Can Any Stationary Iteration Using Linear Information Be Globally Convergent?

1 April 1980

journal article
Published by Association for Computing Machinery (ACM) in Journal of the ACM

Vol. 27 (2) , 263-269
https://doi.org/10.1145/322186.322191

Abstract

All known globally convergent iterations for the solution of a nonlinear operator equation ƒ( x ) = 0 are either nonstationary or use nonlinear information. It is asked whether there exists a globally convergent stationary iteration which uses linear information. It is proved that even if global convergence is defined in a weak sense, there exists no such iteration for as simple a class of problems as the set of all analytic complex functions having only simple zeros. It is conjectured that even for the class of all real polynomials which have real simple zeros there does not exist a globally convergent stationary iteration using linear information.

Keywords

LINEAR INFORMATION
STATIONARY ITERATIONS
NONLINEAR INFORMATION
SIMPLE ZERO
CONVERGENT ITERATION
GLOBALLY CONVERGENT
NONLINEAR EQUATIONS
REAL SIMPLE ZERO
GLOBAL CONVERGENCE
ANALYTIC COMPLEX FUNCTION
STATIONARY ITERATION
NONLINEAR OPERATOR EQUATION
REAL POLYNOMIAL
CONVERGENT STATIONARY ITERATION

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