Steffensen's method

In numerical analysis, Steffensen's method is a root-finding technique similar to Newton's method, named after Johan Frederik Steffensen. Steffensen's method also achieves quadratic convergence, but without using derivatives as Newton's method does. In numerical analysis, Steffensen's method is a root-finding technique similar to Newton's method, named after Johan Frederik Steffensen. Steffensen's method also achieves quadratic convergence, but without using derivatives as Newton's method does. The simplest form of the formula for Steffensen's method occurs when it is used to find the zeros, or roots, of a function f {displaystyle f}  ; that is: to find the value x ⋆ {displaystyle x_{star }} that satisfies f ( x ⋆ ) = 0 {displaystyle f(x_{star })=0}  . Near the solution x ⋆ {displaystyle x_{star }}  , the function f {displaystyle f} is supposed to approximately satisfy − 1 < f ′ ( x ⋆ ) < 0 {displaystyle -1<f'(x_{star })<0}  ; this condition makes f {displaystyle f} adequate as a correction-function for x {displaystyle x} for finding its own solution, although it is not required to work efficiently. For some functions, Steffensen's method can work even if this condition is not met, but in such a case, the starting value x 0   {displaystyle x_{0} } must be very close to the actual solution x ⋆ {displaystyle x_{star }}  , and convergence to the solution may be slow. Given an adequate starting value x 0   {displaystyle x_{0} }  , a sequence of values x 0 ,   x 1 ,   x 2 , … ,   x n , … {displaystyle x_{0}, x_{1}, x_{2},dots , x_{n},dots } can be generated using the formula below. When it works, each value in the sequence is much closer to the solution x ⋆ {displaystyle x_{star }} than the prior value. The value x n   {displaystyle x_{n} } from the current step generates the value x n + 1   {displaystyle x_{n+1} } for the next step, via this formula: for n = 0, 1, 2, 3, ... , where the slope function g ( x n ) {displaystyle g(x_{n})} is a composite of the original function f {displaystyle f} given by the following formula: