Newtons forward | Backward | Raphson Method | Lagrange Interpolation Formula

Newtons forward and Backward Interpolation Formula

Newtons forward | Backward | Raphson Method | Lagrange Interpolation Formula:          

   /*Newtons forward interpolation formula.*/

Newtons forward interpolation:

#include<math.h>
#include<stdio.h>
main()
{
float x[15],y[15],X,p=1,u,h,d;
int i,j,n;
printf("\t\t Output\n");
printf("\nEnter the no. of points: ");
scanf("%d",&n);
printf("\n Enter the value of X at which y is reqd. ");
scanf("%f",&X);
printf("Enter the values of x and y :\n\n");
printf("x\ty=f(x)\n");
for(i=1;i<=n;i++)
scanf("%f\t%f",&x[i],&y[i]);
h=x[2]-x[1];
u=(X-x[1])/h;
d=y[1];
for(i=1;i<=n-1;i++)
{
for(j=1;j<=n-i;j++)
{
y[j]=y[j+1]-y[j];
}
p=p*(u-i+1)/i;
d=d+p*y[1];
}
printf("\n\nThe result of Newtons forward interpolation\n\n");
printf("y(%1.2f)=%3.5f\n",X,d);
}

/*Newtons Backward interpolation formula*/

Newtons Backward interpolation:

#include<math.h>

#include<stdio.h>

main()

{

float x[15],y[15],X,p=1,u,h,d;

int i,j,k,n;

printf("\t\t Output\n");

printf("\nEnter the no. of points: ");

scanf("%d",&n);

printf("enter the value of X at which y is reqd ");

scanf("%f",&X);

printf("Enter the values of x and y :\n");

printf("x\ty=f(x)\n");

for(i=1;i<=n;i++)

scanf("%f\t%f",&x[i],&y[i]);

h=x[2]-x[1];

u=(X-x[n])/h;

d=y[n];

for(i=1;i<=n-1;i=i+1)

{

for(j=n;j>=i+1;j=j-1)

{

y[j]=y[j]-y[j-1];

}

p=p*(u+i-1)/i;

d=d+p*y[n];

}

printf("\n\nThe result of Newtons backward interpolation\n");

printf("y(%f)=%f\n",X,d);

}

/*Lagrange Interpolation Formula*/

Lagrange Interpolation Formula:

#include<math.h>

#include<stdio.h>

main()

{

float x[15],y[15],sum=0.0,X,prod;

int i,j,n;

printf("\t\t Output\n");

printf("\nEnter the no. of points: ");

scanf("%d",&n);

printf("enter the value of X at which y is reqd. ");

scanf("%f",&X);

printf("Enter the values of x and y :\n");

printf("\nx\ty=f(x)\n");

for(i=1;i<=n;i++)

scanf("%f\t%f",&x[i],&y[i]);

for(i=1;i<=n;i++)

{

prod=y[i];

for(j=1;j<=n;j++)

{

if(j!=i)

prod=prod*(X-x[j])/(x[i]-x[j]);

}

sum=sum+prod;

}

printf("\nAt x=%f,\n \n y=%f\n",X,sum);

}

/* Newton Raphson Method*/

Newton Raphson Method:

#include<math.h> #include<stdio.h> float f(float x)

{

return (x*x*x-x-0.1);

}

float f1(float x)

{

return (3*x*x-1);

}

main()

{

float a,b,e,x0,x1;

int n=0;

printf("\n Enter the accuracy: "); scanf("%f",&e);

do

{

printf("\n Enter interval a,b : "); scanf("%f%f",&a,&b);

}while(f(a)*f(b)>0);

x0=(a+b)/2;

printf("\nx0=%f",x0);

do

{

x1=x0-(f(x0)/f1(x0));

x0=x1;

printf("\n n=%d \t x0=%f",n,x0);

n=n+1;

}while(fabs(f(x0))>e);

printf("\n Iteration=%d\tRoot=%f", n,x0);

}