Steffensen's method: Difference between revisions

Revision as of 19:16, 13 February 2025

Revision as of 23:20, 17 March 2025

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This method for finding fixed points of a real-valued function has been generalized for functions <math>F : X \to X </math> that map a [[Banach space]] <math>X </math> onto itself or even more generally <math>F : X \to Y </math> that map from one [[Banach space]] <math>X </math> into another [[Banach space]] <math>Y</math>. The generalized method assumes that a [[Indexed family|family]] of [[Bounded set|bounded]] [[linear operators]] <math>\{ G(u,v): u, v \in X \} </math> associated with <math>u </math> and <math>v </math> can be devised that (locally) satisfies the condition<ref name=Johnson-Scholz-1968/>

:<math>F\left( u \right) - F\left( v \right) = G\left( u, v \right) \bigl[u - v\bigr] </math> {{right|{{nobr| {{grey|eqn. [[Coronis (textual symbol)|{{big|({{math|⸎}})}}]]}} }} }}{{anchor|⸎}}

{{NumBlk|:|<math>F\left( u \right) - F\left( v \right) = G\left( u, v \right) \bigl[u - v\bigr] </math>|{{EquationRef|1}}}}