A class for a multivariate term providing everything what a GiNaC expression of type power or mul with coefficient is and, in addition, stores the variables of the monomial (in lexicographic order). More...

#include <MultivariateTerm.h>

Inheritance diagram for GiNaCRA::MultivariateTerm< monomialOrdering >:

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Collaboration diagram for GiNaCRA::MultivariateTerm< monomialOrdering >:

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Public Member Functions
	MultivariateTerm ()
	MultivariateTerm (const MultivariatePolynomial< monomialOrdering > &p)
	Constructs a multivariate monomial as leading term of the given polynomial.
	MultivariateTerm (const ex &p, const vector< symbol > &l) throw ( invalid_argument )
	Constructs a multivariate term as leading term of the given polynomial.
const MultivariateTerm < monomialOrdering > &	operator= (const MultivariateTerm< monomialOrdering > &)
const MultivariateTerm < monomialOrdering > &	operator= (const ex &)
int	tdegree () const
	The total degree of the term is the sum of degrees of each symbol.
const MultivariateTerm < monomialOrdering >	mul (const MultivariateTerm< monomialOrdering > &o) const throw ( invalid_argument )
	Multiplies two multivariate terms and returns a reference to their sum, allocated on the heap.
void	print (const print_context &c, unsigned level=0) const
const vector< symbol >	Variables () const
	Selects the list of variables of the polynomial (ordered lexicographicly, not by degree!).
const symbol	Variable () const
const list< ex >	CoefficientExs () const
	Selects the list of polynomial coefficients of this polynomial in the order by the constructor.
const list< ex >	MonomialExs () const
	Selects the list of monomials of this polynomial in the order by the constructor.
const bool	operator== (const MultivariatePolynomial< monomialOrdering > &) const
const bool	operator!= (const MultivariatePolynomial< monomialOrdering > &) const
MultivariatePolynomial < monomialOrdering >	diff (const symbol &s, unsigned nth=1) const
MultivariatePolynomial < monomialOrdering >	diff (const vector< symbol >::const_iterator i, unsigned nth=1) const
MultivariatePolynomial < monomialOrdering >	trunc (const symbol &s) const
	Removes the leading term w.r.t.
MultivariatePolynomial < monomialOrdering >	trunc () const
	Removes the leading term.
const MultivariatePolynomial < monomialOrdering >	divideByLcoeff () const
	Normalizes the polynomial e.g.
int	degree (const symbol &s) const
int	degree () const
int	ldegree (const symbol &s) const
	The low-degree is the smallest exponent at the main variable occuring.
int	ldegree () const
	The low-degree is the smallest exponent at the main variable occuring.
int	tdegree (const symbol &s) const
	The total degree of the polynomial w.r.t.
const ex	lterm () const
const ex	lcoeff () const
const ex	lcoeff (const symbol &s) const
const ex	coeff (const symbol &s, const unsigned int &n) const
const ex	coeff (const ex &monomial) const throw ( invalid_argument )
const ex	lmon () const
bool	isZero () const
	Returns true if the univariate polynomial is the zero polynomial, otherwise false.
bool	isConstant () const
	Returns true if the polynomial is constant in the given variables, otherwise false.
const MultivariatePolynomial < monomialOrdering >	reduction (const MultivariatePolynomial< monomialOrdering > &g, const ex &m) const throw ( invalid_argument )
	Computes the reduction of this polynomial w.r.t.
const MultivariatePolynomial < monomialOrdering >	normalForm (const list< MultivariatePolynomial< monomialOrdering > > &gb) const throw ( invalid_argument )
	Computes the normalform of this polynomial modulo gb.
const MultivariatePolynomial < monomialOrdering >	add (const MultivariatePolynomial< monomialOrdering > &o) const throw ( invalid_argument )
	Adds two multivariate polynomials and returns a reference to their sum, allocated on the heap.
const MultivariatePolynomial < monomialOrdering >	minus () const
	Returns the negative value, allocated on the heap.
const MultivariatePolynomial < monomialOrdering >	mul (const MultivariatePolynomial< monomialOrdering > &o) const throw ( invalid_argument )
	Multiplies two multivariate polynomials and returns a reference to their product, allocated on the heap.
const MultivariatePolynomial < monomialOrdering >	pow (const unsigned e) const throw ( invalid_argument )
	Multiplies two multivariate polynomials and returns a reference to their product, allocated on the heap.
const vector< symbol >	mergeVariables (vector< symbol > l) const
	Merges this with the given symbol list into a new symbol list.
Protected Member Functions
const bool	isEqual (const MultivariatePolynomial< monomialOrdering > &o) const

Detailed Description

template<class monomialOrdering>
class GiNaCRA::MultivariateTerm< monomialOrdering >

A class for a multivariate term providing everything what a GiNaC expression of type power or mul with coefficient is and, in addition, stores the variables of the monomial (in lexicographic order).

Author:: Ulrich Loup

Since:: 2010-11-26

Version:: 2011-10-14

See also:: ISBN 0-387-94090-1 and ISBN-13: 978-3642069642

Notation is following http://www.possibility.com/Cpp/CppCodingStandard.html.

Definition at line 41 of file MultivariateTerm.h.

Constructor & Destructor Documentation

template<class monomialOrdering>

GiNaCRA::MultivariateTerm< monomialOrdering >::MultivariateTerm ( )

template<class monomialOrdering>

GiNaCRA::MultivariateTerm< monomialOrdering >::MultivariateTerm ( const MultivariatePolynomial< monomialOrdering > & p )

Constructs a multivariate monomial as leading term of the given polynomial.

Parameters:

p	multivariate polynomial

template<class monomialOrdering>

GiNaCRA::MultivariateTerm< monomialOrdering >::MultivariateTerm	(	const ex &	p,
		const vector< symbol > &	l
	)		throw ( invalid_argument )

Constructs a multivariate term as leading term of the given polynomial.

Parameters:

p	monomial
l	list of the variables of the monomial p

Member Function Documentation

template<class monomialOrdering>

const MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::add ( const MultivariatePolynomial< monomialOrdering > & o ) const throw ( invalid_argument ) [inherited]

Adds two multivariate polynomials and returns a reference to their sum, allocated on the heap.

Parameters:

o

Returns:: sum allocated on the heap

template<class monomialOrdering>

const ex GiNaCRA::MultivariatePolynomial< monomialOrdering >::coeff	(	const symbol &	s,
		const unsigned int &	n
	)		const `[inherited]`

Parameters:

n	the exponent
s	the main symbol

Returns:: coefficient in front of s^n

template<class monomialOrdering>

const ex GiNaCRA::MultivariatePolynomial< monomialOrdering >::coeff ( const ex & monomial ) const throw ( invalid_argument ) [inherited]

Returns:: coefficient in front of the given monomial w.r.t. the given monomial ordering

Exceptions:

invalid_argument in case the given monomial is not a monomial of this polynomial

template<class monomialOrdering>

const list<ex> GiNaCRA::MultivariatePolynomial< monomialOrdering >::CoefficientExs ( ) const [inherited]

Selects the list of polynomial coefficients of this polynomial in the order by the constructor.

Returns:: list of polynomial coefficients of this polynomial

template<class monomialOrdering>

int GiNaCRA::MultivariatePolynomial< monomialOrdering >::degree ( const symbol & s ) const [inherited]

Parameters:

s

Returns:: degree of the univariate polynomial in the given symbol

Referenced by GiNaCRA::MultivariatePolynomialFactory::searchMonomialsUnderTheStaircase().

template<class monomialOrdering>

int GiNaCRA::MultivariatePolynomial< monomialOrdering >::degree ( ) const [inherited]

Returns:: degree of the univariate polynomial in the main symbol

template<class monomialOrdering>

MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::diff	(	const symbol &	s,
		unsigned	nth = `1`
	)		const `[inherited]`

Parameters:

s	symbol
nth

Returns:: nth derivative of the univariate polynomial in the given symbol

Referenced by GiNaCRA::MultivariatePolynomialFactory::specialGroebnerBasis().

template<class monomialOrdering>

MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::diff	(	const vector< symbol >::const_iterator	i,
		unsigned	nth = `1`
	)		const `[inherited]`

Parameters:

i	iterator pointing to the symbol (in the list of symbols of this MultivariatePolynomial)
nth

Returns:: nth derivative of the univariate polynomial in the given symbol

template<class monomialOrdering>

const MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::divideByLcoeff ( ) const [inherited]

Normalizes the polynomial e.g.

if P = c*(x^2+y^2) in D[x,y] it returns 1*(x^2+y^2).

Returns:: normalized polynomial

template<class monomialOrdering>

bool GiNaCRA::MultivariatePolynomial< monomialOrdering >::isConstant ( ) const [inherited]

Returns true if the polynomial is constant in the given variables, otherwise false.

Returns:: true if the polynomial is constant, otherwise false

template<class monomialOrdering>

const bool GiNaCRA::MultivariatePolynomial< monomialOrdering >::isEqual ( const MultivariatePolynomial< monomialOrdering > & o ) const [inline, protected, inherited]

Parameters:

o

Returns:: true in case the other multivariate polynomial equals this

template<class monomialOrdering>

bool GiNaCRA::MultivariatePolynomial< monomialOrdering >::isZero ( ) const [inherited]

Returns true if the univariate polynomial is the zero polynomial, otherwise false.

Returns:: true if the univariate polynomial is the zero polynomial, otherwise false.

template<class monomialOrdering>

const ex GiNaCRA::MultivariatePolynomial< monomialOrdering >::lcoeff ( ) const [inherited]

Returns:: coefficient in front of the monomial at the highest position in the given monomial ordering

template<class monomialOrdering>

const ex GiNaCRA::MultivariatePolynomial< monomialOrdering >::lcoeff ( const symbol & s ) const [inherited]

Parameters:

s	the variable

Returns:: the leading coefficient with respect to a variable x

template<class monomialOrdering>

int GiNaCRA::MultivariatePolynomial< monomialOrdering >::ldegree ( const symbol & s ) const [inherited]

The low-degree is the smallest exponent at the main variable occuring.

Parameters:

s

Returns:: low-degree of the univariate polynomial in the given symbol

template<class monomialOrdering>

int GiNaCRA::MultivariatePolynomial< monomialOrdering >::ldegree ( ) const [inherited]

The low-degree is the smallest exponent at the main variable occuring.

Returns:: low-degree of the univariate polynomial in the main symbol

template<class monomialOrdering>

const ex GiNaCRA::MultivariatePolynomial< monomialOrdering >::lmon ( ) const [inherited]

Returns:: monomial at the highest position according to the monomial ordering

template<class monomialOrdering>

const ex GiNaCRA::MultivariatePolynomial< monomialOrdering >::lterm ( ) const [inherited]

Returns:: lcoeff * lmon

template<class monomialOrdering>

const vector<symbol> GiNaCRA::MultivariatePolynomial< monomialOrdering >::mergeVariables ( vector< symbol > l ) const [inherited]

Merges this with the given symbol list into a new symbol list.

Returns:: new symbol list containing both this and the other symbols

template<class monomialOrdering>

const MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::minus ( ) const [inherited]

Returns the negative value, allocated on the heap.

Returns:: negative value allocated on the heap

template<class monomialOrdering>

const list<ex> GiNaCRA::MultivariatePolynomial< monomialOrdering >::MonomialExs ( ) const [inherited]

Selects the list of monomials of this polynomial in the order by the constructor.

Returns:: list of monomials of this polynomial

template<class monomialOrdering>

const MultivariateTerm<monomialOrdering> GiNaCRA::MultivariateTerm< monomialOrdering >::mul ( const MultivariateTerm< monomialOrdering > & o ) const throw ( invalid_argument )

Multiplies two multivariate terms and returns a reference to their sum, allocated on the heap.

Parameters:

o

Returns:: product allocated on the heap

template<class monomialOrdering>

const MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::mul ( const MultivariatePolynomial< monomialOrdering > & o ) const throw ( invalid_argument ) [inherited]

Multiplies two multivariate polynomials and returns a reference to their product, allocated on the heap.

Parameters:

o

Returns:: product allocated on the heap

template<class monomialOrdering>

const MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::normalForm ( const list< MultivariatePolynomial< monomialOrdering > > & gb ) const throw ( invalid_argument ) [inherited]

Computes the normalform of this polynomial modulo gb.

Parameters:

gb	special Groebner basis of the respective ideal

Returns:: normalform of this polynomial modulo gb

See also:: Algorithm 12.2 in ISBN-13: 978-3642069642

template<class monomialOrdering>

const bool GiNaCRA::MultivariatePolynomial< monomialOrdering >::operator!= ( const MultivariatePolynomial< monomialOrdering > & ) const [inherited]

template<class monomialOrdering>

const MultivariateTerm<monomialOrdering>& GiNaCRA::MultivariateTerm< monomialOrdering >::operator= ( const MultivariateTerm< monomialOrdering > & )

template<class monomialOrdering>

const MultivariateTerm<monomialOrdering>& GiNaCRA::MultivariateTerm< monomialOrdering >::operator= ( const ex & )

Reimplemented from GiNaCRA::MultivariatePolynomial< monomialOrdering >.

Reimplemented in GiNaCRA::MultivariateCoefficient< monomialOrdering >, and GiNaCRA::MultivariateMonomial< monomialOrdering >.

template<class monomialOrdering>

const bool GiNaCRA::MultivariatePolynomial< monomialOrdering >::operator== ( const MultivariatePolynomial< monomialOrdering > & ) const [inherited]

template<class monomialOrdering>

const MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::pow ( const unsigned e ) const throw ( invalid_argument ) [inherited]

Multiplies two multivariate polynomials and returns a reference to their product, allocated on the heap.

Parameters:

e exponent

Returns:: power of this polynomial to e

template<class monomialOrdering>

void GiNaCRA::MultivariateTerm< monomialOrdering >::print	(	const print_context &	c,
		unsigned	level = `0`
	)		const

Reimplemented from GiNaCRA::MultivariatePolynomial< monomialOrdering >.

template<class monomialOrdering>

const MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::reduction	(	const MultivariatePolynomial< monomialOrdering > &	g,
		const ex &	m
	)		const throw ( invalid_argument ) `[inherited]`

Computes the reduction of this polynomial w.r.t.

its monomial m by another polynomial q. This method performs a basic reduction step in computing normal forms modulo a given Groebner basis.

Parameters:

g	basis polynomial
m	monomial of this polynomial w.r.t. which the reduction should take place

Returns:: reduction modulo q w.r.t. m

Exceptions:

invalid_argument in case m is not a monomial of this polynomial

See also:: Definition 4.64 in ISBN-13: 978-3642069642

Referenced by GiNaCRA::MultivariatePolynomialFactory::specialGroebnerBasis().

template<class monomialOrdering>

int GiNaCRA::MultivariateTerm< monomialOrdering >::tdegree ( ) const

The total degree of the term is the sum of degrees of each symbol.

Returns:: total degree of the MultivariateTerm

Reimplemented from GiNaCRA::MultivariatePolynomial< monomialOrdering >.

Referenced by GiNaCRA::MultivariatePolynomialFactory::checkForSpecialGroebnerBasis().

template<class monomialOrdering>

int GiNaCRA::MultivariatePolynomial< monomialOrdering >::tdegree ( const symbol & s ) const [inherited]

The total degree of the polynomial w.r.t.

the variable s is the maximum sum of degrees of each monomial containing the variable s.

Parameters:

s variable

Returns:: total degree of the MultivariatePolynomial

See also:: page 485 in ISBN 0-387-94090-1 and ISBN-13: 978-3642069642

template<class monomialOrdering>

MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::trunc ( const symbol & s ) const [inherited]

Removes the leading term w.r.t.

one variable.

Parameters:

s	the variable

Returns:: the polynomial without leading term

template<class monomialOrdering>

MultivariatePolynomial<monomialOrdering> GiNaCRA::MultivariatePolynomial< monomialOrdering >::trunc ( ) const [inherited]

Removes the leading term.

Returns:: the polynomial without leading term

template<class monomialOrdering>

const symbol GiNaCRA::MultivariatePolynomial< monomialOrdering >::Variable ( ) const [inherited]

Returns:: the (or one of the) symbols of the polynomial with maximal degree

template<class monomialOrdering>

const vector<symbol> GiNaCRA::MultivariatePolynomial< monomialOrdering >::Variables ( ) const [inherited]

Selects the list of variables of the polynomial (ordered lexicographicly, not by degree!).

Returns:: list of symbols

Referenced by GiNaCRA::MultivariatePolynomialFactory::searchMonomialsUnderTheStaircase(), and GiNaCRA::MultivariatePolynomialFactory::synthesizeSpecialGroebnerBasisDegrees().

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

MultivariateTerm.h

Public Member Functions

Protected Member Functions

Detailed Description

template<class monomialOrdering> class GiNaCRA::MultivariateTerm< monomialOrdering >

Constructor & Destructor Documentation

Member Function Documentation

template<class monomialOrdering>
class GiNaCRA::MultivariateTerm< monomialOrdering >