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Two basic types of false position method can be distinguished historically, simple false position and double false position. Simple false position is aimed at solving problems involving direct proportion. Such problems can be written algebraically in the form: determine x such that
The formula below converges quadratically when the function is well-behaved, which implies that the number of additional significant digits found at each step approximately doubles; but the function has to be evaluated twice for each step, so the overall order of convergence of the method with respect to function evaluations rather than with ...
Retrieved from "https://en.wikipedia.org/w/index.php?title=Rule_of_false_position&oldid=906963438"
Bracketing with a super-linear order of convergence as the secant method can be attained with improvements to the false position method (see Regula falsi § Improvements in regula falsi) such as the ITP method or the Illinois method. The recurrence formula of the secant method can be derived from the formula for Newton's method
The false position method, also called the regula falsi method, is similar to the bisection method, but instead of using bisection search's middle of the interval it uses the x-intercept of the line that connects the plotted function values at the endpoints of the interval, that is
Illustration of the false positition method. Created by Jitse Niesen using Xfig. Date: 19 June 2006 (original upload date) Source: No machine-readable source provided. Own work assumed (based on copyright claims). Author: No machine-readable author provided. Jitse Niesen assumed (based on copyright claims). SVG development
The term "method of false position" has consistently been more common than "false position method" or "rule of false position" during the 20-21st centuries, according to Google ngrams. This article should be moved to Method of false position. Comments? --Macrakis 03:26, 19 July 2019 (UTC)
In the general case, for a word with n 1 letters X 1, n 2 letters X 2, ..., n r letters X r, it turns out (after a proper use of the inclusion-exclusion formula) that the answer has the form () , for a certain sequence of polynomials P n, where P n has degree n.