CSITauthority: Numerical Methods

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1. Program to find Rootof a fuction by using Bisection method

/*program to find Rootof a fuction by using Bisection method*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define MAX 100
#define E 0.00001
#define f(x) x*x-4*x-10
void main()
{
 clrscr();
   int count=0;
   float x0,x1,x2,f0,f1,f2;
   printf("Enter the value of x1\n");
   scanf("%f",&x1);
   printf("Enter the value of x2\n");
   scanf("%f",&x2);
   f1=f(x1);
   f2=f(x2);
   if (f1*f2>0)
   printf("Solution does not exit");
   else
   {
 begin:
 x0=(x1+x2)/2;
 f0=f(x0);
 if((f1*f0)<0)
  x2=x0;
      else
      {
 x1=x0;
  f1=f0;
      }
      if((fabs(x2-x1)/x1)<E)
      {
 x0=(x1+x2)/2;
  printf("\nRoor=%f",x0);
  printf("\nFuctional value =%f",count);
  printf("\nIteration steps =%d",count);
      }
      else
      {
 count ++;
  goto begin;
      }
   }
   getch();
}

2. Program to find Rootof a fuction by using Newton Raption method

/*program to find Rootof a fuction by using Newton Raption method*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define MAX 10
#define E 0.0001
#define f(x)  x*x*x-3*x-2
#define fd(x)  3*x*x-3
void main()
{
    clrscr();
    int count=0;
    float x0,x1,fx,fdx;
    printf("\nEnter the initial value of x0\n");
    scanf("%f",&x0);
begin:
    printf("\nStep %d",count+1);
    fx=f(x0);
    printf("\nvalue of fx%d=%f",count,fx);
    fdx=fd(x0);
    printf("\nvalue of fdx%d= %f",count,fdx);
    x1=x0-(fx/fdx);
    if((fabs(x1-x0)/x1)<E)
    {
        printf("\nRoot=%f",x1);
        printf("\nThe fuction value=%f",f(x1));
        printf("\nNo. of Iritation steps=%d",count);
    }
    else
    {
        x0=x1;
        count ++;
        if (count<MAX)
        {
            goto begin;
        }
        else
        {
            printf("It takes more than 10 steps So solution does not coverse");
        }
    }
    getche();
}

2. Program to Rootof a fuction by using Newton Raption method (Alternate)

/*program to find Rootof a fuction by using Newton Raption method*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define MAX 10
#define E 0.0001
#define f(x) x*x-3*x+2
#define fd(x) 2*x-3
void main()
{
    clrscr();
    int count=0;
    float x0,x1,fx,fdx;
    printf("\nEnter the initial value of x0");
    scanf("%f",&x0);
begin:
    fx=f(x0);
    fdx=fd(x0);
    x1=x0-(fx/fdx);
    if((fabs(x1-x0)/x1)<E)
    {
        printf("\nRoot=%f",x1);
        printf("\nThe fuction value=%f",f(x1));
        printf("\nNo. of Iritation steps=%d",count);
    }
    else
    {
        x0=x1;
        count ++;
        if (count<MAX)
        {
            goto begin;
        }
        else
        {
            printf("It takes more than 10 steps So solution does not coverse");
        }
    }
    getche();
}

3. Program to find Root of a fuction by using Secant method

/*program to find Root of a fuction by using Secant method*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define MAX 10
#define E 0.0001
#define f(x) x*x-4*x-10
void main()
{
    clrscr();
    int count=0;
    float x1,x2,x3,f1,f2;
    printf("\nEnter the two value of x1 and x2\n");
    scanf("%f%f",&x1,&x2);
begin:
    f1=f(x1);
    f2=f(x2);
    x3=x2-f2*(x2-x1)/(f2-f1);
    if((fabs(x3-x2)/x2)<E)
    {
        printf("\nRoot=%f",x3);
        printf("\nNo. of Iteration steps=%d",count);
    }
    else
    {
        x1=x2;
        x2=x3;
        f1=f2;
        f2=f(x3);
        count ++;
        if (count<MAX)
        {
            goto begin;
        }
        else
        {
            printf("It solution does not coverge");
        }

    }
    getche();
}

4. Program to find Rootof a fuction by using Fixed point method

/*program to find Rootof a fuction by using Fixed point method*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define MAX 100
#define E 0.001
#define g(x) exp(x)/3
void main()
{
    clrscr();
    int count=0;
    float x0,x1,error;
    printf("\nEnter the initial guess value of x0\n");
    scanf("%f",&x0);
begin:
    x1=g(x0);
    error=fabs(x1-x0);
    if(error<=E)
    {
        printf("\nThe root is %f",x1);
        printf("\nThe error value %f",error);
        printf("\nThe iteration steps is %d",count);
    }
    else
    {
        x0=x1;
        count ++;
        if (count<MAX)
        {
            goto begin;
        }
        else
        {
            printf("It solution does not coverge");
        }

    }
    getch();
}

5. Demonstrate Lagranges Interpolation

//Lagranges interpolation
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define MAX 10
void main()
{
    clrscr();
    int i,j,n;
    float x[MAX],f[MAX],sum=0,pro,xp,fp;
    printf("Enter the number of data points:\n");
    printf("\nn= ");
    scanf("%d",&n);
    for(i=0; i<n; i++)
    {
        printf("\nInput the data points for x[%d]&f[%d]",i,i);
        printf("\nx[%d]= ",i);
        scanf("%f",&x[i]);
        printf("\nf[%d]= ",i);
        scanf("%f",&f[i]);
    }
    printf("\nInput the specied value of x:");
    scanf("%f",&xp);
    for(i=0; i<=n; i++)
    {
        pro=1;
        for(j=0; j<=n; j++)
        {
            if(i!=j)
                pro=pro*(xp-x[j])/(x[i]-x[j]);
        }
        sum=sum+pro*f[i];
    }
    fp=sum;
    printf("\nThe required functional value at %f=%f",xp,fp);
    getch();
}

6. Demonstrate Newton Interpolation

// NEWTON INTERPOLATION
#include<stdio.h>
#include<math.h>
#include<conio.h>
#define MAX 10
void main()
{
    int i,j,n;
    float xp,fp,pro,sum,a[MAX],f[MAX],x[MAX],d[MAX][MAX];
    printf("\nInput the number of data points:");
    printf("\nn=");
    scanf("%d",&n);
    for(i=0; i<n; i++)
    {
        printf("\nInput the data of x[%d]&f[%d]:",i,i);
        printf("\nx[%d]=",i);
        scanf("%f",&x[i]);
        printf("\nf[%d]=",i);
        scanf("%f",&f[i]);
    }
    for(i=1; i<=n; i++)
        d[i][1]=f[i];
    for(j=2; j<=n; j++)
        for(i=1; i<=n-j+1; i++)
            d[i][j]=(d[i+1][j-1]-d[i][j-1])/(x[i+j-1]-x[i]);
    for(j=1; j<=n; j++)
        a[j]=d[1][j];
    printf("\nInput the value of xp point:");
    printf("\nxp=");
    scanf("%f",&xp);
    sum=a[1];
    for(i=2; i<=n; i++)
    {
        pro= 1.0;
        for(j=1; j<=i-1; j++)
            pro=pro*(xp-x[j]);
        sum=sum+a[i]*pro;
    }
    fp=sum;
    printf("\n At xp=%f is %f",xp,fp);
    getch();
}

7. Demonstrate Least Square Method

// Least square method
#include<stdio.h>
#include<math.h>
#include<conio.h>
#define MAX 10
void main()
{
    int i,n;
    float a0,a1;
    float x[MAX],y[MAX],sumx=0.0,sumy=0.0,sumxy=0.0,sumxsq=0.0;
    printf("***Least square method***" );
    printf("\nInput the number of data points:");
    printf("\nn=");
    scanf("%d",&n);
    printf("Enter the data sets one after another:");
    for(i=0; i<n; i++)
    {
        // printf("\nInput the data of x[%d]&f[%d]:",i,i);
        printf("\nx[%d]=",i);
        scanf("%f",&x[i]);
        printf("\ny[%d]=",i);
        scanf("%f",&y[i]);
        sumx+=x[i];
        sumy+=y[i];
        sumxy+=x[i]*y[i];
        sumxsq+=x[i]*x[i];
    }
    a0=(sumy*sumxsq-sumx*sumxy)/(n*sumxsq-pow(sumx,2));
    a1=(n*sumxy-sumx*sumy)/(n*sumxsq-pow(sumx,2));
    printf("The equation of line:\n");
    printf("\n y=%fx+%f",a1,a0);
    getch();
}

8. Program to find Integration of given function by using Trapezoidal rule

/*program to find Integration of given funtion by using Trapezoidal rule*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define f(x) x*x*x+1
void main()
{
    clrscr();
    float a,b,h,It;
    printf("\nEnter initial value of X:");
    scanf("%f",&a);
    printf("\nEnter  Final  value of X:");
    scanf("%f",&b);
    h=(b-a)/2;
    It =h*(f(a)+f(b));
    printf("\nThe integration value of function :%f",It);
    getch();
}

9. Program to find Integration of given tabulated data using Composite Trapezoidal rule*

/*program to find Integration of given tabulated data using  Composit Trapezoidal rule*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define max 15
void main()
{
    clrscr();
    int n,n1,n2,i;
    float a,b,h,sum,ict,x[max],y[max];
    printf("\nEnter the number of data points:");
    printf("\nn=");
    scanf("%d",&n);
    printf("\nInput the values set by set");
    for(i=1; i<=n; i++)
    {
        printf("\nX%i=",i);
        scanf("%f",&x[i]);
        printf("\nF(x%i)=",i);
        scanf("%f",&y[i]);
    }
    printf("\nInput the initial value.");
    printf("\na=");
    scanf("%f",&a);
    printf("\nInput the final value.");
    printf("\nb=");
    scanf("%f",&b);
    h=x[2]-x[1];
    n1= (int)(fabs(a-x[1])/h+1.5);
    n2= (int)(fabs(b-x[1])/h+1.5) ;
    sum=0.0;
    for(i=n1; i<=n2; i++)
        sum =sum+y[i]+y[i+1];
    ict= sum*h/2.0;
    printf("\nIntegral from %f to %f is %f\n",a,b,ict);
    getche();
}

10. Program to find Integration of given funtion by using Composite Trapezoidal rule

/*program to find Integration of given funtion by using composit Trapezoidal rule*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
void main()
{
    clrscr();
    int i,n;
    float a,b,h,sum,Ict;
    float F(float x);
    printf("Enter the initial value of a:");
    scanf("%f",&a);
    printf("\nEnter the Final  value of b: ");
    scanf("%f",&b);
    printf("\nEnter the segment width:");
    scanf("%f",&h);
    n=(b-a)/h;
    sum =(F(a)+F(b))/2.0;
    for(i=1; i<n-1; i++)
    {
        sum= sum +F(a+i*h);
    }
    Ict=sum*h;
    printf("\nThe integration between %f and %f at h=%f is %f",a,b,h,Ict);
    getch();
}
float F(float x)
{
    float f;
    f=1-exp(-x/2.0) ;
    return (f);
}

11. Program to find Integration of given funtion by using Simson's 1/3 rule

/*program to find Integration of given funtion by using Simson's 1/3 rule*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define F(x) 1-exp(-(x)/2.0)
void main()
{
 clrscr();
   int n,m,i;
   float a,b,h,sum,Ics,x,f1,f2,f3;
   printf("Enter initial value of X:");
   printf("\na=");scanf("%f",&a);
   printf("\nEnter  Final  value of X:");
   printf("\nb=");scanf("%f",&b);
   printf("\nEnter number of segments (Even number):");
   printf("\nN=");scanf("%d",&n);
   h=(b-a)/n;
   m=n/2;
   sum =0.0;
   x=a;
   f1=F(x);
   for(i=1;i<m;i++)
   {
    f2=F(x+h);
      f3=F(x+2*h);
      sum=sum+f1+4*f2+f3;
      f1=f3;
      x=x+2*h;
   }
   Ics=sum *h/3.0;
   printf("\nIntegral from %f to %f\n",a,b);
   printf("When h=%f is %f \n",h,Ics);
   getch();
}

12. Program to find Integration of given funtion by using Simson's 1/8 rule

/*program to find Integration of given funtion by using Simson's 1/8 rule*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define F(x) (x*x*x+1)
void main()
{
    clrscr();
    float a,b,h,Is;
    printf("\nEnter initial value of X:");
    scanf("%f",&a);
    printf("\nEnter  Final  value of X:");
    scanf("%f",&b);
    h=(b-a)/3;
    Is=3*h*(F(a)+3*F(a+h)+3*F(a+2*h)+F(b))/8;
    printf("\nIntegral from %f to %f\n",a,b);
    printf("Integration is %f \n",Is);
    getch();
}

13. Program to solve Integration equation by using Gauss Integration Method for n=2 interpolating point

/*program to solve Integration equation by using gauss integration method for
n=2 interpolating point*/
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define f(x) exp(x)
void main()
{
    clrscr();
    int n=2,i;
    float a,b,w[2],z[2],x[2],Ig =0.0;
    printf("\nInput initial value of x:");
    printf("\na=");
    scanf("%f",&a);
    printf("\nInput final value of x:");
    printf("\nb=");
    scanf("%f",&b);
    w[0]=1.0;
    w[1]=1.0;
    z[0]=-0.5773502;
    z[1]=0.5773502;
    for (i=0; i<n; i++)
        x[i]=((b-a)/2)* z[i]+((b+a)/2);
    for(i=0; i<n; i++)
        Ig +=((b-a)/2)*(w[i]*f(z[i]));
    printf("\nIntegral from %f to %f is %f\n",a,b,Ig);
    getche();
}

14. Program to solve Integration equation by using gauss integration method for n=3 interpolating point

/*program to solve Integration equation by using gauss integration method for
n=3 interpolating point*/
#include&lt;stdio.h&gt;
#include&lt;conio.h&gt;
#include&lt;math.h&gt;
#define f(x) exp(x)
void main()
{
    clrscr();
    int n=3,i;
    float a,b,w[3],z[3],x[3],Ig =0.0000;
    printf("\nInput initial value of x:");
    printf("\na=");
    scanf("%f",&amp;a);
    printf("\nInput final value of x:");
    printf("\nb=");
    scanf("%f",&amp;b);
    w[0]=0.55556;
    w[1]=0.88889;
    w[2]=0.55556;
    z[0]=-0.77460;
    z[1]=0.00;
    z[2]=0.77460;
    for (i=0; i&lt;n; i++)
        x[i]=((b-a)/2)* z[i]+((b+a)/2);
    for(i=0; i&lt;n; i++)
        Ig +=((b-a)/2)*(w[i]*f(z[i]));
    printf("\nIntegral from %f to %f is %f\n",a,b,Ig);
    getche();
}

15. Demonstrate Euler Method

//Eluer method
#include <stdio.h>
#include <conio.h>
#include <math.h>
#define f(x,y) 2*y/x
void main()
{
    clrscr();
    int i,n;
    float x,y,h,xp,dy;
    printf("Input initial value of x and y");
    scanf ("%f%f",&x,&y);
    printf("Input X-value at which Y is required :\n");
    scanf("%f",&xp);
    printf("Input  step size:\n");
    scanf("%f",&h);
    n=((xp-x)/h+0.5);
    for (i=1; i<=n; i++)
    {
        dy=h*f(x,y);
        x=x+h;
        y=y+dy;
        printf("x=%f and y=%f\n",x,y);
    }
    printf("\n Value of y at x =%f is %f",x,y);
    getche();
}

16. Demonstrate Heuns Method

// Heuns method
#include<stdio.h>
#include<conio.h>
#include<math.h>
#define max 15
void main()
{
    clrscr();
    int n,i;
    float x,y,xp,h,m1,m2;
    float func(float,float);
    printf("\n***Solution of Heuns method***");
    printf("\nEnter the initial value of x \n");
    scanf("%f",&x);
    printf("\nEnter the initial value of y \n");
    scanf("%f",&y);
    printf("\nInput x at which y is required");
    scanf("%f",&xp);
    printf("\nInput step size");
    scanf("%f",&h);
    n=(int)((xp-x)/h+0.5);
    for(i=1; i<=n; i++)
    {
        m1=func(x,y);
        m2=func(x+h,y+m1*h);
        x=x+h;
        y=y+0.5*h*(m1+m2);
        printf("%5d %10.6f %10.6f\n",i,x,y);
    }
    printf("\nThe value of y at x=%f is %f\n",x,y);
    getch();
}
float func(float x,float y)
{
    float f;
    f=2.0*y/x;
    return(f);
}

17. Demonstrate Runge-Kutta Method

#include <stdio.h>
#include <conio.h>
#include <math.h>
//#define f(x,y) (2.0*y/x)
void main()
{
  clrscr();
  int n,i;
  float x,y,xp,h;
  float m1,m2,m3,m4;
  float func(float , float );
  printf("Input initial value of x and y");
  scanf ("%f%f",&x,&y);
  printf("Input X-value at which Y is required :\n");
  scanf("%f",&xp);
  printf("Input  step size:\n");
  scanf("%f",&h);
  n=(int)((xp-x)/h+0.5);
  for (i=1;i<=n;i++)
  {
    m1=func(x,y);
      m2=func(x+0.5*h,y+0.5+m1*h);
      m3=func(x+0.5*h,y+0.5+m2*h);
      m4=func(x+h,y+m3*h);
      x=x+h;
      y=y+(m1+2.0*m2+2.0*m3+m4)*h/6.0;
      printf("x=%f and y=%f\n",x,y);
  }
  printf("\n Value of y at x=%f is %f",x,y);
    getche();
}
float func(float x, float y)
  {
    float f;
      f=x*x+y*y;
      return(f);
  }

18. Runge-Kutta (Alternate)

#include <stdio.h>
#include <conio.h>
#include <math.h>
#define f(x,y) 2.0*y/x
void main()
{
    clrscr();
    int i,n;
    float x,y,xp,h;
    float m1,m2,m3,m4;
    printf("Input initial value of x and y");
    scanf ("%f%f",&x,&y);
    printf("Input X-value at which Y is required :\n");
    scanf("%f",&xp);
    printf("Input  step size:\n");
    scanf("%f",&h);
    n=(int)((xp-x)/h+0.5);
    for (i=1; i<=n; i++)
    {
        m1=f(x,y);
        m2=f(x+0.5*h,y+(m1*h*0.5));
        m3=f(x+0.5*h,y+(m2*h*0.5));
        m4=f(x+h,y+m3*h);
        x=x+h;
        y=y+((m1+2.0*m2+2.0*m3+m4)*h)/6.0;
        printf("x=%f and y=%f\n",x,y);
    }
    printf("\n Value of y at xp =%f is %f",x,y);
    getche();
}

Saturday, April 1, 2017

Some common Numerical Method Programs for 3rd Semester BSc. CSIT students

Program to find Root of a fuction by using Newton Raphson method

Program to find Root of a fuction by using Bisection method

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