REGULA - FALSI METHOD | ENGINEERING MATHEMATICS | FINDIND ROOTS OF AN EQUATION

INTRODUCTION:

Regula-Falsi method is also used for finding approximate roots of an equation .It is very similar to bisection method.

                If you don't read about bisection method OR forgot the bisection method just click here to read now.

This is also called False Position Method.

THEORY:

As before (in Bisection Method), for a given continuous function f(x) we assume that f (a) and f (b) have opposite signs ( a = lower guess, b = upper guess). This condition satisfies the Balzano's Theorem for continuous function.

Note::Theorem (Bolzano) : If the function f(x) is continuous in [a, b] and f(a)f(b) < 0 (i.e. f(x) has opposite signs signs at a and b) then a value c ∈ (a, b) exists such that f(c) = 0.

Now after this bisection method used the midpoint of the interval [a, b] as the next iterate to converge towards the root of f(x).

A better approximation is obtained if we find the point (c, 0) where the secant line L joining the points (a, f (a)) and (b, f (b)) crosses the x-axis (see the image below). To find the value c, we write down two versions of the

slope m of the line L:

We first use points (a, f (a)) and (b, f (b)) to get equation 1 (below), and then use the points (c, 0) and (b, f (b)) to get equation 2 (below). Equating these two equation we get equation 3 (below) which is easily solved for c to get equation 4 (below):

The three possibilities are the same as before in the bisection method:

• If f (a) and f (c) have opposite signs, a zero lies in [a, c].
• If f (c) and f (b) have opposite signs, a zero lies in [c, b].
• If f (c) = 0, then the zero is c.

A short GIF for how this method works:

Working Rules:

1. Find Points x0 and x1such that x0< x1and f(x0).f( x1) < 0
2. Take the interval [x0, x1] and find next value x2= x0-f( x0).(( x1- x0)/f( x1)-f( x0))
3. If f(x2)=0 then x2is an exact root.
4. If f(x1).f(x2) < 0 then x1= x2
5. If f(x0).f(x2) <0 then x0= x2
6. Repeat the step 2-5 until f(xi)=0 or f(x0)≤ Accuracy

Solve: Find a root of an equation f(x)=x3-x-1 using False Position method

Solution:
Here x3-x-1=0

Let f(x)=x3-x-1

x	0	1	2
f(x)	-1	-1	5

1st iteration :

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

∴ Now, Root lies between x0=1 and x1=2

x2=1-(-1)⋅(2-1)/(5-(-1))

x2=1.16667

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

2nd iteration :[ FROM HERE THIS LEAVES FOR READERS AND HOPE ALL ITERATION WILL BE SOLVED ]

LASTLY WE GOT

[HERE n= No. of Iteration , x0 = value of 1st Guess point and f( x0 )= value of Fuction at x0 ]
Approximate root of the equation x3-x-1=0 using False Position method is 1.32464