A New Broyden rank reduction method to solve large systems of nonlinear equations
DOI:
https://doi.org/10.11121/ijocta.01.2019.00743Keywords:
Large nonlinear systems of equations, Limited memory Broyden method, Singular values thresholding, Rank reduction, Superlinear convergence.Abstract
We propose a modification of limited memory Broyden methods, called dynamical Broyden rank reduction method, to solve high dimensional systems of nonlinear equations. Based on a thresholding process of singular values, the proposed method determines a priori the rank of the reduced update matrix. It significantly reduces the number of singular values decomposition calls of the update matrix during the iterations. Local superlinear convergence of the method is proved and some numerical examples are displayed.
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References
Dennis, J. E. and Schnabel, R. B. (1983). Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall.
Kelley, C. T. (1995). Iterative methods for linear and nonlinear equations. Volume 16 of Frontiers in Applied Mathematics, SIAM, Philadelphia.
Ortega, J. M. and Reinboldt, W. C. (2000). Iterative solution of nonlinear equations in several variables. SIAM, Philadelphia.
Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of Computation, (19), 577-593.
Broyden, C. G., Dennis, J. E. and More, J. J. (1973). On the local and superlinear convergence of quasi-Newton methods. IMA Journal of Applied Mathematics, (12), 223-245.
Byrd, R. H., Nocedal, J. and Schnabel, R. B. (1994). Representations of quasi-Newton matrices and their use in limited memory methods. Mathematical Programming, (63), 129-156.
Van De Rotten, B. and Verduyn Lunel, S. M. (2003). A limited memory Broyden method to solve high dimensional systems of nonlinear equations. Technical Report MI 2003-06, Mathematical Institute, University of Leiden, The Netherlands.
Ziani M. and Guyomarc'h F. (2008). An autoadaptative limited memory Broyden's method to solve systems of nonlinear equations. Applied Mathematics and Computation, (205), 202-211.
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