PL

#include <iostream>
#include <cmath>
#include <fstream>
#include <map>
#include <iomanip>

using namespace std;

ifstream in("input.txt");
ofstream out("output.txt");

int FindN(double Eps) {
    int N = ceil(1 / sqrt(2 * Eps));
    return N;
}


double Y(double x, int N) {
    double a = 1, sum = 0;
    for (double k = 1; k <= N; ++k) {
        sum += cos(x * k) / (k * k);
    }
    return sum;
}
double FindPL(map<double, double> A, double x) {
    double sum = 0;
    for (auto it1 = A.begin(); it1 != A.end(); ++it1) {
        double mul = 1;
        for (auto it2 = A.begin(); it2 != A.end(); ++it2) {
            if (it1->first != it2->first)
                mul *= (x - it2->first) / (it1->first - it2->first);
        }
        sum += it1->second * mul;
    }
    return sum;
}


int main() {
    double Eps = 0.0001, a = 0, b = 1;
    int n = 8;
    double h = (b - a) / n;
    map<double, double> A;
    int N = FindN(Eps);
    for (double i = a; i <= b; i += h) {
        double y = Y(i, N);
        A[i] = y;
    }
    cout << "\nx\t\tf(x)\t\tPL(x)\t\tf(x) - PL(x)\n\n";
    cout << fixed;
    cout.precision(5);
    double max = 0;
    for (auto it = A.begin(); it != A.end(); ++it) {
        cout << it->first << "\t\t" << it->second << "\t\t" << FindPL(A, it->first) << "\t\t" << abs(it->second - FindPL(A, it->first)) << endl;
        if (max < abs(it->second - FindPL(A, it->first))) {
            max = abs(it->second - FindPL(A, it->first));
        }
        ++it;
        if (it == A.end()) {
            break;
        }
        --it;
        if (max < abs(Y(it->first + h / 2, N) - FindPL(A, it->first + h / 2))) {
            max = abs(Y(it->first + h / 2, N) - FindPL(A, it->first + h / 2));
        }
        cout << it->first + h / 2 << "\t\t" << Y(it->first + h / 2, N) << "\t\t" << FindPL(A, it->first + h / 2) << "\t\t" << abs(Y(it->first + h / 2, N) - FindPL(A, it->first + h / 2)) << endl;
    }
    cout << "\n\n" << N << "\t\t" << max;
    return 0;
}