Finding Zeros of Analytic Functions: α Theory for Secant Type Methods
Copyright policy )Finding Zeros of Analytic Functions: α Theory for Secant Type Methods
The author considers the zero-finding problem for an analytic function \(f: E\to F\) between two real or complex Banach spaces. A series of convergence theorems for secant methods requiring only conditions at a point is proved. The radius of robustness of these methods is obtained with applications to the study of the complexity of homotopy methods for approximating roots. Connections with the \(\alpha\)-theory [\textit{M. Shub} and \textit{S. Smale}, J. Am. Math. Soc. 6, No. 2, 459-501 (1993; Zbl 0821.65035)] in terms of data computed at a point \(x_0\) alone are discussed.
- Université de Toulouse France
- Paul Sabatier University France
Statistics and Probability, Numerical Analysis, convergence, Algebra and Number Theory, Control and Optimization, Regula Falsi, Numerical solutions to equations with nonlinear operators, Applied Mathematics, alpha-theory, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, \(\alpha\)-theory, approximate zero, secant method, α-theory, Banach spaces, Iterative procedures involving nonlinear operators, Newton method, homotopy method, regula falsi, complexity
Statistics and Probability, Numerical Analysis, convergence, Algebra and Number Theory, Control and Optimization, Regula Falsi, Numerical solutions to equations with nonlinear operators, Applied Mathematics, alpha-theory, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, \(\alpha\)-theory, approximate zero, secant method, α-theory, Banach spaces, Iterative procedures involving nonlinear operators, Newton method, homotopy method, regula falsi, complexity
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