Large floating point precision

Question

I am trying to summarize a sorted array of positively decreasing floating points. I saw that the best way to sum them up is to start adding numbers from the lowest to the highest. I wrote this code to have an example of this, however the amount that starts with the largest number is more accurate. What for? (of course, the sum 1 / k ^ 2 should be f = 1.644934066848226).

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

int main() {

    double sum = 0;
    int n;
    int e = 0;
    double r = 0;
    double f = 1.644934066848226;
    double x, y, c, b;
    double sum2 = 0;

    printf("introduce n\n");
    scanf("%d", &n);

    double terms[n];

    y = 1;

    while (e < n) {
        x = 1 / ((y) * (y));
        terms[e] = x;
        sum = sum + x;
        y++;
        e++;
    }

    y = y - 1;
    e = e - 1;

    while (e != -1) {
        b = 1 / ((y) * (y));
        sum2 = sum2 + b;
        e--;
        y--;
    }
    printf("sum from biggest to smallest is %.16f\n", sum);
    printf("and its error %.16f\n", f - sum);
    printf("sum from smallest to biggest is %.16f\n", sum2);
    printf("and its error %.16f\n", f - sum2);
    return 0;
}

Answer 1

Your code creates an array double terms[n];on the stack, and this puts a hard limit on the number of iterations that can be performed before your program crashes.

, , . , terms[]:

#include <stdio.h>

int main() {

    double pi2over6 = 1.644934066848226;
    double sum = 0.0, sum2 = 0.0;
    double y;
    int i, n;

    printf("Enter number of iterations:\n");
    scanf("%d", &n);

    y = 1.0;

    for (i = 0; i < n; i++) {
        sum += 1.0 / (y * y);
        y += 1.0;
    }

    for (i = 0; i < n; i++) {
        y -= 1.0;
        sum2 += 1.0 / (y * y);
    }
    printf("sum from biggest to smallest is %.16f\n", sum);
    printf("and its error %.16f\n", pi2over6 - sum);
    printf("sum from smallest to biggest is %.16f\n", sum2);
    printf("and its error %.16f\n", pi2over6 - sum2);
    return 0;

}

, , , - :

Enter number of iterations:
1000000000
sum from biggest to smallest is 1.6449340578345750
and its error 0.0000000090136509
sum from smallest to biggest is 1.6449340658482263
and its error 0.0000000009999996

Answer 2

, .

, Σ1/k² k N N, .. 1/n-1/N ( ), 1/n².

, Σ1/k² k N N, π²/6-1/n ( ) 1/n².

, .