# zbMATH — the first resource for mathematics

Inexact Gauss-Newton like methods for injective-overdetermined systems of equations under a majorant condition. (English) Zbl 1343.65059
The authors consider injective-overdetermined systems of nonlinear equations, i.e., nonlinear systems of the form $$F(x)=0$$, for which the derivative $$F'(x)$$ is injective, as linear operator. For solving such a system they propose an inexact Gauss-Newton iteration which in its iterations replace the exact solution of linear system involving $$F'(x)$$ with an iteratively computed approximation. They prove convergence of the new algorithm together with some applications.

##### MSC:
 65H10 Numerical computation of solutions to systems of equations
Full Text:
##### References:
 [1] Argyros, IK; Hilout, S, Extending the applicability of the Gauss-Newton method under average Lipschitz-type conditions, Numer. Algor., 58, 23-52, (2011) · Zbl 1228.65076 [2] Argyros, IK; Hilout, S, Local convergence analysis of inexact Newton-like methods, J. sci. Appl., 2, 11-18, (2009) · Zbl 1167.65368 [3] Argyros, IK; Hilout, S, On the solution of systems of equations with constant rank derivatives, Numer. Algor., 57, 235-253, (2011) · Zbl 1218.65044 [4] Chen, J, The convergence analysis of inexact Gauss-Newton methods for nonlinear problems, Comput. Optim. Appl., 40, 97-118, (2008) · Zbl 1192.90200 [5] Chen, J; Li, W, Convergence of Gauss-newton’s method and uniqueness of the solution, Appl. Math. Comput., 170, 686-705, (2005) · Zbl 1084.65058 [6] Ferreira, OP, Local convergence of newton’s method under majorant condition, J. Comput. Appl. Math., 235, 1515-1522, (2011) · Zbl 1225.65060 [7] Ferreira, OP; Gonċalves, MLN, Local convergence analysis of inexact Newton-like methods under majorant condition, Comput. Optim. Appl., 48, 1-21, (2011) · Zbl 1279.90195 [8] Ferreira, OP; Gonċalves, MLN; Oliveira, PR, Local convergence analysis of the Gauss-Newton method under a majorant condition, J. Complexity, 27, 111-125, (2011) · Zbl 1216.65070 [9] Ferreira, OP; Gonċalves, MLN; Oliveira, PR, Local convergence analysis of inexact Gauss-Newton like methods under majorant condition, J. Comput. Appl. Math., 236, 2487-2498, (2012) · Zbl 1241.65052 [10] Ferreira, OP; Gonċalves, MLN; Oliveira, PR, Convergence of the Gauss-Newton method for convex composite optimization under a majorant condition, SIAM J. Optim., 23, 1757-1783, (2013) · Zbl 1277.49036 [11] Ferreira, OP; Svaiter, BF, A robust kantorovich’s theorem on the inexact Newton method with relative residual error tolerance, Comput. Optim. Appl., 42, 213-229, (2012) · Zbl 1245.65060 [12] Gonċalves, MLN, Local convergence analysis of the Gauss-Newton method for injective-overdetermined systems of equations under a majorant condition, Comput. Math. Appl., 66, 490-499, (2013) · Zbl 1360.65160 [13] Huang, Z, The convergence ball of newton’s method and the uniqueness ball of equations under Hölder-type continuous derivatives, Comput. Math. Appl., 47, 247-251, (2004) · Zbl 1052.65054 [14] Li, C; Zhang, WH; Jin, XQ, Convergence and uniqueness properties of Gauss-newton’s method, Comput. Math. Appl., 47, 1057-1067, (2004) · Zbl 1076.65053 [15] Martinez, JM; Qi, L, Inexact Newton methods for solving nonsmooth equations, J. Comput. Appl. Math., 60, 127-145, (1999) · Zbl 0833.65045 [16] Morini, B, Convergence behaviour of inexact Newton methods, Math. Comp., 68, 1605-1613, (1999) · Zbl 0933.65050 [17] Porcelli, M, On the convergence of an inexact Gauss-Newton trust-region method for nonlinear least-squares problems with simple bounds, Optim. Letters, 7, 447-465, (2013) · Zbl 1268.90091 [18] Shen, W; Li, C, Kantorovich-type convergence criterion inexact Newton methods, Math. Comp., 68, 1605-1613, (2009) · Zbl 1165.65354 [19] Smale, S.: Newton’s method estimates from data at one point. In: The merging of disciplines: new directions in pure, applied, and computational mathematics (Laramie, Wyo., 1985), pp 185-196. Springer, New York (1986) [20] Stewart, GW, On the continuity of the generalized inverse, SIAM J. Appl. Math., 17, 33-45, (1969) · Zbl 0172.03801 [21] Wedin, PA, Perturbation theory for pseudo-inverses, Nordisk Tidskr. Informationsbehandling (BIT), 13, 217-232, (1973) · Zbl 0263.65047 [22] Wang, X, Convergence of newton’s method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal., 20, 123-134, (2000) · Zbl 0942.65057 [23] Wang, XH; Li, C, Convergence of newton’s method and uniqueness of the solution of equations in Banach spaces. II, Acta Math. Sin. (Engl. Ser.), 19, 405-412, (2003) · Zbl 1027.65078 [24] Xu, X; Li, C, Convergence of newton’s method for systems of equations with constant rank derivatives, J. Comput. Math., 25, 705-718, (2007) · Zbl 1150.49011
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.