Regular Falsi Method : Definition, Working Rules with Example

 Regular Falsi Method:-

       Definition:-

                       In Numarical Analysis, for finding root of Algebraic and transcendental equation, we are use Regular Falsi Method. 

    Working Rules:-

               1:- Given equation is f(x) =0 . 

Find x₀ and x₁ such that f(x₀) < 0 , f(x₁) > 0 . 

∴Root lies between x₀ and x₁ .

           2:- We find two point ( x₀ , f(x₀) ) and (x₁ , f(x₁) ), in f(x) curve.

     Then we join  ( x₀ , f(x₀) ) and (x₁ , f(x₁) ) point by a stragitht line.

    We input y = 0 to find a point which point cross x axis.

  

  Let at x₂ ,cross x asis the stragith line.

   Now , X₂ = [x₀ . f(x₁) - x₁ f(x₂) ] / [ f(x₁) - f(x₀) ]

Now we examine f(x₂) and its sign.

     3:-         If f(x₂) = 0 ,So x₂ is the root of the equation.

   

               

    If f(x₂) > 0, root lies between  x₂ and x₀.

Then process to like 2nd step. 

  

   If f(x₂) < 0, root lies between  x₂ and x₁.

Then process ti like 2nd step. 

Example of Regular Falsi Method:-

Question 1:-

          Find the real root of the equation x³ + 7x + 9 =0 upto 2 decimal place.

Sloution:- 

       

         The given equation is f(x)= x³ + 7x + 9 =0 

Now, we are find x₀ and x₁ such that f(x₀)< 0 and f(x₁) > 0

f(0) = 0+ 0+ 9 = 9 > 0

f( -1) = 1 - 7 + 9 = 3 > 0

f( -2)= -8 -14 +9 = -13 < 0

Root lies on (-1) to (-2)

f(-1.5)= -4.875 < 0

f(-1.4) = -3.544<0

f(-1.1)= -0.037< 0

f(-1.09) = 0.074971 >0

Let x₀= -1.1 and x₁ = -1.09

By Regular Falsi Method new aproximate root x₂ = [x₀ . f(x₁) - x₁ f(x₀) ] / [ f(x₁) - f(x₀) ]

             = [-1.09(-0.037) +  1.1(0.074971)] / [-0.037 - 0.074971]

            = [0.1227981] / [-0.111971]

            = -1.09669557

Now f(-1.09669557)= 0.004090 (Correct uptp 2 decimalplace)

Root of the givrn equation is -1.09669557