MPOLDST Namespace Reference

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Detailed Description

Module for polynomials, which are sequences of monomials.

Functions
template<class OSTREAM, class ITER>
OSTREAM &	lisp (OSTREAM &os, ITER it, const ITER &ed, const Variables &V=Variables::default_)
template<class O, class R>
void	add (R &result, const R &p1, const R &p2)
template<class R, class O>
void	sub (R &result, const R &p1, const R &p2)
template<class O, class R>
void	add (R &p1, const R &p2)
	Inplace addition.
template<class R, class I, class J, class M, class O>
I	add (R &p1, I b1, J e1, const M &m2, const O &o)
template<class R, class S, class C>
void	mul_ext (R &result, const S &a, const C &m)
	Multiplication of a polynomial by a monomial or a scalar.
template<class R, class C>
void	mul_ext_e (std::list< R > &result, const std::list< R > &a, const C &m)
template<class R, class O>
void	mul (R &r, const R &a, const R &b, const O &o)
	Multiplication of two polynomials.
template<class M, class O>
void	mul (std::list< M > &r, const std::list< M > &a, const std::list< M > &b, const O &o)
	Specialisation for list.
template<class R, class O>
void	mul (R &r, typename R::const_iterator b, typename R::const_iterator e, const R &p)
template<class R>
int	lvar (const R &p)
	Index of the leading variable (of maximal index) of a polynomial.
template<class R>
unsigned	nbvar (const R &p)
	Number of variables of a polynomial.
template<class R>
int	degree (const R &p)
	Degree of a polynomial.
template<class R>
int	degree (const R &p, int i)
	Degree of a polynomial with respect to the i th variable..
template<class R>
R::coeff_t &	leadingcoeff (R &a)
template<class R>
R::coeff_t	leadingcoeff (const R &a)
template<class R>
void	div_rem (R &q, R &a, const R &b)
	Divide a in place by b.
template<class POL>
POL::coeff_t	coeffof (const POL &p, const typename POL::monom_t &mono)
template<class R>
void	div_rem_x (R &q, R &a, const R &b0)
	Divide a in place by b, concidering all the monomials.
template<class R>
void	diff (R &r, const R &p, int i)
	Derivative of p with respect to i$^{ th}$ variable put in r.
template<class R>
void	copy (R &r, const R &a)
	Copy of a in r.
template<class R>
R::coeff_t	eval (const R &p, const typename R::coeff_t &x, const typename R::coeff_t &y)
template<class MP>
MP	convert (const MP &P, typename MP::coeff_t x, typename MP::coeff_t y, int ind)
template<class OS, class R, class VARIABLES>
OS &	print (OS &os, const R &P, const VARIABLES &V)
template<class OS, class R>
OS &	print (OS &os, const R &P)
template<class OS, class R>
OS &	print_verbatim (OS &os, const R &P)
template<class MP>
MP	Rotate_yz (const MP &P, typename MP::coeff_t t)
template<class MP>
MP	Rotate_xz (const MP &P, typename MP::coeff_t t)
template<class MP>
MP	Rotate_xy (const MP &P, typename MP::coeff_t t)
template<class MP>
MP	Rotate_xyz (const MP &P, typename MP::coeff_t t)
template<class POL, class C>
POL	shift (typename POL::const_iterator monom, C a, int i)
template<class POL, class C>
POL	shift (const POL &polynom, C x0, C y0, C z0)
template<class POL, class C>
POL	scale (const POL &polynom, C a, C b, C c)
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 T, class MP, class V>
T	eval (const MP &p, const V &v)

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Function Documentation

template<class O, class R>

void MPOLDST::add	(	R &	p1,
		const R &	p2
	)

Inplace addition.

Definition at line 109 of file MPOLDST.m.

template<class POL>

POL::coeff_t MPOLDST::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.

Definition at line 328 of file MPOLDST.m.

Referenced by let::convert().

template<class MP>

MP MPOLDST::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.

Definition at line 439 of file MPOLDST.m.

template<class R>

void MPOLDST::copy	(	R &	r,
		const R &	a
	)

Copy of a in r.

Definition at line 408 of file MPOLDST.m.

References let::copy(), and matrixof::reserve().

template<class R>

int MPOLDST::degree	(	const R &	p,
		int	i
	)

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

Definition at line 280 of file MPOLDST.m.

template<class R>

int MPOLDST::degree

(

const R &

p

)

Degree of a polynomial.

Definition at line 266 of file MPOLDST.m.

References degree().

Referenced by macaulay(), MonomV(), and solve().

template<class R>

void MPOLDST::diff	(	R &	r,
		const R &	p,
		int	i
	)

Derivative of p with respect to i$^{ th}$ variable put in r.

Definition at line 379 of file MPOLDST.m.

Referenced by evaluation().

template<class R>

void MPOLDST::div_rem	(	R &	q,
		R &	a,
		const R &	b
	)

Divide a in place by b.

Definition at line 303 of file MPOLDST.m.

template<class R>

void MPOLDST::div_rem_x	(	R &	q,
		R &	a,
		const R &	b0
	)

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

Definition at line 346 of file MPOLDST.m.

Referenced by Theta().

template<class R>

R::coeff_t MPOLDST::eval	(	const R &	p,
		const typename R::coeff_t &	x,
		const typename R::coeff_t &	y
	)

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

Definition at line 419 of file MPOLDST.m.

template<class R>

int MPOLDST::lvar

(

const R &

p

)

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

Definition at line 251 of file MPOLDST.m.

Referenced by nbvar(), subs(), and swap().

template<class M, class O>

void MPOLDST::mul	(	std::list< M > &	r,
		const std::list< M > &	a,
		const std::list< M > &	b,
		const O &	o
	)

Specialisation for list.

Definition at line 204 of file MPOLDST.m.

References add(), and mul_ext().

template<class R, class O>

void MPOLDST::mul	(	R &	r,
		const R &	a,
		const R &	b,
		const O &	o
	)

Multiplication of two polynomials.

Definition at line 194 of file MPOLDST.m.

template<class R, class S, class C>

void MPOLDST::mul_ext	(	R &	result,
		const S &	a,
		const C &	m
	)

Multiplication of a polynomial by a monomial or a scalar.

Definition at line 171 of file MPOLDST.m.

References VECTOR::apply_mult().

Referenced by mul().

template<class R>

unsigned MPOLDST::nbvar

(

const R &

p

)

Number of variables of a polynomial.

Definition at line 260 of file MPOLDST.m.

References lvar().

Referenced by MPol< C, O, R >::nbvar().

template<class MP>

MP MPOLDST::Rotate_xy	(	const MP &	P,
		typename MP::coeff_t	t
	)

Procedure which stores in Pol the polynomial after making a rotation by t in the x0,x1 plane in 3 dimension. t represents, in the case of rotation, tan(teta/2), with teta is the rotation angle.

Definition at line 669 of file MPOLDST.m.

Referenced by Rotate_xyz().

template<class MP>

MP MPOLDST::Rotate_xyz	(	const MP &	P,
		typename MP::coeff_t	t
	)

Procedure which stores in Pol the polynomial after making a rotation by t in 3 dimension. t represents, in the case of rotation, tan(teta/2), with teta is the rotation angle.

Definition at line 744 of file MPOLDST.m.

References Rotate_xy(), Rotate_xz(), and Rotate_yz().

template<class MP>

MP MPOLDST::Rotate_xz	(	const MP &	P,
		typename MP::coeff_t	t
	)

Procedure which stores in Pol the polynomial after making a rotation by t in the x0,x2 plane in 3 dimension. t represents, in the case of rotation, tan(teta/2), with teta is the rotation angle.

Definition at line 594 of file MPOLDST.m.

Referenced by Rotate_xyz().

template<class MP>

MP MPOLDST::Rotate_yz	(	const MP &	P,
		typename MP::coeff_t	t
	)

Procedure which stores in Pol the polynomial after making a rotation by t in the x1,x2 plane in 3 dimension. t represents, in the case of rotation, tan(teta/2), with teta is the rotation angle.

Definition at line 519 of file MPOLDST.m.

Referenced by Rotate_xyz().

template<class POL, class C>

POL MPOLDST::scale	(	const POL &	polynom,
		C	a,
		C	b,
		C	c
	)

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

Definition at line 850 of file MPOLDST.m.

template<class POL, class C>

POL MPOLDST::shift	(	const POL &	polynom,
		C	x0,
		C	y0,
		C	z0
	)

Carry out a shift by [x0,y0,z0] of 3 first variables for the polynomial passed in parameter

Definition at line 799 of file MPOLDST.m.

template<class POL, class C>

POL MPOLDST::shift	(	typename POL::const_iterator	monom,
		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.

Definition at line 762 of file MPOLDST.m.

template<class MP>

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

Substitute the variable x by the value val.

Definition at line 950 of file MPOLDST.m.

References subs().

template<class MP>

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

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

Definition at line 909 of file MPOLDST.m.

References lvar().

template<class MP, class X>

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

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

Definition at line 885 of file MPOLDST.m.

Referenced by evaluation(), and subs().

template<class MP>

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

Swap the variable x_i and x_j in the polynomial P.

Definition at line 990 of file MPOLDST.m.

References swap().

template<class MP>

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

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

Definition at line 964 of file MPOLDST.m.

References lvar().

Referenced by topology::assign(), baseLocal(), and swap().