Numerical Methods with Computer Programming

Object Oriented
Programming
C++
Assignments

INDEX
Sr. No. Title Date Sign Grade
1 Bisection Method
2 Regula-Falsi Method
3 Secant Method
4 Newton-Raphson Method
5 Gauss-Jordan Method
6 Gauss-Elimination Method
7 Method of Least Squares
8 Euler’s Method
9 Runge-Kutta Method

BISECTION METHOD
Program:
#include<iostream.h>
#include<iomanip.h>
#include<math.h>
float f(float x)
{
return (x*sin(x)-1);
}
void bisect(float *x,float a,float b,int *itr)
{
*x=(a+b)/2;
++(*itr);
cout<<"Iteration No."<<setw(3)<<*itr<<"x="<<setw(7)
<<setprecision(5)<<*x<<endl;
}
int main()
{
int itr=0,maxitr;
float x,a,b,aerr,x1;
cout<<"Enter the value of a=t";
cin>>a;
cout<<"Enter the value of b=t";
cin>>b;

cout<<"Enter the Value of allowed error=t";
cin>>aerr;
cout<<"Enter the value of maximum iterations=t";
cin>>maxitr;
bisect(&x,a,b,&itr);
do
{
if(f(a)*f(x)<0)
b=x;
else
a=x;
bisect(&x1,a,b,&itr);
if(fabs(x1-x)<aerr)
{
cout<<"After "<<itr<<" itrations, Our Desired root is"
<<"="<<setw(6)<<setprecision(4)<<x1<<endl;
return 0;
}
x=x1;
}
while(itr<maxitr);
cout<<"Solution does not converge"<<"iteration not sufficient"
<<endl;
return 1;
}

REGULA-FALSI METHOD
Program:
#include<iomanip.h>
#include<math.h>
float f(float x)
{
return x*x*x-x-5;
}
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<<"x="<<setw(7)
}
int main()
{
int itr=0,maxitr;
float x0,x1,x2,x3,aerr;
cout<<"Enter the value of x0=t";
cin>>x0;
cin>>x1;

cout<<"Enter the value of allowed error=t";
cin>>aerr;
cout<<"Enter the value of maximum iteration=t";
cin>>maxitr;
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,"<<"tOur desired root is="
<<setw(6)<<setprecision(4)<<x3<<endl;
return 0;
}
x2=x3;
}
while(itr<maxitr);
cout<<"Solution does not converge,"
<<"iteration not sufficient"<<endl;
return 1;
}

SECANT METHOD
Program:
#include<iomanip.h>
#include<math.h>
float f(float x)
{
return x*x*x-exp(x);
}
void secant(float *x,float x0,float x1,float fx0,float fx1, int *itr)
{
*x=x1-((x1-x0)/(fx1-fx0))*fx1;
++(*itr);
cout<<"Iteration no."<<setw(3)<<*itr<<"x="<<setw(7)
}
int main()
{
int itr=0,maxitr;
float x0,x1,x2,x3,aerr;
cin>>x0;
cin>>x1;

cout<<"Enter the value of allowed errors=t";
cin>>aerr;
cin>>maxitr;
secant(&x2,x0,x1,f(x0),f(x1),&itr);
do
{
x0=x1;
x1=x2;
secant(&x3,x0,x1,f(x0),f(x1),&itr);
if(fabs(x3-x2)<aerr)
{
cout<<"After "<<itr<<" iterations, Our desired root is="
<<setw(6)<<setprecision(4)<<x3<<endl;
return 0;
}
x2=x3;
}
while(itr<maxitr);
cout<<"Solution does not converge,"
<<"iterations not sufficient"<<endl;
return 1;
}

NEWTON RAPHSON METHOD
Program:
#include<iomanip.h>
#include<math.h>
float f(float x)
{
return cos(x)-x*x;
}
float df(float x)
{
return -sin(x)-2*x;
}
int main()
{
int itr,maxitr;
float h,x0,x1,aerr;
cin>>x0;
cout<<"Enter the value of allowed error=t";
cin>>aerr;
cin>>maxitr;
for(itr=1;itr<=maxitr;itr++)

{
h=f(x0)/df(x0);
x1=x0-h;
cout<<"Iteration no."<<setw(3)<<itr<<"x="<<setw(9)
<<setprecision(6)<<x1<<endl;
if(fabs(h)<aerr)
{
cout<<"After no. "<<setw(3)<<itr
<<" iterations,tOur desired root is="
<<setw(8)<<setprecision(6)<<x1;
return 0;
}
x0=x1;
}
cout<<"Iterations not sufficient,"
<<"Solution does not converge"<<endl;
return 1;
}

GAUSS-JORDAN METHOD
Program:
#include<iomanip.h>
#define N 3
int main()
{
float a[N][N+1],t;
int i,j,k;
cout<<"Enter the matrix elements row wise:-n";
for(i=0;i<N;i++)
{
for(j=0;j<N+1;j++)
{
cin>>a[i][j];
}
}
for(j=0;j<N;j++)
{
for(i=0;i<N;i++)
{
if(i!=j)
{
t=a[i][j]/a[j][j];
for(k=0;k<N+1;k++)
{

a[i][k]-=a[j][k]*t;
}
}
}
}
cout<<"The diagonal matrix is:- n";
for(i=0;i<N;i++)
{
for(j=0;j<N+1;j++)
{
cout<<setw(15)<<setprecision(4)<<a[i][j];
}
cout<<"n";
}
cout<<"n";
cout<<"The solution is:-n";
for(i=0;i<N;i++)
{
cout<<"x["<<i+1<<"]="<<setw(7)<<setprecision(3)
<<a[i][N]/a[i][i]<<endl;
}
return 0;
}

GAUSS ELIMINATION METHOD
Program:
#include<iomanip.h>
#include<math.h>
#define N 3
int main()
{
float a[N][N+1],x[N],t,s;
int i,j,k;
cout<<"Enter the matrix elements row wise:-n";
for(i=0;i<N;i++)
{
for(j=0;j<N+1;j++)
{
cin>>a[i][j];
}
}
for(j=0;j<N-1;j++)
{
for(i=j+1;i<N;i++)
{
t=a[i][j]/a[j][j];
for(k=0;k<N+1;k++)

{
a[i][k]-=a[j][k]*t;
}
}
}
cout<<"The upper triangular matrix is:- n";
for(i=0;i<N;i++)
{
for(j=0;j<N+1;j++)
{
cout<<setw(15)<<setprecision(4)<<a[i][j];
}
cout<<"n";
}
for(i=N-1;i>=0;i--)
{
s=0;
{
for(j=i+1;j<N;j++)
{
s+=a[i][j]*x[j];
}
x[i]=(a[i][N]-s)/a[i][i];
}
}

cout<<"n";
cout<<"The solution is:-n";
for(i=0;i<N;i++)
{
cout<<"x["<<i+1<<"]="<<setw(7)<<setprecision(3)<<x[i]<<endl;
}
return 0;
}
Output:

METHOD OF LEAST SQUARES
Program:
#include<iomanip.h>
int main()
{
float mat[3][4]={{0,0,0,0},{0,0,0,0},{0,0,0,0}};
float t,a,b,c,x,y,xsq;
int i,j,k,n;
cout<<"Enter the no. of pairs of observed values:";
cin>>n;
mat[0][0]=n;
for(i=0;i<n;i++)
{
cout<<"Pair no.:"<<i+1<<"n";
cin>>x>>y;
xsq=x*x;
mat[0][1]+=x;
mat[0][2]+=xsq;
mat[1][2]+=x*xsq;
mat[2][2]+=xsq*xsq;
mat[0][3]+=y;
mat[1][3]+=x*y;
mat[2][3]+=xsq*y;

}
mat[1][1]=mat[0][2];
mat[2][1]=mat[1][2];
mat[1][0]=mat[0][1];
mat[2][0]=mat[1][1];
cout<<"The Matrix is:-n";
for(i=0;i<3;i++)
{
for(j=0;j<4;j++)
{
cout<<setw(10)<<setprecision(4)<<mat[i][j];
}
cout<<"n";
}
for(j=0;j<3;j++)
{
for(i=0;i<3;i++)
{
if(i!=j)
{
t=mat[i][j]/mat[j][j];
for(k=0;k<4;k++)
{
mat[i][k]-=mat[j][k]*t;
}

}
}
}
a=mat[0][3]/mat[0][0];
b=mat[1][3]/mat[1][1];
c=mat[2][3]/mat[2][2];
cout<<setprecision(4)
<<"a="<<setw(5)<<a<<"n"
<<"b="<<setw(5)<<b<<"n"
<<"c="<<setw(5)<<c<<"n";
return 0;
}
Output:

EULER’S METHOD
Program:
#include<iomanip.h>
float df(float x,float y)
{
return (y-x)/(y+x);
}
int main()
{
float x0,y0,h,x,x1,y1;
cout<<"Enter the values of x0,y0,h,x:n";
cin>>x0>>y0>>h>>x;
x1=x0;
y1=y0;
while(1)
{
if(x1>x)
{
return 0;
}
y1+=h*df(x1,y1);
x1+=h;
cout<<"when x="<<setw(3)<<setprecision(2)<<x1<<"t"

<<"y="<<setw(3)<<setprecision(4)<<y1<<"n";
}
}
Output:

RUNGE-KUTTA METHOD
Program:
#include<iomanip.h>
float f(float x,float y)
{
return x+y*y;
}
int main()
{
float x0,y0,h,xn,x,y,k1,k2,k3,k4,k;
cout<<"Enter the values of x0,y0,h,xn:n";
cin>>x0>>y0>>h>>xn;
x=x0;
y=y0;
while(1)
{
if(x==xn)
{
break;
}
k1=h*f(x,y);
k2=h*f(x+h/2,y+k1/2);
k3=h*f(x+h/2,y+k2/2);

k4=h*f(x+h,y+k3);
k=(k1+(k2+k3)*2+k4)/6;
x+=h;
y+=k;
cout<<"When x="<<setprecision(4)<<setw(4)<<x<<setw(8)
<<"y="<<setprecision(4)<<setw(4)<<y<<endl;
}
return 0;
}
Output:

Numerical Methods with Computer Programming

Recommended

More Related Content

What's hot (20)

Similar to Numerical Methods with Computer Programming (20)

More from Utsav Patel (14)

Recently uploaded (20)

Numerical Methods with Computer Programming