To implement newtons forward and backward interpolation formula using C ! !

 code: 

#include <stdio.h>

// Function to calculate the factorial of a number

int factorial(int n) {

  int result = 1;

  for (int i = 1; i <= n; i++) {

    result *= i;

  }

  return result;

}

// Function to calculate the divided difference table

void calculateDividedDifferenceTable(double x[], double y[], int n, double f[][n]) {

  // Initialize the divided differences with y values

  for (int i = 0; i < n; i++) {

    f[i][0] = y[i];

  }

  // Calculate the divided differences

  for (int j = 1; j < n; j++) {

    for (int i = 0; i < n - j; i++) {

      f[i][j] = (f[i + 1][j - 1] - f[i][j - 1]) / (x[i + j] - x[i]);

    }

  }

}

// Function to interpolate using Newton's forward interpolation formula

double newtonForwardInterpolation(double x[], double f[][n], int n, double value) {

  // Calculate the u value

  double u = (value - x[0]) / (x[1] - x[0]);

  // Calculate the interpolated value

  double sum = f[0][0];

  for (int i = 1; i < n; i++) {

    sum += u_cal(u, i) * f[0][i] / factorial(i);

  }

  return sum;

}

// Function to interpolate using Newton's backward interpolation formula

double newtonBackwardInterpolation(double x[], double f[][n], int n, double value) {

  // Calculate the u value

  double u = (value - x[n - 1]) / (x[n - 1] - x[n - 2]);

  // Calculate the interpolated value

  double sum = f[n - 1][0];

  for (int i = 1; i < n; i++) {

    sum += u_cal(u, i) * f[n - 1][i] / factorial(i);

  }

  return sum;

}

// Function to calculate u mentioned in the formula

double u_cal(double u, int n) {

  double temp = u;

  for (int i = 1; i < n; i++) {

    temp *= (u + i);

  }

  return temp;

}

int main() {

  // Number of data points

  int n = 4;

  // x and y values

  double x[] = {1, 2, 3, 4};

  double y[] = {1, 4, 9, 16};

  // Divided difference table

  double f[n][n];

  // Calculate the divided difference table

  calculateDividedDifferenceTable(x, y, n, f);

  // Value to interpolate at

  double value = 2.5;

  // Interpolated value using Newton's forward interpolation formula

  double interpolatedValueForward = newtonForwardInterpolation(x, f, n, value);

  // Interpolated value using Newton's backward interpolation formula

  double interpolatedValueBackward = newtonBackwardInterpolation(x, f, n, value);

  // Print the interpolated values

  printf("Interpolated value using Newton's forward interpolation formula: %lf\n", interpolatedValueForward);

  printf("Interpolated value using Newton's backward interpolation formula: %lf\n", interpolatedValueBackward);

  return 0;

}