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

Basis	Bisection Method	Regula Falsi Method
Definition	In mathematics, the bisection method is a root-finding method that applies to continuous function for which knows two values with opposite signs.	In mathematics, the false position method is a very old method for solving equations with one unknown this method is modified form is still in use.
Simplicity	it is simple to use and easy to implement.	Simple to use as compared to Bisection Method
Computational Efforts	Less as compared to Regula Falsi Method	More as compared to Bisection Method
Iteration required	In the bisection method, if one of the initial guesses is closer to the root, it will take a large number of iterations to reach the root.	Less as compared to Bisection Method. This method can be less precise than bisection – no strict precision is guaranteed.
Convergence	The order of convergence of the bisection method is slow and linear.	This method faster order of convergence than the bisection method.
General Iterative Formula	Formula is : X3 =( X1 + X2)/2	Formula is : X3 = X1(fx2) – x2(fx1)/ f(x2) -f(x1)
Other Names	It is also known as the Bolzano method, Binary chopping method, half Interval method.	It is also known as the False Position method.

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