Steffensen's method: Difference between revisions

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Revision as of 18:31, 10 July 2008

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Steffensen's method finds [[fixed point]]s of a [[Map (mathematics)|mapping]] &fnof;. In the original definition, &fnof; 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]] <math>\{F(x',x''):x', x'' \in X\}</math> of [[Bounded set|bounded]] [[linear operators]] (called ''divided difference'') associated with ''x''<nowiki>'</nowiki> and ''x''"<nowiki>''</nowiki> is known which satisfies

: <math>f(x')- f(x'')=F(x',x'')(x'-x''). \,</math>

Steffensen's method is then very similar to the Newton's method, except that it uses thisthe operatordivided difference<math>F(f(x),x)</math> instead of the derivative <math>Df(x)</math>. It is thus defined by

: <math>x_{k+1} = x_k + [I - F(f(x_k), x_k)]^{-1}(f(x_k) - x_k)., \, </math>

for ''k'' = 1, 2, 3, ...&nbsp;. If the operator ''F'' satisfies

: <math>\|F(x',x'')-F(y',y'')\| \le K \big( \|x'-y'\| + \|x''-y''\| \big) </math>

for some constant ''K'', then the method converges quadratically to a fixed point of &fnof; if the initial approximation <math>x_0</math> is sufficiently goodclose to the desired solution <math>\xi</math>, that satisifies <math>\xi = f(\xi)</math>.

==References==