Computing 62, 129–145 (1999) c  Springer-Verlag 1999 Printed in Austria A Modification of Newton’s Method for Analytic Mappings Having Multiple Zeros P. Kravanja and A. Haegemans, Heverlee Received August 17, 1998 Abstract We propose a modification of Newton’s method for computing multiple roots of systems of analytic equations. Under mild assumptions the iteration converges quadratically. It involves certain constants whose product is a lower bound for the multiplicity of the root. As these constants are usually not known in advance, we devise an iteration in which not only an approximation for the root is refined, but also approximations for these constants. Numerical examples illustrate the effectiveness of our approach. AMS Subject Classification: 65H10. Key Words: Newton’s method, analytic mappings, multiple zeros, Van de Vel’s iteration. 1. Introduction Consider a smooth function f : C → C that has a zero of multiplicity µ at the point z ⋆ . If µ = 1, then Newton’s method converges quadratically to z ⋆ if the initial iterate is sufficiently close to z ⋆ . If µ> 1, then the convergence is only linear. In the latter case, if µ is known in advance, quadratic convergence can be regained by considering the iteration z (p+1) = z (p) − µ f (z (p) ) f ′ (z (p) ) , p = 0, 1, 2, ... . (1) Van de Vel [40, 41] devised an iteration in which not only an approximation for the zero is refined, but also an estimate of its multiplicity. King [20] analysed Van de Vel’s method and proved that its order of convergence is 1.554. He re- arranged the order of the calculations and obtained an iteration that has order of convergence 1.618. This iteration proceeds as follows:    µ (p+1) = u(z (p) ) u(z (p) ) − u(z (p+1) ) µ (p) z (p+2) = z (p+1) − µ (p+1) u(z (p+1) ) (2)