This document contains C code to implement Newton-Gregory forward interpolation and Lagrange interpolation. The Newton-Gregory code calculates the difference table and uses it to interpolate a value for y given a value of x. The Lagrange code uses the given x and y value pairs to calculate the Lagrange polynomials and sum them to find the interpolated y value for a given x.

1. /* Write C program to implement the Newton- Gregory forward interpolation.*/
#include<stdio.h>
#include<conio.h>
#define MaxN 100
#define Order_of_diff 4
void main ()
{
float arr_x[MaxN+1], arr_y[MaxN+1], numerator=1.0, denominator=1.0, x, y, p, h,
diff_table[MaxN+1][Order_of_diff+1];
int i,j,n,k;
clrscr();
printf("Enter the value of n n");
scanf("%d",&n);
printf("Enter the values of x and y");
for(i=0; i<=n; i++)
scanf("%f%f", &arr_x[i], &arr_y[i]);
printf("Enter the value of x at which value of y is to be calculated");
scanf("%f", &x);
h=arr_x[1]-arr_x[0];
for(i=0; i<=n-1; i++)

2. diff_table[i][1]=arr_y[i+1]-arr_y[i];/*Creating the difference table and calculating first order
differences*/
for(j=2; j<=Order_of_diff; j++)/*Calculating higher order differences*/
for(i=0; i<=n-j; i++)
diff_table[i][j]=diff_table[i+1][j-1] - diff_table[i][j-1];
i=0;
while(!(arr_x[i]>x)) /* Finding x0 */
i++;
i--;
p=(x-arr_x[i])/h;
y=arr_y[i];
for (k=1; k<=Order_of_diff; k++)
{
numerator *=p-k+1;
denominator *=k;
y +=(numerator/denominator)*diff_table[i][k];
}
printf("When x=%6.1f, y=%6.2fn",x, y);
getch();
}

3. /* Write C program to implement the Lagrange interpolation.*/
#include<stdio.h>
#include<conio.h>
#define MaxN 90
void main()
{
float arr_x[MaxN+1], arr_y[MaxN+1], numerator, denominator, x, y=0;
int i, j, n;
clrscr();
printf("Enter the value of n: n");
scanf("%d", &n);
printf("Enter the values of x and y: n");
for(i=0; i<=n; i++)
scanf("%f%f", &arr_x[i], &arr_y[i]);
printf("Enter the value of x at which value of y is to be calculated: ");
scanf("%f", &x);
for (i=0; i<=n; i++)
{
numerator=1;
denominator=1;
for (j=0; j<=n; j++)

4. if(j!=i)
{
numerator *= x-arr_x[j];
denominator *= arr_x[i]-arr_x[j];
}
y+=(numerator/denominator)*arr_y[i];
}
printf("When x=%4.1f y=%7.1fn",x,y);
getch();
}