Editing Secant method (section)

==Convergence==

The iterates <math>x_n</math> of the secant method converge to a root of <math>f</math> if the initial values <math>x_0</math> and <math>x_1</math> are sufficiently close to the root and <math>f</math> is well-behaved. When <math>f</math> is twice continuously differentiable and the root in question is a simple root, i.e., it has multiplicity 1, the [[order of convergence]] is the [[golden ratio]] <math>\varphi = (1+\sqrt{5})/2 \approx 1.618.</math><ref name=":0">{{Cite web |last=Chanson |first=Jeffrey R. |date=October 3, 2024 |title=Order of Convergence |url=https://math.libretexts.org/Bookshelves/Applied_Mathematics/Numerical_Methods_(Chasnov)/02%3A_Root_Finding/2.04%3A_Order_of_Convergence |access-date=October 3, 2024 |website=LibreTexts Mathematics}}</ref> This convergence is superlinear but subquadratic.

If the initial values are not close enough to the root or <math>f</math> is not well-behaved, then there is no guarantee that the secant method converges at all. There is no general definition of "close enough", but the criterion for convergence has to do with how "wiggly" the function is on the interval between the initial values. For example, if <math>f</math> is differentiable on that interval and there is a point where <math>f' = 0</math> on the interval, then the algorithm may not converge.