How to find the point on the root locus with a certain damping in Octave

[FlatAssembler]

OK, this C++ program I've made:

Code:

#include <algorithm>
#include <complex>
#include <fstream>
#include <iostream>
#include <vector>

double prigusenje(std::complex<double> s) { return std::cos(std::arg(-s)); }

int main() {
  using namespace std;
  double potrebno_prigusenje = 0.7;
  vector<complex<double>> nule{-0.003515, -1.754},
      polovi{0, -20 * 1.754, -1.754, -0.1986, -0.003515};
  vector<complex<double>> KMK;
  vector<double> argumenti, prigusenja;
  double epsilon = 0.001;
  for (double x = -1; x < 1; x += epsilon)
    for (double y = -4; y < 4; y += epsilon) {
      complex<double> s(x, y);
      complex<double> brojnik = 1, nazivnik = 1;
      for (auto nula : nule)
        brojnik *= s - nula;
      for (auto pol : polovi)
        nazivnik *= s - pol;
      complex<double> Gs = brojnik / nazivnik;
      if (abs(arg(Gs) * (180 / M_PI) - 180) < 3 * epsilon ||
          abs(arg(Gs) * (180 / M_PI) + 180) < 3 * epsilon) {
        KMK.push_back(s);
        argumenti.push_back(arg(Gs) * (180 / M_PI));
        prigusenja.push_back(prigusenje(s));
      }
    }
  ofstream datoteka("KMK.txt");
  datoteka << "x\ty\tphi\tdzeta\n";
  for (int i = 0; i < KMK.size(); i++) {
    complex<double> tocka = KMK[i];
    double argument = argumenti[i];
    double prigusenje = prigusenja[i];
    datoteka << tocka.real() << '\t' << tocka.imag() << '\t' << argument << '\t'
             << prigusenje << '\n';
  }
  datoteka.close();
  complex<double> najbolji = *min_element(
      KMK.begin(), KMK.end(),
      [potrebno_prigusenje](complex<double> prvi, complex<double> drugi) {
        return abs(prigusenje(prvi) - potrebno_prigusenje) <
               abs(prigusenje(drugi) - potrebno_prigusenje);
      });
  cout << "prigusenje = " << prigusenje(najbolji) << endl;
  cout << "frekvencija = " << abs(najbolji.imag()) << endl;
  cout << "x = " << najbolji.real() << endl;
  double brojnik = 1, nazivnik = 1;
  for (auto pol : polovi)
    brojnik *= abs(najbolji - pol);
  for (auto nula : nule)
    nazivnik *= abs(najbolji - nula);
  double pojacanje = brojnik / nazivnik;
  cout << "pojacanje = " << pojacanje << endl;
}

It prints out the following:

Code:

prigusenje = 0.689481
frekvencija = 0.104
x = -0.099
pojacanje = 0.723288

So, allegedly, for the gain of 0.723288, the damping is equal to 0.689481 (very close to the required damping of 0.7), and the frequency is 0.104 rad/sec. How do I check whether that's true?