Data Structures
struct	infinitesimal_symbol_struct
	Type for infinitesimal values. More...
struct	ex_is_lessdeg
	Prototypal ordering on expressions which are terms. More...
struct	ex_is_less_deggrlex
	Graded lexicographic ordering on expressions which are terms. More...
struct	ex_is_less_degrevlex
	Lexicographic degree-reverse ordering on expressions which are terms. More...
struct	ex_is_under_the_staircase
	Relation determining whether the given expression is a monomial under the staircase, which is determined by computing the remainders dividing by the corners of a staircase. More...
struct	expair_is_lesseq
	Prototypal ordering on pairs of expressions which represent terms. More...
class	UnivariateRepresentation
	An implementation of a multi dimensional real algebraic number providing arithmetic and relational operations. More...
Typedefs
typedef GiNaC::infinitesimal_symbol_struct	infinitesimal_symbol
	Type for infinitesimal values.
Enumerations
enum	sign { NEGATIVE_SIGN, ZERO_SIGN, POSITIVE_SIGN }
	type for signs More...
Functions
const infinitesimal_symbol	EPSILON ("€")
const infinitesimal_symbol	DELTA ("ð")
const infinitesimal_symbol	GAMMA ("đ")
const infinitesimal_symbol	ZETA ("¢")
const symbol	X ("X")
	standard variable for standard polynomial objects
const numeric	ZERO (0)
	unique representation of zero
const ex	prod (const lst &l) throw ( invalid_argument )
	Computes the product of a list of expressions.
const ex	lcm (const lst &l) throw ( invalid_argument )
	Computes the least common multiple of a list of expressions.
long	lcm (long a, long b)
	Computes the least common multiple of two machine integers a and b.
long	gcd (long a, long b)
	Computes the greatest common divisor of two integers a and b.
long	numerator (long a, long b)
	Computes the numerator q such that b / a = q / r and q and r are coprime.
long	denominator (long a, long b)
	Computes the numerator r such that b / a = q / r and q and r are coprime.
bool	is_constant (const ex &polynomial, const vector< symbol > &symbolLst)
	Determines whether the given polynomial expression is constant in the given list of variables.
bool	is_rational_polynomial (const ex &p, const symbol &x)
	Tests whether the given expression p is a rational polynomial in x.
bool	is_realalgebraic_polynomial (const ex &p, const symbol &x)
	Tests whether the given expression p is a polynomial in x with real algebraic coefficients, i.e.
bool	is_realalgebraic_term (const ex &p)
	Tests whether the given expression p is a term only containing only numbers returning `true` on `info(info_flags::numeric) && (info(info_flags::rational) \|\| (info(info_flags::real) && info_flags::algebraic))`.
sign	sgn (const numeric &n)
	Determines the sign of the given numeric.
const ex	monpart (const ex &polynomial, const vector< symbol > &symbolLst)
	For internal use only! Computes the monomial underlying the given polynomial in case the polynomial is a single term.
const ex	coeffpart (const ex &polynomial, const vector< symbol > &symbolLst)
	For internal use only! Computes the polynomial coefficient underlying the given polynomial in case the polynomial is a single term.
void	isolateByVariables (const ex &polynomial, const vector< symbol > &symbolLst, ex &coefficient, ex &monomial)
	For internal use only! Determins the monomial and its coefficient of the given polynomial w.r.t.
const GiNaC::ex	rationalize (const GiNaC::ex &p, const vector< GiNaC::symbol > &symbolLst)
	Converts all coefficients of the given rational polynomial in the variables symbolLst to an exact rational numeric, so that is_rational_polynomial returns true.
const GiNaC::ex	rationalize (const GiNaC::ex &p, const GiNaC::symbol &s)
	Converts all coefficients of the given rational polynomial in the variable s to an exact rational numeric, so that is_rational_polynomial returns true.
const GiNaC::numeric	rationalize (const GiNaC::numeric &n)
	Converts the given numeric value to an exact numeric one.
const vector< symbol >	sortVariables (const vector< symbol > &l)
	Sorts the given list of symbols lexicographicly.
bool	symbol_is_lesseq_lex (const symbol &a, const symbol &b)
	Compares two expressions lexicographically by their string representations.
bool	symbol_is_less_lex (const symbol &a, const symbol &b)
	Compares two expressions lexicographically by their string representations.
const map< int, ex >	signedSubresultants (const ex &A, const ex &B, const symbol &sym)
	Returns the signed Subresultant sequence.
const vector< ex >	signedSubresultantsCoefficients (const ex &A, const ex &B, const symbol &sym) throw ( invalid_argument )
	Returns the Signed Subresultant Coefficient sequence.

Detailed Description

Author:: Ulrich Loup

Since:: 2011-04-30

Version:: 2011-04-30

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

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

Typedef Documentation

typedef GiNaC::infinitesimal_symbol_struct GiNaC::infinitesimal_symbol

Type for infinitesimal values.

Enumeration Type Documentation

enum GiNaC::sign

type for signs

Enumerator:

NEGATIVE_SIGN
ZERO_SIGN
POSITIVE_SIGN

Definition at line 99 of file constants.h.

Function Documentation

const ex GiNaC::coeffpart	(	const ex &	polynomial,
		const vector< symbol > &	symbolLst
	)

For internal use only! Computes the polynomial coefficient underlying the given polynomial in case the polynomial is a single term.

Otherwise the method returns 1.

Parameters:

polynomial
symbolLst

Returns:: the (polynomial) coefficient underlying the given polynomial or 1 in case the expression is a sum.

Definition at line 160 of file utilities.cpp.

References isolateByVariables().

const infinitesimal_symbol GiNaC::DELTA ( "ð" )

long GiNaC::denominator	(	long	a,
		long	b
	)

Computes the numerator r such that b / a = q / r and q and r are coprime.

Parameters:

a
b

Returns:: b / gcd(a, b)

Runtime complexity:: O( digits( min(a, b) ) ) + integer division

Definition at line 87 of file utilities.cpp.

References gcd().

Referenced by GiNaCRA::OpenInterval::findSample(), and GiNaCRA::OpenInterval::sample().

const infinitesimal_symbol GiNaC::EPSILON ( "€" )

const infinitesimal_symbol GiNaC::GAMMA ( "đ" )

long GiNaC::gcd	(	long	a,
		long	b
	)

Computes the greatest common divisor of two integers a and b.

Parameters:

a
b

Returns:: gcd(a, b)

Runtime complexity:: O( digits( min(a, b) ) )

Definition at line 68 of file utilities.cpp.

Referenced by denominator(), GiNaCRA::OpenInterval::findSample(), GiNaCRA::UnivariatePolynomial::gcd(), lcm(), and numerator().

bool GiNaC::is_constant	(	const ex &	polynomial,
		const vector< symbol > &	symbolLst
	)

Determines whether the given polynomial expression is constant in the given list of variables.

Parameters:

polynomial
symbolLst

Returns:: the monomial underlying the given polynomial or 1 in case the expression is a sum.

Definition at line 92 of file utilities.cpp.

Referenced by isolateByVariables(), and GiNaCRA::MultivariatePolynomialMR::MultivariatePolynomialMR().

bool GiNaC::is_rational_polynomial	(	const ex &	p,
		const symbol &	x
	)

Tests whether the given expression p is a rational polynomial in x.

Parameters:

p
x

Returns:: false in case the polynomial was found to be not rational, true otherwise

Definition at line 100 of file utilities.cpp.

Referenced by GiNaCRA::RationalUnivariatePolynomial::RationalUnivariatePolynomial().

bool GiNaC::is_realalgebraic_polynomial	(	const ex &	p,
		const symbol &	x
	)

Tests whether the given expression p is a polynomial in x with real algebraic coefficients, i.e.

coefficients being rational or mixed terms with RealAlgebraicNumberIRs.

Parameters:

p
x

Returns:: false in case the polynomial was found to be not real algebraic, true otherwise

Definition at line 113 of file utilities.cpp.

References is_realalgebraic_term().

bool GiNaC::is_realalgebraic_term ( const ex & p )

Tests whether the given expression p is a term only containing only numbers returning true on info(info_flags::numeric) && (info(info_flags::rational) || (info(info_flags::real) && info_flags::algebraic)).

Parameters:

p

Returns:: false in case the term was found to be not real algebraic, true otherwise

Definition at line 123 of file utilities.cpp.

Referenced by is_realalgebraic_polynomial().

void GiNaC::isolateByVariables	(	const ex &	polynomial,
		const vector< symbol > &	symbolLst,
		ex &	coefficient,
		ex &	monomial
	)

For internal use only! Determins the monomial and its coefficient of the given polynomial w.r.t.

the given list of variables, in case the polynomial is a single term. Otherwise the method returns 1.

Parameters:

polynomial
symbolLst
coefficient	refernce to the coefficient to isolate
monomial	reference to the monomial to isolate

Returns:: the monomial underlying the given polynomial or 1 in case the expression is a sum.

Definition at line 168 of file utilities.cpp.

References is_constant().

Referenced by GiNaCRA::Constraint::checkVariables(), coeffpart(), monpart(), and rationalize().

const ex GiNaC::lcm ( const lst & l ) throw ( invalid_argument )

Computes the least common multiple of a list of expressions.

Parameters:

l	list of expressions

Returns:: lcm

Runtime complexity:: linearly (in the length of l) many calls of GiNaC::lcm

Definition at line 52 of file utilities.cpp.

Referenced by GiNaCRA::MultivariatePolynomialFactory::coefficientLcm(), and GiNaCRA::OpenInterval::findSample().

long GiNaC::lcm	(	long	a,
		long	b
	)

Computes the least common multiple of two machine integers a and b.

Parameters:

a
b

Returns:: lcm(a, b)

Runtime complexity:: the complexity of gcd(a, b)

Definition at line 62 of file utilities.cpp.

References gcd().

const ex GiNaC::monpart	(	const ex &	polynomial,
		const vector< symbol > &	symbolLst
	)

For internal use only! Computes the monomial underlying the given polynomial in case the polynomial is a single term.

Otherwise the method returns 1.

Parameters:

polynomial
symbolLst

Returns:: the monomial underlying the given polynomial or 1 in case the expression is a sum.

Definition at line 152 of file utilities.cpp.

References isolateByVariables().

long GiNaC::numerator	(	long	a,
		long	b
	)

Computes the numerator q such that b / a = q / r and q and r are coprime.

Parameters:

a
b

Returns:: a / gcd(a, b)

Runtime complexity:: O( digits( min(a, b) ) ) + one integer division

Definition at line 82 of file utilities.cpp.

References gcd().

Referenced by GiNaCRA::OpenInterval::findSample(), and GiNaCRA::OpenInterval::sample().

const ex GiNaC::prod ( const lst & l ) throw ( invalid_argument )

Computes the product of a list of expressions.

Parameters:

l	list of expressions

Returns:: prod

Definition at line 42 of file utilities.cpp.

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

const GiNaC::ex GiNaC::rationalize	(	const GiNaC::ex &	p,
		const vector< GiNaC::symbol > &	symbolLst
	)

Converts all coefficients of the given rational polynomial in the variables symbolLst to an exact rational numeric, so that is_rational_polynomial returns true.

Parameters:

p
symbolLst

Returns:: p with all coefficients converted to an exact rational numeric

Definition at line 216 of file utilities.cpp.

References isolateByVariables().

Referenced by rationalize().

const GiNaC::ex GiNaC::rationalize	(	const GiNaC::ex &	p,
		const GiNaC::symbol &	s
	)		`[inline]`

Converts all coefficients of the given rational polynomial in the variable s to an exact rational numeric, so that is_rational_polynomial returns true.

Parameters:

p
s

Returns:: p with all coefficients converted to an exact rational numeric

Definition at line 236 of file utilities.cpp.

References rationalize().

const GiNaC::numeric GiNaC::rationalize ( const GiNaC::numeric & n ) [inline]

Converts the given numeric value to an exact numeric one.

Parameters:

n

Returns:: an exact rational numeric of n

Definition at line 241 of file utilities.cpp.

References rationalize().

sign GiNaC::sgn ( const numeric & n )

Determines the sign of the given numeric.

Parameters:

n numeric

Returns:: the sign of the given numeric

Definition at line 147 of file utilities.cpp.

References NEGATIVE_SIGN, POSITIVE_SIGN, and ZERO_SIGN.

Referenced by GiNaCRA::OpenInterval::abs(), and GiNaCRA::Constraint::satisfiedBy().

const map< int, ex > GiNaC::signedSubresultants	(	const ex &	A,
		const ex &	B,
		const symbol &	sym
	)

Returns the signed Subresultant sequence.

The algorithm is originally from "New structure theorem for subresultants" - Henri Lombardi, Marie-Francoise Roy and Mohab Safey El Din. See page 11-12 for the algorithm and page 5 for the notation.

Parameters:

A	the first Polynomial
B	the second Polynomial
sym	the main variable

Runtime complexity:: O( deg(A)*deg(B) )

Returns:: a map from int to ex, where the i-th index corresponds to the i-th subresultant in the subresultant sequence

Definition at line 269 of file utilities.cpp.

Referenced by signedSubresultantsCoefficients().

const vector< ex > GiNaC::signedSubresultantsCoefficients	(	const ex &	A,
		const ex &	B,
		const symbol &	sym
	)		throw ( invalid_argument )

Returns the Signed Subresultant Coefficient sequence.

The vectors contains the i-th coefficient of the i-th element of Signed Subresultant sequence in the i-th position. The Algorithm calls calculateSignedSubresultants for the computation.

Parameters:

A	the first Polynomial
B	the second Polynomial
sym	the main variable

Runtime complexity:: O( deg(A)*deg(B) )

Returns:: a vector of ex with the coefficients

Definition at line 381 of file utilities.cpp.

References signedSubresultants().

const vector< symbol > GiNaC::sortVariables ( const vector< symbol > & l )

Sorts the given list of symbols lexicographicly.

Returns:: new symbol list sorted lexicographicly

Definition at line 246 of file utilities.cpp.

References symbol_is_less_lex().

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

bool GiNaC::symbol_is_less_lex	(	const symbol &	a,
		const symbol &	b
	)

Compares two expressions lexicographically by their string representations.

Parameters:

a
b

Returns:: true in case a is strictly less than b

Definition at line 261 of file utilities.cpp.

Referenced by sortVariables().

bool GiNaC::symbol_is_lesseq_lex	(	const symbol &	a,
		const symbol &	b
	)

Compares two expressions lexicographically by their string representations.

Parameters:

a
b

Returns:: true in case a is less or equal to b

Definition at line 253 of file utilities.cpp.

const symbol GiNaC::X ( "X" )

standard variable for standard polynomial objects

const numeric GiNaC::ZERO ( 0 )

unique representation of zero

Referenced by GiNaCRA::CAD::samples().

const infinitesimal_symbol GiNaC::ZETA ( "¢" )

Data Structures

Typedefs

Enumerations

Functions

Detailed Description

Typedef Documentation

Enumeration Type Documentation

Function Documentation