Question

Obtain the root of 𝒙^𝟑−𝒙−𝟏=𝟎 by Regula Falsi Method
(Take three iteration).

Answer 1

Equation : 𝒙^𝟑−𝒙−𝟏=𝟎 

`therefore f(x) =x^3-x-1`

𝒇(𝟎)=−𝟏<𝟎 𝒂𝒏𝒅 𝒇(𝟏)=−𝟏<𝟎 𝒂𝒏𝒅 𝒇(𝟐)=𝟓>𝟎.

`(x_0,y_0)=(1,-1)`           `(x_1,y_1)=(2,5)`

`x_2=(x_0y_1-x_1y_0)/(y_1-y_0)=`1.2249

𝒇(𝒙_𝟐)=−𝟎.𝟑𝟖𝟕𝟏<𝟎

Next iteration : 

`(x_0,y_0)` = (1.2249,-0.3871)

`(x_1,y_1)` = (1.667,1.9654)

`therefore x_2=(x_0y_1-x_1y_0)/(y_1-y_0)=`1.9654

𝒇(𝒙_𝟐)=−𝟎.𝟏𝟏𝟐𝟕<𝟎

Next iteration :

`(x_0,y_0)` = (1.2976,-0.1127)

`(x_1,y_1)` = (1.667,1.9654)

`therefore x_2=(x_0y_1-x_1y_0)/(y_1-y_0)=`1.3176

The root of given eqn after 3^rd iteration is 1.3176.