zbMATH — the first resource for mathematics

Modified Newton’s method for systems of nonlinear equations with singular Jacobian. (English) Zbl 1159.65050
Authors’ summary: It is well known that Newton’s method for a nonlinear system has quadratic convergence when the Jacobian is a nonsingular matrix in a neighborhood of the solution. Here we present a modification of this method for nonlinear systems whose Jacobian matrix is singular. We prove, under certain conditions, that this modified Newton’s method has quadratic convergence. Moreover, different numerical tests confirm the theoretical results and allow us to compare this variant with the classical Newton’s method.

65H10 Numerical computation of solutions to systems of equations
Full Text: DOI
[1] Bouaricha, A.; Schnabel, R.B., Tensor methods for nonlinear least square problems, SIAM journal on scientific computing, 21, 4, 1199-1221, (1999) · Zbl 0957.65065
[2] Frontini, M.; Sormani, E., Modified newton’s method with third-order convergence and multiple roots, Computational and applied mathematics, 156, 345-354, (2003) · Zbl 1030.65044
[3] Jisheng, K.; Yitian, L.; Xiuhua, W., Efficient continuation Newton-like method for solving systems of non-linear equations, Applied mathematics and computation, 174, 846-853, (2006) · Zbl 1094.65049
[4] Kelley, C.T.; Xue, Z.Q., Inexact Newton methods for singular problems, Optimization methods & software, 2, 249-267, (1993)
[5] Nedzhibov, G.H., An acceleration of iterative processes for solving nonlinear equations, Applied mathematics and computation, 168, 320-332, (2005) · Zbl 1092.65042
[6] Traub, J.F., Iterative methods for the solution of equations, (1982), Chelsea Publishing Company New York · Zbl 0472.65040
[7] Weerakoon, S.; Fernando, T.G.I., A variant of newton’s method with accelerated third-order convergence, Applied mathematics letters, 13, 8, 87-93, (2000) · Zbl 0973.65037
[8] Wu, X., Note on the improvement of newton’s method for systems of nonlinear equations, Applied mathematics and computation, 189, 1476-1479, (2007) · Zbl 1243.65058
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.