graphical explanation

Method of False Position

Regula-Falsi method also called the Method of False Position, closely resembles the Bisection method.
This is the oldest method of finding the real root of an equation.

This method has linear rate of convergence which is faster than the bisection method.

Regula Falsi Method C++ Program

//Regula Falsi method //wikkihut.com #include<iostream> #include<math.h> #include<iomanip> using namespace std; float f(float x) { return cos(x)-x*exp(x); } void regula (float *x, float x0,float x1, float fx0,float fx1, int *itr) { *x=x0-((x1-x0)/(fx1-fx0))*fx0; ++(*itr); cout<<" Iteration No."<<setw(3)<<*itr<<"\t X= "<<setw(7)<<setprecision(5)<<*x<<endl; } int main() { int itr=0, maxitr; float x0,x1,x2,x3,aerr; cout<<"\nEnter the value of x0: "; cin>>x0; cout<<"\nEnter the value of x1: "; cin>>x1; cout<<"\n enter allowed error: "; cin>>aerr; cout<<"\n maximum iterations: "; cin>>maxitr; cout<<endl; regula(&x2,x0,x1,f(x0),f(x1),&itr); do { if(f(x0)*f(x2)<0) x1=x2; else x0=x2; regula(&x3,x0,x1,f(x0),f(x1),&itr); if(fabs(x3-x2)<aerr) { cout<<"After "<<itr<<" iterations, root = "<<x3<<endl; return 0; } x2=x3; } while(itr<maxitr); cout<<"Solution does not converge"<<endl; cout<<"Iterations are not sufficient"<<endl; return 1; }
Code language: C++ (cpp)

Example

Find a real Root of equation f(x)=x3-2x-5 by the method of false position method ( Regula Falsi method ).

let f(x)=x3-2x-5 and we have to find its real root correct to three decimal places.

if we put x=2 and x=3, we find f(2) is negative and f(3) is positive.
it means the root lies between 2 and 3.

therefore, taking
x0 = 2
x1 = 3
f ( x0 ) = -1
and
f ( x1 ) = 16
in equation (2) we get
x2 = 0.0588

Now f(x2) = f( 0.0588 ) = -0.3982

i.e., the root lies between 2.0588 and 3.

Therefore, taking

x0 = 2.0588

x1 = 3

f ( x0 ) = -0.3908

f ( x1 ) = 16 in eq. (2)

we get, x3 = 2.0813

By repeating this process, the successive approximations are

x4=2.0682 x5=2.0915
.
.
.
x7=2.0941 and
x8=2.0943
and therefore the root is 2.094 .

Suggested Read:

Graphical explanation.

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graphical explanation

As in picture we use two points x0 and x1 such that f(x0) and f(x1) have opposite sign,
that is the graph
y=f(x)
cuts the x-axis at least once while going from x0 to x1, which indicates that the root lies between x0 and x1 .

Now we connect the two points f(x0) and f(x1) by a straight line.

The equation of straight line joining these points is :

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This straight line cuts the x-axis at point x2,
this point where the line (1) cuts the x-axis is taken as an approximation to the root.

At this point Y=0 and X=x2, put value of Y and X in (1) we get:

which is an approximation to the root. see more

Find f(x2).
If f(x2) and f(x0) are of opposite signs then we replace x1 by x2 and draw a straight line connecting
f(x2) to f(x0)
to find the new intersection point.

If f(x2) and f(x0) are of the same sign then x0 is replaced by x2 and proceed as before.
In both cases, the new interval of search is smaller than the initial interval and ultimately convergence is guaranteed.

Algorithm for Regula-falsi method:

  • read x0 & x1, two initial guesses.
  • Enter Allowed error and Max iterations.
  • Take x0=1 & x1=3
  • f(x0)=-0.3908 and f(x1)=16, in equation (2) we get x3=2.0813
  • (x0) and f(x1) are not same.
  • f0=f(x0)
  • f1=f(x1)
  • for i=1 to n in steps of 1 do
  • x2=(x0f1-x1f0)/(f1-f0)
  • f2=f(x2)
  • if |f2|<e then
  • begin Write ‘convergent solution’, x2,f2
  • stop end
  • if sign (f2)=sign f(0)
  • then begin x0=x2
  • f0=f2
  • else begin x1=x2
  • f1=f2
  • write ‘Does not converge in n iterations’
  • write x3,f2
  • stop

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