Newton's Backward Interpolation Formula is a method used to interpolate values for a function that is known only at discrete points. Here's a simple C program to demonstrate this formula:

C Program for Newton's Backward Interpolation:

cCopy code
#include <stdio.h>

// Function to calculate factorial
int factorial(int n) {
    int fact = 1;
    for (int i = 1; i <= n; i++)
        fact *= i;
    return fact;
}

// Function to perform backward difference
void backwardDifference(float y[], int n) {
    for (int i = 1; i < n; i++)
        for (int j = n - 1; j >= i; j--)
            y[j] = y[j] - y[j - 1];
}

// Main function demonstrating Newton's Backward Interpolation
int main() {
    int n, i, j;
    float x[20], y[20], X, h, p, sum;
    
    // Input the number of data points
    printf("Enter the number of data points: ");
    scanf("%d", &n);
    
    // Input the data points
    printf("Enter the values of x and y:\n");
    for (i = 0; i < n; i++) {
        printf("x[%d] = ", i);
        scanf("%f", &x[i]);
        printf("y[%d] = ", i);
        scanf("%f", &y[i]);
    }
    
    // Input the value to interpolate
    printf("Enter the value of x for which you want to find y: ");
    scanf("%f", &X);
    
    // Step size (assuming uniform step size)
    h = x[1] - x[0];
    
    // Calculate backward difference table
    backwardDifference(y, n);
    
    // Calculate value of p
    p = (X - x[n - 1]) / h;
    
    // Apply Newton's Backward Interpolation formula
    sum = y[n - 1];  // First term in the series
    for (i = 1; i < n; i++) {
        float term = p;
        for (j = 1; j < i; j++)
            term *= (p + j);
        sum += (term * y[n - 1 - i]) / factorial(i);
    }
    
    // Output the result
    printf("The interpolated value at x = %.2f is y = %.3f\n", X, sum);
    
    return 0;
}

Explanation:

1. Input:

   • The user is prompted to input the number of data points and their corresponding x and y values.
   • Then, the user is asked for the value of X (the point where they want to interpolate the value of y).

2. Backward Differences:

   • The function backwardDifference computes the backward difference table for the given y values.

3. Formula:

   • The main formula used for interpolation is Newton's Backward Interpolation Formula: $P(x) = y_n + \frac{p \Delta y_{n-1}}{1!} + \frac{p(p+1) \Delta^2 y_{n-2}}{2!} + \dots$ Where $p = \frac{X - x_n}{h}$.

4. Factorial:

   • The factorial function is used to divide each term by $i!$.

5. Result:

   • The interpolated value of y at the given X is printed.

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Do you want more details or have any questions on this?

Related Questions:

1. How does Newton's Forward Interpolation differ from the Backward Interpolation?
2. What are the conditions for using Newton’s Backward Interpolation?
3. Can this method be extended to non-uniform grids?
4. How does the backward difference table work?
5. What are alternative methods to Newton's Backward Interpolation?

Tip:

Always ensure the x values are evenly spaced when applying Newton's Backward Interpolation formula for accurate results.