Numerical solution of large sets of algebraic nonlinear equations
Author:
Ph. L. Toint
Journal:
Math. Comp. 46 (1986), 175189
MSC:
Primary 65H10
MathSciNet review:
815839
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Abstract: This paper describes the application of the partitioned updating quasiNewton methods for the solution of highdimensional systems of algebraic nonlinear equations. This concept was introduced and successfully tested in nonlinear optimization of partially separable functions (see [6]). Here its application to the case of nonlinear equations is explored. Nonlinear systems of this nature arise in many largescale applications, including finite elements and econometry. It is shown that the method presents some advantages in efficiency over competing algorithms, and that use of the partially separable structure of the system can lead to significant improvements also in the more classical discrete Newton method.
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 [1]
 C. G. Broyden, "A class of methods for solving nonlinear simultaneous equations," Math. Comp., v. 19, 1965, pp. 577593. MR 0198670 (33:6825)
 [2]
 J. E. Dennis & R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, PrenticeHall, Englewood Cliffs, N. J., 1983. MR 702023 (85j:65001)
 [3]
 A. Griewank & Ph. L. Toint, "Partitioned variable metric updates for large structured optimization problems," Numer. Math., v. 39, 1982, pp. 119137. MR 664541 (83k:65054)
 [4]
 A. Griewank & Ph. L. Toint, "Local convergence analysis for partitioned quasiNewton updates," Numer. Math., v. 39, 1982, pp. 429448. MR 678746 (84b:65062)
 [5]
 A. Griewank & Ph. L. Toint, "On the unconstrained optimization of partially separable functions," in Nonlinear Optimization 1981 (M. J. D. Powell, ed.), Academic Press, New York, 1982. MR 775354
 [6]
 A. Griewank & Ph. L. Toint, "Numerical experiments with partially separable optimization problems," in Numerical Analysis, Proceedings Dundee 1983 (D. F. Griffiths, ed.), Lecture Notes in Math., vol. 1066, SpringerVerlag, Berlin, 1984, pp. 203220. MR 760465 (85h:90102)
 [7]
 E. Marwil, Exploiting Sparsity in NewtonLike Methods, Ph.D. thesis, Cornell University, Ithaca, New York, 1978.
 [8]
 J. J. Moré, B. S. Garbow & K. E. Hillstrom, "Testing unconstrained optimization software," ACM Trans. Math. Software, v. 7(1), 1981, pp. 1741. MR 607350 (83b:90144)
 [9]
 C. C. Paige & M. A. Saunders, "LSQR: An algorithm for sparse linear equations and sparse least squares," ACM. Trans. Math. Software, v. 8(1), 1982, pp. 4371. MR 661121 (83f:65048)
 [10]
 L. K. Schubert, "Modification of a quasiNewton method for nonlinear equations with a sparse Jacobian," Math. Comp., v. 24, 1970, pp. 2730. MR 0258276 (41:2923)
 [11]
 D. F. Shanno, "On variable metric methods for sparse Hessians," Math. Comp., v. 34, 1980, pp. 499514. MR 559198 (81j:65077)
 [12]
 Ph. L. Toint, "On sparse and symmetric matrix updating subject to a linear equation," Math. Comp., v. 31, 1977, pp. 954961. MR 0455338 (56:13577)
 [13]
 O. C. Zienkiewicz, The Finite Element Method, McGrawHill, London, 1977.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608158399
PII:
S 00255718(1986)08158399
Article copyright:
© Copyright 1986
American Mathematical Society
