/// <summary>
/// <para>
/// Returns the least common multiple of the absolute value of two numbers,
/// using the formula <c>lcm(a,b) = (a / gcd(a,b)) * b</c>.
/// </para>
/// Special cases:
/// <list type="bullet">
/// <item>The invocations <c>lcm(Int64.MinValue, n)</c> and
/// <c>lcm(n, Int64.MinValue)</c>, where <c>abs(n)</c> is a
/// power of 2, throw an <c>ArithmeticException</c>, because the result
/// would be 2^63, which is too large for an int value.</item>
/// <item>The result of <c>lcm(0L, x)</c> and <c>lcm(x, 0L)</c> is
/// <c>0L</c> for any <c>x</c>.</item>
/// </list>
/// </summary>
/// <param name="a">Number.</param>
/// <param name="b">Number.</param>
/// <returns>the least common multiple, never negative.</returns>
/// <exception cref="MathArithmeticException"> if the result cannot be represented
/// as a non-negative <c>long</c> value.</exception>
public static long lcm(long a, long b)
{
if (a == 0 || b == 0)
{
return(0);
}
long lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));
if (lcm == Int64.MinValue)
{
throw new MathArithmeticException(new LocalizedFormats("LCM_OVERFLOW_64_BITS"), a, b);
}
return(lcm);
}
/// <summary>
/// <para>
/// Returns the least common multiple of the absolute value of two numbers,
/// using the formula <c>lcm(a,b) = (a / gcd(a,b)) * b</c>.
/// </para>
/// Special cases:
/// <list type="bullet">
/// <item>The invocations <c>lcm(Int32.MinValue, n)</c> and
/// <c>lcm(n, Int32.MinValue)</c>, where <c>abs(n)</c> is a
/// power of 2, throw an <c>ArithmeticException</c>, because the result
/// would be 2^31, which is too large for an int value.</item>
/// <item>The result of <c>lcm(0, x)</c> and <c>lcm(x, 0)</c> is
/// <c>0</c> for any <c>x</c>.</item>
/// </list>
/// </summary>
/// <param name="a">Number.</param>
/// <param name="b">Number.</param>
/// <returns>the least common multiple, never negative.</returns>
/// <exception cref="MathArithmeticException"> if the result cannot be represented as
/// a non-negative <c>int</c> value.</exception>
public static int lcm(int a, int b)
{
if (a == 0 || b == 0)
{
return(0);
}
int lcm = FastMath.abs(ArithmeticUtils.mulAndCheck(a / gcd(a, b), b));
if (lcm == Int32.MinValue)
{
throw new MathArithmeticException(new LocalizedFormats("LCM_OVERFLOW_32_BITS"), a, b);
}
return(lcm);
}
/// <summary>
/// Returns an exact representation of the <a
/// href="http://mathworld.wolfram.com/BinomialCoefficient.html"> Binomial
/// Coefficient</a>, "<c>n choose k</c>", the number of
/// <c>k</c>-element subsets that can be selected from an
/// <c>n</c>-element set.
/// <para>
/// Preconditions:
/// <list type="bullet">
/// <item> <c>0 <= k <= n</c> (otherwise
/// <c>MathIllegalArgumentException</c> is thrown)</item>
/// <item> The result is small enough to fit into a <c>long</c>. The
/// largest value of <c>n</c> for which all coefficients are
/// <c> < Int64.MaxValue</c> is 66. If the computed value exceeds
/// <c>Int64.MaxValue</c> an <c>ArithMeticException</c> is
/// thrown.</item>
/// </list></para>
/// </summary>
/// <param name="n">the size of the set</param>
/// <param name="k">the size of the subsets to be counted</param>
/// <returns><c>n choose k</c></returns>
/// <exception cref="NotPositiveException"> if <c>n < 0</c>.</exception>
/// <exception cref="NumberIsTooLargeException"> if <c>k > n</c>.</exception>
/// <exception cref="MathArithmeticException"> if the result is too large to be
/// represented by a long integer.</exception>
public static long binomialCoefficient(int n, int k)
{
CombinatoricsUtils.checkBinomial(n, k);
if ((n == k) || (k == 0))
{
return(1);
}
if ((k == 1) || (k == n - 1))
{
return(n);
}
// Use symmetry for large k
if (k > n / 2)
{
return(binomialCoefficient(n, n - k));
}
// We use the formula
// (n choose k) = n! / (n-k)! / k!
// (n choose k) == ((n-k+1)*...*n) / (1*...*k)
// which could be written
// (n choose k) == (n-1 choose k-1) * n / k
long result = 1;
if (n <= 61)
{
// For n <= 61, the naive implementation cannot overflow.
int i = n - k + 1;
for (int j = 1; j <= k; j++)
{
result = result * i / j;
i++;
}
}
else if (n <= 66)
{
// For n > 61 but n <= 66, the result cannot overflow,
// but we must take care not to overflow intermediate values.
int i = n - k + 1;
for (int j = 1; j <= k; j++)
{
// We know that (result * i) is divisible by j,
// but (result * i) may overflow, so we split j:
// Filter out the gcd, d, so j/d and i/d are integer.
// result is divisible by (j/d) because (j/d)
// is relative prime to (i/d) and is a divisor of
// result * (i/d).
long d = ArithmeticUtils.gcd(i, j);
result = (result / (j / d)) * (i / d);
i++;
}
}
else
{
// For n > 66, a result overflow might occur, so we check
// the multiplication, taking care to not overflow
// unnecessary.
int i = n - k + 1;
for (int j = 1; j <= k; j++)
{
long d = ArithmeticUtils.gcd(i, j);
result = ArithmeticUtils.mulAndCheck(result / (j / d), i / d);
i++;
}
}
return(result);
}
/// <summary>
/// Returns the <a
/// href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">
/// Stirling number of the second kind</a>, "<c>S(n,k)</c>", the number of
/// ways of partitioning an <c>n</c>-element set into <c>k</c> non-empty
/// subsets.
/// <para>
/// The preconditions are <c>0 <= k <= n</c> (otherwise
/// <c>NotPositiveException</c> is thrown)
/// </para>
/// </summary>
/// <param name="n">the size of the set</param>
/// <param name="k">the number of non-empty subsets</param>
/// <returns><c>S(n,k)</c></returns>
/// <exception cref="NotPositiveException"> if <c>k < 0</c>.</exception>
/// <exception cref="NumberIsTooLargeException"> if <c>k > n</c>.</exception>
/// <exception cref="MathArithmeticException"> if some overflow happens, typically
/// for n exceeding 25 and k between 20 and n-2 (S(n,n-1) is handled specifically
/// and does not overflow)</exception>
public static long stirlingS2(int n, int k)
{
if (k < 0)
{
throw new NotPositiveException <Int32>(k);
}
if (k > n)
{
throw new NumberIsTooLargeException <Int32, Int32>(k, n, true);
}
long[][] stirlingS2;
lock (STIRLING_S2)
{
stirlingS2 = STIRLING_S2;
if (stirlingS2 == null)
{
// the cache has never been initialized, compute the first numbers
// by direct recurrence relation
// as S(26,9) = 11201516780955125625 is larger than Long.MAX_VALUE
// we must stop computation at row 26
int maxIndex = 26;
stirlingS2 = new long[maxIndex][];
stirlingS2[0] = new long[] { 1L };
for (int i = 1; i < stirlingS2.Length; ++i)
{
stirlingS2[i] = new long[i + 1];
stirlingS2[i][0] = 0;
stirlingS2[i][1] = 1;
stirlingS2[i][i] = 1;
for (int j = 2; j < i; ++j)
{
STIRLING_S2[i][j] = j * stirlingS2[i - 1][j] + stirlingS2[i - 1][j - 1];
}
}
// atomically save the cache, thread-safe
STIRLING_S2 = (long[][])stirlingS2.Clone();
}
}
if (n < stirlingS2.Length)
{
// the number is in the small cache
return(stirlingS2[n][k]);
}
else
{
// use explicit formula to compute the number without caching it
if (k == 0)
{
return(0);
}
else if (k == 1 || k == n)
{
return(1);
}
else if (k == 2)
{
return((1L << (n - 1)) - 1L);
}
else if (k == n - 1)
{
return(binomialCoefficient(n, 2));
}
else
{
// definition formula: note that this may trigger some overflow
long sum = 0;
long sign = ((k & 0x1) == 0) ? 1 : -1;
for (int j = 1; j <= k; ++j)
{
sign = -sign;
sum += sign * binomialCoefficient(k, j) * ArithmeticUtils.pow(j, n);
if (sum < 0)
{
// there was an overflow somewhere
throw new MathArithmeticException(new LocalizedFormats("ARGUMENT_OUTSIDE_DOMAIN"), n, 0, stirlingS2.Length - 1);
}
}
return(sum / factorial(k));
}
}
}
