Convergence behavior of Gauss–Newton’s method and extensions of the Smale point estimate theory

The notions of Lipschitz conditions with LL average are introduced to the study of convergence analysis of Gauss–Newton’s method for singular systems of equations. Unified convergence criteria ensuring the convergence of Gauss–Newton’s method for one kind of singular systems of equations with constant rank derivatives are established and unified estimates of radii of convergence balls are also obtained. Applications to some special cases such as the Kantorovich type conditions, γγ-conditions and the Smale point estimate theory are provided and some important known results are extended and/or improved.