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 secant method predates Newton's method by over 3000 years. For finding a zero of a function f, the secant method is defined by the recurrence relation. As can be seen from this formula, two initial values x0 and x1 are required. Ideally, they should be chosen close to the desired zero. 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 of the secant method converge to a root of if the initial values and are sufficiently close to the root. The order of convergence is , where is the golden ratio. In particular, the convergence is super linear, but not quite quadratic. This result only holds under some technical conditions, namely that be twice continuously differentiable and the root in question be simple (i.e., with multiplicity 1). If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval . For example, if is differentiable on that interval and there is a point where on the interval, then the algorithm may not converge. The secant method does not require that the root remain bracketed, like the bisection method does, and hence it does not always converge.