Secant method

The secant method is an iterative algorithm for approximating a function’s zero. It uses the intersection of a secant line through two successive approximations to generate the next one.

Secant line: A line passing through two points on a curve, with slope given by their difference quotient. The method extrapolates each new approximation from a secant line through its latest two points.
Root of a function: An input at which a function’s value is zero. Finding such an input is the method’s direct objective.
Newton's method: A root-finding algorithm that uses function values and derivatives to generate successive approximations. The secant method replaces Newton’s derivative with a difference quotient.
Root-finding algorithm: A computational procedure for approximating solutions of an equation by locating zeros of a function. The secant method is one member of this algorithm family.
Difference quotient: A ratio of changes in a function’s values to changes in its input values. It estimates the derivative from two sampled values in the secant update.
Continuous function: A function whose values change without jumps as its input varies. Continuity near a root helps explain why nearby approximations can remain informative.
Bisection method: A bracketing algorithm that repeatedly halves an interval whose endpoints have opposite function signs. Bisection guarantees convergence under continuity, while the secant method generally does not.
Initial guess: A starting estimate supplied to an iterative numerical algorithm. The method requires two starting approximations, and their choice affects its behavior.
Fixed-point iteration: An iterative method that repeatedly applies a map until its output approaches its input. The secant update can be written as repeated application of an iteration map.
Derivative: The instantaneous rate of change of a function with respect to its input. The secant method estimates this quantity from function values rather than evaluating it.

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Linear approximationCompared with: It replaces Newton’s tangent-based update with a slope estimated from two points.