## Regula FalsiEdit

In numerical analysis, the**false position method**or

**regula falsi method**is a root-finding algorithm that combines features from the bisection method and the secant method.

A method of calculating an unknown quantity by first making an estimate and then using this and the properties of the unknown to obtain it. Also known as rule of false position.

## The MethodEdit

The **Regula–Falsi Method** is a numerical method for estimating the roots of a polynomial **f(x)**. A value **x** replaces the midpoint in the Bisection Method and serves as the new approximation of a root of**f(x)**. The objective is to make convergence faster. Assume that **f(x)** is continuous.

**Algorithm for the Regula–Falsi Method**: Given a continuous function **f(x)**

- Find points
**a**and**b**such that**a < b**and**f(a) * f(b) < 0**. - Take the interval
**[a, b]**and determine the next value of**x**._{1} - If
**f(x**then_{1}) = 0**x**is an exact root, else if_{1}**f(x**then let_{1}) * f(b) < 0**a = x**, else if_{1}**f(a) * f(x**then let_{1}) < 0**b = x**._{1} - Repeat steps 2 & 3 until
**f(x**or_{i}) = 0**|f(x**, where_{i})| £DOA**DOA**stands for**degree of accuracy**.

Observe that :

EC / BC |
= |
E / AB |
---|---|---|

[ x – a ] / [ b – a ] |
= |
[ f(x) – f(a) ] / [ f(b) – f(a) ] |

x – a |
= |
[ b – a ] [ 0 – f(a) ] / [ f(b) – f(a) ] |

x |
= |
a + [ b – a ] [ – f(a) ] / [ f(b) – f(a) ] |

x |
= |
a – [ b – a ] f(a) / [ f(b) – f(a) ] |

Note that the line segment drawn from **f(a)** to **f(b)** is called the **interpolation line**.

Graphically, if the root is in **[ a, x _{i} ]**, then the next interpolation line is drawn between

**( a, f(a) )**and

**( x**; otherwise, if the root is in

_{i}, f(x_{i}) )**[ x**, then the next interpolation line is drawn between

_{i}, b ]**( x**and

_{i}, f(x_{i}) )**(b, f(b))**.