A multivariate polynomial with Integer coefficients using a sparse representation. More...

#include <mzpolynomial.hpp>

Simplified semantic inheritance diagram for SparseMultivariateIntegerPolynomial:

Full inheritance diagram for SparseMultivariateIntegerPolynomial:

Public Member Functions
SparseMultivariateIntegerPolynomial	subresultantGCD (const SparseMultivariateIntegerPolynomial &q) const

std::vector< SparseMultivariateIntegerPolynomial >	subresultantChain (const SparseMultivariateIntegerPolynomial &q, int filled=0) const

SparseMultivariateIntegerPolynomial	resultant (const SparseMultivariateIntegerPolynomial &q) const

	SparseMultivariateIntegerPolynomial ()
	Construct a multivariate polynomial. More...

	SparseMultivariateIntegerPolynomial (int v)
	Construct a multivariate polynomial with specific number of variables. More...

	SparseMultivariateIntegerPolynomial (const Symbol &x)
	htop More...

	SparseMultivariateIntegerPolynomial (const std::string &str)
	Construct a polynomial by parsing the string str.

	SparseMultivariateIntegerPolynomial (const SparseMultivariateIntegerPolynomial &b)
	Copy Constructor. More...

	SparseMultivariateIntegerPolynomial (SparseMultivariateIntegerPolynomial &&b)
	Move Constructor. More...

	SparseMultivariateIntegerPolynomial (const SparseMultivariateRationalPolynomial &b)
	Create a SMZP from an SMQP.

	SparseMultivariateIntegerPolynomial (const Integer &r, int nvar=0)
	Create a SMZP from an Integer.

	SparseMultivariateIntegerPolynomial (const RationalNumber &r, int nvar=0)
	Create a SMZP from a RationalNumber.

	SparseMultivariateIntegerPolynomial (const DenseUnivariateIntegerPolynomial &p)
	Create a SMZP from a univariate rational polynomial. More...

	SparseMultivariateIntegerPolynomial (const SparseUnivariatePolynomial< SparseMultivariateIntegerPolynomial > &s)
	Construct from a SUP<SMZP> polynomial such that the resulting SMZP has main variable equal to the variable of s. More...

	~SparseMultivariateIntegerPolynomial ()
	Destroy the polynomial and underlying node memory.

bool	isZero () const
	Is this polynomial zero. More...

void	zero ()
	Set this polynomial to zero. More...

bool	isOne () const
	Is this polynomial one. More...

void	one ()
	Sets this polynomial to one. More...

bool	isNegativeOne () const
	Is this polynomial negative one. More...

void	negativeOne ()
	Sets this polynomial to -1. More...

int	isConstant () const
	Determine if this polynomial encodes a constant. More...

SparseMultivariateIntegerPolynomial	unitCanonical (SparseMultivariateIntegerPolynomial u, SparseMultivariateIntegerPolynomial v) const
	Obtain the unit normal (a.k.a canonical associate) of an element. More...

SparseMultivariateIntegerPolynomial &	operator= (const SparseMultivariateIntegerPolynomial &b)
	Assign this polynomail to equal the specified.

SparseMultivariateIntegerPolynomial &	operator= (SparseMultivariateIntegerPolynomial &&b)
	Movement assignment: move b to be this polynomail.

SparseMultivariateIntegerPolynomial &	operator= (const Integer &r)
	Assign this polynomial to be equal to the constant ring element.

SparseMultivariateIntegerPolynomial	operator+ (const SparseMultivariateIntegerPolynomial &b) const
	Add two SMQP polynomials together, *this and the specified.

SparseMultivariateIntegerPolynomial	operator+ (SparseMultivariateIntegerPolynomial &&b) const

SparseMultivariateIntegerPolynomial &	operator+= (const SparseMultivariateIntegerPolynomial &b)
	Add the polynomails a and b and return the sum. More...

SparseMultivariateIntegerPolynomial	operator- (const SparseMultivariateIntegerPolynomial &b) const
	Subtract the specified polynomail from *this.

SparseMultivariateIntegerPolynomial	operator- (SparseMultivariateIntegerPolynomial &&b) const

SparseMultivariateIntegerPolynomial	operator- () const
	Subtract the polynomial b from a and return the difference. More...

SparseMultivariateIntegerPolynomial &	operator-= (const SparseMultivariateIntegerPolynomial &b)
	Update *this by subtracting the specified polynomial from it.

SparseMultivariateIntegerPolynomial	operator* (const SparseMultivariateIntegerPolynomial &b) const
	Multiply *this by the specified polynomail.

SparseMultivariateIntegerPolynomial	operator* (SparseMultivariateIntegerPolynomial &&b) const

SparseMultivariateIntegerPolynomial &	operator*= (const SparseMultivariateIntegerPolynomial &b)
	Multiply the polynomials a and b, returning their product. More...

SparseMultivariateIntegerPolynomial	operator/ (const SparseMultivariateIntegerPolynomial &b) const
	Divide *this by the specified polynomial.

SparseMultivariateIntegerPolynomial	operator/ (SparseMultivariateIntegerPolynomial &&b) const

SparseMultivariateIntegerPolynomial &	operator/= (const SparseMultivariateIntegerPolynomial &b)
	Update *this by dividing by the specified polynomial.

SparseMultivariateIntegerPolynomial	operator^ (long long int e) const
	Exponentiate *this by the input exponent integer. More...

SparseMultivariateIntegerPolynomial &	operator^= (long long int e)
	Update *this by exponentiating this to the input integer. More...

bool	operator== (const SparseMultivariateIntegerPolynomial &b) const
	Determine if *this is equal to the specified polynomial. More...

bool	operator!= (const SparseMultivariateIntegerPolynomial &b) const
	Determine if *this is not equal to the specified polynomial. More...

void	print (std::ostream &os) const
	Output the string representation of *this to the input ostream.

void	fromString (const std::string &str)
	Parse a polynomial from the string str and place it in *this.

SparseMultivariateIntegerPolynomial	gcd (const SparseMultivariateIntegerPolynomial &b) const
	Get GCD between *this and b.

SparseMultivariateIntegerPolynomial	primitiveGCD (const SparseMultivariateIntegerPolynomial &b) const
	Get the GCD between *this and b as a primitive polynomial.

Factors< SparseMultivariateIntegerPolynomial >	squareFree () const
	Compute squarefree factorization of *this with respect to all of its variables.

Factors< SparseMultivariateIntegerPolynomial >	squareFree (const std::vector< Symbol > &vars) const
	Compute squarefree factorization of *this with respect to the list of variables, vars.

SparseMultivariateIntegerPolynomial	squareFreePart () const
	Computes the square free part of this polynomail. More...

SparseMultivariateIntegerPolynomial	squareFreePart (std::vector< Symbol > &vars) const
	Computes the square free part of this polynomail. More...

Integer	content () const
	Get the content with respect to all variables. More...

SparseMultivariateIntegerPolynomial	content (const std::vector< Symbol > &v) const
	Get the content of this polynomial with respect to the variables in the input vector v. More...

SparseMultivariateIntegerPolynomial	primitivePart () const
	Get the primitive part with respect to all variables. More...

SparseMultivariateIntegerPolynomial	primitivePart (Integer &content) const
	Get the primitive part with respect to all variables. More...

SparseMultivariateIntegerPolynomial	primitivePart (const std::vector< Symbol > &v) const
	Get the primitive part with respect to the variables in the vector v. More...

SparseMultivariateIntegerPolynomial	primitivePart (const std::vector< Symbol > &v, SparseMultivariateIntegerPolynomial &content) const
	Get the primitive part with respect to the variables in the vector v. More...

SparseMultivariateIntegerPolynomial	initial () const
	Get the leading coefficient of *this with respect to the main variable. More...

Symbol	mainVariable () const
	Get the main variable. More...

int	mainDegree () const
	Get the degree of the main variable. More...

SparseMultivariateIntegerPolynomial	rank () const
	Get the rank of this polynomial. More...

SparseMultivariateIntegerPolynomial	head () const
	Get the head of this polynomial. More...

SparseMultivariateIntegerPolynomial	tail () const
	Get the tail of this polynomial. More...

SparseMultivariateIntegerPolynomial	separant () const
	Get the separant of this polynomial, that is, its derivative with respect to the main variable. More...

int	numberOfVariables () const
	Get the number of variables in this polynomial.

int	numberOfRingVariables () const
	Get the number of variables in this polynomial ring.

Integer	numberOfTerms () const
	Get the number of non-zero terms.

Integer	degree () const
	Total degree.

Integer	degree (const Symbol &) const
	Get the degree of a variable.

Integer	leadingCoefficient () const
	Get the leading coefficient.

void	setLeadingCoefficient (const Integer &it)
	Set the leading coefficient.

Integer	trailingCoefficient () const
	Get the trailing coefficient.

Integer	coefficient (int, const int *) const
	Get a coefficient, given the exponent of each variable.

Integer	coefficient (const std::vector< int > &v) const
	Get the coefficient of this polynomial with respect to the monomial defined by exponent vector exps. More...

void	setCoefficient (int, const int *, const Integer &)
	Set a coefficient, given the exponent of each variable.

void	setCoefficient (const std::vector< int > &v, const Integer &r)
	Set the coefficient of this polynomial for the monomial defined by exponent vector exps to r. More...

void	setRingVariables (const std::vector< Symbol > &)
	Set variables' names. More...

std::vector< Symbol >	ringVariables () const
	Get variable names of all variables available to this polynomial, even those that have zero degree.

std::vector< Symbol >	variables () const
	Get variable names of variables with non-zero degree;.

bool	isEqual (const SparseMultivariateIntegerPolynomial &b) const
	Determine if *this is equal to b. More...

void	differentiate (const Symbol &s, int k)
	Convert current object to its k-th derivative. More...

void	differentiate (const Symbol &s)
	Convert current object to its derivative. More...

SparseMultivariateIntegerPolynomial	derivative (const Symbol &s, int k) const
	Return k-th derivative. More...

SparseMultivariateIntegerPolynomial	derivative (const Symbol &s) const
	Compute derivative. More...

void	integrate (const Symbol &s, int k)
	Convert current object to its k-th integral. More...

void	integrate (const Symbol &s)
	Convert current object to its derivative. More...

SparseMultivariateIntegerPolynomial	integral (const Symbol &s, int k) const
	Return k-th integral. More...

SparseMultivariateRationalPolynomial	rationalIntegral (const Symbol &s, int k) const
	Return k-th integral. More...

SparseMultivariateIntegerPolynomial	integral (const Symbol &s) const
	Compute integral. More...

SparseMultivariateRationalPolynomial	rationalIntegral (const Symbol &s) const
	Return the integral of this, with respect to Symbol s, as a rational number polynomial. More...

SparseMultivariateIntegerPolynomial	evaluate (int n, const Symbol syms, const Integer xs) const
	Evaluate f(x) More...

SparseMultivariateIntegerPolynomial	evaluate (const std::vector< Symbol > &vars, const std::vector< Integer > &values) const
	Evaluate *this polynomial given the variables and their values in vars, and values. More...

bool	divide (const SparseMultivariateIntegerPolynomial &b, SparseMultivariateIntegerPolynomial &q, SparseMultivariateIntegerPolynomial &r) const
	Divide this by polynomial b, returning the quotient and remainder in q and r, respectively. More...

bool	rationalDivide (const SparseMultivariateIntegerPolynomial &b, SparseMultivariateRationalPolynomial &q, SparseMultivariateRationalPolynomial &r) const
	Divide this by polynomial b, return the quotient and remainder in q and r as rational number polynomials. More...

SparseMultivariateIntegerPolynomial	operator% (const SparseMultivariateIntegerPolynomial &b) const
	Get the remainder of *this divided by b.

SparseMultivariateIntegerPolynomial &	operator%= (const SparseMultivariateIntegerPolynomial &b)
	Update this by setting it to the remainder of this divided by b.

SparseMultivariateIntegerPolynomial	pseudoDivide (const SparseMultivariateIntegerPolynomial &b, SparseMultivariateIntegerPolynomial quo=NULL, SparseMultivariateIntegerPolynomial mult=NULL, bool lazy=0) const
	Pseudo divide this by b. More...

SparseMultivariateIntegerPolynomial	operator+ (const mpz_t r) const
	Add *this and a mpz_t.

SparseMultivariateIntegerPolynomial	operator+ (const mpz_class &r) const
	Add *this and the mpz_class r.

SparseMultivariateIntegerPolynomial	operator+ (const Integer &r) const

SparseMultivariateIntegerPolynomial &	operator+= (const mpz_t r)
	Update *this by adding r.

SparseMultivariateIntegerPolynomial &	operator+= (const mpz_class &r)
	Update *this by adding the mpz_class r to it.

SparseMultivariateIntegerPolynomial &	operator+= (const Integer &r)
	Update *this by adding the Integer r to it.

SparseMultivariateIntegerPolynomial	operator- (const mpz_t r) const
	Subtract the mpz_t r from *this.

SparseMultivariateIntegerPolynomial	operator- (const mpz_class &r) const
	Subtract the mpz_class r from *this.

SparseMultivariateIntegerPolynomial	operator- (const Integer &r) const
	Subtract the Integer r from *this.

SparseMultivariateIntegerPolynomial &	operator-= (const mpz_t r)
	Update *this by subtracting mpz_t r.

SparseMultivariateIntegerPolynomial &	operator-= (const mpz_class &r)
	Update *this by subtracting mpz_class r.

SparseMultivariateIntegerPolynomial &	operator-= (const Integer &r)
	Update *this by subtracting Integer r.

SparseMultivariateIntegerPolynomial	operator* (const mpz_t r) const
	Multiply *this by mpz_t r.

SparseMultivariateIntegerPolynomial	operator* (const mpz_class &r) const
	Multiply *this by mpz_class r.

SparseMultivariateIntegerPolynomial	operator* (const Integer &r) const
	Multiply *this by Integer r.

SparseMultivariateIntegerPolynomial &	operator*= (const mpz_t r)
	Update *this by multiplying by mpz_t r.

SparseMultivariateIntegerPolynomial &	operator*= (const mpz_class &r)
	Update *this by multiplying by mpz_class r.

SparseMultivariateIntegerPolynomial &	operator*= (const Integer &r)
	Update *this by multiplying by Integer r.

SparseMultivariateIntegerPolynomial	operator/ (const mpz_t r) const
	Divide *this by mpz_t r.

SparseMultivariateIntegerPolynomial	operator/ (const mpz_class &r) const
	Divide *this by mpz_class r.

SparseMultivariateIntegerPolynomial	operator/ (const Integer &r) const
	Divide *this by Integer r.

SparseMultivariateIntegerPolynomial &	operator/= (const mpz_t r)
	Update *this by dividing by mpz_t r.

SparseMultivariateIntegerPolynomial &	operator/= (const mpz_class &r)
	Update *this by dividing by mpz_class r.

SparseMultivariateIntegerPolynomial &	operator/= (const Integer &r)
	Update *this by dividing by Integer r.

SparseMultivariateIntegerPolynomial	operator[] (int index) const
	Get the polynomial term at index. More...

Symbol	leadingVariable () const
	Get the leading variable, that is, the highest-order variable with positive degree of this polynomial. More...

Integer	leadingVariableDegree () const
	Get the degree of this polynomial w.r.t the leading variable.

bool	isConstantTermZero () const
	Is the contant term zero.

SparseMultivariateIntegerPolynomial	leadingCoefficientInVariable (const Symbol &x, int *e=NULL) const
	Get the leading coefficient w.r.t the input variable 'x'. More...

SparseUnivariatePolynomial< SparseMultivariateIntegerPolynomial >	convertToSUP (const Symbol &x) const
	Convert to a SUP<SMQP> given the variable 'x'. More...

void	negate ()
	Negate all the coefficients of *this. More...

SparseMultivariateIntegerPolynomial	deepCopy () const
	Get a copy of this such that all underlying memory is NOT shared. More...

Factors< SparseMultivariateIntegerPolynomial >	factor () const
	Factors this polynomial into irreducible factors. More...

void	straightLineProgram ()
	SLP representation of the polynomial.

void	printSLP (std::ostream &out=std::cout) const
	Print SLP representation.

void	randomPolynomial (int numvar, int nterms, unsigned long int coefBound, degree_t sparsity, bool includeNeg)
	Change *this to be a random non-zero polynomial. More...

void	randomPolynomial (std::vector< int > maxDegs, unsigned long int coefBound, float sparsity, bool includeNeg)
	Change *this to be a random non-zero polynomial. More...

ExpressionTree	convertToExpressionTree () const
	Convert *this to an ExpressionTree. More...

Static Public Member Functions
static SparseMultivariateIntegerPolynomial	interpolate (const std::vector< std::vector< Integer >> &points, const std::vector< Integer > &vals)
	Find the interpolating polynomial for the integer points and values specified by each vector, respectively. More...

static SparseMultivariateIntegerPolynomial	interpolate (const std::vector< std::vector< RationalNumber >> &points, const std::vector< RationalNumber > &vals)
	Find the interpolating polynomial for the rational number points and values specified by each vector, respectively. More...

Public Attributes
std::vector< SLPZRepresentation >	slp

Friends
class	SparseMultivariateRationalPolynomial

class	MapleInterface

std::istream &	operator>> (std::istream &in, SparseMultivariateIntegerPolynomial &p)
	Parse a polynomial from the in stream. More...

SparseMultivariateIntegerPolynomial	operator+ (const mpz_t r, const SparseMultivariateIntegerPolynomial &b)
	Add mpz_t r and SMQP b.

SparseMultivariateIntegerPolynomial	operator+ (const mpz_class &r, const SparseMultivariateIntegerPolynomial &b)
	Add mpz_class r and SMQP b.

SparseMultivariateIntegerPolynomial	operator- (const mpz_t r, const SparseMultivariateIntegerPolynomial &b)
	Subtract SMQP b from mpz_t r.

SparseMultivariateIntegerPolynomial	operator- (const mpz_class &r, const SparseMultivariateIntegerPolynomial &b)
	Subtract SMQP b from mpz_class r.

SparseMultivariateIntegerPolynomial	operator* (const mpz_t r, const SparseMultivariateIntegerPolynomial &b)
	Multiply mpz_t r and SMQP b.

SparseMultivariateIntegerPolynomial	operator* (const mpz_class &r, const SparseMultivariateIntegerPolynomial &b)
	Multiply mpz_class r and SMQP b.

SparseMultivariateIntegerPolynomial	operator/ (const mpz_t r, const SparseMultivariateIntegerPolynomial &b)
	Divide mpz_t r by SMQP b.

SparseMultivariateIntegerPolynomial	operator/ (const mpz_class &r, const SparseMultivariateIntegerPolynomial &b)
	Divide mpz_class r by SMQP b.

Detailed Description

A multivariate polynomial with Integer coefficients using a sparse representation.

Only non-zero coefficients are encoded.

Constructor & Destructor Documentation

◆ SparseMultivariateIntegerPolynomial() [1/7]

SparseMultivariateIntegerPolynomial::SparseMultivariateIntegerPolynomial ( )

Construct a multivariate polynomial.

◆ SparseMultivariateIntegerPolynomial() [2/7]

SparseMultivariateIntegerPolynomial::SparseMultivariateIntegerPolynomial ( int v )

Construct a multivariate polynomial with specific number of variables.

Parameters

v	Number of variables

◆ SparseMultivariateIntegerPolynomial() [3/7]

SparseMultivariateIntegerPolynomial::SparseMultivariateIntegerPolynomial ( const Symbol & x )

htop

Construct with a variable name such that f(x) = x;

Parameters

x	The variable name

◆ SparseMultivariateIntegerPolynomial() [4/7]

SparseMultivariateIntegerPolynomial::SparseMultivariateIntegerPolynomial ( const SparseMultivariateIntegerPolynomial & b )

Copy Constructor.

Does not reuse underlying memory allocated by b.

Parameters

b	A sparse multivariate polynomial

◆ SparseMultivariateIntegerPolynomial() [5/7]

SparseMultivariateIntegerPolynomial::SparseMultivariateIntegerPolynomial ( SparseMultivariateIntegerPolynomial && b )

Move Constructor.

b: The r-value reference polynomial.

◆ SparseMultivariateIntegerPolynomial() [6/7]

SparseMultivariateIntegerPolynomial::SparseMultivariateIntegerPolynomial ( const DenseUnivariateIntegerPolynomial & p )

Create a SMZP from a univariate rational polynomial.

Parameters

p	A SUZP polynomial.

◆ SparseMultivariateIntegerPolynomial() [7/7]

SparseMultivariateIntegerPolynomial::SparseMultivariateIntegerPolynomial ( const SparseUnivariatePolynomial< SparseMultivariateIntegerPolynomial > & s )

Construct from a SUP<SMZP> polynomial such that the resulting SMZP has main variable equal to the variable of s.

Parameters

s	The SUP<SMZP> polynomial

Member Function Documentation

◆ coefficient()

Integer SparseMultivariateIntegerPolynomial::coefficient ( const std::vector< int > & exps ) const

inlinevirtual

Get the coefficient of this polynomial with respect to the monomial defined by exponent vector exps.

The exponent vector should be defined over all variables in the polynomial ring.

Parameters

exps	the exponent vector

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ content() [1/2]

Integer SparseMultivariateIntegerPolynomial::content ( ) const

Get the content with respect to all variables.

That is, a single integer which is the GCD of all coefficients. The content here is one such that this/content is an integer polynomial with content of 1.

Moreover, the content is one such that the leading coefficient of the corresponding primitive part is positive.

◆ content() [2/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::content ( const std::vector< Symbol > & v ) const

Get the content of this polynomial with respect to the variables in the input vector v.

That is, viewing the polynomial recursively such that the variables in v are the polynomial variables and all other variables belong to the coefficient ring.

In this case, the content does not necessarily make the leading coefficient of the corresponding primitive part positive.

◆ convertToExpressionTree()

ExpressionTree SparseMultivariateIntegerPolynomial::convertToExpressionTree ( ) const

Convert *this to an ExpressionTree.

returns the new ExpressionTree.

◆ convertToSUP()

SparseUnivariatePolynomial<SparseMultivariateIntegerPolynomial> SparseMultivariateIntegerPolynomial::convertToSUP ( const Symbol & x ) const

Convert to a SUP<SMQP> given the variable 'x'.

returns the SUP<SMQP>.

◆ deepCopy()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::deepCopy ( ) const

Get a copy of this such that all underlying memory is NOT shared.

Note, any following arithmetic operations on the returned result will result in possibly making the underlying memory shared again.

◆ derivative() [1/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::derivative	(	const Symbol &	s,
		int	k
	)		const

virtual

Return k-th derivative.

Parameters

s	Symbol to differentiate with respect to
k	Order of the k-th derivative, k > 0

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ derivative() [2/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::derivative ( const Symbol & s ) const

inlinevirtual

Compute derivative.

Parameters

s	Symbol to differentiate with respect to

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ differentiate() [1/2]

void SparseMultivariateIntegerPolynomial::differentiate	(	const Symbol &	s,
		int	k
	)

inlinevirtual

Convert current object to its k-th derivative.

Parameters

s	Symbol to differentiate with respect to
k	Order of the derivative, k > 0

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ differentiate() [2/2]

void SparseMultivariateIntegerPolynomial::differentiate ( const Symbol & s )

inlinevirtual

Convert current object to its derivative.

Parameters

s	Symbol to differentiate with respect to

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ divide()

bool SparseMultivariateIntegerPolynomial::divide	(	const SparseMultivariateIntegerPolynomial &	b,
		SparseMultivariateIntegerPolynomial &	q,
		SparseMultivariateIntegerPolynomial &	r
	)		const

Divide this by polynomial b, returning the quotient and remainder in q and r, respectively.

returns a boolean indicating if the division was exact.

◆ evaluate() [1/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::evaluate	(	int	n,
		const Symbol *	syms,
		const Integer *	xs
	)		const

inlinevirtual

Evaluate f(x)

Parameters

syms	Array of Symbols to evaluate at corresponding xs
xs	Evaluation points

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ evaluate() [2/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::evaluate	(	const std::vector< Symbol > &	vars,
		const std::vector< Integer > &	values
	)		const

virtual

Evaluate *this polynomial given the variables and their values in vars, and values.

vars must be a list of variables which are a (not necessarily proper) subset. vars and values should have matching indices. i.e. the values of vars[i] is values[i].

returns a new SMQP where all variables in vars have been evaluated using the values given.

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ factor()

Factors<SparseMultivariateIntegerPolynomial> SparseMultivariateIntegerPolynomial::factor ( ) const

Factors this polynomial into irreducible factors.

The Factors may include a common numerical (integer) factor. See also, Factors.

◆ head()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::head ( ) const

virtual

Get the head of this polynomial.

That is, the initial multiplied by the rank.

returns the head.

Implements BPASRecursivelyViewedPolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ initial()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::initial ( ) const

virtual

Get the leading coefficient of *this with respect to the main variable.

returns the initial.

Implements BPASRecursivelyViewedPolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ integral() [1/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::integral	(	const Symbol &	s,
		int	k
	)		const

Return k-th integral.

If Symbol s is not a symbol contained in this polynomial then it becomes the main variable of the result and integration proceeds as expected.

If the integral of this polynomial cannot be represented as an integer polynomial then an error occurs. See rationalIntegral().

Parameters

s	Symbol to differentiate with respect to
k	Order of the k-th derivative, k > 0

◆ integral() [2/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::integral ( const Symbol & s ) const

inline

Compute integral.

Parameters

s	Symbol to differentiate with respect to

◆ integrate() [1/2]

void SparseMultivariateIntegerPolynomial::integrate	(	const Symbol &	s,
		int	k
	)

inline

Convert current object to its k-th integral.

Parameters

s	Symbol to differentiate with respect to
k	Order of the derivative, k > 0

◆ integrate() [2/2]

void SparseMultivariateIntegerPolynomial::integrate ( const Symbol & s )

inline

Convert current object to its derivative.

Parameters

s	Symbol to differentiate with respect to

◆ interpolate() [1/2]

static SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::interpolate	(	const std::vector< std::vector< Integer >> &	points,
		const std::vector< Integer > &	vals
	)

static

Find the interpolating polynomial for the integer points and values specified by each vector, respectively.

The points vector is a vector of vectors, where each element of the outer vector represents a multi-dimensional point. Each multi-dimensional point should have the same number of dimensions and in the same order.

An error will occur if the resulting interpolating polynomial cannot be represented as an integer polynomial.

returns the interpolating polynomial.

◆ interpolate() [2/2]

static SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::interpolate	(	const std::vector< std::vector< RationalNumber >> &	points,
		const std::vector< RationalNumber > &	vals
	)

static

Find the interpolating polynomial for the rational number points and values specified by each vector, respectively.

The points vector is a vector of vectors, where each element of the outer vector represents a multi-dimensional point. Each multi-dimensional point should have the same number of dimensions and in the same order.

An error will occur if the resulting interpolating polynomial cannot be represented as an integer polynomial.

returns the interpolating polynomial.

◆ isConstant()

int SparseMultivariateIntegerPolynomial::isConstant ( ) const

Determine if this polynomial encodes a constant.

returns 0 if not a constant, 1 if a constant >= 0, -1 if a negative contstant.

◆ isEqual()

bool SparseMultivariateIntegerPolynomial::isEqual ( const SparseMultivariateIntegerPolynomial & b ) const

Determine if *this is equal to b.

This takes into account the variable ordering on both poylnomials in such a way that the same polynomial under different variable orderings are NOT equal.

◆ isNegativeOne()

bool SparseMultivariateIntegerPolynomial::isNegativeOne ( ) const

Is this polynomial negative one.

returns true iff this polynomial encodes -1.

◆ isOne()

bool SparseMultivariateIntegerPolynomial::isOne ( ) const

Is this polynomial one.

return true iff this polynomial encodes 1.

◆ isZero()

bool SparseMultivariateIntegerPolynomial::isZero ( ) const

Is this polynomial zero.

returns true iff this polynomial encodes 0.

◆ leadingCoefficientInVariable()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::leadingCoefficientInVariable	(	const Symbol &	x,
		int *	e = `NULL`
	)		const

Get the leading coefficient w.r.t the input variable 'x'.

Returns the leading exponent as e.

returns the coefficient as a new SMQP.

◆ leadingVariable()

Symbol SparseMultivariateIntegerPolynomial::leadingVariable ( ) const

Get the leading variable, that is, the highest-order variable with positive degree of this polynomial.

returns the leading variable or the empty string if this polynomial has zero variables.

◆ mainDegree()

int SparseMultivariateIntegerPolynomial::mainDegree ( ) const

virtual

Get the degree of the main variable.

returns the degree.

Implements BPASRecursivelyViewedPolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ mainVariable()

Symbol SparseMultivariateIntegerPolynomial::mainVariable ( ) const

inlinevirtual

Get the main variable.

That is, the highest-ordered variable with positive degree.

returns the main variable.

Implements BPASRecursivelyViewedPolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ negate()

void SparseMultivariateIntegerPolynomial::negate ( )

Negate all the coefficients of *this.

Note, that due to the sharing nature of underling nodes, this may alter the Nodes of other SMQP.

◆ negativeOne()

void SparseMultivariateIntegerPolynomial::negativeOne ( )

Sets this polynomial to -1.

Maintains existing variable ordering.

◆ one()

void SparseMultivariateIntegerPolynomial::one ( )

Sets this polynomial to one.

Maintains existing variable ordering.

◆ operator!=()

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

Determine if *this is not equal to the specified polynomial.

This takes into account the variable ordering on both poylnomials in such a way that the same polynomial under different variable orderings are NOT equal.

◆ operator*=()

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

Multiply the polynomials a and b, returning their product.

Update this by multiplying by the specified polynomail.

◆ operator+=()

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

Add the polynomails a and b and return the sum.

Update *this by adding the specified polynomail to it.

◆ operator-()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::operator- ( ) const

Subtract the polynomial b from a and return the difference.

Unary operation, return *this * -1.

◆ operator==()

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

Determine if *this is equal to the specified polynomial.

This takes into account the variable ordering on both poylnomials in such a way that the same polynomial under different variable orderings are NOT equal.

◆ operator[]()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::operator[] ( int index ) const

Get the polynomial term at index.

Returns 0 if index is beyond the number of terms in this polynomial.

◆ operator^()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::operator^ ( long long int e ) const

Exponentiate *this by the input exponent integer.

Treats negative exponents as positive.

◆ operator^=()

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

Update *this by exponentiating this to the input integer.

Treats negative exponents as positive.

◆ primitivePart() [1/4]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::primitivePart ( ) const

Get the primitive part with respect to all variables.

This is equivalent to this / content().

◆ primitivePart() [2/4]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::primitivePart ( Integer & content ) const

Get the primitive part with respect to all variables.

This is equivalent to this / content().

Simultaneously returns the rational number content in the parameter content.

◆ primitivePart() [3/4]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::primitivePart ( const std::vector< Symbol > & v ) const

Get the primitive part with respect to the variables in the vector v.

returns the primitive part.

◆ primitivePart() [4/4]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::primitivePart	(	const std::vector< Symbol > &	v,
		SparseMultivariateIntegerPolynomial &	content
	)		const

Get the primitive part with respect to the variables in the vector v.

Returns the corresponding content in the content reference. returns the primitive part.

◆ pseudoDivide()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::pseudoDivide	(	const SparseMultivariateIntegerPolynomial &	b,
		SparseMultivariateIntegerPolynomial *	quo = `NULL`,
		SparseMultivariateIntegerPolynomial *	mult = `NULL`,
		bool	lazy = `0`
	)		const

Pseudo divide this by b.

The remainder is returned. if parameter quo is not null then the quotient is returned in quo. if parameter mult is not null then the multiplier is set to the initial of b raised to the power of degree(c, mvar(c)) - degree(b, mvar(c)) + 1.

returns the pseudo remainder.

◆ randomPolynomial() [1/2]

void SparseMultivariateIntegerPolynomial::randomPolynomial	(	int	numvar,
		int	nterms,
		unsigned long int	coefBound,
		degree_t	sparsity,
		bool	includeNeg
	)

Change *this to be a random non-zero polynomial.

numvar: number of variables nterms: number of terms coefBound: limit on the number of bits encoding the coefficients. sparsity: succesive terms are at most sparsity away from each other includeNeg: a bool to say if coefs can be randomly negative or all positive

◆ randomPolynomial() [2/2]

void SparseMultivariateIntegerPolynomial::randomPolynomial	(	std::vector< int >	maxDegs,
		unsigned long int	coefBound,
		float	sparsity,
		bool	includeNeg
	)

Change *this to be a random non-zero polynomial.

The number of variables will be equal to the size of the maxDegs vector. The term whose monomial has exponents equal to those in maxDegs is guaranteed to exist in the resulting polynomial. A sparsity of 0 produces a dense polynomial. A sparsity of 1 produces only one term; one whose monomial has exponents equal to maxDegs.

coefBound: limit on the number of bits encoding the coefficients. sparsity: a percentage of zero-terms between term with maxDegs and the constant term. includeNeg: a bool to say if coefs can be randomly negative or all positive

◆ rank()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::rank ( ) const

virtual

Get the rank of this polynomial.

That is, the main variable raised to the main degree.

returns the rank.

Implements BPASRecursivelyViewedPolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ rationalDivide()

bool SparseMultivariateIntegerPolynomial::rationalDivide	(	const SparseMultivariateIntegerPolynomial &	b,
		SparseMultivariateRationalPolynomial &	q,
		SparseMultivariateRationalPolynomial &	r
	)		const

Divide this by polynomial b, return the quotient and remainder in q and r as rational number polynomials.

This differs from the normal divide function where divisibility of coefficients does not matter as the result belongs to the polynomial ring over the rational numbers.

returns a boolean indicating if the division was exact.

◆ rationalIntegral() [1/2]

SparseMultivariateRationalPolynomial SparseMultivariateIntegerPolynomial::rationalIntegral	(	const Symbol &	s,
		int	k
	)		const

Return k-th integral.

If Symbol s is not a symbol contained in this polynomial then it becomes the main variable of the result and integration proceeds as expected.

This method returns a rational number polynomial and therefore works regardless of the coefficients of this polynomial.

Parameters

s	Symbol to differentiate with respect to
k	Order of the k-th derivative, k > 0

◆ rationalIntegral() [2/2]

SparseMultivariateRationalPolynomial SparseMultivariateIntegerPolynomial::rationalIntegral ( const Symbol & s ) const

Return the integral of this, with respect to Symbol s, as a rational number polynomial.

Parameters

s	the symbol to integrate with respect to returns the integral

◆ separant()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::separant ( ) const

virtual

Get the separant of this polynomial, that is, its derivative with respect to the main variable.

Returns: the separant of this polynomial.

Implements BPASRecursivelyViewedPolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ setCoefficient()

void SparseMultivariateIntegerPolynomial::setCoefficient	(	const std::vector< int > &	v,
		const Integer &	r
	)

inlinevirtual

Set the coefficient of this polynomial for the monomial defined by exponent vector exps to r.

The exponent vector should be defined over all variables in the polynomial ring.

Parameters

exps	the exponent vector
r	the ring element to set as coefficient.

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ setRingVariables()

void SparseMultivariateIntegerPolynomial::setRingVariables ( const std::vector< Symbol > & )

virtual

Set variables' names.

This method can be used to shrink, expand, re-order, and re-name the variables of the polynomial ring in which this polynomial belongs.

Any symbol in the input vector that matches the current variables is considered a re-order. Non-matching symbols are considered a re-name and this renames is done in the order they appear by increasing index. For example: Q[x,y,z] -> Q[y,s,t] will have y reordered to be the first variable, x renamed to s, and z renamed to t.

If the size of the input vector is greater than the current number of variables then the polynomial ring is being expanded. Matching variables from this polynomial are re-ordered as they appear in the input vector. Current variables that have no match in the input vector are re-named in order of increasing index of unused variables from the input vector. For example: Q[x,y,z] -> Q[s,t,z,x] will have y named to s, t a new variable, and z and x re-ordered accordingly.

If the size of the input vector is less than the current number of variables then the polynomial ring is shrink to remove variables. Variables in the input vector that are also found to be in the current polynomial ring are matched and re-ordered as necessary.

It is invalid to remove a variable which is non-zero in this polynomial.

Implements BPASMultivariatePolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ squareFreePart() [1/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::squareFreePart ( ) const

Computes the square free part of this polynomail.

That is, the polynomial of this divided by all square factors. This is with respect to all variables.

◆ squareFreePart() [2/2]

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::squareFreePart ( std::vector< Symbol > & vars ) const

Computes the square free part of this polynomail.

That is, the polynomial of this divided by all square factors. This is with respect to all variables.

◆ tail()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::tail ( ) const

virtual

Get the tail of this polynomial.

That is, This - this.head().

returns the tail.

Implements BPASRecursivelyViewedPolynomial< Integer, SparseMultivariateIntegerPolynomial >.

◆ unitCanonical()

SparseMultivariateIntegerPolynomial SparseMultivariateIntegerPolynomial::unitCanonical	(	SparseMultivariateIntegerPolynomial *	u,
		SparseMultivariateIntegerPolynomial *	v
	)		const

inline

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.

◆ zero()

void SparseMultivariateIntegerPolynomial::zero ( )

Set this polynomial to zero.

Maintains existing variable ordering.

Friends And Related Function Documentation

◆ operator>>

std::istream& operator>>	(	std::istream &	in,
		SparseMultivariateIntegerPolynomial &	p
	)

friend

Parse a polynomial from the in stream.

Exactly one line is parsed.

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

include/IntegerPolynomial/mzpolynomial.hpp

Public Member Functions

Static Public Member Functions

Public Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ SparseMultivariateIntegerPolynomial() [1/7]

◆ SparseMultivariateIntegerPolynomial() [2/7]

◆ SparseMultivariateIntegerPolynomial() [3/7]

◆ SparseMultivariateIntegerPolynomial() [4/7]

◆ SparseMultivariateIntegerPolynomial() [5/7]

◆ SparseMultivariateIntegerPolynomial() [6/7]

◆ SparseMultivariateIntegerPolynomial() [7/7]

Member Function Documentation

◆ coefficient()

◆ content() [1/2]

◆ content() [2/2]

◆ convertToExpressionTree()

◆ convertToSUP()

◆ deepCopy()

◆ derivative() [1/2]

◆ derivative() [2/2]

◆ differentiate() [1/2]

◆ differentiate() [2/2]

◆ divide()

◆ evaluate() [1/2]

◆ evaluate() [2/2]

◆ factor()

◆ head()

◆ initial()

◆ integral() [1/2]

◆ integral() [2/2]

◆ integrate() [1/2]

◆ integrate() [2/2]

◆ interpolate() [1/2]

◆ interpolate() [2/2]

◆ isConstant()

◆ isEqual()

◆ isNegativeOne()

◆ isOne()

◆ isZero()

◆ leadingCoefficientInVariable()

◆ leadingVariable()

◆ mainDegree()

◆ mainVariable()

◆ negate()

◆ negativeOne()

◆ one()

◆ operator!=()

◆ operator*=()

◆ operator+=()

◆ operator-()

◆ operator==()

◆ operator[]()

◆ operator^()

◆ operator^=()

◆ primitivePart() [1/4]

◆ primitivePart() [2/4]

◆ primitivePart() [3/4]

◆ primitivePart() [4/4]

◆ pseudoDivide()

◆ randomPolynomial() [1/2]

◆ randomPolynomial() [2/2]

◆ rank()

◆ rationalDivide()

◆ rationalIntegral() [1/2]

◆ rationalIntegral() [2/2]

◆ separant()

◆ setCoefficient()

◆ setRingVariables()

◆ squareFreePart() [1/2]

◆ squareFreePart() [2/2]

◆ tail()

◆ unitCanonical()

◆ zero()

Friends And Related Function Documentation

◆ operator>>