mmx::sparse Namespace Reference

namespace for representation of polynomials as sequence of monomials More...

Classes
struct	CLASS
struct	dual

Functions
template<class C , class O >
void	neg_exponent (dual< C, O > &p)
template<class C , class O >
void	add (dual< C, O > &res, const dual< C, O > &a, const dual< C, O > &b)
template<class C , class O >
void	add (dual< C, O > &res, const dual< C, O > &a, const C &b)
template<class C , class O >
void	add (dual< C, O > &res, const C &b, const dual< C, O > &a)
template<class C , class O >
void	sub (dual< C, O > &res, const dual< C, O > &a, const dual< C, O > &b)
template<class C , class O >
void	sub (dual< C, O > &res, const dual< C, O > &a, const C &b)
template<class C , class O >
void	sub (dual< C, O > &res, const C &a, const dual< C, O > &b)
template<class C , class O >
void	mul (dual< C, O > &res, const dual< C, O > &a, const dual< C, O > &b)
template<class C , class O >
void	mul (dual< C, O > &res, const dual< C, O > &a, const C &b)
template<class C , class O >
void	mul (dual< C, O > &res, const C &b, const dual< C, O > &a)
template<class C , class O >
void	mul (dual< C, O > &res, const monomial_seq< C, O > &p, const dual< C, O > &l)
template<class C , class O >
void	mul (monomial_seq< C, O > &res, const dual< C, O > &l, const monomial_seq< C, O > &p)
template<class C , class O >
void	div (dual< C, O > &f, const C &c)
template<class C , class O >
void	div (dual< C, O > &r, const dual< C, O > &p, const C &c)
template<class X >
bool	with_plus_sign (const X &x)
template<class OSTREAM , class C >
OSTREAM &	print_dual (OSTREAM &os, const monom< C > &m, const variables &V)
template<class OSTREAM , class C , class O >
OSTREAM &	print (OSTREAM &os, const dual< C, O > &P, const variables &V)
	Output operator. More...
TMPL void	add (Polynomial &result, const Polynomial &p1, const Polynomial &p2)
TMPL void	add (Polynomial &p1, const Polynomial &p2)
	Inplace addition. More...
template<class C , class O , class MONOM , class REP , class I , class J , class M >
I	add (Polynomial &p1, I b1, J e1, const M &m2)
TMPL void	add (Polynomial &p, const typename Polynomial::monom_t &m)
TMPL void	add (Polynomial &p, const C &c)
TMPLX void	add (Polynomial &r, const Polynomial &q, const X &c)
TMPL void	add (Polynomial &r, const C &c, const Polynomial &q)
TMPL void	sub (Polynomial &result, const Polynomial &p1, const Polynomial &p2)
TMPL void	sub (Polynomial &p, const Polynomial &q)
TMPLX void	sub (Polynomial &r, const Polynomial &q, const X &c)
TMPLX void	sub (Polynomial &r, const X &c, const Polynomial &q)
TMPL void	mul (Polynomial &r, const C &c)
	Multiplication of a polynomial by a monomial or a scalar. More...
TMPL void	mul (Polynomial &r, const Polynomial &p, const C &c)
TMPLX void	mul (Polynomial &r, const X &c, const Polynomial &p)
template<class C , class O , class MONOM , class REP , class M >
void	mul_ext (Polynomial &r, const Polynomial &a, const M &m)
	Multiplication of a polynomial by a monomial or a scalar. More...
TMPL void	mul (Polynomial &r, const Polynomial &a, const typename Polynomial::monom_t &m)
	Multiplication of a polynomial by a monomial or a scalar. More...
TMPL void	mul (Polynomial &r, const typename Polynomial::monom_t &m)
	Multiplication of a polynomial by a monomial or a scalar. More...
template<class C , class O , class MONOM , class REP , class M >
void	mul_ext_e (Polynomial &result, const Polynomial &a, const M &m)
TMPL void	mul (Polynomial &r, const Polynomial &a, const Polynomial &b)
	Multiplication of two polynomials. More...
TMPLX void	mul (Polynomial &r, const Polynomial &a, const Polynomial &b, const X &o)
	Specialisation for list. More...
TMPL void	mul_iterator (Polynomial &r, typename Polynomial::const_iterator b, typename Polynomial::const_iterator e, const Polynomial &p)
TMPL void	mul (Polynomial &r, const Polynomial &p)
TMPL void	div (Polynomial &q, const Polynomial &a, const Polynomial &b)
TMPL void	div (Polynomial &r, const Polynomial &b)
TMPL void	div (Polynomial &f, const typename Polynomial::Scalar &c)
TMPL void	div (Polynomial &r, const Polynomial &p, const C &c)
TMPL void	rem (Polynomial &r, const Polynomial &a, const Polynomial &b)
template<class C , class O , class MONOM , class REP , class U >
void	coefficients (Seq< U > &r, const Polynomial &f, int v)
TMPL void	coefficients (Seq< C > &r, const Polynomial &f)
template<class R >
int	lvar (const R &p)
	Index of the leading variable (of maximal index) of a polynomial. More...
template<class R >
unsigned	nbvar (const R &p)
	Number of variables of a polynomial. More...
template<class R >
int	degree (const R &p)
	Degree of a polynomial. More...
template<class R >
int	degree (const R &p, int i)
	Degree of a polynomial with respect to the i th variable.. More...
template<class R >
R::coeff_t &	leadingcoeff (R &a)
template<class R >
R::coeff_t	leadingcoeff (const R &a)
template<class POL >
POL::coeff_t	coeffof (const POL &p, const typename POL::monom_t &mono)
template<class POL >
POL::const_iterator	last_term (const POL &p)
template<class R >
void	div_rem (R &q, R &a, const R &b0)
	Divide a in place by b, concidering all the monomials. More...
TMPL void	diff (Polynomial &r, const Polynomial &p, int i)
	Derivative of p with respect to i th variable put in r. More...
TMPL void	shift (Polynomial &r, const Polynomial &p, const typename Polynomial::monom_t &m)
	Multiply p by a monomial m and put the result in r. More...
TMPL void	copy (Polynomial &r, const Polynomial &a)
	Copy of a in r. More...
TMPL Polynomial::coeff_t	eval (const Polynomial &p, const typename Polynomial::coeff_t &x, const typename Polynomial::coeff_t &y)
	Evaluate the polynomial p for x0=x, x1=y, and the other xi=1. More...
template<class C , class O , class MONOM , class REP , class R , class VCT >
void	eval_at (R &r, const Polynomial &p, const VCT &x)
template<class R , class C , class O , class MONOM , class REP , class X >
void	eval (R &r, const Polynomial &p, const X &x)
	Evaluate the polynomial p for x0=x, x1=y, and the other xi=1. **/. More...
TMPL void	homogenize (Polynomial &r, const Polynomial &p, const Polynomial &v)
TMPL void	homogenize (Polynomial &r, const Polynomial &p, int n, const Polynomial &v)
template<class MP >
MP	convert (const MP &P, typename MP::coeff_t x, typename MP::coeff_t y, int ind)
template<class C , class O , class MONOM , class REP , class VARIABLES >
std::string	to_string (const Polynomial &P, const VARIABLES &V)
template<class OS , class C , class O , class MONOM , class REP , class VARIABLES >
OS &	print (OS &os, const Polynomial &P, const VARIABLES &V)
template<class OS , class C , class O , class MONOM , class REP , class VARIABLES >
OS &	print_as_double (OS &os, const Polynomial &P, const VARIABLES &V)
template<class OS , class C , class O , class MONOM , class REP >
OS &	print (OS &os, const Polynomial &P)
template<class OS , class C , class O , class MONOM , class REP >
OS &	print_verbatim (OS &os, const Polynomial &P)
template<class POL , class C >
POL	shift (typename POL::const_iterator monom, const C &a, int i)
template<class POL , class C >
POL	scale (const POL &p, C a, int v)
template<class T , class MP , class V >
T	eval (const MP &p, const V &v)
template<class R , class MP , class V >
void	eval (R &r, const MP &p, const V &v, unsigned n)
template<class MP , class X >
MP	subs (unsigned var, const X &val, const MP &P)
template<class MP >
MP	subs (const MP &P, int var, typename MP::coeff_t val)
template<class MP >
MP	subs (const MP &P, char *x, typename MP::coeff_t val)
template<class T >
void	print (const T &x)
template<class MP >
MP	swap (const MP &P, int var_i, int var_j)
template<class MP >
MP	swap (const MP &P, char *x_i, char *x_j)
template<class C , class O , class MONOM , class REP >
C	content (const Polynomial &P)

Detailed Description

namespace for representation of polynomials as sequence of monomials

Function Documentation

template<class C , class O >

void mmx::sparse::add ( dual< C, O > &  res, const dual< C, O > &  a, const dual< C, O > &  b  )

inline

template<class C , class O >

void mmx::sparse::add ( dual< C, O > &  res, const dual< C, O > &  a, const C &  b  )

inline

TMPL void mmx::sparse::add	(	Polynomial &	result,
		const Polynomial &	p1,
		const Polynomial &	p2
	)

template<class C , class O >

void mmx::sparse::add ( dual< C, O > &  res, const C &  b, const dual< C, O > &  a  )

inline

TMPL void mmx::sparse::add	(	Polynomial &	p1,
		const Polynomial &	p2
	)

Inplace addition.

template<class C , class O , class MONOM , class REP , class I , class J , class M >

I mmx::sparse::add	(	Polynomial &	p1,
		I	b1,
		J	e1,
		const M &	m2
	)

TMPL void mmx::sparse::add	(	Polynomial &	p,
		const typename Polynomial::monom_t &	m
	)

TMPL void mmx::sparse::add	(	Polynomial &	p,
		const C &	c
	)

TMPLX void mmx::sparse::add	(	Polynomial &	r,
		const Polynomial &	q,
		const X &	c
	)

TMPL void mmx::sparse::add	(	Polynomial &	r,
		const C &	c,
		const Polynomial &	q
	)

template<class C , class O , class MONOM , class REP , class U >

void mmx::sparse::coefficients	(	Seq< U > &	r,
		const Polynomial &	f,
		int	v
	)

TMPL void mmx::sparse::coefficients	(	Seq< C > &	r,
		const Polynomial &	f
	)

template<class POL >

POL::coeff_t mmx::sparse::coeffof	(	const POL &	p,
		const typename POL::monom_t &	mono
	)

Return the coefficient of a monomial m in a polynomial p if it appears in p or 0 otherwise.

template<class C , class O , class MONOM , class REP >

C mmx::sparse::content

(

const Polynomial &

P

)

template<class MP >

MP mmx::sparse::convert	(	const MP &	P,
		typename MP::coeff_t	x,
		typename MP::coeff_t	y,
		int	ind
	)

function which return the equation of the polynomial P hiding the indice ind and evaluating at x and y.

TMPL void mmx::sparse::copy	(	Polynomial &	r,
		const Polynomial &	a
	)

Copy of a in r.

template<class R >

int mmx::sparse::degree

(

const R &

p

)

Degree of a polynomial.

template<class R >

int mmx::sparse::degree	(	const R &	p,
		int	i
	)

Degree of a polynomial with respect to the i th variable..

TMPL void mmx::sparse::diff	(	Polynomial &	r,
		const Polynomial &	p,
		int	i
	)

Derivative of p with respect to i th variable put in r.

template<class C , class O >

void mmx::sparse::div ( dual< C, O > &  f, const C &  c  )

inline

template<class C , class O >

void mmx::sparse::div	(	dual< C, O > &	r,
		const dual< C, O > &	p,
		const C &	c
	)

TMPL void mmx::sparse::div ( Polynomial &  q, const Polynomial &  a, const Polynomial &  b  )

inline

TMPL void mmx::sparse::div ( Polynomial &  r, const Polynomial &  b  )

inline

TMPL void mmx::sparse::div ( Polynomial &  f, const typename Polynomial::Scalar &  c  )

inline

TMPL void mmx::sparse::div	(	Polynomial &	r,
		const Polynomial &	p,
		const C &	c
	)

template<class R >

void mmx::sparse::div_rem	(	R &	q,
		R &	a,
		const R &	b0
	)

Divide a in place by b, concidering all the monomials.

TMPL Polynomial::coeff_t mmx::sparse::eval	(	const Polynomial &	p,
		const typename Polynomial::coeff_t &	x,
		const typename Polynomial::coeff_t &	y
	)

Evaluate the polynomial p for x0=x, x1=y, and the other xi=1.

template<class R , class C , class O , class MONOM , class REP , class X >

void mmx::sparse::eval ( R &  r, const Polynomial &  p, const X &  x  )

inline

Evaluate the polynomial p for x0=x, x1=y, and the other xi=1. **/.

template<class T , class MP , class V >

T mmx::sparse::eval	(	const MP &	p,
		const V &	v
	)

template<class R , class MP , class V >

void mmx::sparse::eval	(	R &	r,
		const MP &	p,
		const V &	v,
		unsigned	n
	)

template<class C , class O , class MONOM , class REP , class R , class VCT >

void mmx::sparse::eval_at	(	R &	r,
		const Polynomial &	p,
		const VCT &	x
	)

TMPL void mmx::sparse::homogenize ( Polynomial &  r, const Polynomial &  p, const Polynomial &  v  )

inline

TMPL void mmx::sparse::homogenize ( Polynomial &  r, const Polynomial &  p, int  n, const Polynomial &  v  )

inline

template<class POL >

POL::const_iterator mmx::sparse::last_term

(

const POL &

p

)

template<class R >

R::coeff_t& mmx::sparse::leadingcoeff

(

R &

a

)

template<class R >

R::coeff_t mmx::sparse::leadingcoeff

(

const R &

a

)

template<class R >

int mmx::sparse::lvar

(

const R &

p

)

Index of the leading variable (of maximal index) of a polynomial.

template<class C , class O >

void mmx::sparse::mul ( dual< C, O > &  res, const dual< C, O > &  a, const dual< C, O > &  b  )

inline

template<class C , class O >

void mmx::sparse::mul ( dual< C, O > &  res, const dual< C, O > &  a, const C &  b  )

inline

template<class C , class O >

void mmx::sparse::mul ( dual< C, O > &  res, const C &  b, const dual< C, O > &  a  )

inline

template<class C , class O >

void mmx::sparse::mul	(	dual< C, O > &	res,
		const monomial_seq< C, O > &	p,
		const dual< C, O > &	l
	)

Compute the product p*l where l is a linear form and p a polynomial. The result is the linear form obtained by applying the inverse variables on the polynomial and by keeping the negative exponents.

template<class C , class O >

void mmx::sparse::mul	(	monomial_seq< C, O > &	res,
		const dual< C, O > &	l,
		const monomial_seq< C, O > &	p
	)

Compute the product l*p where l is a linear form and p a polynomial. The result is the polynomial obtained by applying the inverse variables on the polynomial and by keeping the positive exponents.

TMPL void mmx::sparse::mul	(	Polynomial &	r,
		const C &	c
	)

Multiplication of a polynomial by a monomial or a scalar.

TMPL void mmx::sparse::mul	(	Polynomial &	r,
		const Polynomial &	p,
		const C &	c
	)

TMPLX void mmx::sparse::mul	(	Polynomial &	r,
		const X &	c,
		const Polynomial &	p
	)

TMPL void mmx::sparse::mul ( Polynomial &  r, const Polynomial &  a, const typename Polynomial::monom_t &  m  )

inline

Multiplication of a polynomial by a monomial or a scalar.

TMPL void mmx::sparse::mul ( Polynomial &  r, const typename Polynomial::monom_t &  m  )

inline

Multiplication of a polynomial by a monomial or a scalar.

TMPL void mmx::sparse::mul ( Polynomial &  r, const Polynomial &  a, const Polynomial &  b  )

inline

Multiplication of two polynomials.

TMPLX void mmx::sparse::mul ( Polynomial &  r, const Polynomial &  a, const Polynomial &  b, const X &  o  )

inline

Specialisation for list.

TMPL void mmx::sparse::mul ( Polynomial &  r, const Polynomial &  p  )

inline

template<class C , class O , class MONOM , class REP , class M >

void mmx::sparse::mul_ext ( Polynomial &  r, const Polynomial &  a, const M &  m  )

inline

Multiplication of a polynomial by a monomial or a scalar.

template<class C , class O , class MONOM , class REP , class M >

void mmx::sparse::mul_ext_e ( Polynomial &  result, const Polynomial &  a, const M &  m  )

inline

TMPL void mmx::sparse::mul_iterator ( Polynomial &  r, typename Polynomial::const_iterator  b, typename Polynomial::const_iterator  e, const Polynomial &  p  )

inline

template<class R >

unsigned mmx::sparse::nbvar

(

const R &

p

)

Number of variables of a polynomial.

template<class C , class O >

void mmx::sparse::neg_exponent

(

dual< C, O > &

p

)

template<class OSTREAM , class C , class O >

OSTREAM& mmx::sparse::print	(	OSTREAM &	os,
		const dual< C, O > &	P,
		const variables &	V
	)

Output operator.

The inverse variables are denoted by di. The opposite of their exponents is printed.

template<class OS , class C , class O , class MONOM , class REP , class VARIABLES >

OS& mmx::sparse::print	(	OS &	os,
		const Polynomial &	P,
		const VARIABLES &	V
	)

template<class OS , class C , class O , class MONOM , class REP >

OS& mmx::sparse::print	(	OS &	os,
		const Polynomial &	P
	)

template<class T >

void mmx::sparse::print

(

const T &

x

)

template<class OS , class C , class O , class MONOM , class REP , class VARIABLES >

OS& mmx::sparse::print_as_double	(	OS &	os,
		const Polynomial &	P,
		const VARIABLES &	V
	)

template<class OSTREAM , class C >

OSTREAM& mmx::sparse::print_dual ( OSTREAM &  os, const monom< C > &  m, const variables &  V  )

inline

template<class OS , class C , class O , class MONOM , class REP >

OS& mmx::sparse::print_verbatim	(	OS &	os,
		const Polynomial &	P
	)

TMPL void mmx::sparse::rem ( Polynomial &  r, const Polynomial &  a, const Polynomial &  b  )

inline

template<class POL , class C >

POL mmx::sparse::scale	(	const POL &	p,
		C	a,
		int	v
	)

The variables of a "tri-variate" polynomial are multiplied as follows: x0 = a*x0, x1 = b*x1, x2 = c*x2.

TMPL void mmx::sparse::shift	(	Polynomial &	r,
		const Polynomial &	p,
		const typename Polynomial::monom_t &	m
	)

Multiply p by a monomial m and put the result in r.

template<class POL , class C >

POL mmx::sparse::shift	(	typename POL::const_iterator	monom,
		const C &	a,
		int	i
	)

Function applicable only to the polynomials which has three variables. Carry out a change of variable x_i = x_i + a on the multivariable monomial passed as parameter.

template<class C , class O >

void mmx::sparse::sub ( dual< C, O > &  res, const dual< C, O > &  a, const dual< C, O > &  b  )

inline

template<class C , class O >

void mmx::sparse::sub ( dual< C, O > &  res, const dual< C, O > &  a, const C &  b  )

inline

template<class C , class O >

void mmx::sparse::sub ( dual< C, O > &  res, const C &  a, const dual< C, O > &  b  )

inline

TMPL void mmx::sparse::sub	(	Polynomial &	result,
		const Polynomial &	p1,
		const Polynomial &	p2
	)

TMPL void mmx::sparse::sub	(	Polynomial &	p,
		const Polynomial &	q
	)

TMPLX void mmx::sparse::sub	(	Polynomial &	r,
		const Polynomial &	q,
		const X &	c
	)

TMPLX void mmx::sparse::sub	(	Polynomial &	r,
		const X &	c,
		const Polynomial &	q
	)

template<class MP , class X >

MP mmx::sparse::subs	(	unsigned	var,
		const X &	val,
		const MP &	P
	)

Substitute the variable x_(var) by the value val.

template<class MP >

MP mmx::sparse::subs	(	const MP &	P,
		int	var,
		typename MP::coeff_t	val
	)

Substitute the variable x_(var) by the value val.

template<class MP >

MP mmx::sparse::subs	(	const MP &	P,
		char *	x,
		typename MP::coeff_t	val
	)

Substitute the variable x by the value val.

template<class MP >

MP mmx::sparse::swap	(	const MP &	P,
		int	var_i,
		int	var_j
	)

Swap the variable x_(var_i) and x_(var_j) in the polynomial P.

template<class MP >

MP mmx::sparse::swap	(	const MP &	P,
		char *	x_i,
		char *	x_j
	)

Swap the variable x_i and x_j in the polynomial P.

template<class C , class O , class MONOM , class REP , class VARIABLES >

std::string mmx::sparse::to_string	(	const Polynomial &	P,
		const VARIABLES &	V
	)

template<class X >

bool mmx::sparse::with_plus_sign

(

const X &

x

)