Steffensen's method: Difference between revisions

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In [[numerical analysis]], '''Steffensen's method''' is a [[root-finding method]]. It is similar to [[Newton's method]] and it also achieves [[order of convergence|quadratic convergence]], but it does not use [[derivative]]s. The method is named after [[Johan Frederik Steffensen]].

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In [[numerical analysis]], '''Steffensen's method''', named after [[Johan Frederik Steffensen]], is an iterative process achieving quadratic [[Modes of convergence|convergence]] without employing [[derivative]]s.

==Generalised definition==

Steffensen's method finds [[fixed point]]s of a [[Map (mathematics)|mapping]] ƒ. In the original definition, ƒ was supposed to be a real function, but the method has been generalised for functions ƒ on a [[Banach space]] ''X''.

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

forThe amethod [[Mapassumes (mathematics)|mapping]] &fnof; onthat a [[BanachIndexed spacefamily|family]] ''X'' and ''F''(''x''<nowiki>'</nowiki>,x") a [[Indexed family|family]] of [[Bounded set|bounded]] [[linear operators]] associated with ''x''<nowiki>'</nowiki> and ''x''", havingis known thewhich propertiessatisfies

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

Steffensen's method is then the same as Newton's method, except that it uses this operator instead of the derivative. It is thus defined by

for a [[Map (mathematics)|mapping]] &fnof; on a [[Banach space]] ''X'' and ''F''(''x''<nowiki>'</nowiki>,x") a [[Indexed family|family]] of [[Bounded set|bounded]] [[linear operators]] associated with ''x''<nowiki>'</nowiki> and ''x''", having the properties

: <math>x_{k+1} = x_k + [I - F(x'f(x_k),x'' x_k)(x']^{-x'')=F1}(x'f(x_k) - F(x''x_k). \, </math>

If ''F'' satisfies

and

: <math>|\|F(x,yx')-F(y,zy')|\| \le K_0||K \big( \|x-z)|y\| + K_1\||(x'-y)|'\| + K_1||(y-z\big)||. \,</math>

Thisfor process,some givenconstant a''K'', sufficientlythen goodthe initial approximation,method converges quadratically to a fixed point of &fnof; if the initial approximation <math>x_0</math> is sufficiently good.

==References==

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* "On Steffensen's Method", L. W. Johnson; D. R. Scholz, ''SIAM Journal on Numerical Analysis'', Vol. 5, No. 2. (Jun., 1968), pp. 296-302. Stable URL: [http://links.jstor.org/sici?sici=0036-1429%28196806%295%3A2%3C296%3AOSM%3E2.0.CO%3B2-H]

[[Category:MathematicalRoot-finding analysisalgorithms]]