Steffensen's method

Steffensen's method is an iterative process achieving quadratic convergence without employing derivatives.

Generalised definition

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

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

$F(x',x'')(x'-x'')=F(x')-F(x'')\,$

and

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

This process, given a sufficiently good initial approximation, converges quadratically to a fixed point, provided that $||F(x',x'')||<1$ in some neighborhood of that fixed point.

• 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]