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.

The simplest form of the formula for Steffensen's method occurs when it is used to find the zeros, or roots, of a function ${\displaystyle f}$ ; that is: to find the value ${\displaystyle x_{\star }}$ that satisfies ${\displaystyle f(x_{\star })=0}$ . Near the solution ${\displaystyle x_{\star }}$ , the function ${\displaystyle f}$ is supposed to approximately satisfy ${\displaystyle -1<f'(x_{\star })<0}$ ; this condition makes the function ${\displaystyle f}$ adequate as a correction 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 ${\displaystyle x_{0}\ }$ must be very close to the actual solution ${\displaystyle x_{\star }}$ , and convergence to the solution may be slow.

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