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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<stdio.h>
#include<conio.h>
#include<math.h>
#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",&a);
printf("\nInput final value of x:");
printf("\nb=");
scanf("%f",&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<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();
}
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();
}
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