DenseUnivariateRationalPolynomial Class Reference

A univariate polynomial with RationalNumber coefficients represented densely. More...

#include <urpolynomial.h>

Simplified semantic inheritance diagram for DenseUnivariateRationalPolynomial:

 Full inheritance diagram for DenseUnivariateRationalPolynomial:

Public Member Functions
	DenseUnivariateRationalPolynomial ()
	Construct a polynomial. More...
	DenseUnivariateRationalPolynomial (int s)
	Construct a polynomial with degree. More...
	DenseUnivariateRationalPolynomial (const Integer &e)
	Construct a polynomial with a coefficient. More...
	DenseUnivariateRationalPolynomial (const RationalNumber &e)
	DenseUnivariateRationalPolynomial (const DenseUnivariateRationalPolynomial &b)
	Copy constructor. More...
	~DenseUnivariateRationalPolynomial ()
	Destroy the polynomial. More...
Integer	degree () const
	Get degree of the polynomial. More...
RationalNumber	leadingCoefficient () const
	Get the leading coefficient. More...
RationalNumber	trailingCoefficient () const
Integer	numberOfTerms () const
mpq_class *	coefficients (int k=0) const
	Get coefficients of the polynomial, given start offset. More...
RationalNumber	coefficient (int k) const
	Get a coefficient of the polynomial. More...
void	setCoefficient (int k, const RationalNumber &value)
	Set a coefficient of the polynomial. More...
void	setCoefficient (int k, double value)
Symbol	variable () const
	Get variable's name. More...
void	setVariableName (const Symbol &x)
	Set variable's name. More...
DenseUnivariateRationalPolynomial	unitCanonical (DenseUnivariateRationalPolynomial *u=NULL, DenseUnivariateRationalPolynomial *v=NULL) const
DenseUnivariateRationalPolynomial &	operator= (const DenseUnivariateRationalPolynomial &b)
	Overload operator =. More...
DenseUnivariateRationalPolynomial &	operator= (const RationalNumber &r)
bool	operator!= (const DenseUnivariateRationalPolynomial &b) const
	Overload operator !=. More...
bool	operator== (const DenseUnivariateRationalPolynomial &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...
RationalNumber	content () const
	Content of the polynomial. More...
DenseUnivariateRationalPolynomial	primitivePart () const
DenseUnivariateRationalPolynomial	operator^ (long long int e) const
	Overload operator ^ replace xor operation by exponentiation. More...
DenseUnivariateRationalPolynomial &	operator^= (long long int e)
	Overload operator ^= replace xor operation by exponentiation. More...
DenseUnivariateRationalPolynomial	operator<< (int k) const
	Overload operator << replace by muplitying x^k. More...
DenseUnivariateRationalPolynomial &	operator<<= (int k)
	Overload operator <<= replace by muplitying x^k. More...
DenseUnivariateRationalPolynomial	operator>> (int k) const
	Overload operator >> replace by dividing x^k, and return the quotient. More...
DenseUnivariateRationalPolynomial &	operator>>= (int k)
	Overload operator >>= replace by dividing x^k, and return the quotient. More...
DenseUnivariateRationalPolynomial	operator+ (const DenseUnivariateRationalPolynomial &b) const
	Overload operator +. More...
DenseUnivariateRationalPolynomial &	operator+= (const DenseUnivariateRationalPolynomial &b)
	Overload Operator +=. More...
void	add (const DenseUnivariateRationalPolynomial &b)
	Add another polynomial to itself. More...
DenseUnivariateRationalPolynomial	operator+ (const RationalNumber &c) const
	Overload Operator +. More...
DenseUnivariateRationalPolynomial	operator+ (const mpq_class &c) const
DenseUnivariateRationalPolynomial &	operator+= (const RationalNumber &c)
	Overload Operator +=. More...
DenseUnivariateRationalPolynomial &	operator+= (const mpq_class &c)
DenseUnivariateRationalPolynomial	operator- (const DenseUnivariateRationalPolynomial &b) const
	Subtract another polynomial. More...
DenseUnivariateRationalPolynomial &	operator-= (const DenseUnivariateRationalPolynomial &b)
	Overload operator -=. More...
DenseUnivariateRationalPolynomial	operator- () const
	Overload operator -, negate. More...
void	subtract (const DenseUnivariateRationalPolynomial &b)
	Subtract another polynomial from itself. More...
DenseUnivariateRationalPolynomial	operator- (const RationalNumber &c) const
	Overload operator -. More...
DenseUnivariateRationalPolynomial	operator- (const mpq_class &c) const
DenseUnivariateRationalPolynomial &	operator-= (const RationalNumber &c)
	Overload operator -=. More...
DenseUnivariateRationalPolynomial &	operator-= (const mpq_class &c)
DenseUnivariateRationalPolynomial	operator* (const DenseUnivariateRationalPolynomial &b) const
	Multiply to another polynomial. More...
DenseUnivariateRationalPolynomial &	operator*= (const DenseUnivariateRationalPolynomial &b)
	Overload operator *=. More...
DenseUnivariateRationalPolynomial	operator* (const RationalNumber &e) const
	Overload operator *. More...
DenseUnivariateRationalPolynomial	operator* (const mpq_class &e) const
DenseUnivariateRationalPolynomial	operator* (const sfixn &e) const
DenseUnivariateRationalPolynomial &	operator*= (const RationalNumber &e)
	Overload operator *=. More...
DenseUnivariateRationalPolynomial &	operator*= (const mpq_class &e)
DenseUnivariateRationalPolynomial &	operator*= (const sfixn &e)
	Overload operator *=. More...
DenseUnivariateRationalPolynomial	operator/ (const DenseUnivariateRationalPolynomial &b) const
	Overload operator / ExactDivision. More...
DenseUnivariateRationalPolynomial &	operator/= (const DenseUnivariateRationalPolynomial &b)
	Overload operator /= ExactDivision. More...
DenseUnivariateRationalPolynomial	operator% (const DenseUnivariateRationalPolynomial &b) const
DenseUnivariateRationalPolynomial &	operator%= (const DenseUnivariateRationalPolynomial &b)
DenseUnivariateRationalPolynomial	operator/ (const RationalNumber &e) const
	Overload operator /. More...
DenseUnivariateRationalPolynomial	operator/ (const mpq_class &e) const
DenseUnivariateRationalPolynomial &	operator/= (const RationalNumber &e)
	Overload operator /=. More...
DenseUnivariateRationalPolynomial &	operator/= (const mpq_class &e)
DenseUnivariateRationalPolynomial	monicDivide (const DenseUnivariateRationalPolynomial &b)
	Monic division Return quotient and itself become the remainder. More...
DenseUnivariateRationalPolynomial	monicDivide (const DenseUnivariateRationalPolynomial &b, DenseUnivariateRationalPolynomial *rem) const
	Monic division Return quotient. More...
DenseUnivariateRationalPolynomial	lazyPseudoDivide (const DenseUnivariateRationalPolynomial &b, RationalNumber *c, RationalNumber *d=NULL)
	Lazy pseudo dividsion Return the quotient and itself becomes remainder e is the exact number of division steps. More...
DenseUnivariateRationalPolynomial	lazyPseudoDivide (const DenseUnivariateRationalPolynomial &b, DenseUnivariateRationalPolynomial *rem, RationalNumber *c, RationalNumber *d) const
	Lazy pseudo dividsion Return the quotient e is the exact number of division steps. More...
DenseUnivariateRationalPolynomial	pseudoDivide (const DenseUnivariateRationalPolynomial &b, RationalNumber *d=NULL)
	Pseudo dividsion Return the quotient and itself becomes remainder. More...
DenseUnivariateRationalPolynomial	pseudoDivide (const DenseUnivariateRationalPolynomial &b, DenseUnivariateRationalPolynomial *rem, RationalNumber *d) const
	Pseudo dividsion Return the quotient. More...
DenseUnivariateRationalPolynomial	halfExtendedEuclidean (const DenseUnivariateRationalPolynomial &b, DenseUnivariateRationalPolynomial *g) const
	s * a g (mod b), where g = gcd(a, b) More...
void	diophantinEquationSolve (const DenseUnivariateRationalPolynomial &a, const DenseUnivariateRationalPolynomial &b, DenseUnivariateRationalPolynomial *s, DenseUnivariateRationalPolynomial *t) const
	s*a + t*b = c, where c in the ideal (a,b) More...
void	differentiate (int k)
	Convert current object to its k-th derivative. More...
void	differentiate ()
	Convert current object to its derivative. More...
DenseUnivariateRationalPolynomial	derivative (int k) const
	Return k-th derivative. More...
DenseUnivariateRationalPolynomial	derivative () const
	Compute derivative. More...
void	integrate ()
	Compute the integral with constant of integration 0. More...
DenseUnivariateRationalPolynomial	integral () const
	Compute integral with constant of integration 0. More...
RationalNumber	evaluate (const RationalNumber &x) const
	Evaluate f(x) More...
Integer	evaluate (const Integer &x) const
	Evaluate f(x) More...
template<class LargerRing >
LargerRing	evaluate (const LargerRing &x) const
	Evaluate f(x) More...
bool	isConstantTermZero () const
	Is the least signficant coefficient zero. More...
DenseUnivariateRationalPolynomial	gcd (const DenseUnivariateRationalPolynomial &q, int type) const
	GCD(p, q) More...
DenseUnivariateRationalPolynomial	gcd (const DenseUnivariateRationalPolynomial &q) const
Factors< DenseUnivariateRationalPolynomial >	squareFree () const
	Square free. More...
bool	divideByVariableIfCan ()
	Divide by variable if it is zero. More...
int	numberOfSignChanges ()
	Number of coefficient sign variation. More...
void	reciprocal ()
	Revert coefficients. More...
void	homothetic (int k=1)
	Homothetic operation. More...
void	scaleTransform (int k)
	Scale transform operation. More...
void	negativeVariable ()
	Compute f(-x) More...
void	negate ()
	Compute -f(x) More...
mpz_class	rootBound ()
	Return an integer k such that any positive root alpha of the polynomial satisfies alpha < 2^k. More...
void	taylorShift (int ts=-1)
	Taylor Shift operation by 1. More...
Intervals	positiveRealRootIsolate (mpq_class width, int ts=-1)
	Positive real root isolation for square-free polynomials. More...
void	positiveRealRootIsolate (mpq_class width, Intervals pIs, int ts=-1)
Intervals	realRootIsolate (mpq_class width, int ts=-1)
	Real root isolation for square-free polynomials. More...
void	refineRoot (Interval *a, mpq_class width)
	Refine a root. More...
Intervals	refineRoots (Intervals &a, mpq_class width)
	Refine the roots. More...
void	print (std::ostream &out) const
	Overload stream operator <<. More...
ExpressionTree	convertToExpressionTree () const
Integer	euclideanSize () const
	BPASEuclideanDomain methods.
DenseUnivariateRationalPolynomial	euclideanDivision (const DenseUnivariateRationalPolynomial &b, DenseUnivariateRationalPolynomial *q=NULL) const
DenseUnivariateRationalPolynomial	extendedEuclidean (const DenseUnivariateRationalPolynomial &b, DenseUnivariateRationalPolynomial *s=NULL, DenseUnivariateRationalPolynomial *t=NULL) const
DenseUnivariateRationalPolynomial	quotient (const DenseUnivariateRationalPolynomial &b) const
DenseUnivariateRationalPolynomial	remainder (const DenseUnivariateRationalPolynomial &b) const

Friends
DenseUnivariateRationalPolynomial	operator+ (const mpq_class &c, const DenseUnivariateRationalPolynomial &p)
DenseUnivariateRationalPolynomial	operator- (const mpq_class &c, const DenseUnivariateRationalPolynomial &p)
DenseUnivariateRationalPolynomial	operator* (const mpq_class &c, const DenseUnivariateRationalPolynomial &p)
DenseUnivariateRationalPolynomial	operator* (const sfixn &c, const DenseUnivariateRationalPolynomial &p)
DenseUnivariateRationalPolynomial	operator/ (const mpq_class &c, const DenseUnivariateRationalPolynomial &p)

Detailed Description

A univariate polynomial with RationalNumber coefficients represented densely.

Constructor & Destructor Documentation

DenseUnivariateRationalPolynomial() [1/4]

DenseUnivariateRationalPolynomial::DenseUnivariateRationalPolynomial ( )

inline

Construct a polynomial.

Parameters

d

DenseUnivariateRationalPolynomial() [2/4]

DenseUnivariateRationalPolynomial::DenseUnivariateRationalPolynomial ( int  s )

inline

Construct a polynomial with degree.

Parameters

d	Size of the polynomial

DenseUnivariateRationalPolynomial() [3/4]

DenseUnivariateRationalPolynomial::DenseUnivariateRationalPolynomial ( const Integer &  e )

inline

Construct a polynomial with a coefficient.

Parameters

e	The coefficient

DenseUnivariateRationalPolynomial() [4/4]

DenseUnivariateRationalPolynomial::DenseUnivariateRationalPolynomial ( const DenseUnivariateRationalPolynomial &  b )

inline

Copy constructor.

Parameters

b	A densed univariate rationl polynomial

~DenseUnivariateRationalPolynomial()

DenseUnivariateRationalPolynomial::~DenseUnivariateRationalPolynomial ( )

inline

Destroy the polynomial.

Parameters

Member Function Documentation

add()

void DenseUnivariateRationalPolynomial::add

(

const DenseUnivariateRationalPolynomial &

b

)

Add another polynomial to itself.

Parameters

b	A univariate rational polynomial

coefficient()

RationalNumber DenseUnivariateRationalPolynomial::coefficient ( int  k ) const

inlinevirtual

Get a coefficient of the polynomial.

Parameters

k

Offset

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

coefficients()

mpq_class* DenseUnivariateRationalPolynomial::coefficients ( int  k = 0 ) const

inline

Get coefficients of the polynomial, given start offset.

Parameters

k

Offset

content()

RationalNumber DenseUnivariateRationalPolynomial::content ( ) const

inline

Content of the polynomial.

Parameters

degree()

Integer DenseUnivariateRationalPolynomial::degree ( ) const

inline

Get degree of the polynomial.

Parameters

derivative() [1/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::derivative ( int  k ) const

inlinevirtual

Return k-th derivative.

Parameters

k	k-th derivative, k > 0

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

derivative() [2/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::derivative ( ) const

inlinevirtual

Compute derivative.

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

differentiate() [1/2]

void DenseUnivariateRationalPolynomial::differentiate ( int  k )

virtual

Convert current object to its k-th derivative.

Parameters

k	Order of the derivative, k > 0

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

differentiate() [2/2]

void DenseUnivariateRationalPolynomial::differentiate ( )

inlinevirtual

Convert current object to its derivative.

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

diophantinEquationSolve()

void DenseUnivariateRationalPolynomial::diophantinEquationSolve	(	const DenseUnivariateRationalPolynomial &	a,
		const DenseUnivariateRationalPolynomial &	b,
		DenseUnivariateRationalPolynomial *	s,
		DenseUnivariateRationalPolynomial *	t
	)		const

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()

bool DenseUnivariateRationalPolynomial::divideByVariableIfCan

(

)

Divide by variable if it is zero.

Parameters

evaluate() [1/3]

RationalNumber DenseUnivariateRationalPolynomial::evaluate ( const RationalNumber &  x ) const

virtual

Evaluate f(x)

Parameters

x	Rational evaluation point

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

evaluate() [2/3]

Integer DenseUnivariateRationalPolynomial::evaluate

(

const Integer &

x

)

const

Evaluate f(x)

Parameters

x	Evaluation point

evaluate() [3/3]

template<class LargerRing >

LargerRing DenseUnivariateRationalPolynomial::evaluate ( const LargerRing &  x ) const

inline

Evaluate f(x)

Parameters

x	Evaluation point in a larger ring, i.e. a ring in which the rationals can be embedded

gcd()

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::gcd	(	const DenseUnivariateRationalPolynomial &	q,
		int	type
	)		const

GCD(p, q)

Parameters

q	The other polynomial

halfExtendedEuclidean()

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::halfExtendedEuclidean	(	const DenseUnivariateRationalPolynomial &	b,
		DenseUnivariateRationalPolynomial *	g
	)		const

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

Parameters

b	A univariate polynomial
g	The GCD of a and b

homothetic()

void DenseUnivariateRationalPolynomial::homothetic

(

int

k = 1

)

Homothetic operation.

Parameters

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

integral()

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::integral ( ) const

inline

Compute integral with constant of integration 0.

integrate()

void DenseUnivariateRationalPolynomial::integrate

(

)

Compute the integral with constant of integration 0.

Parameters

Convert

current object to its integral with constant of integration 0

isConstant()

int DenseUnivariateRationalPolynomial::isConstant ( ) const

inline

Is a constant.

Parameters

isConstantTermZero()

bool DenseUnivariateRationalPolynomial::isConstantTermZero

(

)

const

Is the least signficant coefficient zero.

Parameters

isNegativeOne()

bool DenseUnivariateRationalPolynomial::isNegativeOne ( ) const

inline

Is polynomial a constatn -1.

Parameters

isOne()

bool DenseUnivariateRationalPolynomial::isOne ( ) const

inline

Is polynomial a constatn 1.

Parameters

isZero()

bool DenseUnivariateRationalPolynomial::isZero ( ) const

inline

Is zero polynomial.

Parameters

lazyPseudoDivide() [1/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::lazyPseudoDivide ( const DenseUnivariateRationalPolynomial &  b, RationalNumber *  c, RationalNumber *  d = NULL  )

virtual

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< RationalNumber, DenseUnivariateRationalPolynomial >.

lazyPseudoDivide() [2/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::lazyPseudoDivide ( const DenseUnivariateRationalPolynomial &  b, DenseUnivariateRationalPolynomial *  rem, RationalNumber *  c, RationalNumber *  d  ) const

virtual

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< RationalNumber, DenseUnivariateRationalPolynomial >.

leadingCoefficient()

RationalNumber DenseUnivariateRationalPolynomial::leadingCoefficient ( ) const

inline

Get the leading coefficient.

Parameters

monicDivide() [1/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::monicDivide ( const DenseUnivariateRationalPolynomial &  b )

virtual

Monic division Return quotient and itself become the remainder.

Parameters

b	The dividend polynomial

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

monicDivide() [2/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::monicDivide ( const DenseUnivariateRationalPolynomial &  b, DenseUnivariateRationalPolynomial *  rem  ) const

virtual

Monic division Return quotient.

Parameters

b	The dividend polynomial
rem	The remainder polynomial

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

negate()

void DenseUnivariateRationalPolynomial::negate

(

)

Compute -f(x)

Parameters

negativeOne()

void DenseUnivariateRationalPolynomial::negativeOne ( )

inline

Set polynomial to -1.

Parameters

negativeVariable()

void DenseUnivariateRationalPolynomial::negativeVariable

(

)

Compute f(-x)

Parameters

numberOfSignChanges()

int DenseUnivariateRationalPolynomial::numberOfSignChanges

(

)

Number of coefficient sign variation.

Parameters

one()

void DenseUnivariateRationalPolynomial::one ( )

inline

Set polynomial to 1.

Parameters

operator!=()

bool DenseUnivariateRationalPolynomial::operator!= ( const DenseUnivariateRationalPolynomial &  b ) const

inline

Overload operator !=.

Parameters

b	A univariate rational polynoial

operator*() [1/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator*

(

const DenseUnivariateRationalPolynomial &

b

)

const

Multiply to another polynomial.

Parameters

b	A univariate rational polynomial

operator*() [2/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator* ( const RationalNumber &  e ) const

inline

Overload operator *.

Parameters

e	A rational number

operator*=() [1/3]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator*= ( const DenseUnivariateRationalPolynomial &  b )

inline

Overload operator *=.

Parameters

b	A univariate rational polynomial

operator*=() [2/3]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator*=

(

const RationalNumber &

e

)

Overload operator *=.

Parameters

e	A rational number

operator*=() [3/3]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator*=

(

const sfixn &

e

)

Overload operator *=.

Parameters

e	A constant

operator+() [1/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator+

(

const DenseUnivariateRationalPolynomial &

b

)

const

Overload operator +.

Parameters

b	A univariate rational polynomial

operator+() [2/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator+ ( const RationalNumber &  c ) const

inline

Overload Operator +.

Parameters

c	A rational number

operator+=() [1/2]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator+= ( const DenseUnivariateRationalPolynomial &  b )

inline

Overload Operator +=.

Parameters

b	A univariate rational polynomial

operator+=() [2/2]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator+= ( const RationalNumber &  c )

inline

Overload Operator +=.

Parameters

c	A rational number

operator-() [1/3]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator-

(

const DenseUnivariateRationalPolynomial &

b

)

const

Subtract another polynomial.

Parameters

b	A univariate rational polynomial

operator-() [2/3]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator-

(

)

const

Overload operator -, negate.

Parameters

operator-() [3/3]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator- ( const RationalNumber &  c ) const

inline

Overload operator -.

Parameters

c	A rational number

operator-=() [1/2]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator-= ( const DenseUnivariateRationalPolynomial &  b )

inline

Overload operator -=.

Parameters

b	A univariate rational polynomial

operator-=() [2/2]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator-= ( const RationalNumber &  c )

inline

Overload operator -=.

Parameters

c	A rational number

operator/() [1/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator/ ( const DenseUnivariateRationalPolynomial &  b ) const

inline

Overload operator / ExactDivision.

Parameters

b	A univariate rational polynomial

operator/() [2/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator/ ( const RationalNumber &  e ) const

inline

Overload operator /.

Parameters

e	A rational number

operator/=() [1/2]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator/=

(

const DenseUnivariateRationalPolynomial &

b

)

Overload operator /= ExactDivision.

Parameters

b	A univariate rational polynomial

operator/=() [2/2]

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator/=

(

const RationalNumber &

e

)

Overload operator /=.

Parameters

e	A rational number

operator<<()

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator<< ( int  k ) const

virtual

Overload operator << replace by muplitying x^k.

Parameters

k	The exponent of variable, k > 0

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

operator<<=()

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator<<= ( int  k )

inlinevirtual

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

Parameters

k	The exponent of variable, k > 0

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

operator=()

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator= ( const DenseUnivariateRationalPolynomial &  b )

inline

Overload operator =.

Parameters

b	A univariate rational polynoial

operator==()

bool DenseUnivariateRationalPolynomial::operator== ( const DenseUnivariateRationalPolynomial &  b ) const

inline

Overload operator ==.

Parameters

b	A univariate rational polynoial

operator>>()

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator>> ( int  k ) const

virtual

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

Parameters

k	The exponent of variable, k > 0

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

operator>>=()

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::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< RationalNumber, DenseUnivariateRationalPolynomial >.

operator^()

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::operator^

(

long long int

e

)

const

Overload operator ^ replace xor operation by exponentiation.

Parameters

e	The exponentiation, e > 0

operator^=()

DenseUnivariateRationalPolynomial& DenseUnivariateRationalPolynomial::operator^= ( long long int  e )

inline

Overload operator ^= replace xor operation by exponentiation.

Parameters

e	The exponentiation, e > 0

positiveRealRootIsolate()

Intervals DenseUnivariateRationalPolynomial::positiveRealRootIsolate ( mpq_class  width, int  ts = -1  )

inline

Positive real root isolation for square-free polynomials.

Parameters

width

Interval's right - left < width : Taylor Shift option: 0 - CMY; -1 - optimized

print()

void DenseUnivariateRationalPolynomial::print

(

std::ostream &

out

)

const

Overload stream operator <<.

Parameters

out	Stream object
b	A univariate rational polynoial

pseudoDivide() [1/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::pseudoDivide ( const DenseUnivariateRationalPolynomial &  b, RationalNumber *  d = NULL  )

virtual

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< RationalNumber, DenseUnivariateRationalPolynomial >.

pseudoDivide() [2/2]

DenseUnivariateRationalPolynomial DenseUnivariateRationalPolynomial::pseudoDivide ( const DenseUnivariateRationalPolynomial &  b, DenseUnivariateRationalPolynomial *  rem, RationalNumber *  d  ) const

virtual

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< RationalNumber, DenseUnivariateRationalPolynomial >.

realRootIsolate()

Intervals DenseUnivariateRationalPolynomial::realRootIsolate ( mpq_class  width, int  ts = -1  )

inline

Real root isolation for square-free polynomials.

Parameters

width

Interval's right - left < width : Taylor Shift option: 0 - CMY; -1 - optimized

reciprocal()

void DenseUnivariateRationalPolynomial::reciprocal

(

)

Revert coefficients.

Parameters

refineRoot()

void DenseUnivariateRationalPolynomial::refineRoot ( Interval *  a, mpq_class  width  )

inline

Refine a root.

Parameters

a	The root
width	Interval's right - left < width

refineRoots()

Intervals DenseUnivariateRationalPolynomial::refineRoots ( Intervals &  a, mpq_class  width  )

inline

Refine the roots.

a: The roots

Parameters

width

Interval's right - left < width

rootBound()

mpz_class DenseUnivariateRationalPolynomial::rootBound

(

)

Return an integer k such that any positive root alpha of the polynomial satisfies alpha < 2^k.

Parameters

scaleTransform()

void DenseUnivariateRationalPolynomial::scaleTransform

(

int

k

)

Scale transform operation.

Parameters

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

setCoefficient()

void DenseUnivariateRationalPolynomial::setCoefficient ( int  k, const RationalNumber &  value  )

inlinevirtual

Set a coefficient of the polynomial.

Parameters

k	Offset
val	Coefficient

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

setVariableName()

void DenseUnivariateRationalPolynomial::setVariableName ( const Symbol &  x )

inlinevirtual

Set variable's name.

Parameters

x	Varable's name

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

squareFree()

Factors<DenseUnivariateRationalPolynomial> DenseUnivariateRationalPolynomial::squareFree

(

)

const

Square free.

Parameters

subtract()

void DenseUnivariateRationalPolynomial::subtract

(

const DenseUnivariateRationalPolynomial &

b

)

Subtract another polynomial from itself.

Parameters

b	A univariate rational polynomial

taylorShift()

void DenseUnivariateRationalPolynomial::taylorShift

(

int

ts = -1

)

Taylor Shift operation by 1.

Parameters

ts	Algorithm id

variable()

Symbol DenseUnivariateRationalPolynomial::variable ( ) const

inlinevirtual

Get variable's name.

Parameters

Implements BPASUnivariatePolynomial< RationalNumber, DenseUnivariateRationalPolynomial >.

zero()

void DenseUnivariateRationalPolynomial::zero ( )

inline

Zero polynomial.

Parameters

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

• include/RationalNumberPolynomial/urpolynomial.h