Numerical solution of large sets of algebraic nonlinear equations

Toint, Ph. L.

doi:10.1090/S0025-5718-1986-0815839-9

Numerical solution of large sets of algebraic nonlinear equations
HTML articles powered by AMS MathViewer

by Ph. L. Toint PDF

Math. Comp. 46 (1986), 175-189 Request permission

Abstract:

This paper describes the application of the partitioned updating quasi-Newton methods for the solution of high-dimensional 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 large-scale 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.

References

C. G. Broyden, A class of methods for solving nonlinear simultaneous equations, Math. Comp. 19 (1965), 577–593. MR 198670, DOI 10.1090/S0025-5718-1965-0198670-6
John E. Dennis Jr. and Robert B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Prentice Hall Series in Computational Mathematics, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983. MR 702023
A. Griewank and Ph. L. Toint, Partitioned variable metric updates for large structured optimization problems, Numer. Math. 39 (1982), no. 1, 119–137. MR 664541, DOI 10.1007/BF01399316
A. Griewank and Ph. L. Toint, Local convergence analysis for partitioned quasi-Newton updates, Numer. Math. 39 (1982), no. 3, 429–448. MR 678746, DOI 10.1007/BF01407874
A. Griewank and Ph. L. Toint, On the unconstrained optimization of partially separable functions, Nonlinear optimization, 1981 (Cambridge, 1981) NATO Conf. Ser. II: Systems Sci., Academic Press, London, 1982, pp. 301–312. MR 775354
A. Griewank and Ph. L. Toint, Numerical experiments with partially separable optimization problems, Numerical analysis (Dundee, 1983) Lecture Notes in Math., vol. 1066, Springer, Berlin, 1984, pp. 203–220. MR 760465, DOI 10.1007/BFb0099526

Exploiting Sparsity in Newton-Like Methods

Jorge J. Moré, Burton S. Garbow, and Kenneth E. Hillstrom, Testing unconstrained optimization software, ACM Trans. Math. Software 7 (1981), no. 1, 17–41. MR 607350, DOI 10.1145/355934.355936
Christopher C. Paige and Michael A. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Software 8 (1982), no. 1, 43–71. MR 661121, DOI 10.1145/355984.355989
L. K. Schubert, Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian, Math. Comp. 24 (1970), 27–30. MR 258276, DOI 10.1090/S0025-5718-1970-0258276-9
D. F. Shanno, On variable-metric methods for sparse Hessians, Math. Comp. 34 (1980), no. 150, 499–514. MR 559198, DOI 10.1090/S0025-5718-1980-0559198-2
Ph. L. Toint, On sparse and symmetric matrix updating subject to a linear equation, Math. Comp. 31 (1977), no. 140, 954–961. MR 455338, DOI 10.1090/S0025-5718-1977-0455338-4

The Finite Element Method