This class implements a general rank 1 lattice. More...

#include <Rank1Lattice.h>

Inherits LatticeTester::IntLattice< Int, BasInt, Dbl, RedDbl >.

Public Member Functions
	Rank1Lattice (const Int &m, const IntVec &a, int maxDim, LatticeTester::NormType norm=LatticeTester::L2NORM)
	Full constructor.
	Rank1Lattice (const Rank1Lattice< Int, BasInt, Dbl, RedDbl > &Lat)
	Copy constructor.
Rank1Lattice &	operator= (const Rank1Lattice< Int, BasInt, Dbl, RedDbl > &Lat)
	Assigns Lat to this object.
	~Rank1Lattice ()
	Destructor.
std::string	toStringCoef () const
	Returns the firsts components of the vector of multipliers \(a\) as a string.
void	buildBasis (long d)
	Builds the basis in dimension d.
void	dualize ()
	Dualize the matrix.
void	incDim ()
	Increases the dimension by 1.
Public Member Functions inherited from LatticeTester::IntLattice< Int, BasInt, Dbl, RedDbl >
	IntLattice (Int modulo, int k, int maxDim, bool withDual, NormType norm=L2NORM)
	Constructor initializing the primal and the dual basis with the identity matrix.
	IntLattice (const IntLattice< Int, BasInt, Dbl, RedDbl > &Lat)
	Copy constructor that makes a copy of Lat.
void	copy (const IntLattice< Int, BasInt, Dbl, RedDbl > &lattice)
	Copies lattice into this object.
virtual	~IntLattice ()
	Destructor.
void	init ()
	Allocates space to vectors used internally.
int	getOrder () const
	This returns the rank of the lattice.
void	calcLgVolDual2 (double lgm2)
	Computes the logarithm of the normalization factor (m_lgVolDual2) in all dimensions \(\leq\) MaxDim for the lattice.
double	getLgVolDual2 (int i) const
	Gives the log of m^(2i) if i < order, else gives the log of m^(2i)
void	dualize ()
	Exchange the primal basis and the dual basis.
void	fixLatticeNormalization (bool dualF)
	This function is called to fix the normalization constants to get the normalized merit from the shortest distance in the lattice.
virtual void	buildProjection (IntLattice< Int, BasInt, Dbl, RedDbl > *lattice, const Coordinates &proj)
	Builds the basis (and dual basis) of the projection proj for this lattice.
virtual void	buildBasis (int d)
	Builds the basis for the lattice in dimension d.
LatticeTester::Normalizer< RedDbl > *	getNormalizer (NormaType norma, int alpha, bool dualF)
	Creates and returns the normalizer corresponding to criterion norma.
virtual void	setLac (const Lacunary< BasInt > &)
	A utility method to store a vector of indices with lacunary values in subclasses of this one.
virtual std::string	toString () const
	Returns a string describing the lattice.
Public Member Functions inherited from LatticeTester::IntLatticeBasis< Int, BasInt, Dbl, RedDbl >
	IntLatticeBasis (const int dim, NormType norm=L2NORM)
	Constructor initializing the primal basis with the identity matrix.
	IntLatticeBasis (const BasIntMat basis, const int dim, NormType norm=L2NORM)
	Constructor taking all three needed component of an IntLatticeBasis.
	IntLatticeBasis (const BasIntMat primalbasis, const BasIntMat dualbasis, const Int modulo, const int dim, NormType norm=L2NORM)
	Complete constructor.
	IntLatticeBasis (const IntLatticeBasis< Int, BasInt, Dbl, RedDbl > &Lat)
	Copy constructor.
	~IntLatticeBasis ()
	Destructor.
void	kill ()
	Cleans and releases all the memory allocated to this lattice.
void	copyBasis (const IntLatticeBasis< Int, BasInt, Dbl, RedDbl > &lat)
	Copy the lattice lat, except it's NormType and dimension, into this object.
void	copyBasis (const IntLatticeBasis< Int, BasInt, Dbl, RedDbl > &lat, long n)
	Copy the n first elements of the basis of the lattice lat into this object.
void	initVecNorm ()
	Initializes a vector containing the norms of the basis vectors to -1 at all components.
BasIntMat &	getBasis ()
	Returns the basis represented in a matrix.
BasIntMat &	getDualBasis ()
	Returns the dual basis represented in a matrix.
int	getDim () const
	Returns the dimension of the lattice.
NormType	getNorm () const
	Returns the NormType used by this lattice.
Dbl	getVecNorm (const int &i)
	Returns the norm of the i-th vector of the basis.
DblVec	getVecNorm () const
	Returns the norm of each vector of the basis in a vector.
Dbl	getDualVecNorm (const int &i)
	Returns the norm of the i-th vector of the dual basis.
DblVec	getDualVecNorm () const
	Returns the norm of each vector of the dual basis in a vector.
Int	getModulo () const
	Returns the m used for rescaling if it has been defined.
void	setDim (const int &d)
	Sets the dimension of the basis to d.
void	setNorm (const NormType &norm)
	Sets the NormType used by this lattice to norm.
void	setVecNorm (const Dbl &value, const int &i)
	Sets the norm of the i-th component of the basis to value.
void	setDualVecNorm (const Dbl &value, const int &i)
	Sets the norm of the i-th component of the dual basis to value.
bool	withDual () const
	Returns true if a dual has been defined and false otherwise.
void	setDualFlag (bool flag)
	Sets the withDual flag to flag.
void	setNegativeNorm ()
	Sets all the values in the array containing the norms of the basis vectors to -1.
void	setNegativeNorm (const int &i)
	Sets the value of the i-th component in the array containing the norms of the basis vectors to -1.
void	setDualNegativeNorm ()
	Sets all the values in the array containing the norms of the dual basis vectors to -1.
void	setDualNegativeNorm (const int &i)
	Sets the value of the i-th component in the array containing the norms of the dual basis vectors to -1.
void	updateVecNorm ()
	Updates the array containing the basis vectors norms by recomputing them.
void	updateVecNorm (const int &d)
	Updates the array containing the basis vectors norms from the d-th component to the last by recomputing them.
void	updateDualVecNorm ()
	Updates the array containing the dual basis vectors norms by recomputing them.
void	updateDualVecNorm (const int &d)
	Updates the array containing the dual basis vectors norms from the d-th component to the last by recomputing them.
void	updateScalL2Norm (const int i)
	Updates the i-th value of the array containing the norms of the basis vectors by recomputing it using the L2NORM.
void	updateScalL2Norm (const int k1, const int k2)
	Updates the k1-th to the k2-1-th values of the array containing the norms of the basis vectors by recomputing them using the L2NORM.
void	updateDualScalL2Norm (const int i)
	Updates the i-th value of the array containing the norms of the dual basis vectors by recomputing it using the L2NORM.
void	updateDualScalL2Norm (const int k1, const int k2)
	Updates the k1-th to the k2-1-th values of the array containing the norms of the dual basis vectors by recomputing them using the L2NORM.
void	permute (int i, int j)
	Exchanges vectors i and j in the basis.
void	permuteNoDual (int i, int j)
	Exchanges vectors i and j in the basis without changing the dual.
bool	checkDuality ()
	Returns true if the dual basis contained in the object really is the dual of the basis, and false otherwise.
void	sort (int d)
	Sorts the basis vectors with indices greater of equal to \(d\) by increasing length.
void	sortNoDual (int d)
	Sorts the basis vectors with indices greater of equal to \(d\) by increasing length.
std::string	toStringBasis () const
	Returns a string with the primal basis and its norms.
std::string	toStringDualBasis () const
	Returns a string with the dual basis and its norms.
void	write () const
	Writes the lattice and its parameters on standard output.

Protected Member Functions
void	init ()
	Initializes the rank 1 lattice.
Protected Member Functions inherited from LatticeTester::IntLattice< Int, BasInt, Dbl, RedDbl >
virtual void	kill ()
	Cleans and releases all the memory allocated to this lattice.

Protected Attributes
IntVec	m_a
	The multipliers of the rank 1 lattice rule.
Protected Attributes inherited from LatticeTester::IntLattice< Int, BasInt, Dbl, RedDbl >
int	m_order
	The order of the basis.
int	m_maxDim
double *	m_lgVolDual2
	Represente sur dual along the diagonal?
double	m_lgm2
	The logarithm \(\log (m^2)\).
BasIntMat	m_wSI
	The dual basis of the current projection.
BasIntMat	m_vSI
	The primal basis of the current projection.
Int	m_t1
	Working Variables used in MRGLattice.h.
Int	m_t2
Int	m_t3
Protected Attributes inherited from LatticeTester::IntLatticeBasis< Int, BasInt, Dbl, RedDbl >
BasIntMat	m_basis
	Each row of this matrix represents a vector in the basis of the lattice.
BasIntMat	m_dualbasis
	Each row of this matrix represents a vector in the dual basis of the lattice.
int	m_dim
	The dimension of the lattice.
NormType	m_norm
	The NormType used in the reduction and for this lattice.
DblVec	m_vecNorm
	The norm of each vector in the basis.
DblVec	m_dualvecNorm
	The norm of each vector in the dual basis.
BasInt	m_modulo
	The m used for rescaling the lattice.
bool	m_withDual
	If m_withDual is true a dual basis has been specified, otherwise it is false.

Detailed Description

template<typename Int, typename BasInt, typename Dbl, typename RedDbl>
class LatticeTester::Rank1Lattice< Int, BasInt, Dbl, RedDbl >

This class implements a general rank 1 lattice.

A rank 1 lattice is the lattice \(\mathbb{Z}_m^d\) generated by a recurrence of the form

\begin{equation} u_i = (i \mathbf{a} \mod m), \end{equation}

where \(\mathbf{a} \in \mathbb{Z}_m^d\). It can also be seen as a lattice in \([0,1)^d\) by dividing the coordinates of all the points by \(m\). Given \(\mathbf{a}\), the basis of such a lattice is given, quite simply, by

\begin{align*} \mathbf{v}_1 & = a \\ \mathbf{v}_2 & = m \mathbf{e}_2 \\ \vdots & \\ \mathbf{v}_d & = m \mathbf{e}_v \\ \end{align*}

where \(\mathbf{e}_i\) is a vector of 0 with a 1 at the \(i^\text{th}\) coordinate.

A condition that is most wanted when building a rank 1 lattice is that \(\gcd(a_i, m) = 1,\ \forall 1\leq i \leq d\). If this condition is verified, the lattice covers the same number of points, \(m\), in all its projections. Note that if one searches for such a lattice, it is possible, without loss of generality generality, to assume that \(a_1 = 1\).

This class implements or reuses the methods of IntLattice, as well as some new methods to build a basis from a generator.

Constructor & Destructor Documentation

◆ Rank1Lattice()

template<typename Int, typename BasInt, typename Dbl, typename RedDbl>

LatticeTester::Rank1Lattice< Int, BasInt, Dbl, RedDbl >::Rank1Lattice	(	const Int &	m,
		const IntVec &	a,
		int	maxDim,
		LatticeTester::NormType	norm = LatticeTester::L2NORM )

Full constructor.

The lattice can be completely described by the generating vector a, the dimension maxDim, and the modulo m since all the points of the lattice can be obtained by taking

\[ u_i = (i \mathbf{a} \mod m), \]

with \(\mathbf{a}\) being an assortment of the first components of a (a being a vector of dimension maxDim).

This initializes all needed internal components (see IntLattice). In particular, this sets maximal dimension for this object to maxDim. This means that when creating an object of this class, it is important that \(\text{maxDim} \leq \dim(a)\) and ideal that the equality holds. This constructor does not build the basis to allow for more flexibility in the dimension when doing so.

References init(), LatticeTester::IntLattice< Int, BasInt, Dbl, RedDbl >::IntLattice(), and m_a.

Referenced by operator=(), and Rank1Lattice().