Question

Write a function forward_eval that evaluates the interpolating polynomial using Newton’s Forward Difference table. The function prototype and descriptive comments have been provided:

function y = forward_eval(X, T, x)

%FORWARD_EVAL Evaluate Newton's forward difference form of the

%interpolating polynomial

% y = FORWARD_EVAL(X, T, x) returns y = Pn(x), where Pn is the

% interpolating polynomial constructed using the abscissas X and % forward difference table T.

Answer 1

#include <bits/stdc++.h>

using namespace std;

  

// calculating u mentioned in the formula

float u_cal(float u, int n)

{

    float temp = u;

    for (int i = 1; i < n; i++)

        temp = temp * (u - i);

    return temp;

}

  

// calculating factorial of given number n

int fact(int n)

{

    int f = 1;

    for (int i = 2; i <= n; i++)

        f *= i;

    return f;

}

  

int main()

{

    // Number of values given

    int n = 4;

    float x[] = { 45, 50, 55, 60 };

      

    // y[][] is used for difference table

    // with y[][0] used for input

    float y[n][n];

    y[0][0] = 0.7071;

    y[1][0] = 0.7660;

    y[2][0] = 0.8192;

    y[3][0] = 0.8660;

  

    // Calculating the forward difference

    // table

    for (int i = 1; i < n; i++) {

        for (int j = 0; j < n - i; j++)

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

    }

  

    // Displaying the forward difference table

    for (int i = 0; i < n; i++) {

        cout << setw(4) << x[i]

             << "\t";

        for (int j = 0; j < n - i; j++)

            cout << setw(4) << y[i][j]

                 << "\t";

        cout << endl;

    }

  

    // Value to interpolate at

    float value = 52;

  

    // initializing u and sum

    float sum = y[0][0];

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

    for (int i = 1; i < n; i++) {

        sum = sum + (u_cal(u, i) * y[0][i]) /

                                 fact(i);

    }

  

    cout << "\n Value at " << value << " is "

         << sum << endl;

    return 0;

}