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 3000 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 x0 and x1, we construct a line through the points (x0, f(x0)) and (x1, f(x1)), as shown in the picture above. In slope–intercept form, the equation of this line is The root of this linear function, that is the value of x such that y = 0 is We then use this new value of x as x2 and repeat the process, using x1 and x2 instead of x0 and x1. We continue this process, solving for x3, x4, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between xn and xn−1): The iterates x n {displaystyle x_{n}} of the secant method converge to a root of f {displaystyle f} , if the initial values x 0 {displaystyle x_{0}} and x 1 {displaystyle x_{1}} are sufficiently close to the root. The order of convergence is φ, where is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic.