// f(x) = x^3 + cx + d = 0
public static double[] Solve(double c, double d)
{
if (d < 0)
{
return(Solve(c, -d).Reverse().Select(x => - x).ToArray());
}
// 自明解
if (d == 0 && c >= 0)
{
return new[] { 0D }
}
;
// 負の実数解
var x1 = SolveNegative();
// f(x) = (x - x_1) (x^2 + x_1 x + x_1^2 + c)
return(QuadraticEquation0.Solve(1, x1, x1 * x1 + c).Prepend(x1).ToArray());
double SolveNegative()
{
var f = CreateFunction(c, d);
var f1 = CreateDerivative(c);
var x0 = -1D;
while (f(x0) > 0)
{
x0 *= 2;
}
return(NewtonMethod.Solve(f, f1, x0));
}
}
}
// f(x) = ax^2 + bx + c = 0
public static double[] Solve(double a, double b, double c)
{
if (a == 0)
{
throw new ArgumentException("The value must not be 0.", nameof(a));
}
if (a < 0)
{
return(Solve(-a, -b, -c));
}
var det = b * b - 4 * a * c;
if (det.EqualsNearly(0))
{
return new[] { -b / (2 * a) }
}
;
if (det < 0)
{
return(new double[0]);
}
var f = CreateFunction(a, b, c);
var f1 = CreateDerivative(a, b);
var x01 = (-b - Math.Max(det, 1)) / (2 * a);
var x02 = (-b + Math.Max(det, 1)) / (2 * a);
return(new[] { NewtonMethod.Solve(f, f1, x01), NewtonMethod.Solve(f, f1, x02) });
}
}
public static double[] Solve(double c, double d)
{
if (d < 0)
{
return(Solve(c, -d).Reverse().Select(x => - x).ToArray());
}
// 3重解の場合 (c = d = 0) を含む
if (d == 0)
{
return c >= 0 ? new[] { 0D }
}
: new[] { -Sqrt(-c), 0D, Sqrt(-c) };
// この式では誤差が大きくなることがあります。
var det3 = -4 * c * c * c - 27 * d * d;
// 重解の場合
if (det3.EqualsNearly(0, 8))
{
return new[] { -2 * Sqrt(-c / 3), Sqrt(-c / 3) }
}
;
// 負の実数解
var x1 = SolveNegative();
if (det3 < 0)
{
return new[] { x1 }
}
;
// f(x) = (x - x_1) (x^2 + x_1 x + x_1^2 + c)
var sqrt_det2 = Sqrt(-3 * x1 * x1 - 4 * c);
return(new[] { x1, (-x1 - sqrt_det2) / 2, (-x1 + sqrt_det2) / 2 });
double SolveNegative()
{
var f = CubicEquation1.CreateFunction(c, d);
var f1 = CubicEquation1.CreateDerivative(c);
var x0 = -1D;
while (f(x0) > 0)
{
x0 *= 2;
}
return(NewtonMethod.Solve(f, f1, x0));
}
}
}
}
// f(x) = ax^3 + bx^2 + cx + d = 0
public static double[] Solve(double a, double b, double c, double d)
{
if (a == 0)
{
throw new ArgumentException("The value must not be 0.", nameof(a));
}
if (a < 0)
{
return(Solve(-a, -b, -c, -d));
}
var f = CreateFunction(a, b, c, d);
var xc = -b / (3 * a);
var yc = f(xc);
if (yc < 0)
{
return(Solve(a, -b, c, -d).Reverse().Select(x => - x).ToArray());
}
var f1 = CreateDerivative(a, b, c);
// 自明解
if (yc == 0 && f1(xc) >= 0)
{
return new[] { xc }
}
;
// xc より小さい実数解
var x1 = SolveNegative();
var p = a * x1 + b;
var q = p * x1 + c;
return(QuadraticEquation0.Solve(a, p, q).Prepend(x1).ToArray());
double SolveNegative()
{
var x0 = -1D;
while (f(xc + x0) > 0)
{
x0 *= 2;
}
return(NewtonMethod.Solve(f, f1, xc + x0));
}
}
}
public static double[] Solve(double c, double d)
{
if (d < 0)
{
return(Solve(c, -d).Reverse().Select(x => - x).ToArray());
}
// 自明解
if (d == 0 && c >= 0)
{
return new[] { 0D }
}
;
// 負の実数解
var x1 = SolveNegative();
// f(x) = (x - x_1) (x^2 + x_1 x + x_1^2 + c)
var det = -3 * x1 * x1 - 4 * c;
if (det < 0)
{
return new[] { x1 }
}
;
if (det == 0)
{
return new[] { x1, -x1 / 2 }
}
;
return(new[] { x1, (-x1 - Sqrt(det)) / 2, (-x1 + Sqrt(det)) / 2 });
double SolveNegative()
{
var f = CubicEquation1.CreateFunction(c, d);
var f1 = CubicEquation1.CreateDerivative(c);
var x0 = -1D;
while (f(x0) > 0)
{
x0 *= 2;
}
return(NewtonMethod.Solve(f, f1, x0));
}
}
}
}
// f(x) = x^2 + c = 0
public static double[] Solve(double c)
{
if (c > 0)
{
return(new double[0]);
}
if (c == 0)
{
return new[] { 0D }
}
;
var f = CreateFunction(c);
var f1 = CreateDerivative();
var x0 = Math.Max(-c, 1);
var x1 = NewtonMethod.Solve(f, f1, x0);
return(new[] { -x1, x1 });
}
}
