Steffensen's method: Difference between revisions

Revision as of 01:32, 12 July 2008

Revision as of 18:11, 13 July 2008

Line 17:

==Generalised definition==

Steffensen's method can also be used more abstractly, to find the locations a different kind of function <math>f\ </math>, that producefor some input values, <math>x\ </math> produces the same output as input: <math>x = f(x)\ </math>,. Such solutions are called "[[fixed point]]s". Steffensen's method finds [[fixed point]]sMany of asuch [[Mapfunctions (mathematics)|mapping]]can <math>f\be </math>.used Into Steffensen'sfind originaltheir description,own <math>f\solutions </math>by wasrepeatedly supposedrecycling tothe beresult aback realas functioninput, but the methodrate hasof beenconvergence generalisedcan forbe functionsslow, <math>for :the Xfunction can fail \to Xconverge </math>at onall, adepending [[Banachon space]]the <math>X</math>individual function. Steffensen's method accelerates convergence.

Steffensen's method finds [[fixed point]]s of a [[Map (mathematics)|mapping]] <math>f\ </math>. In Steffensen's original description, <math>f\ </math> was supposed to be a real function, but the method has been generalised for functions <math>f : X \to X </math> on a [[Banach space]] <math>X</math>.

The method assumes that a [[Indexed family|family]] of [[Bounded set|bounded]] [[linear operators]] <math>\{L(u,v): u, v \in X\}</math> associated with <math>u\ </math> and <math>v\ </math> are can be found to satisfy the condition<ref>L. W. Johnson; D. R. Scholz (1968) On Steffensen's Method, ''SIAM Journal on Numerical Analysis'' (June 1968), vol.&nbsp;5, no.&nbsp;2., pp. 296-302. Stable URL: [http://links.jstor.org/sici?sici=0036-1429%28196806%295%3A2%3C296%3AOSM%3E2.0.CO%3B2-H]</ref>