False position method: Difference between revisions

Revision as of 16:46, 26 April 2014

This article is about a root-finding algorithm. See all root-finding algorithms

Definition

False position method is a root-finding algorithm that is qualitative similar to the bisection method in that it uses nested intervals based on opposite signs at the endpoints to converge to a root, but is computationally based on the secant method.

Initial exploratory phase

The exploratory phase of the false position method involves finding a pair of input values at which the function has opposite signs. This could be done by running the usual secant method and evaluating at each stage until we get to opposite signs, or by some other means. Once we have found two points in the domain at which the function value has opposite signs, we are ready to begin the false position method proper.

Iterative step

At stage $n$ , we find the largest $k$ for which $f(x_{k})$ has sign opposite to $f(x_{n-1})$ . We then define:

$x_{n}:={\frac {x_{k}f(x_{n-1})-x_{n-1}f(x_{k})}{f(x_{n-1})-f(x_{k})}}$

@@ Line 3: / Line 3: @@
 ==Definition==
-'''False position method''' is a [[root-finding algorithm]] that is qualitative similar to the [[bisection method]] in that it uses nested intervals based on opposite signs at the endpoints to converge to a root, but is computationally based on the [[secant method]]. The idea is to use the secant method to make successive point estimates, and provide the nested interval estimate based on the most recent point and the most recent point prior to that where the function value has opposite sign.
+'''False position method''' is a [[root-finding algorithm]] that is qualitative similar to the [[bisection method]] in that it uses nested intervals based on opposite signs at the endpoints to converge to a root, but is computationally based on the [[secant method]].
+==Initial exploratory phase==
+The exploratory phase of the false position method involves finding a pair of input values at which the function has opposite signs. This could be done by running the usual [[secant method]] and evaluating at each stage until we get to opposite signs, ''or'' by some other means. Once we have found two points in the domain at which the function value has opposite signs, we are ready to begin the false position method ''proper''.
+==Iterative step==
+At stage <math>n</math>, we find the largest <math>k</math> for which <math>f(x_k)</math> has sign opposite to <math>f(x_{n-1})</math>. We then define:
+<math>x_n := \frac{x_kf(x_{n-1}) - x_{n-1}f(x_k)}{f(x_{n-1}) - f(x_k)}</math>