Advantages, Disadvantages and Applications of Regula Falsi Method

Introduction

This is a very simple and old method of obtaining a real root. View the history associated with this method. In mathematics, Regula-Falsi Method or False Position Method is used for solving an equation in one unknown variable. This method is trial and error because in this method we use test (false) values for the unknown variable and then adjust them according to the outcome. This method is sometimes referred to as guess and check method.

Move to Advantages, Disadvantages and Applications of regular falsi method.

Graphical Representation

Here is the graphical representation of this method which is easily understandable.

Regula-Falsi/False Position Method
First of all we search for the points x1 and x2 such that f(x1) and f(x2) are of opposite signs so that there will lie a root of f(x)=0 between x1 and x2. Geometrically these points are A(x1,f(x1)) and B(x2,f(x2)) on the curve of y=f(x). The exact root is at C, where the curve intersects the X-axis. Now join AB. Then AB is chord. We replace the curve AB by chord AB. So that the point of intersection of chord AB with X-axis gives the first approximate position (false position) of the exact root.

Equation of chord AB in two point form is

x-x1 x1-x2
=
y-f(x1) f(x1)-f(x2)

put y=0,
x-x1 x1-x2
=
-f(x1) f(x1)-f(x2)

∴ x=
x1-
f(x1)(x1-x2) f(x1)-f(x2)

∴ x3=
x1f(x1)-x1f(x2)-x1f(x1)+x2f(x1) f(x1)-f(x2)

......(x=x3, the first approximation)

∴ x3=
x1f(x2)-x2f(x1) f(x2)-f(x1)
......(1)

Where f(x1)f(x2)<0)

If further f(x3) and f(x2) are of opposite signs (as shown in the fig) then the root lies between x3 and x2 an we replace x1 by x2 in (1) otherwise we replace x2 by x3 and get next approximation. Thus 1 can be generalized as

∴ xn+1=
xn-1f(xn)-xnf(xn-1) f(xn)-f(xn-1)

where, f(xn-1)f(xn)<0 for each n.

That is why this method called as 'Variable Chord Method'.

Procedure for false position method to find the root of the equation f(x)=0

  1. Choose two initial values x1,x2 (x2>x1) such that f(x1), f(x2) are of opposite signs so that there is a root in between x1 and x2.
  2. Let x3 be the next approximation, now the formula
    ∴ x3=
    x1f(x2)-x2f(x1) f(x2)-f(x1)

  3. Find f(x3). If f(x3).f(x2) < 0 then go to step 4. Otherwise name x1 as x2 and then go to step 4.
  4. Calculate the next successive approximations using the formula,
    ∴ xn+1=
    xn-1f(xn)-xnf(xn-1) f(xn)-f(xn-1)
  5. Stop this process when |xn-xn-1|<ε, where ε is the accuracy required.

Example:

Here is the solved example on regula falsi/false position method.

Problem:

Find the root lying between 1 and 2 of the equation x3-3x+1=0 upto 3 decimal places by Regula Falsi / False Position Method.

Solution:

Note that we are solving for f(x)=0.

f(x)=x3-3x+1 is the given equation.
f(1)=-1,f(2)=3
∴ the root lies between 1 and 2. ....f(1)f(2)<0

The first approximation is

∴ x1=
1xf(2)-2xf(1) f(2)-f(1)
=
1x3-2x(-1) 3-(-1)
= 1.25

Now f(x1)=f(1.25)=-0.7969<0 and f(x2)=3>0

∴ root lies in (1.25,2)

The second approximation is

∴ x2=
1.25xf(2)-2xf(1.25) f(2)-f(1.25)
=
1.25x3-2x(-0.7969) 3-(-0.7969)
= 1.4074

Now f(x2)=f(1.4074)=-0.4345<0 and f(x2)=3>0

∴ root lies in (1.25,2)

The third approximation is

∴ x3=
1.4074xf(2)-2xf(1.4074) f(2)-f(1.4074)
=
1.4074x3-2x(-0.4345) 3-(-0.4345)
= 1.4824

Now f(x3)=f(1.4824)=-0.1896<0 and f(x2)=3>0
∴ root lies in (1.4824,2) by continuing this procedure, we get x4=1.5132, x4=1.525, x6=1.5296, x7=1.5311, x8=1.5317, x9=1.5319 ∴ the correct root = 1.532 (up to 3 decimal places)

Scilab Program For Regula-Falsi / False Position Method

Here is the source code of Scilab program for Regula-Falsi / False Position Method.

deff('y=f(x)','y=x^3-1'); // Define the function 

a=0;b=2; //Determining the initial values such that f(x1)f(x2) < 0

i=1; // set counter to 1;

while(i<=15) //up to 15 iteration
    c=(a*f(b)-b*f(a))/(f(b)-f(a)); // False position formula
    
    if (f(a)*f(c)<0) then
        b=c;
    else
        a=c;
    end
    i=i+1;
    [c,f(c)]
end


/* Output of above code:-
ans  =

   0.25  -0.984375

 ans  =

   0.4657534  -0.8989659

 ans  =

   0.640363  -0.7374097

 ans  =

   0.7699425  -0.5435693

 ans  =

   0.8585771  -0.3670959

 ans  =

   0.9154532  -0.2328002

 ans  =

   0.9503612  -0.1416466

 ans  =

   0.9711796  -0.0839932

 ans  =

   0.9833781  -0.0490414

 ans  =

   0.9904509  -0.0283745

 ans  =

   0.9945266  -0.0163304

 ans  =

   0.9968668  -0.00937

 ans  =

   0.9982078  -0.0053669

 ans  =

   0.9989753  -0.0030709

 ans  =

   0.9994143  -0.0017562

*/

Conclusion from Scilab Program for False Position Method

As we increase the number of iterations we get more accurate values of root.

Advantages

  1. It does not require the derivative calculation.
  2. This method has first order rate of convergence i.e. it is linearly convergent. It always converges.

Disadvantages

Following are the limitations of regula falsi method
  1. As it is trial and error method in some cases it may take large time span to calculate the correct root and thereby slowing down the process.
  2. It is used to calculate only a single unknown in the equation.

Applications

  • The Regula Falsi method is applied to prediction of trace quantities of air pollutants produced by combustion reactions such as those found in industrial point sources.
    (source: wiki)

Examples For Practice

Here are some examples for practice on regula-falsi / false position method.
  1. Find a real root of x3-9x+2=0 up to 3 decimal places by regula-false method.
  2. Find the positive root of x2-log10x-10=0 by false position method.
  3. Find the root of the equation ex-2x=0 which lies between 0 and 1.
  4. Find the real root of the equation xex-1=0, using Regula-falsi method, correct up to 3 decimal places.