Add Binomal, GCD, and Multinomial methods

src/Mathematics.NET/Mathematics.cs

@@ -25,6 +25,10 @@
 // SOFTWARE.
 // </copyright>
 
+using System.Numerics;
+using System.Runtime.CompilerServices;
+using Complex = Mathematics.NET.Core.Complex;
+
 namespace Mathematics.NET;
 
 /// <summary>Provides Mathematics.NET functionality</summary>
@@ -84,4 +88,89 @@ public static class Mathematics
 
     /// <inheritdoc cref="Constants.ZetaOf4"/>
     public static Real ZetaOf4 => Real.ZetaOf4;
+
+    //
+    // Methods
+    //
+
+    // Integer-related
+
+    /// <summary>The binomial coefficient</summary>
+    /// <remarks>This method works with <c>BigInteger</c>. Be aware that this method may take a long time.</remarks>
+    /// <typeparam name="T">A type that implements <see cref="IBinaryInteger{TSelf}"/></typeparam>
+    /// <param name="n">A total of <paramref name="n"/> elements</param>
+    /// <param name="k">An unordered subset of <paramref name="k"/> elements</param>
+    /// <param name="cancellationToken">A cancellation token</param>
+    /// <returns><paramref name="n"/> choose <paramref name="k"/></returns>
+    /// <exception cref="OverflowException"></exception>
+    public static T Binomial<T>(T n, T k, CancellationToken cancellationToken)
+        where T : IBinaryInteger<T>
+    {
+        var result = T.One;
+        checked
+        {
+            T u = T.One;
+            for (T i = T.Zero; i < k; i++)
+            {
+                cancellationToken.ThrowIfCancellationRequested();
+                result *= n - i;
+                u *= i + T.One;
+
+                var gcd = GCD(result, u);
+                if (gcd != T.One)
+                {
+                    result /= gcd;
+                    u /= gcd;
+                }
+            }
+        }
+        return result;
+    }
+
+    /// <summary>Compute the greatest common divisor of two integers.</summary>
+    /// <typeparam name="T">A type that implements <see cref="IBinaryInteger{TSelf}"/></typeparam>
+    /// <param name="p">An integer</param>
+    /// <param name="q">An integer</param>
+    /// <returns>The GCD of the two values</returns>
+    [MethodImpl(MethodImplOptions.AggressiveInlining)]
+    public static T GCD<T>(T p, T q)
+        where T : IBinaryInteger<T>
+    {
+        p = T.Abs(p);
+        q = T.Abs(q);
+        while (p != T.Zero && q != T.Zero)
+        {
+            if (p > q)
+            {
+                p %= q;
+            }
+            else
+            {
+                q %= p;
+            }
+        }
+        return p | q;
+    }
+
+    /// <summary>The multinomial coefficient</summary>
+    /// <param name="α">An array of positive integers</param>
+    /// <param name="cancellationToken">A cancellation token</param>
+    /// <returns>The coefficient associated with a term with powers given by <paramref name="α"/></returns>
+    /// <exception cref="OverflowException"></exception>
+    public static T Multinomial<T>(ReadOnlySpan<T> α, CancellationToken cancellationToken)
+        where T : IBinaryInteger<T>
+    {
+        var result = T.One;
+        checked
+        {
+            var partialSum = α[0];
+            for (int i = 1; i < α.Length; i++)
+            {
+                var element = α[i];
+                partialSum += element;
+                result *= Binomial(partialSum, element, cancellationToken);
+            }
+        }
+        return result;
+    }
 }