Least-Change Sparse Secant Update Methods with Inaccurate Secant Conditions
Pages 760 - 778
Abstract
We investigate the role of the secant or quasi-Newton condition in the sparse Broyden or Schubert update method for solving systems of nonlinear equations whose Jacobians are either sparse, or can be approximated acceptably by conveniently sparse matrices. We develop a theory on perturbations to the secant equation that will still allow a proof of local q-linear convergence. To illustrate the theory, we show how to generalize the standard secant condition to the case when the function difference is contaminated by noise.
References
[1]
P. Barrera, J. E. Dennis, Jr., When to stop making quasi-Newton updates, presented at the Tenth International Symposium on Mathematical Programming, Montreal
[2]
C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp., 19 (1965), 577–593
[3]
C. G. Broyden, The convergence of an algorithm for solving sparse nonlinear systems, Math. Comp., 25 (1971), 285–294
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J. E. Dennis, Jr., Jorge J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp., 28 (1974), 549–560
[5]
J. E. Dennis, Jr., R. B. Schnabel, Least change secant updates for quasi-Newton methods, SIAM Rev., 21 (1979), 443–459
[6]
John E. Dennis, Jr., Robert B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Prentice Hall Series in Computational Mathematics, Prentice Hall Inc., Englewood Cliffs, NJ, 1983xiii+378
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J. E. Dennis, Jr., Homer F. Walker, Convergence theorems for least-change secant update methods, SIAM J. Numer. Anal., 18 (1981), 949–987
[8]
J. E. Dennis, Jr., Homer F. Walker, Inaccuracy in quasi-Newton methods: local improvement theorems, Math. Programming Stud., (1984), 70–85, Rice MASC TR 83-11
[9]
E. S. Marwil, Convergence results for Schubert's method for solving sparse nonlinear equations, SIAM J. Numer. Anal., 16 (1979), 588–604
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J. J. Moré, B. S. Garbow, K. E. Hillstrom, User guide for MINPACK-1, Report, ANL-80-74, Argonne National Labs, 1980
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J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970xx+572
[12]
J. K. Reid, Least squares solution of sparse systems of nonlinear equations by a modified Marquardt algorithm, Proc. NATO Conf. at Cambridge, July 1972, North-Holland, Amsterdam, 1973, 437–445
[13]
L. K. Schubert, Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian, Math. Comp., 24 (1970), 27–30
[14]
T. J. Ypma, The effect of rounding errors on Newtonlike methods, IMA J. Numer. Anal., 3 (1983), 109–118
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Copyright © 1985 Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 August 1985
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