Shub, M. The implicit function theorem revisited. (English) Zbl 0815.65066 IBM J. Res. Dev. 38, No. 3, 259-264 (1994). Newton’s method for solving systems of nonlinear equations, its extensions and properties are nicely summarized in conjunction with the implicit function theorem, homotopy methods and Bézout’s theorem. The convergence of Newton’s method, the complexity of continuation methods and the number of projective Newton steps which is sufficient to find all approximate zeros of a system of homogeneous polynomial equations are described. Reviewer: Z.Mei (Toowoomba) MSC: 65H10 Numerical computation of solutions to systems of equations 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations 26C10 Real polynomials: location of zeros 12Y05 Computational aspects of field theory and polynomials (MSC2010) Keywords:convergence; Newton’s method; homotopy methods; Bézout theorem; complexity; continuation methods; system of homogeneous polynomial equations PDFBibTeX XMLCite \textit{M. Shub}, IBM J. Res. Dev. 38, No. 3, 259--264 (1994; Zbl 0815.65066) Full Text: DOI