Secant Method
Move two initial guesses and step through the secant intercept, comparing convergence with denominator failure and unbracketed iterates.
Free theory articleMove two initial guesses and step through the secant intercept, comparing convergence with denominator failure and unbracketed iterates.
Concept & interpretation
The secant method replaces the analytical derivative used in Newton’s method with a finite difference approximation based on the two most recent iterates. This eliminates the need to compute derivatives while retaining superlinear convergence under right conditions. However, unlike the bisection method, the iterates are not guaranteed to bracket the root.
Experiment
You can trace the geometry of the method interactively in the Secant Method visualization:
- Set the Function to
x^2 - 2and the Initial guesses to1and2. - Adjust the Iteration budget slider to exactly 1. Observe the first amber secant line intersecting the horizontal axis around 1.33.
- Step the Iteration budget slider to 2. The updated secant line connects (4/3, -2/9) to (2, 2) giving a clearer intersection precisely at 1.4.