Skeleton/src/Algebra/ScalarFieldStructure/FieldStructure.cs

using System.Numerics;
using Skeleton.Algebra.Extensions;
using Skeleton.DataStructure.Packs;

namespace Skeleton.Algebra.ScalarFieldStructure;

/// <summary>
/// operations / structure of the field
/// </summary>
public abstract class FieldStructure
{
    
}

/// <inheritdoc />
public abstract class FieldStructure<TScalar> : FieldStructure
{
    
    /// <summary>
    /// return the singleton base on type
    /// </summary>
    /// <returns></returns>
    public static FieldStructure<TScalar> Dispatch()
    {
        if (typeof(TScalar) == typeof(double))
            return (RealFieldStructure.Structure as FieldStructure<TScalar>)!;
        if (typeof(TScalar) == typeof(Complex))
            return (ComplexFieldStructure.Structure as FieldStructure<TScalar>)!;
        return ScalarFieldStructureProvider.GetOperation<TScalar>();
    }

    /// <inheritdoc />
    public FieldStructure()
    {
        StaticAccess<TScalar>.AdditionUnit = AdditionUnit;
        StaticAccess<TScalar>.MultiplicationUnit = MultiplicationUnit;
        StaticAccess<TScalar>.Structure = this;
    }

    /// <summary>
    /// addition in field 
    /// </summary>
    /// <param name="self"></param>
    /// <param name="other"></param>
    /// <returns></returns>
    public abstract TScalar Addition(TScalar self, TScalar other);
    /// <summary>
    /// addition unit of the field<br/>
    /// a + unit = a for any a in field
    /// </summary>
    public abstract TScalar AdditionUnit { get; }
    /// <summary>
    /// multiplication in field
    /// </summary>
    /// <param name="self"></param>
    /// <param name="other"></param>
    /// <returns></returns>
    public abstract TScalar Multiplication(TScalar self, TScalar other);
    /// <summary>
    /// multiplication unit of the field<br/>
    /// unit * a = a for any a in the field except the addition unit
    /// </summary>
    public abstract TScalar MultiplicationUnit { get; }
    /// <summary>
    /// inverse of element under addition<br/>
    /// a + AddInv(a) = unit(addition) for any a in field
    /// </summary>
    /// <param name="self"></param>
    /// <returns></returns>
    public abstract TScalar AdditionInverse(TScalar self);
    /// <summary>
    ///     inverse of element under multiplication<br/>
    ///     a * MulInv(a) = unit(multiplication) for any a in field except the addition unit
    /// </summary>
    /// <param name="self"></param>
    /// <returns></returns>
    public abstract TScalar MultiplicationInverse(TScalar self);
    /// <summary>
    /// a + AddInv(b)
    /// </summary>
    /// <param name="self"></param>
    /// <param name="other"></param>
    /// <returns></returns>
    public virtual TScalar Subtraction(TScalar self, TScalar other)
        => Addition(self, AdditionInverse(other));

    /// <summary>
    /// a * MulInv(b)
    /// </summary>
    /// <param name="self"></param>
    /// <param name="other"></param>
    /// <returns></returns>
    public virtual TScalar Division(TScalar self, TScalar other)
        => Multiplication(self, MultiplicationInverse(other));
    /// <summary>
    /// conjugate of the scalar
    /// </summary>
    /// <param name="self"></param>
    /// <returns></returns>
    public abstract TScalar Conj(TScalar self);
    /// <summary>
    /// if two elements are close to each other in the field
    /// </summary>
    /// <param name="self"></param>
    /// <param name="other"></param>
    /// <param name="absTol"></param>
    /// <param name="relTol"></param>
    /// <returns></returns>
    public abstract bool IsEqualApprox(TScalar self, TScalar other, double? absTol=null, double? relTol = null);

    /// <summary>
    /// tolerance to check if two scalars are close
    /// </summary>
    public double AbsoluteTolerance { get; set; } = 1E-8d;

    /// <summary>
    /// 
    /// </summary>
    public double RelativeTolerance { get; set; } = 1E-5d;

    /// <summary>
    /// if calculation in field won't introduce error, override this and leave it empty
    /// otherwise, provide method to reduce the error here
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual TScalar Fix(TScalar x)
    {
        if (IsEqualApprox(x, AdditionUnit))
            return AdditionUnit;
        if (IsEqualApprox(x, MultiplicationUnit))
            return MultiplicationUnit;
        return x;
    }

    /// <summary>
    /// homomorphism from scalar field to real<br/>
    /// ignore if not applicable
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual double AsReal(TScalar x) => IsEqualApprox(x, AdditionUnit) ? 0 : 1;
    /// <summary>
    /// conjugated square xx* when field is real/complex
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public double ConjX2(TScalar x) => AsReal(Multiplication(x, Conj(x)));

    /// <summary>
    /// homomorphism from real to scalar<br/>
    /// ignore if not applicable
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual TScalar FromReal(double x) => x.IsEqualApprox(0) ? AdditionUnit : MultiplicationUnit;
    /// <summary>
    /// readable string representation of data
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public string Representation(TScalar x) => RawRepresentation(Fix(x));

    /// <summary>
    /// representation of the original data, without fix
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual string RawRepresentation(TScalar x) => $"{x}";

    /// <summary>
    /// code to construct object in python
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public string PythonRepresentation(TScalar x) => RawPythonRepresentation(Fix(x));

    /// <summary>
    /// code to construct scalar object in python<br/>
    /// ignore if not applicable
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual string RawPythonRepresentation(TScalar x) => "";
    /// <summary>
    /// code to construct the object in c#
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public string CSharpRepresentation(TScalar x) => RawCSharpRepresentation(Fix(x));
    /// <summary>
    /// C# string for the original data
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public abstract string RawCSharpRepresentation(TScalar x);

    /// <summary>
    /// dump the object as string which can be restored exactly<br/>
    /// ignore if not applicable<br/>
    /// if override can not contain following string as substring<br/>
    /// &lt;D&gt; &lt;/D&gt; &lt;C&gt; &lt;/C&gt; &lt;CSPLIT/&gt;<br/>
    /// &lt;VECTOR&gt; &lt;VSPLIT/&gt; &lt;/VECTOR&gt;<br/>
    /// &lt;MATRIX&gt; &lt;MSPLIT/&gt; &lt;/MSPLIT&gt;
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual string DumpString(TScalar x) => "";

    /// <summary>
    /// method to restore a value from dump string<br/>
    /// ignore if not applicable
    /// </summary>
    /// <param name="dumpString"></param>
    /// <returns></returns>
    public virtual TScalar Resolve(string dumpString) => AdditionUnit;
    /// <summary>
    /// max diff from zero
    /// </summary>
    /// <param name="a"></param>
    /// <returns></returns>
    public abstract double MaxError(TScalar a);

    /// <summary>
    /// homomorphism from field to complex<br/>
    /// ignore if not applicable
    /// </summary>
    /// <param name="a"></param>
    /// <returns></returns>
    public virtual Complex ComplexCast(TScalar a) => IsEqualApprox(a, AdditionUnit) ? 0 : 1;
    

    /// <summary>
    /// K x R -> K<br/>
    /// ignore if not applicable
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <returns></returns>
    public virtual TScalar RealMul(TScalar a, double b) => a;

    /// <summary>
    /// for y in field, find x such that x*x = y
    /// </summary>
    /// <param name="a"></param>
    /// <returns></returns>
    public abstract TScalar SquareRoot(TScalar a);

    /// <summary>
    /// Absolute value
    /// </summary>
    /// <param name="a"></param>
    /// <returns></returns>
    public TScalar Abs(TScalar a) => SquareRoot(Multiplication(a, Conj(a)));
    /// <summary>
    /// a x^2 + b x + c = unit(add)
    /// </summary>
    /// <param name="a"></param>
    /// <param name="b"></param>
    /// <param name="c"></param>
    /// <returns></returns>
    public QuadraticRoots<TScalar> QuadraticRoot(TScalar a, TScalar b, TScalar c)
    {
        TScalar invB = AdditionInverse(b);
        TScalar b2 = Multiplication(b, b);
        TScalar ax2 = Addition(a, a);
        TScalar ac = Multiplication(a, c);
        TScalar acx2 = Addition(ac, ac);
        TScalar acx4 = Addition(acx2, acx2);
        TScalar x1 = Division(Addition(invB, SquareRoot(Subtraction(b2, acx4))), ax2);
        TScalar x2 = Division(Subtraction(invB, SquareRoot(Subtraction(b2, acx4))), ax2);
        return new QuadraticRoots<TScalar>(x1, x2);
    }

    internal TScalar WilkinsonShift(TScalar a1, TScalar b1, TScalar b2, TScalar a2)
    {
        QuadraticRoots<TScalar> es = QuadraticRoot(
            AdditionUnit,
            AdditionInverse(Addition(a1, a2)),
            Subtraction(Multiplication(a1, a2), Multiplication(b1, b2))
        );
        TScalar k = a2;
        TScalar s = Addition(Addition(Abs(a1), Abs(b1)), Addition(Abs(b2), Abs(a2)));
        if (IsEqualApprox(s, AdditionUnit))
            return k;
        TScalar q = Multiplication(Division(b1, s), Division(b2, s));

        if (IsEqualApprox(q, AdditionUnit))
            return k;
        TScalar p = Division(
            Subtraction(Division(a1, s), Division(a2, s)),
            Addition(MultiplicationUnit, MultiplicationUnit)
        );
        TScalar r = SquareRoot(Addition(Multiplication(p, p), q));
        Complex cp = ComplexCast(p);
        Complex cr = ComplexCast(r);
        if (cp.Real * cr.Real + cp.Imaginary * cr.Imaginary < 0)
            r = AdditionInverse(r);
        k = Subtraction(k, Multiplication(s, Division(q, Addition(p, r))));
        return k;
    }

    /// <summary>
    /// homomorphism from complex to this field
    /// </summary>
    /// <param name="a"></param>
    /// <returns></returns>
    public virtual TScalar FromComplex(Complex a) => AdditionUnit;


    /// <summary>
    /// log of the element in the field
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual TScalar Log(TScalar x) => AdditionUnit;

    /// <summary>
    /// exp of the element in the field
    /// </summary>
    /// <param name="x"></param>
    /// <returns></returns>
    public virtual TScalar Exp(TScalar x) => MultiplicationUnit;
}