DenseUnivariatePolynomial< Field > Class Template Reference

A univariate polynomial with arbitrary BPASField coefficients represented densely. More...

#include <dupolynomial.h>

Simplified semantic inheritance diagram for DenseUnivariatePolynomial< Field >:

 Full inheritance diagram for DenseUnivariatePolynomial< Field >:

Public Member Functions
	DenseUnivariatePolynomial ()
	Construct a polynomial. More...
	DenseUnivariatePolynomial (int s)
	Construct a polynomial with degree. More...
	DenseUnivariatePolynomial (Field e)
	DenseUnivariatePolynomial (const DenseUnivariatePolynomial< Field > &b)
	Copy constructor. More...
	~DenseUnivariatePolynomial ()
	Destroy the polynomial. More...
Integer	degree () const
	Get degree of the polynomial. More...
Field	leadingCoefficient () const
	Get the leading coefficient. More...
Field	trailingCoefficient () const
Integer	numberOfTerms () const
Field *	coefficients (int k=0)
	Get coefficients of the polynomial, given start offset. More...
Field	coefficient (int k) const
	Get a coefficient of the polynomial. More...
void	setCoefficient (int k, const Field &value)
	Set a coefficient of the polynomial. More...
Symbol	variable () const
	Get variable's name. More...
void	setVariableName (const Symbol &x)
	Set variable's name. More...
DenseUnivariatePolynomial< Field > &	operator= (const DenseUnivariatePolynomial< Field > &b)
	Overload operator =. More...
DenseUnivariatePolynomial< Field > &	operator= (const Field &f)
bool	operator!= (const DenseUnivariatePolynomial< Field > &b) const
	Overload operator !=. More...
bool	operator== (const DenseUnivariatePolynomial< Field > &b) const
	Overload operator ==. More...
bool	isZero () const
	Is zero polynomial. More...
void	zero ()
	Zero polynomial. More...
bool	isOne () const
	Is polynomial a constatn 1. More...
void	one ()
	Set polynomial to 1. More...
bool	isNegativeOne () const
	Is polynomial a constatn -1. More...
void	negativeOne ()
	Set polynomial to -1. More...
int	isConstant () const
	Is a constant. More...
DenseUnivariatePolynomial< Field >	unitCanonical (DenseUnivariatePolynomial< Field > *u, DenseUnivariatePolynomial< Field > *v) const
	Obtain the unit normal (a.k.a canonical associate) of an element. More...
Field	content () const
	Content of the polynomial. More...
DenseUnivariatePolynomial< Field >	primitivePart () const
DenseUnivariatePolynomial< Field >	operator^ (long long int e) const
	Overload operator ^ replace xor operation by exponentiation. More...
DenseUnivariatePolynomial< Field > &	operator^= (long long int e)
	Overload operator ^= replace xor operation by exponentiation. More...
DenseUnivariatePolynomial< Field >	operator<< (int k) const
	Overload operator << replace by muplitying x^k. More...
DenseUnivariatePolynomial< Field > &	operator<<= (int k)
	Overload operator <<= replace by muplitying x^k. More...
DenseUnivariatePolynomial< Field >	operator>> (int k) const
	Overload operator >> replace by dividing x^k, and return the quotient. More...
DenseUnivariatePolynomial< Field > &	operator>>= (int k)
	Overload operator >>= replace by dividing x^k, and return the quotient. More...
DenseUnivariatePolynomial< Field >	operator+ (const DenseUnivariatePolynomial< Field > &b) const
	Overload operator +. More...
DenseUnivariatePolynomial< Field > &	operator+= (const DenseUnivariatePolynomial< Field > &b)
	Overload Operator +=. More...
void	add (const DenseUnivariatePolynomial< Field > &b)
	Add another polynomial to itself. More...
DenseUnivariatePolynomial< Field >	operator+ (const Field &c) const
	Overload Operator +. More...
DenseUnivariatePolynomial< Field > &	operator+= (const Field &c)
	Overload Operator +=. More...
DenseUnivariatePolynomial< Field >	operator- (const DenseUnivariatePolynomial< Field > &b) const
	Subtract another polynomial. More...
DenseUnivariatePolynomial< Field > &	operator-= (const DenseUnivariatePolynomial< Field > &b)
	Overload operator -=. More...
DenseUnivariatePolynomial< Field >	operator- () const
	Overload operator -, negate. More...
void	subtract (const DenseUnivariatePolynomial< Field > &b)
	Subtract another polynomial from itself. More...
DenseUnivariatePolynomial< Field >	operator- (const Field &c) const
	Overload operator -. More...
DenseUnivariatePolynomial< Field > &	operator-= (const Field &c)
	Overload operator -=. More...
bool	SRGFNcmp (mpz_class &p)
	Helper function to compare a given prime number with manually computed Generalized Fermat Numbers. More...
DenseUnivariatePolynomial< Field >	operator* (const DenseUnivariatePolynomial< Field > &b) const
	Multiply to another polynomial. More...
void	FFT (SmallPrimeField *field, int K, int e, SmallPrimeField &omega)
void	FFT (BigPrimeField *field, int K, int e, BigPrimeField &omega)
void	FFT (GeneralizedFermatPrimeField *field, int K, int e, GeneralizedFermatPrimeField &omega)
void	IFFT (SmallPrimeField *field, int K, int e, SmallPrimeField &omega)
void	IFFT (BigPrimeField *field, int K, int e, BigPrimeField &omega)
void	IFFT (GeneralizedFermatPrimeField *field, int K, int e, GeneralizedFermatPrimeField &omega)
DenseUnivariatePolynomial< Field > &	operator*= (const DenseUnivariatePolynomial< Field > &b)
	Overload operator *=. More...
DenseUnivariatePolynomial< Field >	operator* (const Field &e) const
	Overload operator *. More...
DenseUnivariatePolynomial< Field > &	operator*= (const Field &e)
	Overload operator *=. More...
DenseUnivariatePolynomial< Field >	operator/ (const DenseUnivariatePolynomial< Field > &b) const
	Overload operator / ExactDivision. More...
DenseUnivariatePolynomial< Field > &	operator/= (const DenseUnivariatePolynomial< Field > &b)
	Overload operator /= ExactDivision. More...
DenseUnivariatePolynomial< Field >	operator/ (const Field &e) const
	Overload operator /. More...
DenseUnivariatePolynomial< Field > &	operator/= (const Field &e)
	Overload operator /=. More...
DenseUnivariatePolynomial< Field >	monicDivide (const DenseUnivariatePolynomial< Field > &b)
	Monic division Return quotient and itself become the remainder. More...
DenseUnivariatePolynomial< Field >	monicDivide (const DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > *rem) const
	Monic division Return quotient. More...
DenseUnivariatePolynomial< Field >	lazyPseudoDivide (const DenseUnivariatePolynomial< Field > &b, Field *c, Field *d)
	Lazy pseudo dividsion Return the quotient and itself becomes remainder e is the exact number of division steps. More...
DenseUnivariatePolynomial< Field >	lazyPseudoDivide (const DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > *rem, Field *c, Field *d) const
	Lazy pseudo dividsion Return the quotient e is the exact number of division steps. More...
DenseUnivariatePolynomial< Field >	pseudoDivide (const DenseUnivariatePolynomial< Field > &b, Field *d=NULL)
	Pseudo dividsion Return the quotient and itself becomes remainder. More...
DenseUnivariatePolynomial< Field >	pseudoDivide (const DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > *rem, Field *d) const
	Pseudo dividsion Return the quotient. More...
DenseUnivariatePolynomial< Field >	halfExtendedEuclidean (const DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > *g)
	s * a g (mod b), where g = gcd(a, b) More...
void	diophantinEquationSolve (const DenseUnivariatePolynomial< Field > &a, const DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > *s, DenseUnivariatePolynomial< Field > *t)
	s*a + t*b = c, where c in the ideal (a,b) More...
void	differentiate (int k)
	Compute k-th differentiate. More...
void	differentiate ()
	Differentiate this polynomial, setting itself to its derivative.
DenseUnivariatePolynomial< Field >	derivative (int k) const
	Obtain the kth derivative of this polynomial. More...
DenseUnivariatePolynomial< Field >	derivative () const
	Obtain the derivative of this polynomial. More...
DenseUnivariatePolynomial< Field >	integrate () const
	Compute the integral with constant of integration 0. More...
Field	evaluate (const Field &x) const
	Evaluate f(x) More...
bool	isConstantTermZero () const
	Is the least signficant coefficient zero. More...
DenseUnivariatePolynomial< Field >	gcd (const DenseUnivariatePolynomial< Field > &q, int type) const
	GCD(p, q) More...
DenseUnivariatePolynomial< Field >	gcd (const DenseUnivariatePolynomial< Field > &q) const
	Get GCD of *this and other. More...
Factors< DenseUnivariatePolynomial< Field > >	squareFree () const
	Square free. More...
bool	divideByVariableIfCan ()
	Divide by variable if it is zero. More...
void	reciprocal ()
	Number of coefficient sign variation. More...
void	homothetic (int k)
	Homothetic operation. More...
void	scaleTransform (int k)
	Scale transform operation. More...
void	negativeVariable ()
	Compute f(-x) More...
void	negate ()
	Compute -f(x) More...
void	print (std::ostream &out) const
	Overload stream operator <<. More...
ExpressionTree	convertToExpressionTree () const
	Convert this to an expression tree. More...
Integer	euclideanSize () const
	Euclidean domain methods.
DenseUnivariatePolynomial< Field >	euclideanDivision (const DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > *q=NULL) const
	Perform the eucldiean division of *this and b. More...
DenseUnivariatePolynomial< Field >	extendedEuclidean (const DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > *s=NULL, DenseUnivariatePolynomial< Field > *t=NULL) const
	Perform the extended euclidean division on *this and b. More...
DenseUnivariatePolynomial< Field >	quotient (const DenseUnivariatePolynomial< Field > &b) const
	Get the quotient of *this and b. More...
DenseUnivariatePolynomial< Field >	remainder (const DenseUnivariatePolynomial< Field > &b) const
	Get the remainder of *this and b. More...
DenseUnivariatePolynomial< Field >	operator% (const DenseUnivariatePolynomial< Field > &b) const
	Get the remainder of *this and b;. More...
DenseUnivariatePolynomial< Field > &	operator%= (const DenseUnivariatePolynomial< Field > &b)
	Assign *this to be the remainder of *this and b. More...
DenseUnivariatePolynomial< Field >	Reverse () const
	Reverse a polynomial (based on Modern Computer Algebra , Newton Iteration Chapter 9) More...
DenseUnivariatePolynomial< Field >	NewtonIterationInversion (int l)
	Does Newton Iteration to compute the Reverse Inverse of a polynomial (based on Modern Computer Algebra , Newton Iteration Chapter 9) More...
void	NewtonDivisionQuotient (DenseUnivariatePolynomial< Field > &b, DenseUnivariatePolynomial< Field > &q, DenseUnivariatePolynomial< Field > &r, int l)
	Fast Division (based on Modern Computer Algebra , Newton Iteration Chapter 9) More...

Public Attributes
mpz_class	characteristic

Static Public Attributes
static mpz_class	p1
static mpz_class	p2
static mpz_class	p3
static mpz_class	p4
static mpz_class	p5
static mpz_class	p6
static mpz_class	p7

Friends
DenseUnivariatePolynomial< Field >	operator+ (Field c, DenseUnivariatePolynomial< Field > p)
DenseUnivariatePolynomial< Field >	operator- (const Field &c, const DenseUnivariatePolynomial< Field > &p)
DenseUnivariatePolynomial< Field >	operator* (const Field &c, const DenseUnivariatePolynomial< Field > &p)
DenseUnivariatePolynomial< Field >	operator/ (const Field &c, const DenseUnivariatePolynomial< Field > &p)

Detailed Description

template<class Field>
class DenseUnivariatePolynomial< Field >

A univariate polynomial with arbitrary BPASField coefficients represented densely.

This class is templated by a Field which should be a BPASField.

Constructor & Destructor Documentation

DenseUnivariatePolynomial() [1/3]

template<class Field>

DenseUnivariatePolynomial< Field >::DenseUnivariatePolynomial ( )

inline

Construct a polynomial.

Parameters

d

DenseUnivariatePolynomial() [2/3]

template<class Field>

DenseUnivariatePolynomial< Field >::DenseUnivariatePolynomial ( int  s )

inline

Construct a polynomial with degree.

Parameters

d	Size of the polynomial

DenseUnivariatePolynomial() [3/3]

template<class Field>

DenseUnivariatePolynomial< Field >::DenseUnivariatePolynomial ( const DenseUnivariatePolynomial< Field > &  b )

inline

Copy constructor.

Parameters

b	A dense univariate rational polynomial

~DenseUnivariatePolynomial()

template<class Field>

DenseUnivariatePolynomial< Field >::~DenseUnivariatePolynomial ( )

inline

Destroy the polynomial.

Parameters

Member Function Documentation

add()

template<class Field>

void DenseUnivariatePolynomial< Field >::add ( const DenseUnivariatePolynomial< Field > &  b )

inline

Add another polynomial to itself.

Parameters

b	A univariate rational polynomial

coefficient()

template<class Field>

Field DenseUnivariatePolynomial< Field >::coefficient ( int  k ) const

inlinevirtual

Get a coefficient of the polynomial.

Parameters

k

Offset

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

coefficients()

template<class Field>

Field* DenseUnivariatePolynomial< Field >::coefficients ( int  k = 0 )

inline

Get coefficients of the polynomial, given start offset.

Parameters

k

Offset

content()

template<class Field>

Field DenseUnivariatePolynomial< Field >::content ( ) const

inline

Content of the polynomial.

Parameters

convertToExpressionTree()

template<class Field>

ExpressionTree DenseUnivariatePolynomial< Field >::convertToExpressionTree ( ) const

inlinevirtual

Convert this to an expression tree.

returns an expression tree describing *this.

Implements ExpressionTreeConvert.

degree()

template<class Field>

Integer DenseUnivariatePolynomial< Field >::degree ( ) const

inline

Get degree of the polynomial.

Parameters

derivative() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::derivative ( int  ) const

inlinevirtual

Obtain the kth derivative of this polynomial.

Parameters

k	the number of times to differentiate.

Returns

the kth derivative.

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

derivative() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::derivative ( ) const

inlinevirtual

Obtain the derivative of this polynomial.

Returns

the derivative.

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

differentiate()

template<class Field>

void DenseUnivariatePolynomial< Field >::differentiate ( int  k )

inlinevirtual

Compute k-th differentiate.

Parameters

k	k-th differentiate, k > 0

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

diophantinEquationSolve()

template<class Field>

void DenseUnivariatePolynomial< Field >::diophantinEquationSolve ( const DenseUnivariatePolynomial< Field > &  a, const DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > *  s, DenseUnivariatePolynomial< Field > *  t  )

inline

s*a + t*b = c, where c in the ideal (a,b)

Parameters

a	A univariate polynomial b: A univariate polynomial
s	Either s = 0 or degree(s) < degree(b)
t

divideByVariableIfCan()

template<class Field>

bool DenseUnivariatePolynomial< Field >::divideByVariableIfCan ( )

inline

Divide by variable if it is zero.

Parameters

euclideanDivision()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::euclideanDivision ( const DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > *  q = NULL  ) const

inlinevirtual

Perform the eucldiean division of *this and b.

Returns the remainder. If q is not NULL, then returns the quotient in q.

Parameters

	b	the divisor.
[out]	q	a pointer to store the quotient in.

Returns

the remainder.

Implements BPASEuclideanDomain< DenseUnivariatePolynomial< Field > >.

evaluate()

template<class Field>

Field DenseUnivariatePolynomial< Field >::evaluate ( const Field &  x ) const

inlinevirtual

Evaluate f(x)

Parameters

x	Evaluation point

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

extendedEuclidean()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::extendedEuclidean ( const DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > *  s = NULL, DenseUnivariatePolynomial< Field > *  t = NULL  ) const

inlinevirtual

Perform the extended euclidean division on *this and b.

Returns the GCD. If s and t are not NULL, returns the bezout coefficients in them.

Parameters

	b	the divisor.
[out]	s	the bezout coefficient of this.
[out]	t	the bezout coefficient of b.

Returns

the remainder.

Implements BPASEuclideanDomain< DenseUnivariatePolynomial< Field > >.

gcd() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::gcd ( const DenseUnivariatePolynomial< Field > &  q, int  type  ) const

inline

GCD(p, q)

Parameters

q	The other polynomial

gcd() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::gcd ( const DenseUnivariatePolynomial< Field > &  other ) const

inlinevirtual

Get GCD of *this and other.

Parameters

other

the other element to get a gcd with.

Returns

the gcd.

Implements BPASGCDDomain< DenseUnivariatePolynomial< Field > >.

halfExtendedEuclidean()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::halfExtendedEuclidean ( const DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > *  g  )

inline

s * a g (mod b), where g = gcd(a, b)

Parameters

b	A univariate polynomial
g	The GCD of a and b

homothetic()

template<class Field>

void DenseUnivariatePolynomial< Field >::homothetic ( int  k )

inline

Homothetic operation.

Parameters

k	> 0: 2^(k*d) * f(2^(-k)*x);

integrate()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::integrate ( ) const

inline

Compute the integral with constant of integration 0.

Parameters

isConstant()

template<class Field>

int DenseUnivariatePolynomial< Field >::isConstant ( ) const

inline

Is a constant.

Parameters

isConstantTermZero()

template<class Field>

bool DenseUnivariatePolynomial< Field >::isConstantTermZero ( ) const

inline

Is the least signficant coefficient zero.

Parameters

isNegativeOne()

template<class Field>

bool DenseUnivariatePolynomial< Field >::isNegativeOne ( ) const

inline

Is polynomial a constatn -1.

Parameters

isOne()

template<class Field>

bool DenseUnivariatePolynomial< Field >::isOne ( ) const

inlinevirtual

Is polynomial a constatn 1.

Parameters

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

isZero()

template<class Field>

bool DenseUnivariatePolynomial< Field >::isZero ( ) const

inlinevirtual

Is zero polynomial.

Parameters

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

lazyPseudoDivide() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::lazyPseudoDivide ( const DenseUnivariatePolynomial< Field > &  b, Field *  c, Field *  d  )

inlinevirtual

Lazy pseudo dividsion Return the quotient and itself becomes remainder e is the exact number of division steps.

Parameters

b	The dividend polynomial
c	The leading coefficient of b to the power e
d	That to the power deg(a) - deg(b) + 1 - e

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

lazyPseudoDivide() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::lazyPseudoDivide ( const DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > *  rem, Field *  c, Field *  d  ) const

inlinevirtual

Lazy pseudo dividsion Return the quotient e is the exact number of division steps.

Parameters

b	The divident polynomial
rem	The remainder polynomial
c	The leading coefficient of b to the power e
d	That to the power deg(a) - deg(b) + 1 - e

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

leadingCoefficient()

template<class Field>

Field DenseUnivariatePolynomial< Field >::leadingCoefficient ( ) const

inline

Get the leading coefficient.

Parameters

monicDivide() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::monicDivide ( const DenseUnivariatePolynomial< Field > &  b )

inlinevirtual

Monic division Return quotient and itself become the remainder.

Parameters

b	The dividend polynomial

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

monicDivide() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::monicDivide ( const DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > *  rem  ) const

inlinevirtual

Monic division Return quotient.

Parameters

b	The dividend polynomial
rem	The remainder polynomial

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

negate()

template<class Field>

void DenseUnivariatePolynomial< Field >::negate ( )

inline

Compute -f(x)

Parameters

negativeOne()

template<class Field>

void DenseUnivariatePolynomial< Field >::negativeOne ( )

inline

Set polynomial to -1.

Parameters

negativeVariable()

template<class Field>

void DenseUnivariatePolynomial< Field >::negativeVariable ( )

inline

Compute f(-x)

Parameters

NewtonDivisionQuotient()

template<class Field>

void DenseUnivariatePolynomial< Field >::NewtonDivisionQuotient ( DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > &  q, DenseUnivariatePolynomial< Field > &  r, int  l  )

inline

Fast Division (based on Modern Computer Algebra , Newton Iteration Chapter 9)

Parameters

b	divisor
q	(out) qoutient
r	(out) remainder
fieldVal	prime field
l	number of newton iteration

NewtonIterationInversion()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::NewtonIterationInversion ( int  l )

inline

Does Newton Iteration to compute the Reverse Inverse of a polynomial (based on Modern Computer Algebra , Newton Iteration Chapter 9)

Parameters

l	number of newton iteration (n - m + 1)

one()

template<class Field>

void DenseUnivariatePolynomial< Field >::one ( )

inlinevirtual

Set polynomial to 1.

Parameters

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator!=()

template<class Field>

bool DenseUnivariatePolynomial< Field >::operator!= ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Overload operator !=.

Parameters

b	A univariate rational polynoial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator%()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator% ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Get the remainder of *this and b;.

Parameters

b	the divisor

Returns

the remainder

Implements BPASEuclideanDomain< DenseUnivariatePolynomial< Field > >.

operator%=()

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator%= ( const DenseUnivariatePolynomial< Field > &  b )

inlinevirtual

Assign *this to be the remainder of *this and b.

Parameters

b	the divisor

Returns

this after assignment.

Implements BPASEuclideanDomain< DenseUnivariatePolynomial< Field > >.

operator*() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator* ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Multiply to another polynomial.

Parameters

b	A univariate rational polynomial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator*() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator* ( const Field &  e ) const

inline

Overload operator *.

Parameters

e	A rational number

operator*=() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator*= ( const DenseUnivariatePolynomial< Field > &  b )

inlinevirtual

Overload operator *=.

Parameters

b	A univariate rational polynomial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator*=() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator*= ( const Field &  e )

inline

Overload operator *=.

Parameters

e	A rational number

operator+() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator+ ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Overload operator +.

Parameters

b	A univariate rational polynomial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator+() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator+ ( const Field &  c ) const

inline

Overload Operator +.

Parameters

c	A rational number

operator+=() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator+= ( const DenseUnivariatePolynomial< Field > &  b )

inlinevirtual

Overload Operator +=.

Parameters

b	A univariate rational polynomial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator+=() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator+= ( const Field &  c )

inline

Overload Operator +=.

Parameters

c	A rational number

operator-() [1/3]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator- ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Subtract another polynomial.

Parameters

b	A univariate rational polynomial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator-() [2/3]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator- ( ) const

inlinevirtual

Overload operator -, negate.

Parameters

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator-() [3/3]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator- ( const Field &  c ) const

inline

Overload operator -.

Parameters

c	A rational number

operator-=() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator-= ( const DenseUnivariatePolynomial< Field > &  b )

inlinevirtual

Overload operator -=.

Parameters

b	A univariate rational polynomial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator-=() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator-= ( const Field &  c )

inline

Overload operator -=.

Parameters

c	A rational number

operator/() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator/ ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Overload operator / ExactDivision.

Parameters

b	A univariate rational polynomial

Implements BPASIntegralDomain< DenseUnivariatePolynomial< Field > >.

operator/() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator/ ( const Field &  e ) const

inline

Overload operator /.

Parameters

e	A rational number

operator/=() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator/= ( const DenseUnivariatePolynomial< Field > &  b )

inlinevirtual

Overload operator /= ExactDivision.

Parameters

b	A univariate rational polynomial

Implements BPASIntegralDomain< DenseUnivariatePolynomial< Field > >.

operator/=() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator/= ( const Field &  e )

inline

Overload operator /=.

Parameters

e	A rational number

operator<<()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator<< ( int  k ) const

inlinevirtual

Overload operator << replace by muplitying x^k.

Parameters

k	The exponent of variable, k > 0

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

operator<<=()

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator<<= ( int  k )

inlinevirtual

Overload operator <<= replace by muplitying x^k.

Parameters

k	The exponent of variable, k > 0

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

operator=()

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator= ( const DenseUnivariatePolynomial< Field > &  b )

inlinevirtual

Overload operator =.

Parameters

b	A univariate rational polynoial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator==()

template<class Field>

bool DenseUnivariatePolynomial< Field >::operator== ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Overload operator ==.

Parameters

b	A univariate rational polynoial

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator>>()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator>> ( int  k ) const

inlinevirtual

Overload operator >> replace by dividing x^k, and return the quotient.

Parameters

k	The exponent of variable, k > 0

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

operator>>=()

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator>>= ( int  k )

inlinevirtual

Overload operator >>= replace by dividing x^k, and return the quotient.

Parameters

k	The exponent of variable, k > 0

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

operator^()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::operator^ ( long long int  e ) const

inlinevirtual

Overload operator ^ replace xor operation by exponentiation.

Parameters

e	The exponentiation, e > 0

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

operator^=()

template<class Field>

DenseUnivariatePolynomial<Field>& DenseUnivariatePolynomial< Field >::operator^= ( long long int  e )

inlinevirtual

Overload operator ^= replace xor operation by exponentiation.

Parameters

e	The exponentiation, e > 0

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

print()

template<class Field>

void DenseUnivariatePolynomial< Field >::print ( std::ostream &  out ) const

inlinevirtual

Overload stream operator <<.

Parameters

out	Stream object
b	A univariate rational polynoial

Reimplemented from BPASRing< DenseUnivariatePolynomial< Field > >.

pseudoDivide() [1/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::pseudoDivide ( const DenseUnivariatePolynomial< Field > &  b, Field *  d = NULL  )

inlinevirtual

Pseudo dividsion Return the quotient and itself becomes remainder.

Parameters

b	The divident polynomial
d	The leading coefficient of b to the power deg(a) - deg(b) + 1

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

pseudoDivide() [2/2]

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::pseudoDivide ( const DenseUnivariatePolynomial< Field > &  b, DenseUnivariatePolynomial< Field > *  rem, Field *  d  ) const

inlinevirtual

Pseudo dividsion Return the quotient.

Parameters

b	The divident polynomial
rem	The remainder polynomial
d	The leading coefficient of b to the power deg(a) - deg(b) + 1

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

quotient()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::quotient ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Get the quotient of *this and b.

Parameters

b	the divisor

Returns

the quotient

Implements BPASEuclideanDomain< DenseUnivariatePolynomial< Field > >.

reciprocal()

template<class Field>

void DenseUnivariatePolynomial< Field >::reciprocal ( )

inline

Number of coefficient sign variation.

Parameters

Revert	coefficients

remainder()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::remainder ( const DenseUnivariatePolynomial< Field > &  b ) const

inlinevirtual

Get the remainder of *this and b.

Parameters

b	the divisor

Returns

the remainder

Implements BPASEuclideanDomain< DenseUnivariatePolynomial< Field > >.

Reverse()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::Reverse ( ) const

inline

Reverse a polynomial (based on Modern Computer Algebra , Newton Iteration Chapter 9)

Parameters

scaleTransform()

template<class Field>

void DenseUnivariatePolynomial< Field >::scaleTransform ( int  k )

inline

Scale transform operation.

Parameters

k	> 0: f(2^k*x)

setCoefficient()

template<class Field>

void DenseUnivariatePolynomial< Field >::setCoefficient ( int  k, const Field &  value  )

inlinevirtual

Set a coefficient of the polynomial.

Parameters

k	Offset
val	Coefficient

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

setVariableName()

template<class Field>

void DenseUnivariatePolynomial< Field >::setVariableName ( const Symbol &  x )

inlinevirtual

Set variable's name.

Parameters

x	Varable's name

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

squareFree()

template<class Field>

Factors<DenseUnivariatePolynomial<Field> > DenseUnivariatePolynomial< Field >::squareFree ( ) const

inlinevirtual

Square free.

Parameters

Implements BPASGCDDomain< DenseUnivariatePolynomial< Field > >.

SRGFNcmp()

template<class Field>

bool DenseUnivariatePolynomial< Field >::SRGFNcmp ( mpz_class &  p )

inline

Helper function to compare a given prime number with manually computed Generalized Fermat Numbers.

Parameters

p	prime number

subtract()

template<class Field>

void DenseUnivariatePolynomial< Field >::subtract ( const DenseUnivariatePolynomial< Field > &  b )

inline

Subtract another polynomial from itself.

Parameters

b	A univariate rational polynomial

unitCanonical()

template<class Field>

DenseUnivariatePolynomial<Field> DenseUnivariatePolynomial< Field >::unitCanonical ( DenseUnivariatePolynomial< Field > *  u, DenseUnivariatePolynomial< Field > *  v  ) const

inlinevirtual

Obtain the unit normal (a.k.a canonical associate) of an element.

If either parameters u, v, are non-NULL then the units are returned such that b = ua, v = u^-1. Where b is the unit normal of a, and is the returned value.

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

variable()

template<class Field>

Symbol DenseUnivariatePolynomial< Field >::variable ( ) const

inlinevirtual

Get variable's name.

Parameters

Implements BPASUnivariatePolynomial< Field, DenseUnivariatePolynomial< Field > >.

zero()

template<class Field>

void DenseUnivariatePolynomial< Field >::zero ( )

inlinevirtual

Zero polynomial.

Parameters

Implements BPASRing< DenseUnivariatePolynomial< Field > >.

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

• include/RingPolynomial/dupolynomial.h