Integral Equations Appl. 9 (1985), no. 1, 25-43.

14.
C. T. Kelley and E. W. Sachs, A quasi-Newton method for elliptic boundary value problems, SIAM J. Numer. Anal. 24 (1987), no. 3, 516-531. crossref(new window)

15.
C. T. Kelley and E. W. Sachs, A pointwise quasi-Newton method for unconstrained optimal control problems, Numer. Math. 55 (1989), no. 2, 159-176. crossref(new window)

16.
C. T. Kelley and E. W. Sachs, A new proof of superlinear convergence for Broyden's method in Hilbert space, SIAM J. Optim. 1 (1991), no. 1, 146-150. crossref(new window)

17.
G. G.Magaril-Il'yaev and V.M. Tikhomirov, Convex Analysis, Theory and Applications, American Mathematical Society, Providence, RI, 2003.

18.
O. Mangasarian, Equivalence of the complementarity problem to a system of nonlinear equations, SIAM J. Appl. Math. 31 (1976), no. 1, 89-92. crossref(new window)

19.
G. Pimbley, Positive solutions of a quadratic integral equation, Arch. Ration. Mech. Anal. 24 (1967), 107-127. crossref(new window)

20.
L. Qi, On superlinear convergence of quasi-Newton methods for nonsmooth equations, Oper. Res. Lett. 20 (1997), no. 5, 223-228. crossref(new window)

21.
E.W. Sachs, Broyden's method in Hilbert space, Math. Program. 35 (1986), no. 1, 71-82. crossref(new window)

22.
W.-H. Yu, A quasi-Newton method in infinite-dimensional spaces and its application for solving a parabolic inverse problem, J. Comput. Math. 16 (1998), no. 4, 305-318.