Steffensen's method

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

Generalised definition

Steffensen's method finds fixed points of a mapping ƒ. In the original definition, ƒ was supposed to be a real function, but the method has been generalised for functions ${\displaystyle f:X\to X}$  on a Banach space ${\displaystyle X}$ .

The method assumes that a family ${\displaystyle \{F(x',x''):x',x''\in X\}}$  of bounded linear operators (called divided difference) associated with x' and x'' is known which satisfies

${\displaystyle f(x')-f(x'')=F(x',x'')(x'-x'').\,}$ 

Steffensen's method is then very similar to the Newton's method, except that it uses the divided difference${\displaystyle F(f(x),x)}$  instead of the derivative ${\displaystyle Df(x)}$ . It is thus defined by

${\displaystyle x_{k+1}=x_{k}+[I-F(f(x_{k}),x_{k})]^{-1}(f(x_{k})-x_{k}),\,}$ 

for k = 1, 2, 3, ... . If the operator F satisfies

${\displaystyle \|F(x',x'')-F(y',y'')\|\leq K{\big (}\|x'-y'\|+\|x''-y''\|{\big )}}$ 

for some constant K, then the method converges quadratically to a fixed point of ƒ if the initial approximation ${\displaystyle x_{0}}$  is sufficiently close to the desired solution ${\displaystyle \xi }$ , that satisifies ${\displaystyle \xi =f(\xi )}$ .

References

• "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: [1]