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# Difference Between Bisection Method and Regula Falsi Method

The bisection method is used for finding the roots of equations of non-linear equations of the form f(x) = 0 is based on the repeated application of the intermediate value property. Let f(x) is continuous function in the closed interval [x1,x2], if f(x1), f(x2) are of opposite signs , then there is at least one root α in the interval (x1,x2), such that f(α) = 0.

#### 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

### Regula Falsi Method:

Regula Falsi is one of the oldest methods to find the real root of an equation f(x) = 0 and closely resembles with Bisection method. It requires less computational effort as we need to evaluate only one function per iteration.

#### Formula

`X3 = X1(fX2) - X2(fX1)/ f(X2) -f(X1)`

#### Example

Problem: Find a root of an equation f(x)=x3-x-1

Solution:

Given equation, x3-x-1=0

let x  = 0, 1, 2

In 1st iteration :

f(1)=-1<0 and f(2)=5>0

Root lies between these two points x0=1 and x1=2

x2=x0-f(x0)

= x1-x0

f(x1)-f(x0)

x2=1-(-1)⋅

= 2-1

= 5-(-1)

x2=1.16667

f(x2)=f(1.16667)=-0.5787<0

In 2nd iteration :

f(1.16667)=-0.5787<0 and f(2)=5>0

Root lies between these two points x0=1.16667 and x1=2

x3=x0-f(x0)

x1-x0

f(x1)-f(x0)

x3=1.16667-(-0.5787)

2-1.16667

5-(-0.5787)

x3=1.25311

f(x3)=f(1.25311)=-0.28536<0

In 3rd iteration :

f(1.25311)=-0.28536<0 and f(2)=5>0

Root lies between these two points x0=1.25311 and x1=2

x4=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x4=1.25311-(-0.28536)⋅

2-1.25311

5-(-0.28536)

x4=1.29344

f(x4)=f(1.29344)=-0.12954<0

In 4th iteration :

f(1.29344)=-0.12954<0 and f(2)=5>0

Root lies between these two points x0=1.29344 and x1=2

x5=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x5=1.29344-(-0.12954)⋅

2-1.29344

5-(-0.12954)

x5=1.31128

f(x5)=f(1.31128)=-0.05659<0

In 5th iteration :

f(1.31128)=-0.05659<0 and f(2)=5>0

Root lies between these two points x0=1.31128 and x1=2

x6=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x6=1.31128-(-0.05659)⋅

2-1.31128

5-(-0.05659)

x6=1.31899

f(x6)=f(1.31899)=-0.0243<0

In 6th iteration :

f(1.31899)=-0.0243<0 and f(2)=5>0

Root lies between these two points x0=1.31899 and x1=2

x7=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x7=1.31899-(-0.0243)⋅

2-1.31899

5-(-0.0243)

x7=1.32228

f(x7)=f(1.32228)=-0.01036<0

In 7th iteration :

f(1.32228)=-0.01036<0 and f(2)=5>0

Root lies between these two points x0=1.32228 and x1=2

x8=x0-f(x0)⋅

x1-x0

f(x1)-f(x0)

x8=1.32228-(-0.01036)⋅

2-1.32228

5-(-0.01036)

x8=1.32368

The approximate root of the equation x3-x-1=0 using the Regula Falsi method is 1.32368

### Differences between Bisection Method and Regula False Method

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