Regula falsi

Regula falsi is an iterative root-finding method that estimates a zero from the x-intercept of a secant through two interval endpoints, retaining endpoints that bracket a root.

Secant method: A root-finding method that repeatedly uses secant-line intercepts without requiring a bracketing interval. It shares the secant estimate but sacrifices regula falsi’s maintained root bracket.
Root of a function: An input at which a function’s value is zero. Finding such an input is the method’s direct objective.
Ancient Egyptian mathematics: The mathematical practices developed in ancient Egypt, known chiefly through surviving papyri and monuments. The Rhind Mathematical Papyrus contains an early false-position procedure.
Nonlinear equation: An equation in which the unknown appears through a nonlinear function. Regula falsi can approximate solutions by applying it to a function equal to zero.
Modified regula falsi: A family of changes to regula falsi that reduce stagnation by altering endpoint weights or update rules. These variants keep the bracket while improving the original method’s progress.
Intermediate value theorem: A theorem stating that a continuous function takes every intermediate value between two values it attains. Opposite endpoint signs guarantee a root inside the interval when the function is continuous.
Bracketing method: A root-finding approach that maintains an interval known to contain a root. Regula falsi belongs to this family because it preserves a sign-changing interval.
Rhind Mathematical Papyrus: An ancient Egyptian mathematical text copied by the scribe Ahmes around 1650 BCE. Its problems include calculations using a method resembling false position.
Numerical solution of ordinary differential equations: Computational methods for approximating solutions to ordinary differential equations. Implicit time-stepping often requires solving nonlinear equations that regula falsi can bracket.
Illinois algorithm: A modified false-position method that scales function values to prevent one endpoint from remaining fixed too long. It is a prominent remedy for regula falsi’s slow convergence.

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Bolzano's theoremRelated: It uses the same guaranteed root bracket as Bolzano's theorem.