Secant method: explain and derive

The Secant method finds a real root of f(x) = 0 by using a straight line (secant) through two recent points on the curve. It needs two starting values but no derivative, and it does not need the root to stay bracketed.

Graphical mechanism

 y
 |           / f(x1)
 |          /*  (x1, f1)
 |        _/
 |  x2   /
 |---+--/-------- x
 |   | * (x0, f0)
 |   root

1. Take two initial approximations x0 and x1 and the points P0(x0, f(x0)) and P1(x1, f(x1)) on the curve.
2. Draw the straight line (secant) through P0 and P1. The point where it cuts the x-axis is the next approximation x2.
3. Now use x1 and x2 (discard x0), draw the secant again and find x3.
4. Repeat until two successive values agree to the required accuracy. The cut points move toward the root where the curve crosses the x-axis.

Derivation of the formula

The line through (x0, f0) and (x1, f1) is

y − f1 = [(f1 − f0)/(x1 − x0)] (x − x1)

Where it cuts the x-axis, y = 0 and x = x2:

−f1 = [(f1 − f0)/(x1 − x0)] (x2 − x1)
x2 = x1 − f1 (x1 − x0)/(f1 − f0)

Equivalent form: x2 = (x0 f1 − x1 f0) / (f1 − f0). In general:

x(n+1) = x(n) − f(x(n)) · [x(n) − x(n−1)] / [f(x(n)) − f(x(n−1))]

This is Newton-Raphson with the derivative f′(x(n)) replaced by the slope of the secant. Its order of convergence is 1.618.