Miscalculation of Lagrange interpolation formula for higher degree

Question

I am approximating Runge’s function using Lagrange’s interpolation formula for 50 interpolation points. I have written the following program to do this, but I am getting the wrong value for x= -0.992008. That wrong value is 4817543.091313, but it should be 5197172.55933613. I have got this value from the following link: Link The code used are as follows:

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

double
runge(double x)
{
    return (1 / (1 + (25 * x * x)));
}

double
ab(double x)
{
    if (x < 0)
        return -1 * x;
    return x;
}

double
lag_func(double x, double *y_i, double *x_i, int n)
{
    double ex = 0.0;

    for (int i = 0; i <= n; i++) {
        double numer = 1.0,
            denom = 1.0,
            prod = 1.0;

        for (int j = 0; j <= n; j++) {
            if (i != j) {
                numer = (x - x_i[j]);
                denom = (x_i[i] - x_i[j]);
                prod *= numer / denom;
            }
        }
        ex += (prod) * y_i[i];
    }
    return ex;
}

int
main()
{
    int n;

    scanf("%d", &n);
    double y_i[n + 1],
     x_i[n + 1];

    for (int i = 0; i < n + 1; i++) {
        x_i[i] = ((2 * (double) i) / (double) n) - 1;
        y_i[i] = runge(x_i[i]);
    }

    printf("%lf\n", lag_func(-0.992008, y_i, x_i, n));
    return 0;
}

Answer 1

The web site is rounding its Runge coefficients to six digits. Given the magnitudes of the terms involved, up to 3.9978•10¹¹, this introduces multiple errors up to around 2•10⁵.

This can be seen by inserting y_i[i] = round(y_i[i] * 1e6) / 1e6; after y_i[i] = runge(x_i[i]);. Then the output of the program is 5197172.558199, matching the web site’s inaccurate result.

The web site is wrong; the result of the code in the question is better.