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Successive column correction algorithms for solving sparse nonlinear systems of equations

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Abstract

This paper presents two algorithms for solving sparse nonlinear systems of equations: the CM-successive column correction algorithm and a modified CM-successive column correction algorithm. Aq-superlinear convergence theorem and anr-convergence order estimate are given for both algorithms. Some numerical results and the detailed comparisons with some previously established algorithms show that the new algorithms have some promise of being very effective in practice.

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References

  1. C.G. Broyden, “A class of methods for solving nonlinear simultaneous equations,”Mathematics of Computation 19 (1965) 577–593.

    Google Scholar 

  2. C.G. Broyden, “The convergence ofan algorithm for solving sparse nonlinear systems,”Mathematics of Computation 25 (1971) 285–294.

    Google Scholar 

  3. C.G. Broyden, J.E. Dennis, Jr. and J.J. Moré, “On the local and superlinear convergence of quasi-Newton methods,”Journal of the Institute of Mathematics and its Application 12 (1973) 223–246.

    Google Scholar 

  4. T.F. Coleman and J.J. Moré, “Estimation of sparse Jacobian and graph coloring problems,”SIAM Journal on Numerical Analysis 20 (1983) 187–209.

    Google Scholar 

  5. A.R. Curtis, M.J.D. Powell and J.K. Reid, “On the estimation of sparse Jacobian matrices,”IMA Journal of Applied Mathematics 13 (1974) 117–119.

    Google Scholar 

  6. J.E. Dennis, Jr. and J.J. Moré, “A characterization of superlinear convergence and its application to quasi-Newton methods,”Mathematics of Computation 28 (1974) 549–560.

    Google Scholar 

  7. J.E. Dennis, Jr. and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1983).

    Google Scholar 

  8. Guocheng Feng and Guangye Li, “Column-update quasi-Newton method,”Numerical Mathematics Journal of Chinese Universities 5 (1983) 139–147.

    Google Scholar 

  9. Guangye Li, “The secant/finite difference algorithm for solving sparse nonlinear systems of equations,” Technical Report 86-1, Mathematical Sciences Department, Rice University (Houston, TX, 1986), to appear inSIAM Journal on Numerical Analysis.

    Google Scholar 

  10. E. Marwil, “Convergence results for Schubert's method for solving sparse nonlinear equations,”SIAM Journal on Numerical Analysis 16 (1979) 588–604.

    Google Scholar 

  11. J.J. Moré and M.Y. Cosnard, “Numerical solution of nonlinear equations,”ACM Transactions on Mathematical Software 5 (1979) 64–85.

    Google Scholar 

  12. J.J. Moré, B.S. Garbow and K.E. Hillstrom, “Testing unconstrained optimization software,”ACM Transactions on Mathematical Software 7 (1981) 17–41.

    Google Scholar 

  13. J.M. Ortega and W.C. Rheinboldt,Iterative solution of nonlinear equations in several variables (Academic Press, New York and London, 1970).

    Google Scholar 

  14. E. Polak, “A modified secant method for unconstrained minimization,” Memorandum No. ERLM373, Electronics Research Lab, College of Engineering, University of California (Berkeley, CA, 1973).

    Google Scholar 

  15. M.J.D. Powell and PH.L. Toint, “On the estimation of sparse Hessian matrices,”SIAM Journal on Numerical Analysis 16 (1979) 1060–1074.

    Google Scholar 

  16. J.K. Reid, “Least squares solution of sparse systems of non-linear equations by a modified Marquardt algorithm,”Proceedings of the NATO Conference at Cambridge (North Holland, Amsterdam, 1972) pp. 437–445.

    Google Scholar 

  17. H.H. Rosenbrock, “An automatic method for finding the greatest or least value of a function,”Computer Journal 3 (1960) 175–184.

    Google Scholar 

  18. L.K. Schubert, “Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian,”Mathematics of Computation 24 (1970) 27–30.

    Google Scholar 

  19. E. Spedicato, “Computational experience with quasi-Newton algorithms for minimization problems of moderately large size,” Report CISE-N-175, Segrate (Milano, 1975).

    Google Scholar 

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Authors and Affiliations

  1. Computer Center, Jilin University, Changchun, Jilin, People's Republic of China

    Guangye Li

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  1. Guangye Li
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Additional information

This research was partially supported by contracts and grants: DOE DE-AS05-82ER1-13016, AFOSR 85-0243 at Rice University, Houston, U.S.A. and Natural Sciences and Engineering Research Council of Canada grant A-8639.

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Li, G. Successive column correction algorithms for solving sparse nonlinear systems of equations. Mathematical Programming 43, 187–207 (1989). https://doi.org/10.1007/BF01582289

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  • DOI: https://doi.org/10.1007/BF01582289

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