Steffensen's method: Difference between revisions

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Steffensen's method can also be used to find an input <math>\ x = x_\star\ </math> for a different kind of function <math>\ F\ </math> that produces output the same as its input: <math>\ x_\star = F(x_\star)\ </math> for the special value <math>\ x_\star ~.</math> Solutions like <math>\ x_\star\ </math> are called ''[[fixed point (mathematics)|fixed point]]s''. Many of these functions can be used to find their own solutions by repeatedly recycling the result back as input, but the rate of convergence can be slow, or the function can fail to converge at all, depending on the individual function. Steffensen's method accelerates this convergence, to make it [[quadratic convergence|quadratic]].

ForAs orientationan example to orient the reader, a pseudo-version of the rootfixed-point function <math>\ fF\ </math> andcan thebe fixed-pointconcocted functionsfrom area simplyroot relatedfunction <math>\ f\ </math> simply by <math>\ F(x) = x + \varepsilon f(x)\ ,</math> where <math>\ \varepsilon\ </math> is some scalar constant small enough in magnitude to make <math>\ F\ </math> stable under iteration, but large enough for the [[non-linearity]] of the function <math>\ f\ </math> to be appreciable. All issues of a more general [[Banach space]] vs. basic [[real numbers]] being momentarily ignored for the sake of the comparison.

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>\ \bigl\{\ G(u,v): u, v \in X\ \bigr\}\ </math> associated with <math>\ u\ </math> and <math>\ v\ </math> can be devised that (locally) satisfies the condition<ref name=Johnson-Scholz-1968/>

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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 merely 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