Secant Method

Summary

The secant method approximates Newton’s step by replacing $f^{\prime }(x_n)$ with a finite difference built from the last two iterates. It needs two starts and no derivative evaluations.

Prerequisites

• Continuous $f$ (smoothness helps local analysis)
• Optional background: Newton-Raphson Method

Problem Type

Solve $f(x)=0$ when $f^{\prime }$ is unavailable or expensive.

Method Definition

Given $x_{n-1}$ and $x_n$ with $f(x_n)\ne f(x_{n-1})$ ,

This is the root of the secant line through $(x_{n-1},f(x_{n-1}))$ and $(x_n,f(x_n))$ .^[1]

Assumptions / Requirements

• Two distinct initial guesses near a simple root
• $f(x_n)\ne f(x_{n-1})$ at each step
• Local theory typically assumes $f\in C^2$ near the root with $f^{\prime }(r)\ne 0$

Algorithm

1. Choose $x_0,x_1$ and tolerance $\epsilon$ .
2. While not stopped:
   • If $|f(x_n)-f(x_{n-1})|$ is extremely small relative to the step scale, abort (ill-conditioned update). A tiny denominator is not the same as having found a root.
   • Compute $x_{n+1}$ from the formula.
   • Stop if $|x_{n+1}-x_n|<\epsilon$ or $|f(x_{n+1})|<\epsilon$ .

Convergence

Local order is the golden ratio $\phi \approx 1.618$ (superlinear) for simple roots under standard conditions. Global convergence is not guaranteed.

Error / Accuracy

Monitor both the step $|x_{n+1}-x_n|$ and residual $|f(x_{n+1})|$ .

Worked Example

$f(x)=x^3-2x-5$ , starts $x_0=2$ , $x_1=2.5$ .

Continuing:

The root is $r\approx 2.094551$ .

Common Failure Modes

• Bad starts far from any root
• Division by nearly zero when $f(x_n)\approx f(x_{n-1})$
• Cycling or divergence without a bracket safeguard

Connections

• Newton-Raphson Method
• False Position Method
• Root Finding

References