Quadratic integers. More...

#include <quadraticinteger.h>

Inheritance diagram for QuadraticInteger:

Collaboration diagram for QuadraticInteger:

Public Member Functions
	QuadraticInteger (const int &iVal)

	QuadraticInteger (const QuadraticInteger &qi)

	QuadraticInteger (int a, int b)

AlgebraicInteger *	copy () const
	Create a copy of an algebraic integer. More...

AlgebraicInteger *	aiCopyToInteger (const int &n) const
	Create an algebraic integer from an integer. More...

virtual void	set (const int &n)
	Assign an integer to the algebraic integer. More...

virtual void	set (AlgebraicInteger *ai)
	Assign an algebraic integer to the algebraic integer. More...

virtual void	removeSquareFactors ()
	Remove all the square factors of the algebraic integer.

virtual bool	bIsInvertible () const
	Check whether the number is invertible. More...

virtual bool	bIsSquareOfIvertible () const
	Check whether the number is the square of an invertible element. More...

virtual void	gcd (const AlgebraicInteger *ai)
	this becomes the gcd of this and the parameter More...

virtual bool	bIsDivisbleBy (const AlgebraicInteger *) const
	Check whether *this is divisible by the parameter. More...

virtual void	divideBy (const AlgebraicInteger *ai)
	Performs the division of this by the parameter Remark: we do not check that the this is divisible by *ai. If it is not the case, the division will be an approximation! More...

virtual bool	divideByIfDivisible (const AlgebraicInteger *ai)

virtual void	multiplyBy (const int &n)
	Performs the multiplication of *this by the parameter. More...

virtual void	multiplyBy (const AlgebraicInteger *ai)
	Performs the multiplication of *this by the parameter. More...

virtual void	add (const AlgebraicInteger *ai)
	Performs the addition of *this by the parameter. More...

virtual void	substract (const AlgebraicInteger *ai)
	Substracts the parameter from *this. More...

virtual void	opp ()
	Computes the opp.

virtual bool	bIsLessThan (const int &n) const
	Check whether *this < n.

virtual bool	bIsLessThan (const long int &n) const

virtual bool	bIsLessThan (const AlgebraicInteger &ai) const
	Check whether *this < ai.

virtual bool	bIsLessOEThan (const AlgebraicInteger &ai) const
	Check whether *this <= ai.

virtual bool	bIsGreaterThan (const int &n) const
	Check whether *this > n.

virtual bool	bIsGreaterThan (const long int &n) const

virtual bool	bIsGreaterOEThan (const int &n) const
	Check whether *this >= n.

virtual bool	bIsEqualTo (const AlgebraicInteger &ai) const
	Check whether *this == ai.

virtual bool	bIsEqualTo (const int &n) const
	Check whether *this == n.

ostream &	print (ostream &) const
	Print the number.

virtual string	to_string (const string &strFormat="generic", const bool &bProtect=false) const
	Convert to string. More...

virtual double	to_double () const
	Convert to double (approximation)

virtual string	get_classname () const
	Return the type (string)

long int	floor () const

void	conjugate ()

long int	iNorm () const

long int	iTrace () const

int	iValuation (const QuadraticInteger &qi)

vector< QuadraticInteger >	qiPrimeFactors () const

map< QuadraticInteger, unsigned int >	qiPrimeDecomposition () const

bool	bIsAssociateTo (QuadraticInteger qi2)

QuadraticInteger &	operator= (const QuadraticInteger &)

QuadraticInteger &	operator/= (QuadraticInteger const &qi)

QuadraticInteger &	*operator=** (QuadraticInteger const &qi)

QuadraticInteger	operator+ (const QuadraticInteger &) const

QuadraticInteger	operator* (const QuadraticInteger &) const

QuadraticInteger	operator- (const QuadraticInteger &) const

QuadraticInteger	operator- () const

bool	operator> (const QuadraticInteger &)

Public Member Functions inherited from AlgebraicInteger
bool	operator== (const AlgebraicInteger &ai) const
	Check whether *this == ai.

bool	operator!= (const int &n) const
	Check whether *this != n.

bool	operator!= (const AlgebraicInteger &ai) const
	Check whether *this != ai.

Static Public Member Functions
static bool	bIsDAdmissible (const unsigned int &d)

static void	set_d (const unsigned int &d)

static vector< QuadraticInteger >	qiFactorsRationalPrime (const unsigned int &iPrime, bool bWithMultiplicities=false)

static array< long int, 2 >	iPellEquation (const unsigned int &iPrime)

static long int	iSQRT_quotient (const QuadraticInteger &qiNum, const QuadraticInteger &qiDen)

static long int	iSQRTsup_quotient (const QuadraticInteger &qiNum, const QuadraticInteger &qiDen)

Public Attributes
long int	a
	First component of the quadratic integer (in the usual Z-basis)

long int	b
	Second component of the quadratic integer (in the usual Z-basis)

Static Public Attributes
static int	d = 0
	We work in Q[ sqrt d ].

static bool	bIsOneMod4 = false
	True if d is 1 mod 4, false otherwise.

static int	iDiscriminant = 0
	Discriminant of the quadratic field.

static map< unsigned int, vector< unsigned int > >	iPellMinimalSolution
	Solutions for the Pell equations (used to compute the decompositions of rational prime numbers)

static map< unsigned int, vector< long int > >	iFundamentalUnits
	Fundamental units.

Detailed Description

Quadratic integers.

Member Function Documentation

void QuadraticInteger::add ( const AlgebraicInteger * ai )

virtual

Performs the addition of *this by the parameter.

Parameters

ai(	const AlgebraicInteger* ) Pointer to the summand

Implements AlgebraicInteger.

AlgebraicInteger * QuadraticInteger::aiCopyToInteger ( const int & n ) const

virtual

Create an algebraic integer from an integer.

Parameters

n(	const int ): Integer

Returns: Pointer to the algebraic integer created

Implements AlgebraicInteger.

bool QuadraticInteger::bIsDivisbleBy ( const AlgebraicInteger * ai ) const

virtual

Check whether *this is divisible by the parameter.

Parameters

ai(	const AlgebraicInteger* ) Pointer to the divisor

Returns: True if *this is divisible by the parameter

Implements AlgebraicInteger.

bool QuadraticInteger::bIsInvertible ( ) const

virtual

Check whether the number is invertible.

Returns: True if invertible or not

Implements AlgebraicInteger.

bool QuadraticInteger::bIsSquareOfIvertible ( ) const

virtual

Check whether the number is the square of an invertible element.

Returns: True if invertible or not

Implements AlgebraicInteger.

AlgebraicInteger * QuadraticInteger::copy ( ) const

virtual

Create a copy of an algebraic integer.

Returns: Pointer to the algebraic integer created

Implements AlgebraicInteger.

void QuadraticInteger::divideBy ( const AlgebraicInteger * ai )

virtual

Performs the division of *this by the parameter Remark: we do not check that the *this is divisible by *ai. If it is not the case, the division will be an approximation!

Parameters

ai(	const AlgebraicInteger* ) Pointer to the divisor

Implements AlgebraicInteger.

void QuadraticInteger::gcd ( const AlgebraicInteger * ai )

virtual

*this becomes the gcd of *this and the parameter

Parameters

ai(	const AlgebraicInteger* ) The other algebraic integer

Implements AlgebraicInteger.

void QuadraticInteger::multiplyBy ( const int & n )

virtual

Performs the multiplication of *this by the parameter.

Parameters

ai(	const AlgebraicInteger* ) Pointer to the multiplicand

Implements AlgebraicInteger.

void QuadraticInteger::multiplyBy ( const AlgebraicInteger * ai )

virtual

Performs the multiplication of *this by the parameter.

Parameters

ai(	const AlgebraicInteger* ) Pointer to the multiplicand

Implements AlgebraicInteger.

void QuadraticInteger::set ( const int & n )

virtual

Assign an integer to the algebraic integer.

Parameters

n(	const int& ) Integer

Implements AlgebraicInteger.

void QuadraticInteger::set ( AlgebraicInteger * ai )

virtual

Assign an algebraic integer to the algebraic integer.

Parameters

ai(	AlgebraicInteger* ) Pointer to the algebraic integer

Implements AlgebraicInteger.

void QuadraticInteger::substract ( const AlgebraicInteger * ai )

virtual

Substracts the parameter from *this.

Parameters

ai(	const AlgebraicInteger* ) Pointer to the subtrahend

Implements AlgebraicInteger.

std::string QuadraticInteger::to_string	(	const string &	strFormat = `"generic"`,
		const bool &	bProtect = `false`
	)		const

virtual

Convert to string.

Parameters

strFormat(	const string& ): Format for the output; can be: generic, mathematica, pari, latex
bProtect(	const bool& ): If true, avoid certain characters (for example if the string is used in for a file name)

Implements AlgebraicInteger.

The documentation for this class was generated from the following files:

quadraticinteger.h
quadraticinteger.cpp

Public Member Functions

Static Public Member Functions

Public Attributes

Static Public Attributes

Detailed Description

Member Function Documentation