Secant method
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.