solutions method

@override List<Complex> solutions()

Finds the roots (the solutions) of the associated P(x) = 0 equation.

Implementation

@override
List<Complex> solutions() {
  final fb = b / a;
  final fc = c / a;
  final fd = d / a;
  final fe = e / a;

  final q1 = (fc * fc) -
      (fb * fd * const Complex.fromReal(3)) +
      (fe * const Complex.fromReal(12));
  final q2 = (fc.pow(3) * const Complex.fromReal(2)) -
      (fb * fc * fd * const Complex.fromReal(9)) +
      (fd.pow(2) * const Complex.fromReal(27)) +
      (fb.pow(2) * fe * const Complex.fromReal(27)) -
      (fc * fe * const Complex.fromReal(72));
  final q3 = (fb * fc * const Complex.fromReal(8)) -
      (fd * const Complex.fromReal(16)) -
      (fb.pow(3) * const Complex.fromReal(2));
  final q4 = (fb.pow(2) * const Complex.fromReal(3)) -
      (fc * const Complex.fromReal(8));

  var temp = (q2 * q2 / const Complex.fromReal(4)) - (q1.pow(3));
  final q5 = (temp.sqrt() + (q2 / const Complex.fromReal(2))).pow(1.0 / 3.0);
  final q6 = ((q1 / q5) + q5) / const Complex.fromReal(3);
  temp = (q4 / const Complex.fromReal(12)) + q6;
  final q7 = temp.sqrt() * const Complex.fromReal(2);
  temp = ((q4 * const Complex.fromReal(4)) / const Complex.fromReal(6)) -
      (q6 * const Complex.fromReal(4)) -
      (q3 / q7);

  final solutions = [
    (fb.negate - q7 - temp.sqrt()) / const Complex.fromReal(4),
    (fb.negate - q7 + temp.sqrt()) / const Complex.fromReal(4),
  ];

  temp = ((q4 * const Complex.fromReal(4)) / const Complex.fromReal(6)) -
      (q6 * const Complex.fromReal(4)) +
      (q3 / q7);

  solutions
    ..add((fb.negate + q7 - temp.sqrt()) / const Complex.fromReal(4))
    ..add((fb.negate + q7 + temp.sqrt()) / const Complex.fromReal(4));

  return solutions;
}