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Secant method

The secant method resembles Newton's method, but uses a secant line through two points instead of the tangent, so the derivative is not needed. It converges a bit slower than Newton (order ≈1.618\approx 1.618), but is useful when f′f' is hard to compute.

xn+1=xn−f(xn) xn−xn−1f(xn)−f(xn−1)x_{n+1} = x_n - f(x_n)\,\frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}the iteration formula, uses the two previous points

Symbols

xn, xn−1x_n,\ x_{n-1}the two most recent approximations

Example

f(x)=x2−3f(x) = x^2 - 3, x0=1x_0 = 1, x1=2x_1 = 2:

x2=2−1⋅2−11−(−2)≈1.6667x_2 = 2 - 1\cdot\frac{2-1}{1-(-2)} \approx 1.6667.

The secant method needs two starting points, not just one, since it uses a line between them.
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Part of Numerical Methods: Solving equations.