论文信息 - On the Global Convergence of Broyden''s Method

On the Global Convergence of Broyden''s Method

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 from a uniformly linearly indepenent 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.

J. J. Moré | J. Trangenstein

[1] C. G. Broyden. A Class of Methods for Solving Nonlinear Simultaneous Equations , 1965 .

[2] M. Powell. A New Algorithm for Unconstrained Optimization , 1970 .

[3] C. G. Broyden. The convergence of single-rank quasi-Newton methods , 1970 .

[4] James M. Ortega,et al. Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[5] P. Gill,et al. Quasi-Newton Methods for Unconstrained Optimization , 1972 .

[6] John E. Dennis,et al. On the Local and Superlinear Convergence of Quasi-Newton Methods , 1973 .

[7] J. J. Moré,et al. A Characterization of Superlinear Convergence and its Application to Quasi-Newton Methods , 1973 .

[8] H. Schwetlick,et al. Über die Realisierung und Konvergenz von Mehrschrittverfahren zur iterativen Lösung nichtlinearer Gleichungen , 1974 .

[9] J. J. Moré,et al. Quasi-Newton Methods, Motivation and Theory , 1974 .

[10] S. W. Thomas. Sequential estimation techniques for quasi-newton algorithms. , 1975 .

[11] M. Powell. CONVERGENCE PROPERTIES OF A CLASS OF MINIMIZATION ALGORITHMS , 1975 .