Implement IRootFunctions for Complex

[hjrb]

System.Numerics.Complex does not implement the IRootFunctions
The n-th root of a complex number is well defined.
see e http://mathonline.wikidot.com/nth-roots-of-complex-numbers
This would allow to solve a quadratic equation with complex factors a, b, c using generic math.

public static OneOf<(T, T), T, All, None> SolveQuadraticEquation<T>(T a, T b, T c)
		where T: ISignedNumber<T>,  IRootFunctions<T>
	{
		if (T.IsZero(a))
		{
			if (T.IsZero(b)) {
				if (T.IsZero(c)) return new All();
				return new None();
			}
			return -c / b;
		}	
		var d = b * b - T.CreateChecked(4.0) * a * c;
		var a2 = T.CreateChecked(2.0) * a;
		var t = T.Sqrt(d);
		return ((-b+t)/a2,(-b-t)/a2);
	}

As a matter of fact the IRootFunctions is IMHO badly designed. Compute the square root of a double value yields 2 results - not one.
The n-th root of a complex number is a list of n complex numbers.
It would be better to define a new mathematically correct interface IRoots:

public interface IRoots]<TSelf>
        : IFloatingPointConstants<TSelf>
        where TSelf : IRoots<TSelf>?
    {
        /// <summary> Computes the three cube-root of a value.</summary>
        /// <param name="x">The value whose cube-root is to be computed.</param>
        /// <returns>The cube-roots of <paramref name="x" />.</returns>
        static abstract (TSelf,TSelf,TSelf)Cbrt(TSelf x);
 
        /// <summary>Computes the n-th root of a value.</summary>
        /// <param name="x">The value whose <paramref name="n" />-th root is to be computed.</param>
        /// <param name="n">The degree of the root to be computed.</param>
        /// <returns>The <paramref name="n" />-th root of <paramref name="x" />.</returns>
        static abstract IEnumerable<TSelf> RootN(TSelf x, int n);
 
        /// <summary>Computes the square-root of a value.</summary>
        /// <param name="x">The value whose square-root is to be computed.</param>
        /// <returns>The square-root of <paramref name="x" />.</returns>
        static abstract (TSelf,TSelf) Sqrt(TSelf x);
    }
}

I removed the hypotenuse thing - that is not generic at all.

[KalleOlaviNiemitalo]

Which types would implement IRoots<TSelf>? Would it be only System.Numerics.Complex?

If Double implemented IRoots<Double>, then the implementation of static abstract (TSelf,TSelf,TSelf)Cbrt(TSelf x) there would have to return (Double, Double, Double) rather than (Complex, Complex, Complex), so Cbrt(8.0) would presumably be (2.0, 2.0, 2.0), which doesn't seem right. So I don't think Double could implement it.

[hjrb]

double, float, decimal, rationals, all integer types, complex - even defined for symmetric matrix https://en.wikipedia.org/wiki/Square_root_of_a_matrix

[tannergooding]

There are no plans to implement individual interfaces on Complex at this time.

This should rather be part of a more comprehensive proposal covering Complex<T> where T : IFloatingPointIeee754<T> and a new IComplexNumber<T> where T : IFloatingPointIeee754<T>.

If someone would like to open such a more comprehensive proposal, I'd be happy to take it to API review; otherwise, I expect I'll get to this in .NET 11