/// POWER INTEGER
///
/// <summary>
/// Performs the quick mathematical power operation.
/// Works only for integer exponent values.
///
/// WARNING! The power value here should be an integer number,
/// that is, the calculator method 'integerPower(power)' should
/// return the same value as power. Otherwise, the result
/// would be unpredictable AND INCORRECT.
/// </summary>
/// <param name="number">The number to raise to the power.</param>
/// <param name="power">The INTEGER exponent of the power.</param>
/// <returns>The number raised to the integer power.</returns>
public static T PowerInteger_Generic(T number, T power)
{
power = calc.intPart(power);
if (power == Numeric <T, C> .Zero)
{
return(calc.fromInt(1));
}
else if (power < Numeric <T, C> .Zero)
{
if (number <= Numeric <T, C> .Zero)
{
throw new ArgumentException("Cannot raise a non-positive number to a negative power.");
}
return(calc.div(calc.fromInt(1), PowerInteger_Generic(number, calc.negate(power))));
}
// Ноль в любой степени будет ноль:
if (calc.eqv(number, calc.zero))
{
return(calc.zero);
}
Numeric <T, C> res = calc.fromInt(1); // результат возведения в степень
Numeric <T, C> copy = calc.getCopy(number); // изменяемая копия (переданное число может быть ссылочным типом)
T two = calc.fromInt(2);
T one = calc.fromInt(1);
while (power > Numeric <T, C> .Zero)
{
// Если остаток от деления на 2 равен 1
if (calc.eqv(WhiteMath <T, C> .Modulus(power, two), one))
{
res *= copy;
}
copy *= copy;
power = calc.intPart(calc.div(power, two));
}
return(res);
}
/// <summary>
/// Нормализует число, делит числитель и знаменатель на НОД,
/// делает числитель положительным.
/// </summary>
private void normalize()
{
T gcd = WhiteMath <T, C> .GreatestCommonDivisor(WhiteMath <T, C> .Abs(num), WhiteMath <T, C> .Abs(denom));
if (calc.eqv(gcd, calc.zero))
{
return; // for infinity-checking cases
}
num = calc.div(num, gcd);
denom = calc.div(denom, gcd);
// Если знаменатель меньше нуля
if (calc.mor(calc.zero, denom))
{
num = calc.negate(num);
denom = calc.negate(denom);
}
}
public static bool operator >(Rational <T, C> one, Rational <T, C> two)
{
if (one is NotANumber || two is NotANumber)
{
return(false);
}
else if (one is Positive_Infinity || two is Negative_Infinity)
{
return(true);
}
else if (two is Positive_Infinity || one is Negative_Infinity)
{
return(false);
}
T denomLcm = WhiteMath <T, C> .LowestCommonMultiple(one.denom, two.denom, WhiteMath <T, C> .GreatestCommonDivisor(one.denom, two.denom));
return(calc.mor(calc.mul(one.num, calc.div(denomLcm, one.denom)), calc.mul(two.num, calc.div(denomLcm, two.denom))));
}
///-----------------------------------
///----ARITHMETIC OPERATORS-----------
///-----------------------------------
public static Rational <T, C> operator +(Rational <T, C> one, Rational <T, C> two)
{
if (one is Infinities)
{
if (two is Infinities)
{
if (one.GetType().Equals(two.GetType()))
{
return(one);
}
else
{
return(NaN);
}
}
return(one);
}
else if (two is Infinities)
{
return(two + one);
}
Rational <T, C> tmp = new Rational <T, C>();
tmp.denom = WhiteMath <T, C> .LowestCommonMultiple(one.denom, two.denom, WhiteMath <T, C> .GreatestCommonDivisor(one.denom, two.denom));
tmp.num = calc.sum(calc.mul(one.num, calc.div(tmp.denom, one.denom)), calc.mul(two.num, calc.div(tmp.denom, two.denom)));
if (checkInf(ref tmp))
{
tmp.normalize();
}
return(tmp);
}
public static Dictionary <T, List <T> > RootsOfUnity(T module, IEnumerable <T> rootDegrees, BoundedInterval <T, C>?searchInterval = null)
{
Contract.Requires <NonIntegerTypeException>(Numeric <T, C> .Calculator.isIntegerCalculator, "This method supports only integral types.");
Contract.Requires <ArgumentNullException>(module != null, "module");
Contract.Requires <ArgumentOutOfRangeException>(module > Numeric <T, C> ._1, "The module should be more than 1.");
Contract.Requires <ArgumentOutOfRangeException>(Contract.ForAll <T>(rootDegrees, (x => x >= Numeric <T, C> ._1 && x < (Numeric <T, C>)module)), "All of the root degrees specified should be located inside the [1; N-1] interval.");
if (!searchInterval.HasValue)
{
searchInterval = new BoundedInterval <T, C>(Numeric <T, C> .Zero, module - Numeric <T, C> ._1, true, true);
}
Numeric <T, C> lowerBound = searchInterval.Value.LeftBound;
Numeric <T, C> upperBound = searchInterval.Value.RightBound;
if (!searchInterval.Value.IsLeftInclusive)
{
lowerBound++;
}
if (!searchInterval.Value.IsRightInclusive)
{
upperBound--;
}
// Contract.Requires<ArgumentOutOfRangeException>(lowerBound > Numeric<T, C>.Zero && upperBound < module - Numeric<T, C>.CONST_1, "The search interval should be located inside the [0; N-1] interval.");
// Нам нужны только уникальные значения
// поэтому сгенерируем множество
ISet <Numeric <T, C> > rootDegreeSet = new HashSet <Numeric <T, C> >();
foreach (T degree in rootDegrees)
{
rootDegreeSet.Add(degree);
}
// -----------------------------
bool evenModule = calc.isEven(module);
// Если нижняя граница четная, и модуль четный - по любому взаимно не простые.
// Значит число - делитель нуля, и не может быть корнем из единицы ни при каких условиях.
// Прибавляем единицу.
if (calc.isEven(lowerBound) && evenModule)
{
lowerBound++;
}
Dictionary <T, List <T> > result = new Dictionary <T, List <T> >();
for (Numeric <T, C> current = lowerBound; current <= upperBound; current++)
{
// Если не взаимно просты с модулем - по-любому не может быть
// примитивным корнем.
if (WhiteMath <T, C> .GreatestCommonDivisor(current, module) != Numeric <T, C> ._1)
{
goto ENDING;
}
// Теперь занимаемся тестированием.
Numeric <T, C> currentPower = Numeric <T, C> ._1;
Numeric <T, C> tmp = current;
while (true)
{
if (tmp == Numeric <T, C> ._1)
{
// Проверить степень корня
if (rootDegreeSet.Contains(currentPower))
{
if (!result.ContainsKey(currentPower))
{
result.Add(currentPower, new List <T>());
}
List <T> currentDegreeRootList = result[currentPower];
currentDegreeRootList.Add(current);
}
goto ENDING;
}
else if (tmp == Numeric <T, C> .Zero)
{
goto ENDING;
}
tmp = (tmp * tmp) % module;
++currentPower;
}
ENDING:
// Если четный модуль - надо перескочить через два.
if (evenModule)
{
current++;
}
}
return(result);
}
public static Rational <T, C> operator -(Rational <T, C> one, Rational <T, C> two)
{
if (one is Infinities || two is Infinities)
{
if (one is NotANumber || two is NotANumber)
{
return(NaN);
}
else if (one is Positive_Infinity)
{
if (two is Infinities)
{
if (two is Negative_Infinity)
{
return(one);
}
else
{
return(NaN);
}
}
return(one);
}
else if (two is Positive_Infinity)
{
if (one is Infinities)
{
if (one is Negative_Infinity)
{
return(one);
}
else
{
return(NaN);
}
}
return(NegativeInfinity);
}
else if (one is Negative_Infinity)
{
if (two is Infinities)
{
if (two is Positive_Infinity)
{
return(one);
}
else
{
return(NaN);
}
}
return(one);
}
else if (two is Negative_Infinity)
{
if (one is Infinities)
{
if (one is Positive_Infinity)
{
return(one);
}
else
{
return(NaN);
}
}
return(PositiveInfinity);
}
}
Rational <T, C> tmp = new Rational <T, C>();
tmp.denom = WhiteMath <T, C> .LowestCommonMultiple(one.denom, two.denom, WhiteMath <T, C> .GreatestCommonDivisor(one.denom, two.denom));
tmp.num = calc.dif(calc.mul(one.num, calc.div(tmp.denom, one.denom)), calc.mul(two.num, calc.div(tmp.denom, two.denom)));
if (checkInf(ref tmp))
{
tmp.normalize();
}
return(tmp);
}
/// <summary>
/// Finds the greatest common divisor of two integer-like numbers
/// using the simple Euclid algorithm.
///
/// The calculator for the numbers is recommended to provide reasonable implementation
/// of the remainder operation (%) - otherwise, the function
/// Modulus from whiteMath class will be used.
///
/// Will work with floating-point type numbers only if they are integers;
/// In case of division errors the result will be rounded and thus not guaranteed.
/// </summary>
/// <param name="one"></param>
/// <param name="two"></param>
/// <returns></returns>
public static T GreatestCommonDivisor(T one, T two)
{
Contract.Requires <ArgumentException>(one != Numeric <T, C> .Zero && two != Numeric <T, C> .Zero, "None of the numbers may be zero.");
Contract.Ensures(Contract.Result <T>() > Numeric <T, C> .Zero);
Contract.Ensures((Numeric <T, C>)one % Contract.Result <T>() == Numeric <T, C> .Zero);
Contract.Ensures((Numeric <T, C>)two % Contract.Result <T>() == Numeric <T, C> .Zero);
// T может быть ссылочным типом, поэтому необходимо
// предостеречь объекты от изменения
// а также отсечь знаки по необходимости
T oneTmp;
T twoTmp;
if (calc.mor(calc.zero, one))
{
oneTmp = calc.negate(one);
}
else
{
oneTmp = calc.getCopy(one);
}
if (calc.mor(calc.zero, two))
{
twoTmp = calc.negate(two);
}
else
{
twoTmp = calc.getCopy(two);
}
try
{
while (!calc.eqv(oneTmp, calc.zero) && !calc.eqv(twoTmp, calc.zero))
{
if (calc.mor(oneTmp, twoTmp))
{
oneTmp = calc.rem(oneTmp, twoTmp);
}
else
{
twoTmp = calc.rem(twoTmp, oneTmp);
}
}
}
catch // calculator throws exception - working with fp's.
{
oneTmp = calc.intPart(oneTmp);
twoTmp = calc.intPart(twoTmp);
while (!calc.eqv(oneTmp, calc.zero) && !calc.eqv(twoTmp, calc.zero))
{
if (calc.mor(oneTmp, twoTmp))
{
oneTmp = WhiteMath <T, C> .Round(WhiteMath <T, C> .Modulus(oneTmp, twoTmp));
}
else
{
twoTmp = WhiteMath <T, C> .Round(WhiteMath <T, C> .Modulus(twoTmp, oneTmp));
}
}
}
return(calc.sum(oneTmp, twoTmp));
}
/// <summary>
/// Returns the value of the Jacobi symbol for
/// the two numbers, that is, the multiplication product
/// of Legendre symbols with numerators all equal to the Jacobi symbol's
/// numerator and denominators taken from the factorization
/// of Jacobi symbol's denominator.
/// </summary>
/// <param name="num">The numerator of the Jacobi symbol.</param>
/// <param name="denom">The denominator of the Jacobi symbol. Should be odd and positive.</param>
/// <returns>The value of the Jacobi symbol for <paramref name="num"/> and <paramref name="denom"/>.</returns>
public static int JacobiSymbol(T num, T denom)
{
Contract.Requires <NonIntegerTypeException>(Numeric <T, C> .Calculator.isIntegerCalculator, "The method works only for integer numeric types.");
Contract.Requires <ArgumentException>(denom > Numeric <T, C> .Zero && !Numeric <T, C> .Calculator.isEven(denom), "The denominator of the Jacobi symbol should be odd and positive.");
bool minus = false; // флаг минуса
Numeric <T, C> x = num;
Numeric <T, C> y = denom;
if (y == Numeric <T, C> ._1)
{
return(1);
}
if (WhiteMath <T, C> .GreatestCommonDivisor(x, y) != Numeric <T, C> ._1)
{
return(0);
}
if (x < Numeric <T, C> .Zero)
{
// Надо домножить на (-1)^((y-1)/2)
// Эта величина равна -1 тогда и только тогда, когда floor(y/2) - четное число.
if ((y / Numeric <T, C> ._2).Even)
{
minus ^= true;
}
x = -x;
}
// На этом шаге ни в числителе,
// ни в знаменателе не осталось отрицательных чисел.
while (true)
{
// (x; y) = (x mod y; y) ------------------
if (x > y)
{
x = x % y;
}
// ---------- избавляемся от четности -----
int t = 0;
while (x.Even)
{
// Надо домножить на (-1)^((y^2 - 1)/8)
// Эта величина равна -1 тогда и только тогда, когда y имеет остатки 3 или 5 при делении на 8
++t;
x /= Numeric <T, C> ._2;
}
if (t % 2 != 0)
{
Numeric <T, C> rem = y % Numeric <T, C> ._8;
if (rem == Numeric <T, C> ._3 || rem == Numeric <T, C> ._5)
{
minus ^= true;
}
}
// ----------------------------------------
// --- Если x - единица, то надо вернуться.
// ----------------------------------------
if (x == Numeric <T, C> ._1)
{
return(minus ? -1 : 1);
}
// ----------------------------------------------------
// -- x и y на этом этапе гарантированно взаимно просты
// -- и нечетны, поэтому можно использовать свойство (8) из твоей тетрадочки
// ----------------------------------------------------
if (x < y)
{
if (x % Numeric <T, C> ._4 == Numeric <T, C> ._3 &&
y % Numeric <T, C> ._4 == Numeric <T, C> ._3)
{
minus ^= true;
}
Numeric <T, C> tmp = x;
x = y;
y = tmp;
}
}
}