Secant method

In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite difference approximation of Newton's method. However, the method was developed independently of Newton's method, and predates it by over 3,000 years.

The secant method is defined by the recurrence relation

As can be seen from the recurrence relation, the secant method requires two initial values, x₀ and x₁, which should ideally be chosen to lie close to the root.

Starting with initial values $x 0$ and $x 1$ , we construct a line through the points $(x 0, f (x 0))$ and $(x 1, f (x 1))$ , as demonstrated in the picture on the right. In slope-intercept form, this line has the equation

We find the root of this line – the value of $x$ such that $y = 0$ – by solving the following equation for $x$ :

The solution is

We then use this new value of $x$ as $x 2$ and repeat the process using $x 1$ and $x 2$ instead of $x 0$ and $x 1$ . We continue this process, solving for $x 3$ , $x 4$ , etc., until we reach a sufficiently high level of precision (a sufficiently small difference between $x n$ and $x n - 1$ ).

The iterates $x_{n}$ of the secant method converge to a root of $f$ , if the initial values $x_{0}$ and $x_{1}$ are sufficiently close to the root. The order of convergence is α, where

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