Numerical solution of large sets of algebraic nonlinear equations

Author:
Ph. L. Toint

Journal:
Math. Comp. **46** (1986), 175-189

MSC:
Primary 65H10

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815839-9

MathSciNet review:
815839

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**C. G. Broyden,*A class of methods for solving nonlinear simultaneous equations*, Math. Comp.**19**(1965), 577–593. MR**0198670**, https://doi.org/10.1090/S0025-5718-1965-0198670-6**[2]**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****[3]**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**, https://doi.org/10.1007/BF01399316**[4]**A. Griewank and Ph. L. Toint,*Local convergence analysis for partitioned quasi-Newton updates*, Numer. Math.**39**(1982), no. 3, 429–448. MR**678746**, https://doi.org/10.1007/BF01407874**[5]**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****[6]**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**, https://doi.org/10.1007/BFb0099526**[7]**E. Marwil,*Exploiting Sparsity in Newton-Like Methods*, Ph.D. thesis, Cornell University, Ithaca, New York, 1978.**[8]**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**, https://doi.org/10.1145/355934.355936**[9]**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**, https://doi.org/10.1145/355984.355989**[10]**L. K. Schubert,*Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian*, Math. Comp.**24**(1970), 27–30. MR**0258276**, https://doi.org/10.1090/S0025-5718-1970-0258276-9**[11]**D. F. Shanno,*On variable-metric methods for sparse Hessians*, Math. Comp.**34**(1980), no. 150, 499–514. MR**559198**, https://doi.org/10.1090/S0025-5718-1980-0559198-2**[12]**Ph. L. Toint,*On sparse and symmetric matrix updating subject to a linear equation*, Math. Comp.**31**(1977), no. no 140, 954–961. MR**0455338**, https://doi.org/10.1090/S0025-5718-1977-0455338-4**[13]**O. C. Zienkiewicz,*The Finite Element Method*, McGraw-Hill, London, 1977.

Retrieve articles in *Mathematics of Computation*
with MSC:
65H10

Retrieve articles in all journals with MSC: 65H10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815839-9

Article copyright:
© Copyright 1986
American Mathematical Society