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Newton-like method with modification of the right-hand-side vector

Authors: Natasa Krejic and Zorana Luzanin
Journal: Math. Comp. 71 (2002), 237-250
MSC (2000): Primary 65H10
Published electronically: May 9, 2001
MathSciNet review: 1862997
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Abstract | References | Similar Articles | Additional Information


This paper proposes a new Newton-like method which defines new iterates using a linear system with the same coefficient matrix in each iterate, while the correction is performed on the right-hand-side vector of the Newton system. In this way a method is obtained which is less costly than the Newton method and faster than the fixed Newton method. Local convergence is proved for nonsingular systems. The influence of the relaxation parameter is analyzed and explicit formulae for the selection of an optimal parameter are presented. Relevant numerical examples are used to demonstrate the advantages of the proposed method.

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  • 1. I. D. L. Bogle, J. D. Perkins, A new sparsity preserving quasi-Newton update for solving nonlinear equations, SIAM J. Sci. Statist. Comp., 11 (1990), pp. 621-630. MR 91f:65099
  • 2. K. M. Brown, A quadratically convergent Newton-like method based upon Gaussian elimination, SIAM J. Numer. Anal., 6 (1969), pp.560-569. MR 41:7834
  • 3. J. C. P. Bus, Numerical solution of systems of nonlinear equations, Tract 122, Mathematisch Centrum, Amsterdam, 1980. MR 81m:65072
  • 4. R. S. Dembo, S. C. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), pp.400-408. MR 83b:65056
  • 5. J. E. Dennis and E. S. Marwil, Direct secant updates of matrix factorizations, Math. Comp., 38 (1982), pp.459-474. MR 83d:65159
  • 6. J. E. Dennis, Jr. and R. B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Prentice Hall, Englewood Cliffs, NJ, 1983. MR 85j:65001
  • 7. L. C. W. Dixon, On the impact of automatic differentiation on the relative performance of parallel truncated Newton and variable metric algorithms, SIAM J. Optim. 1 (1991), 475-486. MR 93c:90079
  • 8. S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), pp.16-32. MR 96k:65037
  • 9. P.A. Farrell, J.J.H. Miller, E. O'Riordan, G. I. Shishkin, A uniformly convergent finite difference scheme for a singularly perturbed semilinear equation, SIAM J. Numer. Anal. 33(3), (1996), 1135-1149. MR 97b:65086
  • 10. G. H. Golub, C. F. van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 1983. MR 85h:65063
  • 11. M. A. Gomes-Ruggiero, J. M. Martínez and A. C. Moretti, Comparing algorithms for solving sparse nonlinear system of equations, SIAM J. Sci. Comput., 13 (1992), pp.459-483. MR 92j:65077
  • 12. D. Herceg, N. Krejic, Z. Luzanin, Quasi-Newton's method with correction, Novi Sad J. Math., 26(1), 1996, pp. 115-127. MR 98h:65021
  • 13. D. Herceg, N. Krejic, On a numerical method for discrete analogues of boundary value problems, Nonlinear Analysis, Theory, Methods & Applications, 30,1 (1997), 9-15. CMP 98:06
  • 14. L. Luksan, Inexact trust region method for large sparse systems of nonlinear equations, JOTA, 81(3), 1994, pp. 569-590. MR 95b:65072
  • 15. L. Luksan, J. Vlcek, Computational experience with globally convergent descent methods for large sparse systems of nonlinear equations, Optimization Methods and Software 8 (1998), 201-224. MR 99a:65071
  • 16. Z. Luzanin, Global and local convergence of modifications of Newton method, Ph.D. Thesis, Institute of Mathematics, Faculty of Science, University of Novi Sad, 1997.
  • 17. C. T. Kelley, Iterative methods for linear and nonlinear equations, SIAM, Philadelphia, 1995. MR 96d:65002
  • 18. J. M. Martínez, A Family of quasi-Newton methods for nonlinear equations with direct secant updates of matrix factorizations, SIAM J. Numer. Anal. 27 (1990), pp.1034-1049. MR 91h:65084
  • 19. J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970. MR 42:8686
  • 20. I. K. Schubert, Modification of a quasi-Newton method for nonlinear equation with a sparse Jacobian, Math. Comp. 24 (1970), pp.27-30. MR 41:2923

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Additional Information

Natasa Krejic
Affiliation: Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Yugoslavia

Zorana Luzanin
Affiliation: Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Yugoslavia

Keywords: Nonlinear systems, Newton method, chord method
Received by editor(s): June 22, 1998
Received by editor(s) in revised form: August 22, 1999, and March 29, 2000
Published electronically: May 9, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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