Steffensen's method: Difference between revisions

Revision as of 02:09, 18 June 2025

Revision as of 02:11, 18 June 2025

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

The operator <math>G</math> is roughly equivalent to a [[Matrix (mathematics)|matrix]] whose entries are all functions of [[vector (mathematics)|vector]] [[Argument of a function|arguments]] <math>\ u\ </math> and <math>\ v ~.</math> Refer again back to the [[Function of a real variable|simple function]] <math>\ f\ ,</math> given in the first section, where the function simplymerely takes in and puts out real numbers: There, the function <math>\ g\ </math> is a ''[[divided difference]]''. In the generalized form here, the operator <math>\ G\ </math> is the analogue of a divided difference for use in the [[Banach space]].

If division is possible in the [[Banach space]], then the linear operator <math>\ G\ </math> can be obtained from