Creating a Math library using Generics in C#

Question

Is there any feasible way of using generics to create a Math library that does not depend on the base type chosen to store data?

In other words, let's assume I want to write a Fraction class. The fraction can be represented by two ints or two doubles or whatnot. The important thing is that the basic four arithmetic operations are well defined. So, I would like to be able to write Fraction<int> frac = new Fraction<int>(1,2) and/or Fraction<double> frac = new Fraction<double>(0.1, 1.0).

Unfortunately there is no interface representing the four basic operations (+,-,*,/). Has anybody found a workable, feasible way of implementing this?

Solution 1

Here is a way to abstract out the operators that is relatively painless.

    abstract class MathProvider<T>
    {
        public abstract T Divide(T a, T b);
        public abstract T Multiply(T a, T b);
        public abstract T Add(T a, T b);
        public abstract T Negate(T a);
        public virtual T Subtract(T a, T b)
        {
            return Add(a, Negate(b));
        }
    }

    class DoubleMathProvider : MathProvider<double>
    {
        public override double Divide(double a, double b)
        {
            return a / b;
        }

        public override double Multiply(double a, double b)
        {
            return a * b;
        }

        public override double Add(double a, double b)
        {
            return a + b;
        }

        public override double Negate(double a)
        {
            return -a;
        }
    }

    class IntMathProvider : MathProvider<int>
    {
        public override int Divide(int a, int b)
        {
            return a / b;
        }

        public override int Multiply(int a, int b)
        {
            return a * b;
        }

        public override int Add(int a, int b)
        {
            return a + b;
        }

        public override int Negate(int a)
        {
            return -a;
        }
    }

    class Fraction<T>
    {
        static MathProvider<T> _math;
        // Notice this is a type constructor.  It gets run the first time a
        // variable of a specific type is declared for use.
        // Having _math static reduces overhead.
        static Fraction()
        {
            // This part of the code might be cleaner by once
            // using reflection and finding all the implementors of
            // MathProvider and assigning the instance by the one that
            // matches T.
            if (typeof(T) == typeof(double))
                _math = new DoubleMathProvider() as MathProvider<T>;
            else if (typeof(T) == typeof(int))
                _math = new IntMathProvider() as MathProvider<T>;
            // ... assign other options here.

            if (_math == null)
                throw new InvalidOperationException(
                    "Type " + typeof(T).ToString() + " is not supported by Fraction.");
        }

        // Immutable impementations are better.
        public T Numerator { get; private set; }
        public T Denominator { get; private set; }

        public Fraction(T numerator, T denominator)
        {
            // We would want this to be reduced to simpilest terms.
            // For that we would need GCD, abs, and remainder operations
            // defined for each math provider.
            Numerator = numerator;
            Denominator = denominator;
        }

        public static Fraction<T> operator +(Fraction<T> a, Fraction<T> b)
        {
            return new Fraction<T>(
                _math.Add(
                  _math.Multiply(a.Numerator, b.Denominator),
                  _math.Multiply(b.Numerator, a.Denominator)),
                _math.Multiply(a.Denominator, b.Denominator));
        }

        public static Fraction<T> operator -(Fraction<T> a, Fraction<T> b)
        {
            return new Fraction<T>(
                _math.Subtract(
                  _math.Multiply(a.Numerator, b.Denominator),
                  _math.Multiply(b.Numerator, a.Denominator)),
                _math.Multiply(a.Denominator, b.Denominator));
        }

        public static Fraction<T> operator /(Fraction<T> a, Fraction<T> b)
        {
            return new Fraction<T>(
                _math.Multiply(a.Numerator, b.Denominator),
                _math.Multiply(a.Denominator, b.Numerator));
        }

        // ... other operators would follow.
    }

If you fail to implement a type that you use, you will get a failure at runtime instead of at compile time (that is bad). The definition of the MathProvider<T> implementations is always going to be the same (also bad). I would suggest that you just avoid doing this in C# and use F# or some other language better suited to this level of abstraction.

Edit: Fixed definitions of add and subtract for Fraction<T>. Another interesting and simple thing to do is implement a MathProvider that operates on an abstract syntax tree. This idea immediately points to doing things like automatic differentiation: http://conal.net/papers/beautiful-differentiation/

Solution 2

Here's a subtle problem that comes with generic types. Suppose an algorithm involves division, say Gaussian elimination to solve a system of equations. If you pass in integers, you'll get a wrong answer because you'll carry out integer division. But if you pass in double arguments that happen have integer values, you'll get the right answer.

The same thing happens with square roots, as in Cholesky factorization. Factoring an integer matrix will go wrong, whereas factoring a matrix of doubles that happen to have integer values will be fine.