secant method

An open method that replaces Newton's tangent line with the line through the last two iterates, trading the need for f' for one extra stored point.

The secant method approximates a root by replacing Newton's method's tangent line at a single point with the secant line through the two most recent iterates $(x_{n-1},f(x_{n-1}))$ and $(x_n,f(x_n))$ , then takes that line's $x$ -intercept as $x_{n+1}$ : $x_{n+1}=x_n-\frac{f(x_n)(x_n-x_{n-1})}{f(x_n)-f(x_{n-1})}.$ It needs two starting values instead of one and, unlike the method of false position, does not maintain a bracket -- the older of the two points is always dropped, regardless of sign.

Examples

Given $x_0=1,\,x_1=2$ with $f(x)=x^2-3$ : $f(1)=-2$ , $f(2)=1$ , so $x_2=2-\dfrac{1(2-1)}{1-(-2)}=2-\dfrac13=\dfrac53\approx1.667$ , already close to $\sqrt3\approx1.732$ .