The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for singular systems of equations with constant rank derivatives. Applications to two special and important cases: the classical Lipschitz condition and the Smale's assumption, are provided; the latter, in particular, extends and improves the corresponding result due to Dedieu and Kim in [J.P. Dedieu, M. Kim, Newton's method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002) 187-209]. © 2008 Elsevier Inc. All rights reserved.
Xu, X., & Li, C. (2008). Convergence criterion of Newton’s method for singular systems with constant rank derivatives. Journal of Mathematical Analysis and Applications, 345(2), 689–701. https://doi.org/10.1016/j.jmaa.2008.04.009