Secant method

In numerical analysis, the secant method is a root-finding algorithm derived from Newton's method.

Newton's method is defined by this recurrence relation:

$x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}$

The derivative of f can be defined as

$f'(x)=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}$

so

$f'(x)\approx {\frac {f(x+h)-f(x)}{h}}$

If we let x = x_n and h = x_n-1 - x_n, we have

$f'(x_{n})\approx {\frac {f(x_{n-1})-f(x_{n})}{x_{n-1}-x_{n}}}$

Replacing the derivative of f, in Newton's method, with this approximation gives the secant method:

$x_{n+1}=x_{n}-{\frac {x_{n}-x_{n-1}}{f(x_{n})-f(x_{n-1})}}f(x_{n})$

The secant method requires two initial values: x₀ and x₁.