/// <summary>
/// Return real roots of the equation: x^3 + c2 x^2 + c1 x + c0 = 0
/// Double and triple solutions are returned as replicated values.
/// Imaginary and non existing solutions are returned as NaNs.
/// </summary>
public static Triple <double> RealRootsOfNormed(
double c2, double c1, double c0)
{
// ------ eliminate quadric term (x = y - c2/3): x^3 + p x + q = 0
double d = c2 * c2;
double p3 = 1 / 3.0 * /* p */ (-1 / 3.0 * d + c1);
double q2 = 1 / 2.0 * /* q */ ((2 / 27.0 * d - 1 / 3.0 * c1) * c2 + c0);
double p3c = p3 * p3 * p3;
double shift = 1 / 3.0 * c2;
d = q2 * q2 + p3c;
if (d < 0) // casus irreducibilis: three real solutions
{
double phi = 1 / 3.0 * Fun.Acos(-q2 / Fun.Sqrt(-p3c));
double t = 2 * Fun.Sqrt(-p3);
double r0 = t * Fun.Cos(phi) - shift;
double r1 = -t *Fun.Cos(phi + Constant.Pi / 3.0) - shift;
double r2 = -t *Fun.Cos(phi - Constant.Pi / 3.0) - shift;
return(Triple.CreateAscending(r0, r1, r2));
}
// else if (Fun.IsTiny(q2)) // one triple root
// { // too unlikely for
// double r = -1/3.0 * c2; // special handling
// return Triple.Create(r, r, r); // to pay off
// }
d = Fun.Sqrt(d); // one single and one double root
double uav = Fun.Cbrt(d - q2) - Fun.Cbrt(d + q2);
double s0 = uav - shift, s1 = -0.5 * uav - shift;
return(s0 < s1?Triple.Create(s0, s1, s1)
: Triple.Create(s1, s1, s0));
}
/// <summary>
/// One real root of the equation: x^3 + c2 x^2 + c1 x + c0 = 0.
/// </summary>
public static double OneRealRootOfNormed(
double c2, double c1, double c0
)
{
// ------ eliminate quadric term (x = y - c2/3): x^3 + p x + q = 0
double d = c2 * c2;
double p3 = 1 / 3.0 * /* p */ (-1 / 3.0 * d + c1);
double q2 = 1 / 2.0 * /* q */ ((2 / 27.0 * d - 1 / 3.0 * c1) * c2 + c0);
double p3c = p3 * p3 * p3;
d = q2 * q2 + p3c;
if (d < 0) // -------------- casus irreducibilis: three real roots
{
return(2 * Fun.Sqrt(-p3) * Fun.Cos(1 / 3.0
* Fun.Acos(-q2 / Fun.Sqrt(-p3c))) - 1 / 3.0 * c2);
}
d = Fun.Sqrt(d); // one triple root or a single and a double root
return(Fun.Cbrt(d - q2) - Fun.Cbrt(d + q2) - 1 / 3.0 * c2);
}
/// <summary>
/// Return real roots of the equation: x^3 + p x + q = 0
/// Double and triple solutions are returned as replicated values.
/// Imaginary and non existing solutions are returned as NaNs.
/// </summary>
public static Triple <double> RealRootsOfDepressed(
double p, double q)
{
double p3 = 1 / 3.0 * p, q2 = 1 / 2.0 * q;
double p3c = p3 * p3 * p3, d = q2 * q2 + p3c;
if (d < 0) // ---------- casus irreducibilis: three real solutions
{
double phi = 1 / 3.0 * Fun.Acos(-q2 / Fun.Sqrt(-p3c));
double t = 2 * Fun.Sqrt(-p3);
double r0 = t * Fun.Cos(phi);
double r1 = -t *Fun.Cos(phi + Constant.Pi / 3.0);
double r2 = -t *Fun.Cos(phi - Constant.Pi / 3.0);
return(Triple.CreateAscending(r0, r1, r2));
}
d = Fun.Sqrt(d); // one triple root or a single and a double root
double s0 = Fun.Cbrt(d - q2) - Fun.Cbrt(d + q2);
double s1 = -0.5 * s0;
return(s0 < s1?Triple.Create(s0, s1, s1)
: Triple.Create(s1, s1, s0));
}