On the worst-case arithmetic complexity of approximating zeros of polynomials
Journal of Complexity
Complexity and real computation
Complexity and real computation
Condition number analysis for sparse polynomial systems
FoCM '97 Selected papers of a conference on Foundations of computational mathematics
Convergence and Complexity of Newton Iteration for Operator Equations
Journal of the ACM (JACM)
Finding a cluster of zeros of univariate polynomials
Journal of Complexity
On the convergence and applications of Newton-like methods for analytic operators
The Korean Journal of Computational & Applied Mathematics
On Location and Approximation of Clusters of Zeros of Analytic Functions
Foundations of Computational Mathematics
On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One
Foundations of Computational Mathematics
Extending the Newton-Kantorovich hypothesis for solving equations
Journal of Computational and Applied Mathematics
Improved generalized differentiability conditions for Newton-like methods
Journal of Complexity
Local convergence of Newton's method under majorant condition
Journal of Computational and Applied Mathematics
A unifying theorem for Newton's method on spaces with a convergence structure
Journal of Complexity
Local convergence analysis of the Gauss-Newton method under a majorant condition
Journal of Complexity
Extended sufficient semilocal convergence for the Secant method
Computers & Mathematics with Applications
Improved local convergence of Newton's method under weak majorant condition
Journal of Computational and Applied Mathematics
Majorizing sequences for iterative methods
Journal of Computational and Applied Mathematics
Local convergence analysis of inexact Gauss-Newton like methods under majorant condition
Journal of Computational and Applied Mathematics
Weaker conditions for the convergence of Newton's method
Journal of Complexity
Majorizing sequences for iterative procedures in Banach spaces
Journal of Complexity
Journal of Complexity
Unified majorizing sequences for Traub-type multipoint iterative procedures
Numerical Algorithms
Expanding the applicability of Newton's method using Smale's α-theory
Journal of Computational and Applied Mathematics
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General local convergence theorems with order of convergence r=1 are provided for iterative processes of the type x"n"+"1=Tx"n, where T:D@?X-X is an iteration function in a metric space X. The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schroder iteration for multiple zeros of polynomials and analytic functions. The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equations in Banach spaces. The new results extend and improve some results of [K. Docev, Uber Newtonsche Iterationen, C. R. Acad. Bulg. Sci. 36 (1962) 695-701; J.F. Traub, H. Wozniakowski, Convergence and complexity of Newton iteration for operator equations, J. Assoc. Comput. Mach. 26 (1979) 250-258; S. Smale, Newton's method estimates from data at one point, in: R.E. Ewing, K.E. Gross, C.F. Martin (Eds.), The Merging of Disciplines: New Direction in Pure, Applied, and Computational Mathematics, Springer, New York, 1986, pp. 185-196; P. Tilli, Convergence conditions of some methods for the simultaneous computation of polynomial zeros, Calcolo 35 (1998) 3-15; X.H. Wang, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000) 123-134; I.K. Argyros, J.M. Gutierrez, A unified approach for enlarging the radius of convergence for Newton's method and applications, Nonlinear Funct. Anal. Appl. 10 (2005) 555-563; M. Giusti, G. Lecerf, B. Salvy, J.-C. Yakoubsohn, Location and approximation of clusters of zeros of analytic functions, Found. Comput. Math. 5 (3) (2005) 257-311], and others.