Convergence criterion of Newton's method for singular systems with constant rank derivatives

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Abstract

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].

Keywords

Newton's method
Singular system
Lipschitz condition with L-average
Moore–Penrose inverse
Convergence criterion

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Supported in part by the Education Ministry of Zhejiang Province (Grant No. 20060492) for the first author, and the National Natural Science Foundation of China (Grant Nos. 10671175 and 10731060) and Program for New Century Excellent Talents in University for the second author.