The column-updating method for solving nonlinear equations in Hilbert space

M. A. Gomes-Ruggiero; J. M. Martínez

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (1992)

  • Volume: 26, Issue: 2, page 309-330
  • ISSN: 0764-583X

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Gomes-Ruggiero, M. A., and Martínez, J. M.. "The column-updating method for solving nonlinear equations in Hilbert space." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 26.2 (1992): 309-330. <http://eudml.org/doc/193665>.

@article{Gomes1992,
author = {Gomes-Ruggiero, M. A., Martínez, J. M.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {column-updating method; nonlinear operator equations; Hilbert spaces; local superlinear convergence; numerical comparison},
language = {eng},
number = {2},
pages = {309-330},
publisher = {Dunod},
title = {The column-updating method for solving nonlinear equations in Hilbert space},
url = {http://eudml.org/doc/193665},
volume = {26},
year = {1992},
}

TY - JOUR
AU - Gomes-Ruggiero, M. A.
AU - Martínez, J. M.
TI - The column-updating method for solving nonlinear equations in Hilbert space
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 1992
PB - Dunod
VL - 26
IS - 2
SP - 309
EP - 330
LA - eng
KW - column-updating method; nonlinear operator equations; Hilbert spaces; local superlinear convergence; numerical comparison
UR - http://eudml.org/doc/193665
ER -

References

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  1. [1] R. H. BARTELS and G. H. GOLUB, The Simplex Method of Linear Programming using LU decomposition, Comm. ACM12 (1969) 266-268. Zbl0181.19104
  2. [2] C. G. BROYDEN, A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19 1965) 577-593. Zbl0131.13905MR198670
  3. [3] C. G. BROYDEN, The convergence of an algorithm for solving sparse nonlinear Systems, Math. Comp. 25 (1971) 285-294. Zbl0227.65038MR297122
  4. [4] C. G. BROYDEN, J. E. DENNIS and J. J. MORÉ, On the local and superlinear convergence of quasi-Newton methods, J. Inst. Math. Appl. 12 (1973) 223-246. Zbl0282.65041MR341853
  5. [5] J. E. DENNIS, Toward a unified convergence theory for Newton-like methods, in L. B. Rall, ed., Nonlinear functional analysis and applications, Academic Press, New York, London, 1971, pp. 425-472. Zbl0276.65029MR278556
  6. [6] J. E. DENNIS and J. J. MORÉ, A charactenzation of superlinear convergence and its application to quasi-Newton methods, Math. Comp. 28 (1974) 543-560. Zbl0282.65042MR343581
  7. [7] J. E. DENNIS and R. B. SCHNABEL, Numerical methods for unconstrained optimization and nonlinear equations, Prentice Hall, Englewood Cliffs, New Jersey, 1983. Zbl0579.65058MR702023
  8. [8] J. E. DENNIS and R. B. SCHNABEL, A View of Unconstrained Optimization, to appear in Handbook m Operations Research and Management Science, Vol.1, Optimization, G. L. Nemhauser, AHG Rinnooy Kan, M. J. Tood, eds., North Holland, Amsterdam 1989. MR1105100
  9. [9] J. E. DENNIS and H. F. WALKER, Convergence theorems for least-change secant update methods, SIAM J. Numer. Anal. 18 (1981), 949-987. Zbl0527.65032MR638993
  10. [10] I. S. DUFF, A. M. ERISMAN and J. K. REID, Direct methods for sparse matrices, Clarendon Press, Oxford, 1986. Zbl0604.65011MR892734
  11. [11] A. GEORGE and E. NG, Symbolic factorization for sparse Gaussian elimination with partial pivoting, SIAM J. Sci. Statist. Comput. 8 (1987), 877-898. Zbl0632.65021MR911061
  12. [12] G. H. GOLUB and Ch. F. VAN LOAN, Matrix Computations, John Hopkins, Baltimore, 1983. Zbl0559.65011MR733103
  13. [13] W. A. GRUVER and E. SACHS, algorithmic methods in optimal control, Pitman, Boston, London, Melbourne, 1981. Zbl0456.49001MR604361
  14. [14] L. V. KANTOROVICH and G. P. AKILOV, Functional analysis in normed spaces, MacMillan, New York, 1964. Zbl0127.06104MR213845
  15. [15] T. KATO, Perturbation theory for linear operators, Springer Verlag, New York, 1966. Zbl0148.12601MR203473
  16. [16] A. KOLMOGOROFF and S. FOMIN, Elements of the Theory of Functions and Functional Analysis, Izdat. Moscow Univ., Moscow, 1954. Zbl0501.46001
  17. [17] J. M. MARTINEZ, A quasi-Newton method with modification of one column periteration, Computing 33 (1984), 353-362. Zbl0546.90102MR773934
  18. [18] J. M. MARTÍNEZ, A new family of quasi-Newton methods with direct secant updates of matrix factorizations, SIAM J. Numer. Anal. 27 (1990), 1034-1049. Zbl0702.65053MR1051122
  19. [19] E. S. MARWIL, Convergence results for Schubert's method for solving sparse nonlinear equations, SIAM J. Numer. Anal. 16 (1979), 588-604. Zbl0453.65033MR537273
  20. [20] H. MATTHIES and G. STRANG, The solution of nonlinear finite element equations, Internat. J. Numer. Methods in Engrg. 14 (1979), 1613-1626. Zbl0419.65070MR551801
  21. [21] J. M. ORTEGA and W. C. RHEINBOLDT, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970. Zbl0241.65046MR273810
  22. [22] E. SACHS, Convergence rates of quasi-Newton algorithms for some nonsmooth optimization problems, SIAM J. Control Optim. 23 (1985), 401-418. Zbl0571.90083MR784577
  23. [23] E. SACHS, Broyden's method in Hilbert space, Math. Programming 35 (1986), 71-82. Zbl0598.90080MR842635
  24. [24] L. K. SCHUBERT, Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian, Math. Comp. 24 (1970), 27-30. Zbl0198.49402MR258276
  25. [25] L. K. SCHUBERT, An interval arithmetic approach for the construction of an almost globally convergence method for the solution of the nonlinear Poisson equation on the unit square, SIAM J. Sci. Statist. Comput. 5 (1984), 427-452. Zbl0539.65076MR740859
  26. [26] H. SCHWETLICK, Numerische Lösung nichtlinearer Gleichungen, Berlin : Deutscher Verlag der Wissenschaften, 1978. Zbl0402.65028MR519682
  27. [27] Ph. L. TOINT, Numerical solution of large sets of algebraic nonlinear equations, Math. Comp. 16 (1986), 175-189. Zbl0614.65058MR815839

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