Secant method

From Calculus

This article is about a root-finding algorithm. See all root-finding algorithms

Definition

The secant method is a root-finding algorithm that makes successive point estimates for the value of a root of a continuous function. In general, the secant method is not guaranteed to converge towards a root, but under some conditions, it does. A slight variant of this method, called the false position method, functions very similarly to the bisection method.

Summary

Item Value
Initial condition we are given a function, typically a continuous function, and two initial guesses and for roots of the function.
Iterative step At stage () we compute in terms of .
Convergence rate The order of convergence is the golden ratio
Computational tools needed Function evaluation at particular points, multiplication, subtraction, and division
Termination We may terminate based on known bounds on the size of the derivative and the function value coming in sufficient proximity to the actual value.

Initial condition

The secant method requires two initial guesses for the root, say and .

Iterative step

For , we define as the following affine combination of the previous two guesses and :

We can therefore think of as an affine linear combination of and with the following respective coefficients:

Geometrically, this can be interpreted as follows: we make a line through the points and in the -plane, and define as the -coordinate of the intersection of this line with the -axis.

Convergence rate