原文:【原创】开源Math.NET基础数学类库使用(09)相关数论函数使用
本博客所有文章分类的总目录：http://www.cnblogs.com/asxinyu/p/4288836.html
开源Math.NET基础数学类库使用总目录：http://www.cnblogs.com/asxinyu/p/4329737.html
前言
数论就是指研究整数性质的一门理论。数论=算术。不过通常算术指数的计算，数论指数的理论。整数的基本元素是素数，所以数论的本质是对素数性质的研究。它是与平面几何同样历史悠久的学科。它大致包括代数数论、解析数论、计算数论等等。
Math.NET也包括了很多数论相关的函数，这些函数都是静态的，可以直接调用，如判断是否奇数，判断幂，平方数，最大公约数等等。同时部分函数已经作为扩展方法，可以直接在对象中使用。
如果本文资源或者显示有问题，请参考 本文原文地址：http://www.cnblogs.com/asxinyu/p/4301097.html
1.数论函数类Euclid
Math.NET包括的数论函数除了一部分大家日常接触到的，奇数，偶数，幂，平方数，最大公约数，最小公倍数等函数，还有一些欧几里得几何的函数，当然也是比较简单的，这些问题的算法有一些比较简单，如最大公约数，最小公倍数等，都有成熟的算法可以使用，对于使用Math.NET的人来说，不必要了解太小，当然对于需要搞清楚原理的人来说，学习Math.NET的架构或者实现，是可以参考的。所以这里先给出Math.NET关于数论函数类Euclid的源代码：
/// <summary>
/// 整数数论函数
/// Integer number theory functions.
/// </summary>
public static class Euclid
{
/// <summary>
/// Canonical Modulus. The result has the sign of the divisor.
/// </summary>
public static double Modulus(double dividend, double divisor)
{
return ((dividend%divisor) + divisor)%divisor;
} /// <summary>
/// Canonical Modulus. The result has the sign of the divisor.
/// </summary>
public static float Modulus(float dividend, float divisor)
{
return ((dividend%divisor) + divisor)%divisor;
} /// <summary>
/// Canonical Modulus. The result has the sign of the divisor.
/// </summary>
public static int Modulus(int dividend, int divisor)
{
return ((dividend%divisor) + divisor)%divisor;
} /// <summary>
/// Canonical Modulus. The result has the sign of the divisor.
/// </summary>
public static long Modulus(long dividend, long divisor)
{
return ((dividend%divisor) + divisor)%divisor;
} #if !NOSYSNUMERICS
/// <summary>
/// Canonical Modulus. The result has the sign of the divisor.
/// </summary>
public static BigInteger Modulus(BigInteger dividend, BigInteger divisor)
{
return ((dividend%divisor) + divisor)%divisor;
}
#endif /// <summary>
/// Remainder (% operator). The result has the sign of the dividend.
/// </summary>
public static double Remainder(double dividend, double divisor)
{
return dividend%divisor;
} /// <summary>
/// Remainder (% operator). The result has the sign of the dividend.
/// </summary>
public static float Remainder(float dividend, float divisor)
{
return dividend%divisor;
} /// <summary>
/// Remainder (% operator). The result has the sign of the dividend.
/// </summary>
public static int Remainder(int dividend, int divisor)
{
return dividend%divisor;
} /// <summary>
/// Remainder (% operator). The result has the sign of the dividend.
/// </summary>
public static long Remainder(long dividend, long divisor)
{
return dividend%divisor;
} #if !NOSYSNUMERICS
/// <summary>
/// Remainder (% operator). The result has the sign of the dividend.
/// </summary>
public static BigInteger Remainder(BigInteger dividend, BigInteger divisor)
{
return dividend%divisor;
}
#endif /// <summary>
/// Find out whether the provided 32 bit integer is an even number.
/// </summary>
/// <param name="number">The number to very whether it's even.</param>
/// <returns>True if and only if it is an even number.</returns>
public static bool IsEven(this int number)
{
return (number & 0x1) == 0x0;
} /// <summary>
/// Find out whether the provided 64 bit integer is an even number.
/// </summary>
/// <param name="number">The number to very whether it's even.</param>
/// <returns>True if and only if it is an even number.</returns>
public static bool IsEven(this long number)
{
return (number & 0x1) == 0x0;
} /// <summary>
/// Find out whether the provided 32 bit integer is an odd number.
/// </summary>
/// <param name="number">The number to very whether it's odd.</param>
/// <returns>True if and only if it is an odd number.</returns>
public static bool IsOdd(this int number)
{
return (number & 0x1) == 0x1;
} /// <summary>
/// Find out whether the provided 64 bit integer is an odd number.
/// </summary>
/// <param name="number">The number to very whether it's odd.</param>
/// <returns>True if and only if it is an odd number.</returns>
public static bool IsOdd(this long number)
{
return (number & 0x1) == 0x1;
} /// <summary>
/// Find out whether the provided 32 bit integer is a perfect power of two.
/// </summary>
/// <param name="number">The number to very whether it's a power of two.</param>
/// <returns>True if and only if it is a power of two.</returns>
public static bool IsPowerOfTwo(this int number)
{
return number > && (number & (number - )) == 0x0;
} /// <summary>
/// Find out whether the provided 64 bit integer is a perfect power of two.
/// </summary>
/// <param name="number">The number to very whether it's a power of two.</param>
/// <returns>True if and only if it is a power of two.</returns>
public static bool IsPowerOfTwo(this long number)
{
return number > && (number & (number - )) == 0x0;
} /// <summary>
/// Find out whether the provided 32 bit integer is a perfect square, i.e. a square of an integer.
/// </summary>
/// <param name="number">The number to very whether it's a perfect square.</param>
/// <returns>True if and only if it is a perfect square.</returns>
public static bool IsPerfectSquare(this int number)
{
if (number < )
{
return false;
} int lastHexDigit = number & 0xF;
if (lastHexDigit > )
{
return false; // return immediately in 6 cases out of 16.
} if (lastHexDigit == || lastHexDigit == || lastHexDigit == || lastHexDigit == )
{
int t = (int)Math.Floor(Math.Sqrt(number) + 0.5);
return (t * t) == number;
} return false;
} /// <summary>
/// Find out whether the provided 64 bit integer is a perfect square, i.e. a square of an integer.
/// </summary>
/// <param name="number">The number to very whether it's a perfect square.</param>
/// <returns>True if and only if it is a perfect square.</returns>
public static bool IsPerfectSquare(this long number)
{
if (number < )
{
return false;
} int lastHexDigit = (int)(number & 0xF);
if (lastHexDigit > )
{
return false; // return immediately in 6 cases out of 16.
} if (lastHexDigit == || lastHexDigit == || lastHexDigit == || lastHexDigit == )
{
long t = (long)Math.Floor(Math.Sqrt(number) + 0.5);
return (t * t) == number;
} return false;
} /// <summary>
/// Raises 2 to the provided integer exponent (0 <= exponent < 31).
/// </summary>
/// <param name="exponent">The exponent to raise 2 up to.</param>
/// <returns>2 ^ exponent.</returns>
/// <exception cref="ArgumentOutOfRangeException"/>
public static int PowerOfTwo(this int exponent)
{
if (exponent < || exponent >= )
{
throw new ArgumentOutOfRangeException("exponent");
} return << exponent;
} /// <summary>
/// Raises 2 to the provided integer exponent (0 <= exponent < 63).
/// </summary>
/// <param name="exponent">The exponent to raise 2 up to.</param>
/// <returns>2 ^ exponent.</returns>
/// <exception cref="ArgumentOutOfRangeException"/>
public static long PowerOfTwo(this long exponent)
{
if (exponent < || exponent >= )
{
throw new ArgumentOutOfRangeException("exponent");
} return ((long)) << (int)exponent;
} /// <summary>
/// Find the closest perfect power of two that is larger or equal to the provided
/// 32 bit integer.
/// </summary>
/// <param name="number">The number of which to find the closest upper power of two.</param>
/// <returns>A power of two.</returns>
/// <exception cref="ArgumentOutOfRangeException"/>
public static int CeilingToPowerOfTwo(this int number)
{
if (number == Int32.MinValue)
{
return ;
} const int maxPowerOfTwo = 0x40000000;
if (number > maxPowerOfTwo)
{
throw new ArgumentOutOfRangeException("number");
} number--;
number |= number >> ;
number |= number >> ;
number |= number >> ;
number |= number >> ;
number |= number >> ;
return number + ;
} /// <summary>
/// Find the closest perfect power of two that is larger or equal to the provided
/// 64 bit integer.
/// </summary>
/// <param name="number">The number of which to find the closest upper power of two.</param>
/// <returns>A power of two.</returns>
/// <exception cref="ArgumentOutOfRangeException"/>
public static long CeilingToPowerOfTwo(this long number)
{
if (number == Int64.MinValue)
{
return ;
} const long maxPowerOfTwo = 0x4000000000000000;
if (number > maxPowerOfTwo)
{
throw new ArgumentOutOfRangeException("number");
} number--;
number |= number >> ;
number |= number >> ;
number |= number >> ;
number |= number >> ;
number |= number >> ;
number |= number >> ;
return number + ;
} /// <summary>
/// Returns the greatest common divisor (<c>gcd</c>) of two integers using Euclid's algorithm.
/// </summary>
/// <param name="a">First Integer: a.</param>
/// <param name="b">Second Integer: b.</param>
/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
public static long GreatestCommonDivisor(long a, long b)
{
while (b != )
{
var remainder = a%b;
a = b;
b = remainder;
} return Math.Abs(a);
} /// <summary>
/// Returns the greatest common divisor (<c>gcd</c>) of a set of integers using Euclid's
/// algorithm.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
public static long GreatestCommonDivisor(IList<long> integers)
{
if (null == integers)
{
throw new ArgumentNullException("integers");
} if (integers.Count == )
{
return ;
} var gcd = Math.Abs(integers[]); for (var i = ; (i < integers.Count) && (gcd > ); i++)
{
gcd = GreatestCommonDivisor(gcd, integers[i]);
} return gcd;
} /// <summary>
/// Returns the greatest common divisor (<c>gcd</c>) of a set of integers using Euclid's algorithm.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
public static long GreatestCommonDivisor(params long[] integers)
{
return GreatestCommonDivisor((IList<long>)integers);
} /// <summary>
/// Computes the extended greatest common divisor, such that a*x + b*y = <c>gcd</c>(a,b).
/// </summary>
/// <param name="a">First Integer: a.</param>
/// <param name="b">Second Integer: b.</param>
/// <param name="x">Resulting x, such that a*x + b*y = <c>gcd</c>(a,b).</param>
/// <param name="y">Resulting y, such that a*x + b*y = <c>gcd</c>(a,b)</param>
/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
/// <example>
/// <code>
/// long x,y,d;
/// d = Fn.GreatestCommonDivisor(45,18,out x, out y);
/// -> d == 9 && x == 1 && y == -2
/// </code>
/// The <c>gcd</c> of 45 and 18 is 9: 18 = 2*9, 45 = 5*9. 9 = 1*45 -2*18, therefore x=1 and y=-2.
/// </example>
public static long ExtendedGreatestCommonDivisor(long a, long b, out long x, out long y)
{
long mp = , np = , m = , n = ; while (b != )
{
long rem;
#if PORTABLE
rem = a % b;
var quot = a / b;
#else
long quot = Math.DivRem(a, b, out rem);
#endif
a = b;
b = rem; var tmp = m;
m = mp - (quot*m);
mp = tmp; tmp = n;
n = np - (quot*n);
np = tmp;
} if (a >= )
{
x = mp;
y = np;
return a;
} x = -mp;
y = -np;
return -a;
} /// <summary>
/// Returns the least common multiple (<c>lcm</c>) of two integers using Euclid's algorithm.
/// </summary>
/// <param name="a">First Integer: a.</param>
/// <param name="b">Second Integer: b.</param>
/// <returns>Least common multiple <c>lcm</c>(a,b)</returns>
public static long LeastCommonMultiple(long a, long b)
{
if ((a == ) || (b == ))
{
return ;
} return Math.Abs((a/GreatestCommonDivisor(a, b))*b);
} /// <summary>
/// Returns the least common multiple (<c>lcm</c>) of a set of integers using Euclid's algorithm.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
public static long LeastCommonMultiple(IList<long> integers)
{
if (null == integers)
{
throw new ArgumentNullException("integers");
} if (integers.Count == )
{
return ;
} var lcm = Math.Abs(integers[]); for (var i = ; i < integers.Count; i++)
{
lcm = LeastCommonMultiple(lcm, integers[i]);
} return lcm;
} /// <summary>
/// Returns the least common multiple (<c>lcm</c>) of a set of integers using Euclid's algorithm.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
public static long LeastCommonMultiple(params long[] integers)
{
return LeastCommonMultiple((IList<long>)integers);
} #if !NOSYSNUMERICS
/// <summary>
/// Returns the greatest common divisor (<c>gcd</c>) of two big integers.
/// </summary>
/// <param name="a">First Integer: a.</param>
/// <param name="b">Second Integer: b.</param>
/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
public static BigInteger GreatestCommonDivisor(BigInteger a, BigInteger b)
{
return BigInteger.GreatestCommonDivisor(a, b);
} /// <summary>
/// Returns the greatest common divisor (<c>gcd</c>) of a set of big integers.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
public static BigInteger GreatestCommonDivisor(IList<BigInteger> integers)
{
if (null == integers)
{
throw new ArgumentNullException("integers");
} if (integers.Count == )
{
return ;
} var gcd = BigInteger.Abs(integers[]); for (int i = ; (i < integers.Count) && (gcd > BigInteger.One); i++)
{
gcd = GreatestCommonDivisor(gcd, integers[i]);
} return gcd;
} /// <summary>
/// Returns the greatest common divisor (<c>gcd</c>) of a set of big integers.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Greatest common divisor <c>gcd</c>(list of integers)</returns>
public static BigInteger GreatestCommonDivisor(params BigInteger[] integers)
{
return GreatestCommonDivisor((IList<BigInteger>)integers);
} /// <summary>
/// Computes the extended greatest common divisor, such that a*x + b*y = <c>gcd</c>(a,b).
/// </summary>
/// <param name="a">First Integer: a.</param>
/// <param name="b">Second Integer: b.</param>
/// <param name="x">Resulting x, such that a*x + b*y = <c>gcd</c>(a,b).</param>
/// <param name="y">Resulting y, such that a*x + b*y = <c>gcd</c>(a,b)</param>
/// <returns>Greatest common divisor <c>gcd</c>(a,b)</returns>
/// <example>
/// <code>
/// long x,y,d;
/// d = Fn.GreatestCommonDivisor(45,18,out x, out y);
/// -> d == 9 && x == 1 && y == -2
/// </code>
/// The <c>gcd</c> of 45 and 18 is 9: 18 = 2*9, 45 = 5*9. 9 = 1*45 -2*18, therefore x=1 and y=-2.
/// </example>
public static BigInteger ExtendedGreatestCommonDivisor(BigInteger a, BigInteger b, out BigInteger x, out BigInteger y)
{
BigInteger mp = BigInteger.One, np = BigInteger.Zero, m = BigInteger.Zero, n = BigInteger.One; while (!b.IsZero)
{
BigInteger rem;
BigInteger quot = BigInteger.DivRem(a, b, out rem);
a = b;
b = rem; BigInteger tmp = m;
m = mp - (quot*m);
mp = tmp; tmp = n;
n = np - (quot*n);
np = tmp;
} if (a >= BigInteger.Zero)
{
x = mp;
y = np;
return a;
} x = -mp;
y = -np;
return -a;
} /// <summary>
/// Returns the least common multiple (<c>lcm</c>) of two big integers.
/// </summary>
/// <param name="a">First Integer: a.</param>
/// <param name="b">Second Integer: b.</param>
/// <returns>Least common multiple <c>lcm</c>(a,b)</returns>
public static BigInteger LeastCommonMultiple(BigInteger a, BigInteger b)
{
if (a.IsZero || b.IsZero)
{
return BigInteger.Zero;
} return BigInteger.Abs((a/BigInteger.GreatestCommonDivisor(a, b))*b);
} /// <summary>
/// Returns the least common multiple (<c>lcm</c>) of a set of big integers.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
public static BigInteger LeastCommonMultiple(IList<BigInteger> integers)
{
if (null == integers)
{
throw new ArgumentNullException("integers");
} if (integers.Count == )
{
return ;
} var lcm = BigInteger.Abs(integers[]); for (int i = ; i < integers.Count; i++)
{
lcm = LeastCommonMultiple(lcm, integers[i]);
} return lcm;
} /// <summary>
/// Returns the least common multiple (<c>lcm</c>) of a set of big integers.
/// </summary>
/// <param name="integers">List of Integers.</param>
/// <returns>Least common multiple <c>lcm</c>(list of integers)</returns>
public static BigInteger LeastCommonMultiple(params BigInteger[] integers)
{
return LeastCommonMultiple((IList<BigInteger>)integers);
}
#endif
}
2.Euclid类的使用例子
上面已经看到源码，也提到了，Euclid作为静态类，其中的很多静态方法都可以直接作为扩展方法使用。这里看看几个简单的例子：
// 1. Find out whether the provided number is an even number
Console.WriteLine(@"1.判断提供的数字是否是偶数");
Console.WriteLine(@"{0} 是偶数 = {1}. {2} 是偶数 = {3}", , Euclid.IsEven(), , .IsEven());
Console.WriteLine(); // 2. Find out whether the provided number is an odd number
Console.WriteLine(@"2.判断提供的数字是否是奇数");
Console.WriteLine(@"{0} 是奇数 = {1}. {2} 是奇数 = {3}", , .IsOdd(), , Euclid.IsOdd());
Console.WriteLine(); // 3. Find out whether the provided number is a perfect power of two
Console.WriteLine(@"2.判断提供的数字是否是2的幂");
Console.WriteLine(@"{0} 是2的幂 = {1}. {2} 是2的幂 = {3}", , .IsPowerOfTwo(), , Euclid.IsPowerOfTwo());
Console.WriteLine(); // 4. Find the closest perfect power of two that is larger or equal to 97
Console.WriteLine(@"4.返回大于等于97的最小2的幂整数");
Console.WriteLine(.CeilingToPowerOfTwo());
Console.WriteLine(); // 5. Raise 2 to the 16
Console.WriteLine(@"5. 2的16次幂");
Console.WriteLine(.PowerOfTwo());
Console.WriteLine(); // 6. Find out whether the number is a perfect square
Console.WriteLine(@"6. 判断提供的数字是否是平方数");
Console.WriteLine(@"{0} 是平方数 = {1}. {2} 是平方数 = {3}", , .IsPerfectSquare(), , Euclid.IsPerfectSquare());
Console.WriteLine(); // 7. Compute the greatest common divisor of 32 and 36
Console.WriteLine(@"7. 返回32和36的最大公约数");
Console.WriteLine(Euclid.GreatestCommonDivisor(, ));
Console.WriteLine(); // 8. Compute the greatest common divisor of 492, -984, 123, 246
Console.WriteLine(@"8. 返回的最大公约数：492、-984、123、246");
Console.WriteLine(Euclid.GreatestCommonDivisor(, -, , ));
Console.WriteLine(); // 9. Compute the least common multiple of 16 and 12
Console.WriteLine(@"10. 计算的最小公倍数：16和12");
Console.WriteLine(Euclid.LeastCommonMultiple(, ));
Console.WriteLine();
结果如下：
.判断提供的数字是否是偶数
是偶数 = False. 是偶数 = True .判断提供的数字是否是奇数
是奇数 = True. 是奇数 = False .判断提供的数字是否是2的幂
是2的幂 = False. 是2的幂 = True .返回大于等于97的最小2的幂整数 . 2的16次幂 . 判断提供的数字是否是平方数
是平方数 = False. 是平方数 = True . 返回32和36的最大公约数 . 返回的最大公约数：、-、、 . 计算的最小公倍数：16和12
3.资源
源码下载：http://www.cnblogs.com/asxinyu/p/4264638.html
如果本文资源或者显示有问题，请参考 本文原文地址：http://www.cnblogs.com/asxinyu/p/4301097.html