## A hybrid algorithm for solving sparse nonlinear systems of equations

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- by J. E. Dennis and Guang Ye Li PDF
- Math. Comp.
**50**(1988), 155-166 Request permission

## Abstract:

This paper presents a hybrid algorithm for solving sparse nonlinear systems of equations. The algorithm is based on dividing the columns of the Jacobian into two parts and using different algorithms on each part. The hybrid algorithm incorporates advantages of both component algorithms by exploiting the special structure of the Jacobian to obtain a good approximation to the Jacobian, using as little effort as possible. A Kantorovich-type analysis and a locally*q*-superlinear convergence result for this algorithm are given.

## 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 - C. G. Broyden,
*The convergence of an algorithm for solving sparse nonlinear systems*, Math. Comp.**25**(1971), 285–294. MR**297122**, DOI 10.1090/S0025-5718-1971-0297122-5 - Thomas F. Coleman and Jorge J. Moré,
*Estimation of sparse Jacobian matrices and graph coloring problems*, SIAM J. Numer. Anal.**20**(1983), no. 1, 187–209. MR**687376**, DOI 10.1137/0720013
A. R. Curtis, M. J. D. Powell & J. K. Reíd, "On the estimation of sparse Jacobian matrices," - J. E. Dennis Jr. and Jorge J. Moré,
*Quasi-Newton methods, motivation and theory*, SIAM Rev.**19**(1977), no. 1, 46–89. MR**445812**, DOI 10.1137/1019005 - 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**
Guangye Li, - Earl Marwil,
*Convergence results for Schubert’s method for solving sparse nonlinear equations*, SIAM J. Numer. Anal.**16**(1979), no. 4, 588–604. MR**537273**, DOI 10.1137/0716044 - J. M. Ortega and W. C. Rheinboldt,
*Iterative solution of nonlinear equations in several variables*, Academic Press, New York-London, 1970. MR**0273810**
J. K. Reíd, - 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

*J. Inst. Math. Appl.*, v. 13, 1974, pp. 117-119.

*The Secant/Finite Difference Algorithm for Solving Sparse Nonlinear Systems of Equations*, Technical Report 86-1, Math Sciences Dept., Rice Univ., 1986.

*Least Squares Solution of Sparse Systems of Non-linear Equations by a Modified Marquardt Algorithm*, Proceedings of the NATO Conf. at Cambridge, July 1972, North-Holland, Amsterdam, pp. 437-445.

## Additional Information

- © Copyright 1988 American Mathematical Society
- Journal: Math. Comp.
**50**(1988), 155-166 - MSC: Primary 65H10
- DOI: https://doi.org/10.1090/S0025-5718-1988-0917823-5
- MathSciNet review: 917823