Secant method

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.

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PLANETCALC, Secant method

Secant method

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Secant method

The secant method can be thought of as a finite difference approximation of Newton's method. where derivative is replaced by secant line.

We use the root of secant line (the value of x such that y=0) as root approximation for function f.

Suppose we have starting values x0 and x1, with function values f(x0) and f(x1).
The secant line has equation

\frac{y - f(x_1)}{f(x_1)-f(x_0)}=\frac{x - x_1}{x_1-x_0}

The root of secant line (where у=0) hence

x = x_1 - \frac{x_1 - x_0}{f(x_1)-f(x_0)}f(x_1)

This is recurrence relation for secant method. Graphical interpretation can be seen below.



The secant method does not require that the root remain bracketed like the bisection method does (see below), and hence it does not always converge.



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.

The tolerance condition can be either:

f(x_k)< \epsilon — function value is less than ε.

\left|x_k-x_{k-1}\right| < \epsilon — difference between two subsequent хk is less than ε.

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