A matrix-free quasi-Newton method for solving large-scale nonlinear systems. (English) Zbl 1231.65091

Summary: One of the widely used methods for solving a nonlinear system of equations is the quasi-Newton method. The basic idea underlining this type of method is to approximate the solution of Newton’s equation by means of approximating the Jacobian matrix via quasi-Newton update. Application of quasi-Newton methods for large scale problems requires, in principle, vast computational resource to form and store an approximation to the Jacobian matrix of the underlying problem. Hence, this paper proposes an approximation for Newton-step based on the update of approximation requiring a computational effort similar to that of matrix-free settings. It is made possible by approximating the Jacobian into a diagonal matrix using the least-change secant updating strategy, commonly employed in the development of quasi-Newton methods. Under suitable assumptions, local convergence of the proposed method is proved for nonsingular systems. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with Newton’s, Chord Newton’s and Broyden’s methods.


65H10 Numerical computation of solutions to systems of equations
Full Text: DOI


[1] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058
[2] Kelley, C.T., Iterative methods for linear and nonlinear equations, (1995), SIAM Philadelphia, PA · Zbl 0832.65046
[3] Broyden, C.G., A class of methods for solving nonlinear simultaneous equations, Math. comp., 19, 577-593, (1965) · Zbl 0131.13905
[4] Broyden, C.G., Quasi-Newton methods and their applications to function minimization, Math. comp., 21, 368-381, (1967) · Zbl 0155.46704
[5] Wang, X.; Kou, J.; Gu, C., A new modified secant-like method for solving nonlinear equations, Comput. math. appl., 60, 1633-1638, (2010) · Zbl 1202.65064
[6] Thukral, R., Introduction to a Newton-type method for solving nonlinear equations, Appl. math. comput., 195, 663-668, (2008) · Zbl 1154.65034
[7] Özban, A.Y., Some new variants of newton’s method, Appl. math. lett., 17, 677-682, (2004) · Zbl 1065.65067
[8] Hassan, M.A.; Leong, W.J.; Farid, M., A new gradient method via quasi-Cauchy relation which guarantees descent, J. comput. appl. math., 230, 300-305, (2009) · Zbl 1179.65067
[9] Dennis, J.E.; Schnabel, R.B., Least change secant updates for quasi-Newton methods, SIAM rev., 21, 443-459, (1979) · Zbl 0424.65020
[10] Dennis, J.E.; Wolkowicz, H., Sizing and least change secant methods, SIAM J. numer. anal., 30, 1291-1313, (1993) · Zbl 0802.65081
[11] Broyden, C.G.; Dennis, J.E.; Morè, J.J., On the local and superlinear convergence of quasi-Newton methods, J. inst. math. appl., 12, 223-246, (1973) · Zbl 0282.65041
[12] Lam, B., On the convergence of a quasi-Newton method for sparse nonlinear systems, Math. comp., 32, 447-451, (1978) · Zbl 0385.65027
[13] Gomes-Ruggiero, M.A.; Martínez, J.M.; Moretti, A.C., Comparing algorithms for solving sparse nonlinear system of equation, SIAM J. sci. comput., 13, 459-483, (1992) · Zbl 0752.65039
[14] Liu, H.; Ni, Q., Incomplete Jacobian Newton method for nonlinear equation, Comput. math. appl., 56, 218-227, (2008) · Zbl 1145.65315
[15] Lukšan, L., Inexact trust region method for large sparse systems of nonlinear equations, J. optim. theory appl., 81, 569-590, (1994) · Zbl 0803.65071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.