Disadvantages of Bisection Method
- Convergence is relatively slow (linear convergence rate).
- Does not exploit the function’s values within the interval for faster convergence.
Formula
X2= (X0 + X1) / 2
Example
Problem: Find a root of an equation f(x)=x3-x-1
Solution:
Given equation f(x)=x3-x-1
let x = 0, 1, 2
In 1st iteration :
f(1)=-1<0 and f(2)=5>0
Root lies between these two points 1 and 2
x0=1+2/2 = 1.5
f(x0)=f(1.5)=0.875>0
In 2nd iteration :
f(1)=-1<0 and f(1.5)=0.875>0
Root lies between these two points 1 and 1.5
x1=1+1.5/2 =1.25
f(x1)=f(1.25)=-0.29688<0
In 3rd iteration :
f(1.25)=-0.29688<0 and f(1.5)=0.875>0
Root lies between these two points 1.25 and 1.5
x2=1.25+1.5/2 = 1.375
f(x2)=f(1.375)=0.22461>0
In 4th iteration :
f(1.25)=-0.29688<0 and f(1.375)=0.22461>0
Root lies between these two points 1.25 and 1.375
x3=1.25+1.375/2=1.3125
f(x3)=f(1.3125)=-0.05151<0
In 5th iteration :
f(1.3125)=-0.05151<0 and f(1.375)=0.22461>0
Root lies between these two points 1.3125 and 1.375
x4=1.3125+1.375/2=1.34375
f(x4)=f(1.34375)=0.08261>0
In 6th iteration :
f(1.3125)=-0.05151<0 and f(1.34375)=0.08261>0
Root lies between these two points 1.3125 and 1.34375
x5=1.3125+1.34375/2=1.32812
f(x5)=f(1.32812)=0.01458>0
In 7th iteration :
f(1.3125)=-0.05151<0 and f(1.32812)=0.01458>0
Root lies between these two points 1.3125 and 1.32812
x6=1.3125+1.32812/2 =1.32031
f(x6)=f(1.32031)=-0.01871<0
In 8th iteration :
f(1.32031)=-0.01871<0 and f(1.32812)=0.01458>0
Root lies between these two points 1.32031 and 1.32812
x7=1.32031+1.32812/2=1.32422
f(x7)=f(1.32422)=-0.00213<0
In 9th iteration :
f(1.32422)=-0.00213<0 and f(1.32812)=0.01458>0
Root lies between these two points 1.32422 and 1.32812
x8=1.32422+1.32812/2=1.32617
f(x8)=f(1.32617)=0.00621>0
In 10th iteration :
f(1.32422)=-0.00213<0 and f(1.32617)=0.00621>0
Root lies between these two points 1.32422 and 1.32617
x9=1.32422+1.32617/2=1.3252
f(x9)=f(1.3252)=0.00204>0
In 11th iteration :
f(1.32422)=-0.00213<0 and f(1.3252)=0.00204>0
Root lies between these two points 1.32422 and 1.3252
x10=1.32422+1.3252/2=1.32471
f(x10)=f(1.32471)=-0.00005<0
The approximate root of the equation x3-x-1=0 using the Bisection method is 1.32471
Difference Between Bisection Method and Regula Falsi Method
The bisection method is used to find the roots of non-linear equations of the form f(x) = 0 non-linear equations based on the repeated application of the intermediate value property. Let f(x) be a continuous function in the closed interval [x1,x2], if f(x1), and f(x2) are of opposite signs, then there is at least one root α in the interval (x1,x2), such that f(α) = 0.
Table of Content
- Bisection Method
- Advantages of Bisection Method
- Disadvantages of Bisection Method
- Formula
- Regula Falsi Method
- Advantages of Regula Falsi Method
- Disadvantages of Regula Falsi Method
- Formula
- Differences between Bisection Method and Regula False Method