/// <summary>
/// Verifica se o módulo do número especificado consiste numa potência perfeita.
/// </summary>
/// <param name="data">O número a testar.</param>
/// <returns>Verdadeiro caso o número seja uma potência perfeita e falso no caso contrário.</returns>
public bool Run(int data)
{
var innerData = Math.Abs(data);
if (innerData == 0)
{
return(true);
}
else if (innerData == 1)
{
return(true);
}
else
{
var maximumTestValue = (int)Math.Floor(Math.Log(innerData) / Math.Log(2));
var primeNumbersIterator = this.primeNumbersIteratorFactory.CreatePrimeNumberIterator(
maximumTestValue + 1);
foreach (var prime in primeNumbersIterator)
{
var root = (int)Math.Floor(Math.Pow(innerData, 1.0 / prime));
var power = MathFunctions.Power(root, prime, this.integerDomain);
if (power == innerData)
{
return(true);
}
}
return(false);
}
}
/// <summary>
/// Determina a potência do elemento.
/// </summary>
/// <param name="left">O elemento.</param>
/// <param name="right">O expoente.</param>
/// <returns>O resultado da potência.</returns>
/// <exception cref="MathematicsException">
/// Se nenhum objecto responsável pelas operações sobre inteiros foi providenciado.
/// </exception>
protected virtual ObjectType Power(
ObjectType left,
ObjectType right)
{
if (this.integerNumber == null)
{
throw new MathematicsException("No integer number arithmetic operations class was provided.");
}
else
{
return(MathFunctions.Power(left, right, this.ring, this.integerNumber));
}
}
/// <summary>
/// Obtém o valor da estimativa para o valor máximo dos números primos a serem gerados.
/// </summary>
/// <param name="polynomialNorm">A norma do polinómio.</param>
/// <param name="degree">O grau do polinómio.</param>
/// <returns>O valor para o número primo máximo.</returns>
private CoeffType ComputeGamma(CoeffType polynomialNorm, int degree)
{
var result = this.integerNumber.Successor(
this.integerNumber.MapFrom(degree));
var doubleDegree = 2 * degree;
result = MathFunctions.Power(result, doubleDegree, this.integerNumber);
var temp = MathFunctions.Power(polynomialNorm, doubleDegree + 1, this.integerNumber);
result = this.integerNumber.Multiply(result, temp);
var doubleLogResult = this.logarithmAlgorithm.Run(result) * 2;
doubleLogResult = Math.Ceiling(doubleLogResult);
doubleLogResult = 2 * doubleLogResult * Math.Log(doubleLogResult);
return(this.integerNumber.MapFrom((int)Math.Floor(doubleLogResult)));
}
/// <summary>
/// Calcula o produto de um dicionário com os factores e respectivas potências.
/// </summary>
/// <param name="factors">O dicionário com os factores e respectivas potências.</param>
/// <returns>O valor do produto.</returns>
private int ComputeProduct(Dictionary <int, int> factors)
{
var result = 1;
foreach (var kvp in factors)
{
if (kvp.Value == 1)
{
result *= kvp.Key;
}
else if (kvp.Value == 2)
{
result *= kvp.Key * kvp.Key;
}
else
{
result *= MathFunctions.Power(kvp.Key, kvp.Value, this.integerEuclideanDomain);
}
}
return(result);
}
/// <summary>
/// Establece os parâmetros internos com base nos argumentos externos.
/// </summary>
/// <param name="integerPart">A parte inteira da raiz.</param>
/// <param name="rootNumberFactorization">A factorização do radicando.</param>
/// <param name="multipliable">O objecto responsável pela multiplicação de coeficientes.</param>
/// <exception cref="ArgumentNullException">Se algum dos argumentos for nulo.</exception>
private void SetupParameters(
ObjectType integerPart,
Dictionary <ObjectType, int> rootNumberFactorization,
IMultiplication <ObjectType> multipliable)
{
if (integerPart == null)
{
throw new ArgumentNullException("integerPart");
}
else if (rootNumberFactorization == null)
{
throw new ArgumentNullException("rootNumberFactorization");
}
else if (multipliable == null)
{
throw new ArgumentNullException("multipliable");
}
else
{
this.integerPart = integerPart;
this.rootNumberFactorization = new Dictionary <ObjectType, int>(rootNumberFactorization.Comparer);
foreach (var fact in rootNumberFactorization)
{
var factDegree = fact.Value;
var factSquaredValue = factDegree / 2;
var factRem = factDegree % 2;
this.integerPart = multipliable.Multiply(
this.integerPart,
MathFunctions.Power(fact.Key, factSquaredValue, multipliable));
if (factRem == 1)
{
rootNumberFactorization.Add(fact.Key, 1);
}
}
}
}
/// <summary>
/// Obtém o último sub-resultante entre dois polinómios.
/// </summary>
/// <param name="first">O primeiro polinómio.</param>
/// <param name="second">O segundo polinómio.</param>
/// <param name="domain">O domínio responsável pelas operações sobre os coeficientes.</param>
/// <returns>O polinómio que constitui o último sub-resultante.</returns>
/// <exception cref="ArgumentNullException">Se algum dos argumentos for nulo.</exception>
public UnivariatePolynomialNormalForm <CoeffType> Run(
UnivariatePolynomialNormalForm <CoeffType> first,
UnivariatePolynomialNormalForm <CoeffType> second,
IEuclidenDomain <CoeffType> domain)
{
if (first == null)
{
throw new ArgumentNullException("first");
}
else if (second == null)
{
throw new ArgumentNullException("second");
}
else if (domain == null)
{
throw new ArgumentNullException("domain");
}
else
{
var pseudoDomain = new UnivarPolynomPseudoDomain <CoeffType>(first.VariableName, domain);
var innerFirstPol = first;
var innerSecondPol = second;
if (innerSecondPol.Degree > innerFirstPol.Degree)
{
var swap = innerFirstPol;
innerFirstPol = innerSecondPol;
innerSecondPol = swap;
}
if (!pseudoDomain.IsAdditiveUnity(innerSecondPol))
{
var firstLeadingCoeff = innerFirstPol.GetLeadingCoefficient(pseudoDomain.CoeffsDomain);
var tempDegree = innerFirstPol.Degree - innerSecondPol.Degree;
var temp = innerSecondPol;
innerSecondPol = pseudoDomain.Rem(innerFirstPol, innerSecondPol);
innerSecondPol = innerSecondPol.ApplyQuo(
firstLeadingCoeff,
pseudoDomain.CoeffsDomain);
if (tempDegree % 2 == 0)
{
innerSecondPol = innerSecondPol.GetSymmetric(pseudoDomain.CoeffsDomain);
}
innerFirstPol = temp;
// Ciclo relativo à determinação do primeiro valor de c
if (!pseudoDomain.IsAdditiveUnity(innerSecondPol))
{
firstLeadingCoeff = innerFirstPol.GetLeadingCoefficient(pseudoDomain.CoeffsDomain);
var c = MathFunctions.Power(firstLeadingCoeff, tempDegree, pseudoDomain.CoeffsDomain);
tempDegree = innerFirstPol.Degree - innerSecondPol.Degree;
temp = innerSecondPol;
innerSecondPol = pseudoDomain.Rem(innerFirstPol, innerSecondPol);
var den = MathFunctions.Power(c, tempDegree, pseudoDomain.CoeffsDomain);
innerSecondPol = innerSecondPol.ApplyQuo(
pseudoDomain.CoeffsDomain.Multiply(firstLeadingCoeff, den),
pseudoDomain.CoeffsDomain);
if (tempDegree % 2 == 0)
{
innerSecondPol = innerSecondPol.GetSymmetric(pseudoDomain.CoeffsDomain);
}
innerFirstPol = temp;
// Início do ciclo para obter a sequência de sub-resultantes
while (!pseudoDomain.IsAdditiveUnity(innerSecondPol))
{
firstLeadingCoeff = innerFirstPol.GetLeadingCoefficient(pseudoDomain.CoeffsDomain);
c = pseudoDomain.CoeffsDomain.Quo(
MathFunctions.Power(firstLeadingCoeff, tempDegree, pseudoDomain.CoeffsDomain),
MathFunctions.Power(c, tempDegree - 1, pseudoDomain.CoeffsDomain));
tempDegree = innerFirstPol.Degree - innerSecondPol.Degree;
temp = innerSecondPol;
innerSecondPol = pseudoDomain.Rem(innerFirstPol, innerSecondPol);
den = MathFunctions.Power(c, tempDegree, pseudoDomain.CoeffsDomain);
innerSecondPol = innerSecondPol.ApplyQuo(
pseudoDomain.CoeffsDomain.Multiply(firstLeadingCoeff, den),
pseudoDomain.CoeffsDomain);
if (tempDegree % 2 == 0)
{
innerSecondPol = innerSecondPol.GetSymmetric(pseudoDomain.CoeffsDomain);
}
innerFirstPol = temp;
}
}
}
return(innerFirstPol);
}
}
/// <summary>
/// Obtém o quociente e o resto da pseudo-divisão entre dois polinómios.
/// </summary>
/// <param name="dividend">O dividendo.</param>
/// <param name="divisor">O divisor.</param>
/// <returns>O quociente e o resto.</returns>
/// <exception cref="ArgumentNullException">Se algum dos argumentos for nulo.</exception>
/// <exception cref="ArgumentException">
/// Se algum dos polinómio contiver uma variável diferente da estipulada para o
/// domínio corrente.
/// </exception>
public DomainResult <UnivariatePolynomialNormalForm <CoeffType> > GetQuotientAndRemainder(
UnivariatePolynomialNormalForm <CoeffType> dividend,
UnivariatePolynomialNormalForm <CoeffType> divisor)
{
if (dividend == null)
{
throw new ArgumentNullException("dividend");
}
else if (divisor == null)
{
throw new ArgumentNullException("divisor");
}
else if (divisor.VariableName != divisor.VariableName)
{
throw new ArgumentException("Polynomials must share the same variable name in order to be operated.");
}
else
{
if (divisor.Degree > dividend.Degree)
{
return(new DomainResult <UnivariatePolynomialNormalForm <CoeffType> >(
new UnivariatePolynomialNormalForm <CoeffType>(this.variableName),
dividend));
}
else
{
var remainderSortedCoeffs = dividend.GetOrderedCoefficients(Comparer <int> .Default);
var divisorSorteCoeffs = divisor.GetOrderedCoefficients(Comparer <int> .Default);
var quotientCoeffs = new UnivariatePolynomialNormalForm <CoeffType>(this.variableName);
var remainderLeadingDegree = remainderSortedCoeffs.Keys[remainderSortedCoeffs.Keys.Count - 1];
var divisorLeadingDegree = divisorSorteCoeffs.Keys[divisorSorteCoeffs.Keys.Count - 1];
var divisorLeadingCoeff = divisorSorteCoeffs.Values[divisorSorteCoeffs.Values.Count - 1];
var multiplyNumber = MathFunctions.Power(
divisorLeadingCoeff,
remainderLeadingDegree - divisorLeadingDegree + 1,
this.ring);
var temporaryRemainderCoeffs = new SortedList <int, CoeffType>(Comparer <int> .Default);
foreach (var kvp in remainderSortedCoeffs)
{
temporaryRemainderCoeffs.Add(
kvp.Key,
this.ring.Multiply(kvp.Value, multiplyNumber));
}
remainderSortedCoeffs = temporaryRemainderCoeffs;
while (remainderLeadingDegree >= divisorLeadingDegree && remainderSortedCoeffs.Count > 0)
{
var remainderLeadingCoeff = remainderSortedCoeffs[remainderLeadingDegree];
var differenceDegree = remainderLeadingDegree - divisorLeadingDegree;
var factor = this.coeffsDomain.Quo(remainderLeadingCoeff, divisorLeadingCoeff);
quotientCoeffs = quotientCoeffs.Add(factor, differenceDegree, this.ring);
remainderSortedCoeffs.Remove(remainderLeadingDegree);
for (int i = 0; i < divisorSorteCoeffs.Keys.Count - 1; ++i)
{
var currentDivisorDegree = divisorSorteCoeffs.Keys[i];
var currentCoeff = this.ring.Multiply(
divisorSorteCoeffs[currentDivisorDegree],
factor);
currentDivisorDegree += differenceDegree;
var addCoeff = default(CoeffType);
if (remainderSortedCoeffs.TryGetValue(currentDivisorDegree, out addCoeff))
{
addCoeff = this.ring.Add(
addCoeff,
this.ring.AdditiveInverse(currentCoeff));
if (this.ring.IsAdditiveUnity(addCoeff))
{
remainderSortedCoeffs.Remove(currentDivisorDegree);
}
else
{
remainderSortedCoeffs[currentDivisorDegree] = addCoeff;
}
}
else
{
remainderSortedCoeffs.Add(
currentDivisorDegree,
this.ring.AdditiveInverse(currentCoeff));
}
}
if (remainderSortedCoeffs.Count > 0)
{
remainderLeadingDegree = remainderSortedCoeffs.Keys[remainderSortedCoeffs.Keys.Count - 1];
}
else
{
remainderLeadingDegree = 0;
}
}
var remainder = new UnivariatePolynomialNormalForm <CoeffType>(
remainderSortedCoeffs,
this.variableName,
this.ring);
return(new DomainResult <UnivariatePolynomialNormalForm <CoeffType> >(
quotientCoeffs,
remainder));
}
}
}
/// <summary>
/// Averigua se o número especificado é primo.
/// </summary>
/// <param name="data">O número.</param>
/// <returns>Verdadeiro caso o número seja primo e falso caso este seja composto.</returns>
public bool Run(int data)
{
if (this.integerNumber.IsAdditiveUnity(data))
{
return(false);
}
else if (
this.integerNumber.IsMultiplicativeUnity(data) ||
this.integerNumber.IsMultiplicativeUnity(this.integerNumber.AdditiveInverse(data)))
{
return(false);
}
else
{
var innerData = this.integerNumber.GetNorm(data);
var two = this.integerNumber.MapFrom(2);
if (this.integerNumber.Equals(innerData, two))
{
return(true);
}
else if (this.perfectPowerTest.Run(data))
{
return(false);
}
else
{
var log = Math.Log(innerData) / Math.Log(2);
var r = this.EvaluateLimitValue(innerData, log);
r = this.totientFunctionAlg.Run(r);
var i = two;
for (; this.integerNumber.Compare(i, r) <= 0; i = this.integerNumber.Successor(i))
{
var gcd = MathFunctions.GreatCommonDivisor(innerData, i, this.integerNumber);
if (this.integerNumber.Compare(gcd, this.integerNumber.MultiplicativeUnity) > 0 &&
this.integerNumber.Compare(gcd, innerData) < 0)
{
return(false);
}
}
var limit = (int)Math.Floor(Math.Sqrt(r) * log);
var modularField = this.modularFactory.CreateInstance(innerData);
var terms = new Dictionary <int, int>();
var modularPolynomialRing = new AuxAksModArithmRing <int>(
r,
"x",
modularField);
for (int j = 1; j <= limit; ++j)
{
terms.Clear();
terms.Add(0, j);
terms.Add(1, 1);
var polynomial = new UnivariatePolynomialNormalForm <int>(
terms,
"x",
modularField);
terms.Clear();
terms.Add(0, j);
terms.Add(innerData, 1);
var comparisionPol = new UnivariatePolynomialNormalForm <int>(
terms,
"x",
modularField);
var poweredPol = MathFunctions.Power(polynomial, innerData, modularPolynomialRing);
if (!poweredPol.Equals(modularPolynomialRing.GetReduced(comparisionPol)))
{
return(false);
}
}
return(true);
}
}
}
/// <summary>
/// Processa o polinómio determinando os respectivos factores.
/// </summary>
/// <remarks>
/// Os factores constantes são ignorados, os factores lineares são anexados ao resultado e os factores
/// cujos graus são superiores são retornados para futuro processamento. Se o polinómio a ser processado
/// for irredutível, é adicionado ao resultado.
/// </remarks>
/// <param name="polynom">O polinómio a ser processado.</param>
/// <param name="result">O contentor dos factores sucessivamente processados.</param>
/// <param name="integerModule">O objecto responsável pelas operações sobre inteiros.</param>
/// <param name="polynomField">O objecto responsável pelas operações sobre os polinómios.</param>
/// <param name="inverseAlgorithm">O algoritmo inverso.</param>
/// <returns></returns>
List <UnivariatePolynomialNormalForm <CoeffType> > Process(
UnivariatePolynomialNormalForm <CoeffType> polynom,
List <UnivariatePolynomialNormalForm <CoeffType> > result,
IModularField <CoeffType> integerModule,
UnivarPolynomEuclideanDomain <CoeffType> polynomField,
LagrangeAlgorithm <UnivariatePolynomialNormalForm <CoeffType> > inverseAlgorithm)
{
var resultPol = new List <UnivariatePolynomialNormalForm <CoeffType> >();
if (polynom.Degree < 2)
{
result.Add(polynom);
}
else
{
var module = new ModularBachetBezoutField <UnivariatePolynomialNormalForm <CoeffType> >(
polynom,
inverseAlgorithm);
var degree = polynom.Degree;
var arrayMatrix = new ArrayMathMatrix <CoeffType>(degree, degree, integerModule.AdditiveUnity);
arrayMatrix[0, 0] = integerModule.AdditiveUnity;
var pol = new UnivariatePolynomialNormalForm <CoeffType>(
integerModule.MultiplicativeUnity,
1,
polynom.VariableName,
integerModule);
var integerModuleValue = this.integerNumber.ConvertToInt(integerModule.Module);
pol = MathFunctions.Power(pol, integerModuleValue, module);
foreach (var term in pol)
{
arrayMatrix[term.Key, 1] = term.Value;
}
var auxPol = pol;
for (int i = 2; i < degree; ++i)
{
auxPol = module.Multiply(auxPol, pol);
foreach (var term in auxPol)
{
arrayMatrix[term.Key, i] = term.Value;
}
}
for (int i = 1; i < degree; ++i)
{
var value = arrayMatrix[i, i];
value = integerModule.Add(
value,
integerModule.AdditiveInverse(integerModule.MultiplicativeUnity));
arrayMatrix[i, i] = value;
}
var emtpyMatrix = new ZeroMatrix <CoeffType>(degree, 1, integerModule);
var linearSystemSolution = this.linearSystemSolver.Run(arrayMatrix, emtpyMatrix);
var numberOfFactors = linearSystemSolution.VectorSpaceBasis.Count;
if (numberOfFactors < 2)
{
result.Add(polynom);
}
else
{
var hPol = default(UnivariatePolynomialNormalForm <CoeffType>);
var linearSystemCount = linearSystemSolution.VectorSpaceBasis.Count;
for (int i = 0; i < linearSystemCount; ++i)
{
var currentBaseSolution = linearSystemSolution.VectorSpaceBasis[i];
var rowsLength = currentBaseSolution.Length;
for (int j = 1; j < rowsLength; ++j)
{
if (!integerModule.IsAdditiveUnity(currentBaseSolution[j]))
{
hPol = this.GetPolynomial(currentBaseSolution, integerModule, polynom.VariableName);
j = rowsLength;
}
if (hPol != null)
{
j = rowsLength;
}
}
if (hPol != null)
{
i = linearSystemCount;
}
}
for (int i = 0, k = 0; k < numberOfFactors && i < integerModuleValue; ++i)
{
var converted = this.integerNumber.MapFrom(i);
var currentPol = MathFunctions.GreatCommonDivisor(
polynom,
hPol.Subtract(converted, integerModule),
polynomField);
var currentDegree = currentPol.Degree;
if (currentDegree == 1)
{
result.Add(currentPol);
++k;
}
else if (currentDegree > 1)
{
resultPol.Add(currentPol);
++k;
}
}
}
}
return(resultPol);
}
/// <summary>
/// Determina a potência do polinómio.
/// </summary>
/// <param name="left">O polinómio.</param>
/// <param name="right">O expoente.</param>
/// <returns>O resultado da potência.</returns>
/// <exception cref="MathematicsException">Se a operação falhar por vários motivos.</exception>
protected virtual ParsePolynomialItem <T> Power(ParsePolynomialItem <T> left, ParsePolynomialItem <T> right)
{
var result = new ParsePolynomialItem <T>();
if (left.ValueType == EParsePolynomialValueType.COEFFICIENT)
{
if (right.ValueType == EParsePolynomialValueType.COEFFICIENT)
{
var degree = this.conversion.DirectConversion(right.Coeff);
result.Coeff = MathFunctions.Power(left.Coeff, degree, this.ring);
}
else if (right.ValueType == EParsePolynomialValueType.INTEGER)
{
result.Coeff = MathFunctions.Power(left.Coeff, right.Degree, this.ring);
}
else if (right.ValueType == EParsePolynomialValueType.POLYNOMIAL)
{
if (right.Polynomial.IsValue)
{
var exponent = this.conversion.DirectConversion(right.Polynomial.GetAsValue(this.ring));
result.Coeff = MathFunctions.Power(left.Coeff, exponent, this.ring);
}
else
{
throw new MathematicsException("Polynomial exponents aren't allowed.");
}
}
}
else if (left.ValueType == EParsePolynomialValueType.INTEGER)
{
if (right.ValueType == EParsePolynomialValueType.COEFFICIENT)
{
var rightConversion = this.conversion.DirectConversion(right.Coeff);
result.Degree = MathFunctions.Power(left.Degree, rightConversion, this.integerRing);
}
else if (right.ValueType == EParsePolynomialValueType.INTEGER)
{
result.Degree = MathFunctions.Power(left.Degree, right.Degree, this.integerRing);
}
else if (right.ValueType == EParsePolynomialValueType.POLYNOMIAL)
{
if (right.Polynomial.IsValue)
{
var exponent = this.conversion.DirectConversion(right.Polynomial.GetAsValue(this.ring));
result.Degree = MathFunctions.Power(left.Degree, exponent, this.integerRing);
}
else
{
throw new MathematicsException("Polynomial exponents aren't allowed.");
}
}
}
else if (left.ValueType == EParsePolynomialValueType.POLYNOMIAL)
{
if (right.ValueType == EParsePolynomialValueType.COEFFICIENT)
{
var exponent = this.conversion.DirectConversion(right.Coeff);
result.Polynomial = left.Polynomial.Power(exponent, this.ring);
}
else if (right.ValueType == EParsePolynomialValueType.INTEGER)
{
result.Polynomial = left.Polynomial.Power(right.Degree, this.ring);
}
else if (right.ValueType == EParsePolynomialValueType.POLYNOMIAL)
{
if (right.Polynomial.IsValue)
{
var exponent = this.conversion.DirectConversion(right.Polynomial.GetAsValue(this.ring));
result.Polynomial = left.Polynomial.Power(exponent, this.ring);
}
else
{
throw new MathematicsException("Polynomial exponents aren't allowed.");
}
}
}
return(result);
}
/// <summary>
/// Executa o algoritmo que permite obter uma factorização livre de quadrados.
/// </summary>
/// <param name="polynomial">O polinómio de entrada.</param>
/// <returns>A factorização livre de quadrados.</returns>
/// <exception cref="ArgumentNullException">
/// Se o argumento for nulo.
/// </exception>
public SquareFreeFactorizationResult <Fraction <CoeffType>, CoeffType> Run(
UnivariatePolynomialNormalForm <Fraction <CoeffType> > polynomial)
{
if (polynomial == null)
{
throw new ArgumentNullException("polynomial");
}
else
{
var independentCoeff = this.fractionField.MultiplicativeUnity;
var result = new Dictionary <int, UnivariatePolynomialNormalForm <CoeffType> >();
var currentDegree = 1;
var polynomDomain = new UnivarPolynomEuclideanDomain <Fraction <CoeffType> >(
polynomial.VariableName,
this.fractionField);
var lagAlg = new LagrangeAlgorithm <CoeffType>(this.integerNumber);
var dataDerivative = polynomial.GetPolynomialDerivative(this.fractionField);
var gcd = this.GreatCommonDivisor(polynomial, dataDerivative, polynomDomain);
if (gcd.Degree == 0)
{
var lcm = this.GetDenominatorLcm(polynomial, lagAlg);
if (this.integerNumber.IsMultiplicativeUnity(lcm))
{
var integerPol = this.GetIntegerPol(polynomial);
result.Add(currentDegree, integerPol);
}
else
{
independentCoeff = new Fraction <CoeffType>(
this.integerNumber.MultiplicativeUnity,
lcm,
this.integerNumber);
var multipliable = new Fraction <CoeffType>(
lcm,
this.integerNumber.MultiplicativeUnity,
this.integerNumber);
var multipliedPol = polynomial.Multiply(independentCoeff, this.fractionField);
var integerPol = this.GetIntegerPol(multipliedPol);
result.Add(currentDegree, integerPol);
}
}
else
{
var polyCoffactor = polynomDomain.Quo(polynomial, gcd);
var nextGcd = this.GreatCommonDivisor(gcd, polyCoffactor, polynomDomain);
var squareFreeFactor = polynomDomain.Quo(polyCoffactor, nextGcd);
polyCoffactor = gcd;
gcd = nextGcd;
if (squareFreeFactor.Degree > 0)
{
var lcm = this.GetDenominatorLcm(squareFreeFactor, lagAlg);
if (this.integerNumber.IsMultiplicativeUnity(lcm))
{
var integerPol = this.GetIntegerPol(squareFreeFactor);
result.Add(currentDegree, integerPol);
}
else if (this.fractionField.IsAdditiveUnity(independentCoeff))
{
independentCoeff = new Fraction <CoeffType>(
this.integerNumber.MultiplicativeUnity,
MathFunctions.Power(lcm, currentDegree, this.integerNumber),
this.integerNumber);
var multipliable = new Fraction <CoeffType>(
lcm,
this.integerNumber.MultiplicativeUnity,
this.integerNumber);
var multipliedPol = squareFreeFactor.Multiply(independentCoeff, this.fractionField);
var integerPol = this.GetIntegerPol(multipliedPol);
result.Add(currentDegree, integerPol);
}
else
{
var multiplicationCoeff = new Fraction <CoeffType>(
this.integerNumber.MultiplicativeUnity,
MathFunctions.Power(lcm, currentDegree, this.integerNumber),
this.integerNumber);
independentCoeff = this.fractionField.Multiply(independentCoeff, multiplicationCoeff);
var multipliable = new Fraction <CoeffType>(
lcm,
this.integerNumber.MultiplicativeUnity,
this.integerNumber);
var multipliedPol = squareFreeFactor.Multiply(independentCoeff, this.fractionField);
var integerPol = this.GetIntegerPol(multipliedPol);
result.Add(currentDegree, integerPol);
}
}
else
{
var value = squareFreeFactor.GetAsValue(this.fractionField);
if (!this.fractionField.IsMultiplicativeUnity(value))
{
if (this.fractionField.IsMultiplicativeUnity(independentCoeff))
{
independentCoeff = value;
}
else
{
independentCoeff = this.fractionField.Multiply(independentCoeff, value);
}
}
}
++currentDegree;
while (gcd.Degree > 0)
{
polyCoffactor = polynomDomain.Quo(polyCoffactor, gcd);
nextGcd = MathFunctions.GreatCommonDivisor(gcd, polyCoffactor, polynomDomain);
squareFreeFactor = polynomDomain.Quo(gcd, nextGcd);
gcd = nextGcd;
if (squareFreeFactor.Degree > 0)
{
var lcm = this.GetDenominatorLcm(squareFreeFactor, lagAlg);
if (this.integerNumber.IsMultiplicativeUnity(lcm))
{
var integerPol = this.GetIntegerPol(squareFreeFactor);
result.Add(currentDegree, integerPol);
}
else if (this.fractionField.IsAdditiveUnity(independentCoeff))
{
independentCoeff = new Fraction <CoeffType>(
this.integerNumber.MultiplicativeUnity,
MathFunctions.Power(lcm, currentDegree, this.integerNumber),
this.integerNumber);
var multipliable = new Fraction <CoeffType>(
lcm,
this.integerNumber.MultiplicativeUnity,
this.integerNumber);
var multipliedPol = squareFreeFactor.Multiply(independentCoeff, this.fractionField);
var integerPol = this.GetIntegerPol(multipliedPol);
result.Add(currentDegree, integerPol);
}
else
{
var multiplicationCoeff = new Fraction <CoeffType>(
this.integerNumber.MultiplicativeUnity,
MathFunctions.Power(lcm, currentDegree, this.integerNumber),
this.integerNumber);
independentCoeff = this.fractionField.Multiply(independentCoeff, multiplicationCoeff);
var multipliable = new Fraction <CoeffType>(
lcm,
this.integerNumber.MultiplicativeUnity,
this.integerNumber);
var multipliedPol = squareFreeFactor.Multiply(multipliable, this.fractionField);
var integerPol = this.GetIntegerPol(multipliedPol);
result.Add(currentDegree, integerPol);
}
}
else
{
var value = squareFreeFactor.GetAsValue(this.fractionField);
if (!this.fractionField.IsMultiplicativeUnity(value))
{
if (this.fractionField.IsMultiplicativeUnity(independentCoeff))
{
independentCoeff = value;
}
else
{
independentCoeff = this.fractionField.Multiply(independentCoeff, value);
}
}
}
++currentDegree;
}
var cofactorValue = polyCoffactor.GetAsValue(this.fractionField);
if (!this.fractionField.IsMultiplicativeUnity(cofactorValue))
{
if (this.fractionField.IsMultiplicativeUnity(independentCoeff))
{
independentCoeff = cofactorValue;
}
else
{
independentCoeff = this.fractionField.Multiply(independentCoeff, cofactorValue);
}
}
}
return(new SquareFreeFactorizationResult <Fraction <CoeffType>, CoeffType>(
independentCoeff,
result));
}
}
/// <summary>
/// Obtém os valores da solução.
/// </summary>
/// <param name="solution">A solução do sistema modular.</param>
/// <param name="matrixList">A matriz.</param>
/// <param name="primesList">A lista dos números primos da base.</param>
/// <param name="innerData">O número que está a ser factorizado.</param>
/// <returns>Os factores.</returns>
private Tuple <NumberType, NumberType> GetSolution(
LinearSystemSolution <int> solution,
List <int[]> matrixList,
List <int> primesList,
NumberType innerData)
{
var innerDataModularField = this.modularFieldFactory.CreateInstance(innerData);
var countFactors = primesList.Count - 1;
foreach (var solutionBase in solution.VectorSpaceBasis)
{
var firstValue = this.integerNumber.MultiplicativeUnity;
var factorsCount = new Dictionary <int, int>();
for (int i = 0; i < matrixList.Count; ++i)
{
var currentMatrixLine = matrixList[i];
if (solutionBase[i] == 1)
{
firstValue = innerDataModularField.Multiply(
firstValue,
this.integerNumber.GetNorm(this.integerNumber.MapFrom(currentMatrixLine[currentMatrixLine.Length - 1])));
for (int j = 0; j < countFactors; ++j)
{
if (currentMatrixLine[j] != 0)
{
var countValue = 0;
if (factorsCount.TryGetValue(primesList[j], out countValue))
{
countValue += currentMatrixLine[j];
factorsCount[primesList[j]] = countValue;
}
else
{
factorsCount.Add(primesList[j], currentMatrixLine[j]);
}
}
}
}
}
var secondValue = this.integerNumber.MultiplicativeUnity;
foreach (var factorCountKvp in factorsCount)
{
var primePower = MathFunctions.Power(
this.integerNumber.MapFrom(factorCountKvp.Key),
factorCountKvp.Value / 2,
innerDataModularField);
secondValue = innerDataModularField.Multiply(
secondValue,
primePower);
}
if (!this.integerNumber.Equals(firstValue, secondValue))
{
var firstFactor = MathFunctions.GreatCommonDivisor(
innerData,
this.integerNumber.GetNorm(this.integerNumber.Add(firstValue, this.integerNumber.AdditiveInverse(secondValue))),
this.integerNumber);
if (!this.integerNumber.IsMultiplicativeUnity(firstFactor))
{
var secondFactor = this.integerNumber.Quo(innerData, firstFactor);
return(Tuple.Create(firstFactor, secondFactor));
}
}
}
return(Tuple.Create(this.integerNumber.MultiplicativeUnity, innerData));
}
/// <summary>
/// Determina o resíduo quadrático de um número módulo o número primo ímpar especificado.
/// </summary>
/// <remarks>
/// Se o módulo não for um número primo ímpar, os resultados estarão errados. Esta verificação
/// não é realizada sobre esse módulo.
/// </remarks>
/// <param name="number">O número.</param>
/// <param name="primeModule">O número primo que servirá de módulo.</param>
/// <returns>A lista com os dois resíduos.</returns>
/// <exception cref="ArgumentException">
/// Se o módulo não for superior a dois, se o módulo for par ou se a solução não existir.
/// </exception>
public List <NumberType> Run(NumberType number, NumberType primeModule)
{
var two = this.integerNumber.MapFrom(2);
if (this.integerNumber.Compare(primeModule, two) < 0)
{
throw new ArgumentException("The prime module must be a number greater than two.");
}
if (this.integerNumber.IsAdditiveUnity(this.integerNumber.Rem(primeModule, two)))
{
throw new ArgumentException("The prime module must be an even number.");
}
else if (!this.integerNumber.IsMultiplicativeUnity(this.legendreJacobiSymAlg.Run(number, primeModule)))
{
throw new ArgumentException("Solution doesn't exist.");
}
else
{
var innerNumber = this.integerNumber.Rem(number, primeModule);
var firstStepModule = this.integerNumber.Predecessor(primeModule);
var power = this.integerNumber.AdditiveUnity;
var remQuoResult = this.integerNumber.GetQuotientAndRemainder(firstStepModule, two);
while (this.integerNumber.IsAdditiveUnity(remQuoResult.Remainder))
{
power = this.integerNumber.Successor(power);
firstStepModule = remQuoResult.Quotient;
remQuoResult = this.integerNumber.GetQuotientAndRemainder(firstStepModule, two);
}
var modularIntegerField = this.modularFieldFactory.CreateInstance(primeModule);
if (this.integerNumber.IsMultiplicativeUnity(power))
{
var tempPower = this.integerNumber.Successor(primeModule);
tempPower = this.integerNumber.Quo(tempPower, this.integerNumber.MapFrom(4));
var value = MathFunctions.Power(number, tempPower, modularIntegerField, this.integerNumber);
var result = new List <NumberType>()
{
value,
this.integerNumber.Add(primeModule, this.integerNumber.AdditiveInverse(value))
};
return(result);
}
else
{
var nonQuadraticResidue = this.FindNonQuadraticResidue(primeModule);
var poweredNonQuadraticResult = MathFunctions.Power(
nonQuadraticResidue,
firstStepModule,
modularIntegerField,
this.integerNumber);
var innerPower = this.integerNumber.Successor(firstStepModule);
innerPower = this.integerNumber.Quo(innerPower, two);
var result = MathFunctions.Power(innerNumber, innerPower, modularIntegerField, this.integerNumber);
var temp = MathFunctions.Power(innerNumber, firstStepModule, modularIntegerField, this.integerNumber);
while (!this.integerNumber.IsMultiplicativeUnity(temp))
{
var lowestIndex = this.FindLowestIndex(temp, power, modularIntegerField);
var aux = this.SquareValue(
poweredNonQuadraticResult,
this.integerNumber.Add(power, this.integerNumber.AdditiveInverse(this.integerNumber.Successor(lowestIndex))),
modularIntegerField);
result = modularIntegerField.Multiply(result, aux);
aux = modularIntegerField.Multiply(aux, aux);
temp = modularIntegerField.Multiply(temp, aux);
poweredNonQuadraticResult = aux;
power = lowestIndex;
}
return(new List <NumberType>()
{
result,
this.integerNumber.Add(primeModule, this.integerNumber.AdditiveInverse(result))
});
}
}
}
/// <summary>
/// Determina o corte de um número sobre as potências de um número primo.
/// </summary>
/// <param name="value">O valor do qual se pretende obter o corte.</param>
/// <param name="prime">O número primo.</param>
/// <param name="firstExponent">O expoente inferior do corte.</param>
/// <param name="secondExponent">O expoente superior do corte.</param>
/// <returns>O resultado do corte.</returns>
private CoeffType ComputeTwoSidedCut(
CoeffType value,
CoeffType prime,
CoeffType firstExponent,
CoeffType secondExponent)
{
var difference = this.integerNumber.Add(
secondExponent,
this.integerNumber.AdditiveInverse(firstExponent));
if (this.integerNumber.Compare(firstExponent, difference) < 0)
{
// Cálculo da primeira potência.
var firstExponentFactor = MathFunctions.Power(
prime,
firstExponent,
this.integerNumber,
this.integerNumber);
// Cálculo da segunda potência com base na primeira.
var increment = this.integerNumber.Add(
firstExponent,
this.integerNumber.AdditiveInverse(difference));
var differenceExponentFactor = MathFunctions.Power(
prime,
increment,
this.integerNumber,
this.integerNumber);
differenceExponentFactor = this.integerNumber.Multiply(
differenceExponentFactor,
firstExponentFactor);
var result = this.integerNumber.Rem(value, firstExponentFactor);
result = this.integerNumber.Add(
value,
this.integerNumber.AdditiveInverse(result));
result = this.integerNumber.Quo(result, firstExponentFactor);
result = this.integerNumber.Rem(result, differenceExponentFactor);
return(result);
}
else
{
// Cálculo da primeira potência.
var differenceExponentFactor = MathFunctions.Power(
prime,
difference,
this.integerNumber,
this.integerNumber);
// Cálculo da segunda potência com base na primeira.
var increment = this.integerNumber.Add(
difference,
this.integerNumber.AdditiveInverse(firstExponent));
var firstExponentFactor = MathFunctions.Power(
prime,
increment,
this.integerNumber,
this.integerNumber);
firstExponentFactor = this.integerNumber.Multiply(
firstExponentFactor,
firstExponentFactor);
var result = this.integerNumber.Rem(value, firstExponentFactor);
result = this.integerNumber.Add(
value,
this.integerNumber.AdditiveInverse(result));
result = this.integerNumber.Quo(result, firstExponentFactor);
result = this.integerNumber.Rem(result, differenceExponentFactor);
return(result);
}
}
/// <summary>
/// Permite calcular a aproximação de Mignotte relativamente à factorização de polinómios.
/// </summary>
/// <remarks>
/// A aproximaçáo calculada consiste num limite superior para os coeficientes de um polinómio
/// módulo a potência de um número primo caso este seja factor do polinómio proposto.
/// </remarks>
/// <param name="polynomialNorm">A norma do polinómio.</param>
/// <param name="leadingCoeff">O coeficiente principal do polinómio.</param>
/// <param name="degree">O grau do polinómio.</param>
/// <returns>O valor da aproximação.</returns>
private CoeffType ComputeMignotteApproximation(
CoeffType polynomialNorm,
CoeffType leadingCoeff,
int degree)
{
var result = MathFunctions.Power(this.integerNumber.MapFrom(2), degree, this.integerNumber);
result = this.integerNumber.Multiply(result, leadingCoeff);
result = this.integerNumber.Multiply(result, polynomialNorm);
// Utilização da expansão em série para a determinação do valor inteiro da raiz quadrada.
// a_1=1, b_1=2, a_{n+1}=-(2n+1)a_n, b_{n+1}=2(n+1)b_n
var integerSquareRoot = this.integerNumber.MapFrom(this.integerSquareRootAlgorithm.Run(degree + 1));
var difference = this.integerNumber.Multiply(integerSquareRoot, integerSquareRoot);
var squared = this.integerNumber.Multiply(integerSquareRoot, integerSquareRoot);
difference = this.integerNumber.Add(
this.integerNumber.MapFrom(degree + 1),
this.integerNumber.AdditiveInverse(squared));
if (this.integerNumber.IsAdditiveUnity(difference))
{
result = this.integerNumber.Multiply(result, integerSquareRoot);
}
else
{
var db = this.integerNumber.MapFrom(2);
var numMult = this.integerNumber.MultiplicativeUnity;
var denMult = this.integerNumber.MultiplicativeUnity;
var sign = true;
var numeratorCoeff = this.integerNumber.Multiply(result, difference);
var denominatorCoeff = db;
denominatorCoeff = this.integerNumber.Multiply(denominatorCoeff, integerSquareRoot);
var gcd = MathFunctions.GreatCommonDivisor(numeratorCoeff, denominatorCoeff, this.integerNumber);
if (!this.integerNumber.IsMultiplicativeUnity(gcd))
{
numeratorCoeff = this.integerNumber.Quo(numeratorCoeff, gcd);
denominatorCoeff = this.integerNumber.Quo(denominatorCoeff, gcd);
}
result = this.integerNumber.Multiply(result, integerSquareRoot);
if (this.integerNumber.Compare(numeratorCoeff, denominatorCoeff) > 0)
{
// Adicionar a primeira parcela da expansão em série.
var divide = this.integerNumber.Quo(numeratorCoeff, denominatorCoeff);
if (!sign)
{
divide = this.integerNumber.AdditiveInverse(divide);
}
result = this.integerNumber.Add(result, divide);
// Actualiza com a nova informação.
sign = !sign;
denMult = this.integerNumber.Successor(denMult);
numeratorCoeff = this.integerNumber.Multiply(numeratorCoeff, difference);
numeratorCoeff = this.integerNumber.Multiply(numeratorCoeff, numMult);
denominatorCoeff = this.integerNumber.Multiply(denominatorCoeff, denMult);
denominatorCoeff = this.integerNumber.Multiply(denominatorCoeff, squared);
denominatorCoeff = this.integerNumber.Multiply(denominatorCoeff, db);
gcd = MathFunctions.GreatCommonDivisor(numeratorCoeff, denominatorCoeff, this.integerNumber);
if (!this.integerNumber.IsMultiplicativeUnity(gcd))
{
numeratorCoeff = this.integerNumber.Quo(numeratorCoeff, gcd);
denominatorCoeff = this.integerNumber.Quo(denominatorCoeff, gcd);
}
while (this.integerNumber.Compare(numeratorCoeff, denominatorCoeff) > 0)
{
// Adicionar a primeira parcela da expansão em série.
divide = this.integerNumber.Quo(numeratorCoeff, denominatorCoeff);
if (!sign)
{
divide = this.integerNumber.AdditiveInverse(divide);
}
result = this.integerNumber.Add(result, divide);
// Actualiza com a nova informação.
sign = !sign;
numMult = this.integerNumber.Successor(this.integerNumber.Successor(numMult));
denMult = this.integerNumber.Successor(denMult);
numeratorCoeff = this.integerNumber.Multiply(numeratorCoeff, difference);
numeratorCoeff = this.integerNumber.Multiply(numeratorCoeff, numMult);
denominatorCoeff = this.integerNumber.Multiply(denominatorCoeff, denMult);
denominatorCoeff = this.integerNumber.Multiply(denominatorCoeff, squared);
denominatorCoeff = this.integerNumber.Multiply(denominatorCoeff, db);
this.integerNumber.Multiply(denominatorCoeff, db);
gcd = MathFunctions.GreatCommonDivisor(numeratorCoeff, denominatorCoeff, this.integerNumber);
if (!this.integerNumber.IsMultiplicativeUnity(gcd))
{
numeratorCoeff = this.integerNumber.Quo(numeratorCoeff, gcd);
denominatorCoeff = this.integerNumber.Quo(denominatorCoeff, gcd);
}
}
}
if (sign)
{
result = this.integerNumber.Successor(result);
}
}
return(result);
}
/// <summary>
/// Permite determinar a factorização racional de um polinómio.
/// </summary>
/// <param name="polynomial">O polinómio.</param>
/// <returns>O resultado da factorização.</returns>
public PolynomialFactorizationResult <Fraction <CoeffType>, CoeffType> Run(
UnivariatePolynomialNormalForm <Fraction <CoeffType> > polynomial)
{
if (polynomial == null)
{
throw new ArgumentNullException("polynomial");
}
else
{
var squareFreeResult = this.squareFreeAlg.Run(polynomial);
var independentCoeff = squareFreeResult.IndependentCoeff;
var factorizationDictionary =
new Dictionary <int, List <UnivariatePolynomialNormalForm <CoeffType> > >();
foreach (var squareFreeFactorKvp in squareFreeResult.Factors)
{
if (squareFreeFactorKvp.Value.Degree == 1)
{
factorizationDictionary.Add(
squareFreeFactorKvp.Key,
new List <UnivariatePolynomialNormalForm <CoeffType> >()
{
squareFreeFactorKvp.Value
});
}
else
{
var currentPolynomial = squareFreeFactorKvp.Value;
var polynomialLeadingCoeff = currentPolynomial.GetLeadingCoefficient(this.integerNumber);
var polynomialNorm = this.GetPolynomialNorm(currentPolynomial);
var bound = this.ComputeMignotteApproximation(
polynomialNorm,
polynomialLeadingCoeff,
polynomial.Degree);
var primeNumbersBound = this.ComputeGamma(polynomialNorm, polynomial.Degree);
var primeNumbersIterator = this.primeNumbersIteratorFactory.CreatePrimeNumberIterator(
primeNumbersBound);
var primeNumbersEnumerator = primeNumbersIterator.GetEnumerator();
var discriminant = this.resultantAlg.Run(
currentPolynomial,
currentPolynomial.GetPolynomialDerivative(this.integerNumber));
var currentPrime = this.integerNumber.MultiplicativeUnity;
var status = 0;
while (status == 0)
{
if (primeNumbersEnumerator.MoveNext())
{
currentPrime = primeNumbersEnumerator.Current;
if (!this.integerNumber.IsAdditiveUnity(
this.integerNumber.Rem(polynomialLeadingCoeff, currentPrime)) &&
!this.integerNumber.IsAdditiveUnity(this.integerNumber.Rem(
discriminant,
currentPrime)))
{
status = 1;
}
}
else
{
status = -1;
}
}
if (status == 1) // Encontrou um número primo sobre o qual é possível proceder à elevação.
{
var factorList = new List <UnivariatePolynomialNormalForm <CoeffType> >();
var iterationsNumber = this.GetHenselLiftingIterationsNumber(bound, currentPrime);
var coeff = this.FactorizePolynomial(
currentPolynomial,
currentPrime,
bound,
iterationsNumber,
factorList);
var multiplicationCoeff = this.fractionField.InverseConversion(coeff);
multiplicationCoeff = MathFunctions.Power(
multiplicationCoeff,
squareFreeFactorKvp.Key,
this.fractionField);
independentCoeff = this.fractionField.Multiply(independentCoeff, multiplicationCoeff);
factorizationDictionary.Add(
squareFreeFactorKvp.Key,
factorList);
}
else
{
factorizationDictionary.Add(
squareFreeFactorKvp.Key,
new List <UnivariatePolynomialNormalForm <CoeffType> >()
{
squareFreeFactorKvp.Value
});
}
}
}
return(new PolynomialFactorizationResult <Fraction <CoeffType>, CoeffType>(
independentCoeff,
factorizationDictionary));
}
}