Falsi

Falsi, in mathematical root finding, most commonly refers to the Regula falsi or false position method. It is a bracketing algorithm for locating a real root of a continuous function f on an interval [a, b] where f(a) and f(b) have opposite signs. The method uses the x-intercept of the secant line through (a, f(a)) and (b, f(b)) as an approximate root. This intercept is c = (a f(b) - b f(a)) / (f(b) - f(a)). If f(c) = 0 we are done; otherwise the interval is updated: if f(a) and f(c) have opposite signs, set b = c; else set a = c. The procedure is repeated until a stopping criterion is met, such as |b - a| < tolerance or |f(c)| < tolerance.

Properties: The Regula falsi maintains a and b with f(a)f(b) < 0, guaranteeing a root in the interval by the intermediate value theorem. It generally converges faster than the bisection method for well-behaved functions, but, unlike the secant method, it does not require derivative information and brackets the root to maintain robustness. A known drawback is stagnation, where one endpoint ceases to move, causing slow convergence when f has a shallow slope near one endpoint. To address this, variants adjust the weighting of endpoints after each iteration.

Variants: The Illinois algorithm modifies the function value at a stagnating endpoint to reduce its weight, helping progress. The Pegasus method uses a scaling of the function values to accelerate convergence while preserving bracketing. These methods are widely taught in numerical analysis and often serve as simple, robust ways to locate roots when derivatives are unavailable.

Historical context: The name Regula falsi is Latin for the false position method. The technique traces to early false position methods used in antiquity and was formalized in the 17th–18th centuries in European calculus literature; the term has since become standard in numerical analysis.