Using generics for calculations

// Copyright (c) 2004, R�diger Klaehn
// All rights reserved.
// 
// Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
// 
//    * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
//    * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
//    * Neither the name of lambda computing nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
using System;

namespace Lambda.Generic.Arithmetic
{
    #region Unconstrained Interfaces

    public interface IConversionProvider<T>
    {
        T ConvertFrom(ulong t);
        T ConvertFrom(long t);
        T ConvertFrom(double t);
    }

    public interface IConstant<T>
    {
        T Value { get; }
    }

    public interface IAdder<T>
    {
        T Add(T a, T b);
    }

    public interface ISubtracter<T>
    {
        T Subtract(T a, T b);
    }

    public interface IZeroProvider<T>
    {
        T Zero { get; }
    }

    public interface INegater<T>
    {
        T Negate(T a);
    }

    public interface IMultiplier<T>
    {
        T Multiply(T a, T b);
    }

    public interface IDivider<T>
    {
        T Divide(T a, T b);
    }

    public interface IOneProvider<T>
    {
        T One { get; }
    }

    public interface IInverter<T>
    {
        T Invert(T a);
    }

    public interface IHasRoot<T>
    {
        T Sqrt(T a);
        T Sqr(T a);
    }

    public interface IMinValueProvider<T>
    {
        T MinValue { get; }
    }

    public interface IMaxValueProvider<T>
    {
        T MaxValue { get; }
    }

    public interface IEpsilonProvider<T>
    {
        T Epsilon { get; }
    }

    #endregion
    #region Constrained Interfaces
    public interface IComparer<T>
    {
        int Compare(T a, T b);
        bool Equals(T a, T b);
        int GetHashCode(T a);
        T Abs(T a);
        T Max(T a, T b);
        T Min(T a, T b);
    }

    public interface ILimitProvider<T> :
        IMaxValueProvider<T>,
        IMinValueProvider<T>,
        IEpsilonProvider<T>
    { }

    public interface IBinaryMath<T>
    {
        T And(T a, T b);
        T Or(T a, T b);
        T Xor(T a, T b);
        T Not(T a);
    }

    /// <summary>
    /// This defines a Group with the operation +, the neutral element Zero,
    /// the inverse Negate and an operation - that is defined in terms of the inverse.
    /// 
    /// Axioms that have to be satisified by the operations:
    /// Commutativity of addition: Add(a,b)=Add(b,a) for all a,b in T
    /// Associativity of addition: Add(Add(a,b),c)=Add(a,Add(b,c))
    /// Inverse of addition: Add(a,Negate(a))==Zero
    /// Subtraction: Subtract(a,b)==Add(a,Negate(b))
    /// Neutral element: Add(Zero,a)==a for all a in T
    /// </summary>
    public interface IAdditionGroup<T> :
        IAdder<T>,
        ISubtracter<T>,
        INegater<T>,
        IZeroProvider<T>
    { }

    public interface ITrigonometry<T>
    {
        T Asin(T a);
        T Acos(T a);
        T Atan(T a);
        T Atan2(T a, T b);
        T Sin(T a);
        T Cos(T a);
        T Tan(T a);
    }

    public interface IExp<T>
    {
        T Exp(T a);
        T Log(T a);
    }

    /// <summary>
    /// This defines a Group with the operation *, the neutral element One,
    /// the inverse Inverse and an operation / that is defined in terms of the inverse.
    /// 
    /// Axioms that have to be satisified by the operations:
    /// Commutativity of multiplication: Multiply(a,b)=Multiply(b,a) for all a,b in T
    /// Associativity of multiplication: Multiply(Multiply(a,b),c)=Multiply(a,Multiply(b,c))
    /// Inverse of multiplication: Multiply(a,Inverse(a))==One for all a in T
    /// Divison: Divide(a,b)==Multiply(a,Inverse(b)) for all a in T
    /// Neutral element: Multiply(One,a)==a for all a in T
    /// </summary>
    public interface IMultiplicationGroup<T> :
        IMultiplier<T>,
        IDivider<T>,
        IInverter<T>,
        IOneProvider<T>
    { }

    /// <summary>
    /// This defines a Ring with the operations +,*        
    /// 
    /// Axioms that have to be satisified by the operations:
    /// The group axioms for +
    /// Associativity of *: a * (b*c) = (a*b) * c
    /// Neutral element of *: Multiply(One,a)==a for all a in T
    /// Distributivity: 
    ///     a * (b+c) = (a*b) + (a*c) 
    ///     (a+b) * c = (a*c) + (b*c) 
    /// </summary>
    public interface IRing<T> :
        IAdditionGroup<T>,
        IMultiplier<T>,
        IOneProvider<T>
    { }

    /// <summary>
    /// This defines a Field with the operations +,-,*,/      
    /// /// 
    /// Axioms that have to be satisified by the operations:
    /// The group axioms for +
    /// The group axioms for *
    /// Associativity: a * (b*c) = (a*b) * c
    /// Distributivity: 
    ///     a * (b+c) = (a*b) + (a*c) 
    ///     (a+b) * c = (a*c) + (b*c) 
    /// </summary>
    public interface IField<T> :
        IAdditionGroup<T>,
        IMultiplicationGroup<T>
    { }

    public interface IMath<T> :
        IAdder<T>,
        ISubtracter<T>,
        IMultiplier<T>,
        IDivider<T>,
        IOneProvider<T>,
        IZeroProvider<T>,
        IHasRoot<T>
    {
    }

    /// <summary>
    /// Use this interface for unsigned types such as byte,ushort,uint,ulong.
    /// Unsigned types have no subtraction or inverse. I have added ISubtracter
    /// anyway since it might be useful in many situations.
    /// </summary>
    public interface IUnsignedMath<T> :
        IMath<T>,
        IComparer<T>,
        ILimitProvider<T>,
        IBinaryMath<T>,
        IConversionProvider<T>
    {
    }

    /// <summary>
    /// Use this interface for signed types such as sbyte,short,int,long.
    /// signed types have a division, but no inverse.
    /// Strictly speaking, signed types do not support IHasRoot since e.g. 
    /// Sqrt(Negate(One)) is not element of T.
    /// But it would be very inconvenient to leave it out. If you do not want
    /// to support IHasRoot, just throw a NotImplementedException.
    /// </summary>
    public interface ISignedMath<T> :
        IMath<T>,
        IComparer<T>,
        ILimitProvider<T>,
        IBinaryMath<T>,
        IRing<T>,
        IConversionProvider<T>
    {
    }

    /// <summary>
    /// Use this interface for rational types such as float,double,decimal.
    /// Rational types have an inverse.
    /// Strictly speaking, rational types do not support IHasRoot since e.g. 
    /// Sqrt(Negate(One)) is not element of T.
    /// But it would be very inconvenient to leave it out. If you do not want
    /// to support IHasRoot, just throw a NotImplementedException.
    /// </summary>
    public interface IRationalMath<T> :
        IMath<T>,
        IComparer<T>,
        ILimitProvider<T>,
        IField<T>,
        ITrigonometry<T>,
        IExp<T>,
        IConversionProvider<T>
    {
    }

    /// <summary>
    /// Use this interface for types such as complex numbers that satisfy 
    /// the field axioms but do not have a natural order.
    /// complex numbers of course do support IHasRoot.
    /// </summary>
    public interface IComplexMath<T> :
        IField<T>,
        IHasRoot<T>
    {
        T ConvertFrom(double x);
    }
    #endregion
}