Steffensen's method

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In numerical analysis, Steffensen's method, named after Johan Frederik Steffensen, is an iterative process achieving quadratic convergence without employing derivatives.

Generalised definition

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

for a mapping ƒ on a Banach space X and F(x',x") a family of bounded linear operators associated with x' and x", having the properties

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

and

${\displaystyle ||F(x,y)-F(y,z)||\leq K_{0}||(x-z)||+K_{1}||(x-y)||+K_{1}||(y-z)||.\,}$ 

This process, given a sufficiently good initial approximation, converges quadratically to a fixed point.

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]