/// <summary>
/// Solves the equation \f$a0 + a1 x + a2 x^2 + a3 x^3 = 0\f$, returning all real values of
/// \f$x\f$ for which the equation is true. See https://en.wikipedia.org/wiki/Cubic_equation for the
/// algorithm used.
/// </summary>
public static IEnumerable <double> Solve(double a0, double a1, double a2, double a3)
{
DebugUtil.AssertFinite(a3, nameof(a3));
DebugUtil.AssertFinite(a2, nameof(a2));
DebugUtil.AssertFinite(a1, nameof(a1));
DebugUtil.AssertFinite(a0, nameof(a0));
if (Math.Abs(a3) <= 0.005)
{
foreach (double v in QuadraticFunction.Solve(a0, a1, a2))
{
CubicFunction f = new CubicFunction(a0, a1, a2, a3);
yield return(f.NewtonRaphson(v));
}
yield break;
}
double delta0 = a2 * a2 - 3.0 * a3 * a1;
double delta1 = 2.0 * a2 * a2 * a2 - 9.0 * a3 * a2 * a1 + 27.0 * a3 * a3 * a0;
Complex p1 = Complex.Sqrt(delta1 * delta1 - 4.0 * delta0 * delta0 * delta0);
// The sign we choose in the next equation is arbitrary. To prevent a divide-by-zero down the line, if p2 is
// zero, we must choose the opposite sign to make it nonzero:
Complex p2 = delta1 + p1;
Complex c = Complex.Pow(0.5 * p2, (1.0 / 3.0));
Complex xi = -0.5 + 0.5 * Complex.Sqrt(-3.0);
List <Complex> roots = new List <Complex>
{
-1.0 / (3.0 * a3) * (a2 + c + delta0 / c),
-1.0 / (3.0 * a3) * (a2 + xi * c + delta0 / (xi * c)),
-1.0 / (3.0 * a3) * (a2 + xi * xi * c + delta0 / (xi * xi * c)),
};
foreach (Complex root in roots)
{
if (Math.Abs(root.Imaginary) <= 0.05)
{
DebugUtil.AssertFinite(root.Real, nameof(root.Real));
yield return(root.Real);
}
}
}
/// <summary>
/// Solves the equation \f$d + c x + b x^2 + a x^3 = 0\f$, returning all real values of
/// \f$x\f$ for which the equation is true. See https://en.wikipedia.org/wiki/Cubic_equation for the
/// algorithm used.
/// </summary>
public static IEnumerable <double> Solve(double d, double c, double b, double a)
{
DebugUtil.AssertFinite(a, nameof(a));
DebugUtil.AssertFinite(b, nameof(b));
DebugUtil.AssertFinite(c, nameof(c));
DebugUtil.AssertFinite(d, nameof(d));
if (Math.Abs(a) <= 0.005f)
{
foreach (double v in QuadraticFunction.Solve(d, c, b))
{
yield return(v);
}
yield break;
}
double delta0 = b * b - 3.0 * a * c;
double delta1 = 2.0 * b * b * b - 9.0 * a * b * c + 27.0 * a * a * d;
Complex p1 = Complex.Sqrt(delta1 * delta1 - 4.0 * delta0 * delta0 * delta0);
// The sign we choose in the next equation is arbitrary. To prevent a divide-by-zero down the line, if p2 is
// zero, we must choose the opposite sign to make it nonzero:
Complex p2 = delta1 + p1;
Complex C = Complex.Pow(0.5 * p2, (1.0 / 3.0));
Complex xi = -0.5 + 0.5 * Complex.Sqrt(-3.0);
List <Complex> roots = new List <Complex>
{
-1.0 / (3.0 * a) * (b + C + delta0 / C),
-1.0 / (3.0 * a) * (b + xi * C + delta0 / (xi * C)),
-1.0 / (3.0 * a) * (b + xi * xi * C + delta0 / (xi * xi * C)),
};
foreach (Complex root in roots)
{
if (Math.Abs(root.Imaginary) <= 0.05)
{
DebugUtil.AssertFinite(root.Real, nameof(root.Real));
yield return(root.Real);
}
}
}
/// <summary>
/// Solves a general equation \f$a_0 + a_1 x + a_2 x^2 = 0\f$, returning all real values of
/// \f$x\f$ for which the equation is true using the 'abc-formula'.
/// </summary>
public static IEnumerable <double> Solve(double a0, double a1, double a2)
{
DebugUtil.AssertFinite(a0, nameof(a0));
DebugUtil.AssertFinite(a1, nameof(a1));
DebugUtil.AssertFinite(a2, nameof(a2));
if (Math.Abs(a2) <= 0.005)
{
if (Math.Abs(a1) <= 0.005)
{
// There are no roots found:
yield break;
}
else
{
// There is a single root, found from solving the linear equation with a1=0. We find a point close
// to the root using the linear approximation, after which we approach the true solution using the
// Newton-Raphson method:
QuadraticFunction f = new QuadraticFunction(a0, a1, a2);
yield return(f.NewtonRaphson(-a0 / a1));
}
yield break;
}
double x1 = 0.5 * (-a1 + Math.Sqrt(a1 * a1 - 4.0 * a0 * a2)) / a2;
double x2 = 0.5 * (-a1 - Math.Sqrt(a1 * a1 - 4.0 * a0 * a2)) / a2;
if (Double.IsNaN(x1) == false)
{
yield return(x1);
}
if (Double.IsNaN(x2) == false)
{
yield return(x2);
}
}
/// <summary>
/// Solves the equation \f$a_0 + a_1 x + a_2 x^2 = 0\f$, returning all real values of
/// \f$x\f$ for which the equation is true using the 'abc-formula' using the parameters
/// in this instance of QuadraticFunction.
/// </summary>
public IEnumerable <float> Roots()
{
return(QuadraticFunction.Solve(a0, a1, a2));
}