GAUSS BACKWARD INTERPOLATION FORMULA IN C & FULL EXPLANATION

GAUSS backward Interpolation is the technique of estimating the value of a function for any intermediate value of the independent variable, while the process of computing the value of the function outside the given range is called extrapolation.
it is mainly used for the unequal difference in values of x

GAUSS  backword interpolation formula

#include<stdio.h>
#include<math.h>


int main()
{
int n;                          // no. of terms.
int i,j;                        // Loop variables
float ax[10];                   // 'X' array limit 9
float ay[10];                   // 'Y' array limit 9
float x;                        // User Query for what value of X
float y=0;                      // Calculated value for coressponding X.
float h;                        // Calc. Section
float p;                        // Calc. Section
float diff[20][20];             // to store Y
float y1,y2,y3,y4;              // Formulae variables.

printf("\t\t !! GAUSS' FORWARD INTERPOLATION FORMULA !! ");
// Input section.
printf("\n\n Enter the no. of terms -> ");
scanf("%d",&n);

// Input Sequel for array X
printf("\n\n Enter the value in the form of x -> ");
// Input loop for X.
for(i=0;i<n;i++)
   {
   printf("\n Enter the value of x%d -> ",i+1);
   scanf("%f",&ax[i]);
   }

// Input sequel for array Y.
printf("\n\n Enter the value in the form of y -> ");
// Input loop for Y.
for(i=0;i<n;i++)
   {
   printf("\n Enter the value of y%d -> ",i+1);
   scanf("%f",&ay[i]);
   }

// Inputting the required value quarry
printf("\n\n Enter the value of x for ");
printf("\n which u want the value of y -> ");
scanf("%f",&x);

// Calculation and processing section.
h=ax[1]-ax[0];
for(i=0;i<n-1;i++)
   diff[i][1]=ay[i+1]-ay[i];

for(j=2;j<=4;j++)
   for(i=0;i<n-j;i++)
      diff[i][j]=diff[i+1][j-1]-diff[i][j-1];

i=0;
do
 {
 i++;
 }
while(ax[i]<x);

i--;
p=(x-ax[i])/h;
y1=p*diff[i-1][1];
y2=p*(p+1)*diff[i-1][2]/2;
y3=(p+1)*p*(p-1)*diff[i-2][3]/6;
y4=(p+2)*(p+1)*p*(p-1)*diff[i-3][4]/24;

// Taking sum
y=ay[i]+y1+y2+y3+y4;

// Outut Section
printf("\n When x = %6.4f , y = %6.4f",x,y);

// Invoke user watch halt function
printf("\n\n\n\t\t\t !! PRESS ENTER TO EXIT !! ");
return 0;
}

OUTPUT 

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