falseposition

False position, also known as regula falsi, is a bracketing method for locating a root of a real-valued function f on an interval [a, b] where f(a) and f(b) have opposite signs. If f is continuous on [a, b], a root exists in the interval by the intermediate value theorem. The method uses the x-intercept of the line through (a, f(a)) and (b, f(b)) as a new approximation: x = b - f(b) * (b - a) / (f(b) - f(a)). If f(x) = 0, x is the root; otherwise the endpoint whose function value has the same sign as f(x) is replaced by x, producing a new bracketing interval.

The procedure repeats until a stopping criterion is met, such as |f(x)| below a tolerance, |b - a| below a tolerance, or a maximum number of iterations. The method requires only evaluations of f and does not rely on derivatives. In practice, it often converges faster than bisection but does not guarantee uniform reduction of the interval. In some problem setups, one endpoint may remain fixed for many iterations, leading to slow convergence or stagnation.

Variants such as the Illinois algorithm and Pegasus method adjust the update rule to mitigate stagnation while retaining the basic idea of using a secant line to approximate the root. False position is a classic tool in numerical analysis for solving equations with a sign-changing bracketing interval and is commonly contrasted with the secant method and bisection for its balance of simplicity and robustness.