STIRLING INTERPOLATION FORMULA IN C++ WITH EXPLANATION

Stirling Interploation

Stirling Approximation or Stirling Interpolation Formula is an interpolation technique, which is used to obtain the value of a function at an intermediate point within the range of a discrete set of known data points .

Stirling Formula is obtained by taking the average or mean of the Gauss Forward and Gauss Backward Formula . Both the Gauss Forward and Backward formula are formulas for obtaining the value of the function near the middle of the tabulated set .

Striling formula




#include <iostream>
#include<math.h>
using namespace std;


// Function to calculate value using  
// Stirling formula 
void Stirling(float x[], float fx[], float x1, 
                                    int n) 
{ 
    float h, a, u, y1 = 0, N1 = 1, d = 1, 
    N2 = 1, d2 = 1, temp1 = 1, temp2 = 1, 
    k = 1, l = 1, delta[n][n]; 
    
    int i, j, s; 
    h = x[1] - x[0]; 
    s = floor(n / 2); 
    a = x[s]; 
    u = (x1 - a) / h; 
  
    // Preparing the forward difference 
    // table 
    for (i = 0; i < n - 1; ++i) { 
        delta[i][0] = fx[i + 1] - fx[i]; 
    } 
    for (i = 1; i < n - 1; ++i) { 
        for (j = 0; j < n - i - 1; ++j) { 
            delta[j][i] = delta[j + 1][i - 1] 
                          - delta[j][i - 1]; 
        } 
    } 
  
    // Calculating f(x) using the Stirling  
    // formula 
    y1 = fx[s]; 
  
    for (i = 1; i <= n - 1; ++i) { 
        if (i % 2 != 0) { 
            if (k != 2) { 
                temp1 *= (pow(u, k) -  
                          pow((k - 1), 2)); 
            }  
            else { 
                temp1 *= (pow(u, 2) -  
                          pow((k - 1), 2)); 
            } 
            ++k; 
            d *= i; 
            s = floor((n - i) / 2); 
            y1 += (temp1 / (2 * d)) *  
                   (delta[s][i - 1] +  
                   delta[s - 1][i - 1]); 
        }  
        else { 
            temp2 *= (pow(u, 2) -  
                      pow((l - 1), 2)); 
            ++l; 
            d *= i; 
            s = floor((n - i) / 2); 
            y1 += (temp2 / (d)) * 
                  (delta[s][i - 1]); 
        } 
    } 
  
    cout<<y1;
} 
  
// Driver main function 
int main() 
{ 
    int n; 
    n = 5; 
    float x[] = { 0, 0.5, 1.0, 1.5, 2.0 }; 
    float fx[] = { 0, 0.191, 0.341, 0.433, 
                             0.477 }; 
  
    // Value to calculate f(x) 
    float x1 = 1.22; 
  
    Stirling(x, fx, x1, n); 
    return 0; 
} 

OUTPUT 

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this progaran is running through the CODECHEF IDE

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