Improved local convergence analysis of the Gauss-Newton method under a majorant condition

Abstract

We present a local convergence analysis of Gauss-Newton method for solving nonlinear least square problems. Using more precise majorant conditions than in earlier studies such as Chen (Comput Optim Appl 40:97-118, 2008), Chen and Li (Appl Math Comput 170:686-705, 2005), Chen and Li (Appl Math Comput 324:13811394, 2006), Ferreira (J Comput Appl Math 235:1515-1522, 2011), Ferreira and Goncalves (Comput Optim Appl 48:1-21, 2011), Ferreira and Goncalves (J Complex 27(1):111-125, 2011), Li et al. (J Complex 26:268-295, 2010), Li et al. (Comput Optim Appl 47:1057-1067, 2004), Proinov (J Complex 25:38-62, 2009), Ewing, Gross, Martin (eds.) (The merging of disciplines:new directions in pure, applied and computational mathematics 185-196, 1986), Traup (Iterative methods for the solution of equations, 1964), Wang (J Numer Anal 20:123-134, 2000), we provide a larger radius of convergence; tighter error estimates on the distances involved and a clearer relationship between the majorant function and the associated least squares problem. Moreover, these advantages are obtained under the same computational cost.