Definition:Rule of False Position
Definition
The rule of false position is a method of successive approximations to obtain the root of an equation $\map f x = 0$ from an initial estimate or estimates of that root.
Simple Position
The method of simple position is a category of the rule of false position in which a single estimate $a_0$ is made, and an iteration of the form:
- $a_{n + 1} = \map g {a_n}$
is used, for $n = 1, 2, \ldots$
Double Position
The method of double position is a category of the rule of false position in which two estimates $a_0$ and $b_0$ are made such that $\map f {a_0}$ and $\map f {b_0}$ are both close to zero but of opposite sign.
These estimates are then used as starting values in the formula:
- $a_{n + 1} = a_n - \dfrac {\paren {b_n - a_n} \map f {a_n} } {\map f {b_n} - \map f {a_n} }$
where, for $n = 0, 1, 2, \ldots$, $b_{n + 1}$ is chosen from $a_n$ and $b_n$ so that $\map f {b_{n + 1} }$ is of opposite sign to $\map f {a_{n + 1} }$.
Also known as
The rule of false position is also known by its Latin name regula falsi.
Also see
- Results about the rule of false position can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): false position (rule of) (regula falsi)${}$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): false position, rule of (regula falsi)