LatticeTester::Rank1Lattice< Int, Real > Class Template Reference

This subclass of IntLatticeExt defines a general rank 1 lattice rule in \(d\) dimensions, whose points \(\mathbb{u}_i\) are defined by. More...

#include <latticetester/Rank1Lattice.h>

Inheritance diagram for LatticeTester::Rank1Lattice< Int, Real >:

Public Member Functions
	Rank1Lattice (const Int &m, const IntVec &aa, NormType norm=L2NORM)
	This constructor takes as input the modulus m, the generating vector aa, and the norm used to measure the vector lengths.
	Rank1Lattice (const Int &m, const Int &a, int64_t maxDim, NormType norm=L2NORM)
	Constructor for the special case of a Korobov lattice.
	Rank1Lattice (const Int &m, int64_t maxDim, NormType norm=L2NORM)
	This constructor does not specify the generating vector.
	Rank1Lattice (const Rank1Lattice< Int, Real > &Lat)
	Copy constructor.
Rank1Lattice &	operator= (const Rank1Lattice< Int, Real > &Lat)
	Assigns Lat to this object.
	~Rank1Lattice ()
	Destructor.
void	setaa (const IntVec &aa)
	Sets the generating vector to aa.
void	seta (const Int &a)
	Sets this lattice to a Korobov lattice with multiplier a.
void	buildBasis (int64_t dim)
	Builds a basis in dim dimensions.
void	buildDualBasis (int64_t dim)
	Builds only an m-dual lower triangular basis (and not the primal) directly in dim dimensions.
void	incDimBasis ()
	Increases the current dimension of the primal basis by 1 and updates the basis.
void	incDimDualBasis ()
	Increases the current dimension of only the m-dual basis by 1.
void	buildProjection (IntLattice< Int, Real > &projLattice, const Coordinates &proj) override
	This method overrides its namesake in IntLattice.
void	buildProjectionDual (IntLattice< Int, Real > &projLattice, const Coordinates &proj) override
	Similar to buildProjection, but builds a basis for the m-dual of the projection.
std::string	toStringCoef () const
	Returns the first dim components of the generating vector \(\ba\) as a string, where dim is the current lattice dimension.
Public Member Functions inherited from LatticeTester::IntLatticeExt< Int, Real >
	IntLatticeExt (Int m, int64_t maxDim, NormType norm=L2NORM)
	A constructor that initializes the primal and dual bases with the identity matrix.
	IntLatticeExt (const IntLatticeExt< Int, Real > &lat)
	Copy constructor that makes a copy of lat.
void	copy (const IntLatticeExt< Int, Real > &lat)
	Makes a copy of lattice into this object.
virtual	~IntLatticeExt ()
	Destructor.
virtual std::string	toString () const
	Returns a string describing this lattice.
Public Member Functions inherited from LatticeTester::IntLattice< Int, Real >
	IntLattice (const Int m, const int64_t maxDim, NormType norm=L2NORM)
	Constructs a lattice whose basis is the identity, in maxDim dimensions, with the specified norm type, and the scaling factor m.
	IntLattice (const IntMat basis, const Int m, const int64_t maxDim, NormType norm=L2NORM)
	Similar to the previous constructor, except that the primal basis is given in basis, which must be an IntMat object of size maxDim by maxDim.
	IntLattice (const IntMat primalbasis, const IntMat dualbasis, const Int m, const int64_t maxDim, NormType norm=L2NORM)
	Constructs a lattice with the given basis and given m-dual basis for the given m, in maxDim dimensions, and with the specified norm type.
	IntLattice (const IntLattice< Int, Real > &lat)
	Copy constructor.
virtual	~IntLattice ()
	Destructor.
void	copyLattice (const IntLattice< Int, Real > &lat)
	Makes a deep copy of the lattice lat into this (existing) object.
void	overwriteLattice (const IntLattice< Int, Real > &lat, long dim, long dimdual)
virtual void	buildProjectionLLL (IntLattice< Int, Real > &projLattice, const Coordinates &proj, double delta=0.5)
	Constructs a set of generating vectors for the projection of the present lattice, over the set of coordinates determined by proj, and builds a basis for that projection using an LLL procedure from BasisConstruction.
IntMat &	getBasis ()
	Returns the IntMat object that contains the basis of this lattice.
void	setBasis (const IntMat basis, const Int m, const int64_t dim, NormType norm=L2NORM)
	Changes the basis to 'basis', the modulus to 'm', and the current dimension to 'dim'.
void	setBasis (const IntMat basis, const int64_t dim)
	Changes only the 'basis' and the current dimension to 'dim'.
IntMat &	getDualBasis ()
	Returns the m-dual basis represented in a matrix.
int64_t	getMaxDim () const
	Returns the maximal dimension for this lattice.
int64_t	getDim () const
	Returns the current dimension of the primal lattice, which is the dimension of the basis vectors, and also usually the number of independent vectors in the basis.
int64_t	getDimDual () const
	Returns the current dimension of the m-dual lattice.
NormType	getNormType () const
	Returns the NormType used by this lattice.
Real	getVecNorm (const int64_t &i)
	Returns the norm (squared in case of the L^2 norm) of the i-th vector of the basis, with the index i starting at 0.
RealVec	getVecNorm () const
	Returns the norm (squared in case of the L^2 norm) of each basis vector, in a vector.
Real	getDualVecNorm (const int64_t &i)
	Returns the norm (squared in case of the L^2 norm) of the i-th vector of the m-dual basis.
RealVec	getDualVecNorm () const
	Returns the norm (squared in case of the L^2 norm) of each vector of the m-dual basis, in a vector.
Int	getModulus () const
	Returns the scaling factor m.
void	setDim (const int64_t dim)
	Sets the dimension of the primal basis to dim.
void	setDimDual (const int64_t dim)
	Sets the dimension of the primal basis to dim.
void	setNormType (const NormType &norm)
	Sets the NormType used by this lattice to norm.
void	setVecNorm (const Real &value, const int64_t &i)
	Sets the norm of the i-th component of the basis to value, which is assumed to be the correct value.
void	setDualVecNorm (const Real &value, const int64_t &i)
	Sets the norm of the i-th component of the m-dual basis to value, which is assumed to be the correct value.
void	setNegativeNorm ()
	Sets to -1 the first dim values in the array containing the norms of the basis vectors, where dim is the current dimension of the lattice.
void	setNegativeNorm (const int64_t &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, to indicate that these norms are no longer up to date.
void	setDualNegativeNorm (const int64_t &i)
	Sets the value of the i-th component in the array containing the norms of the m-dual basis vectors to -1.
void	updateVecNorm ()
	Updates the array containing the norms of the basis vectors by recomputing them.
void	updateVecNorm (const int64_t &d)
	Updates the array containing the norm of the basis vectors from the d-th component to the last, by recomputing them.
void	updateSingleVecNorm (const int64_t &d, const int64_t &c)
	Updates the 'd'-th entry of the array containing the basis vectors norms.
void	updateDualVecNorm ()
	Updates the array containing the m-dual basis vectors norms by recomputing them.
void	updateSingleDualVecNorm (const int64_t &d, const int64_t &c)
	Updates the 'd'-th entry of the array containing the m-dual basis vectors norms.
void	updateDualVecNorm (const int64_t &d)
	Updates the array containing the m-dual basis vectors norms from the d-th component to the last by recomputing them.
void	updateDualVecNorm (const int64_t &d, const int64_t &c)
	Updates the array containing the m-dual basis vectors norms from the d-th component to the last by recomputing them.
void	updateScalL2Norm (const int64_t i)
	Updates the i-th value of the array containing the square norms of the basis vectors by recomputing it using the L2NORM.
void	updateScalL2Norm (const int64_t k1, const int64_t k2)
	Updates the k1-th to the k2-1-th values of the array containing the square norms of the basis vectors by recomputing them using the L2NORM.
void	updateDualScalL2Norm (const int64_t i)
	Updates the i-th value of the array containing the square norms of the m-dual basis vectors by recomputing it using the L2NORM.
void	updateDualScalL2Norm (const int64_t k1, const int64_t k2)
	Updates the k1-th to the k2-1-th values of the array containing the square norms of the m-dual basis vectors by recomputing them using the L2NORM.
void	permute (int64_t i, int64_t j)
	Exchanges vectors i and j in the basis.
void	permuteDual (int64_t i, int64_t j)
	Exchanges vectors i and j in the m-dual basis without changing the primal.
void	dualize ()
	Exchanges the primal and m-dual bases, their dimensions, and vector norms.
bool	checkDuality ()
	Returns true iff the m-dual basis contained in the object really is the m-dual of the current primal basis.
void	sortBasis (int64_t d)
	Sorts the primal basis vectors with indices greater of equal to d by increasing length.
void	sortDualBasis (int64_t d)
	Sorts the m-dual basis vectors with indices greater of equal to d by increasing length.
std::string	toStringBasis () const
	Returns a string that contains the primal basis vectors and their norms.
std::string	toStringDualBasis () const
	Returns a string with the m-dual basis vectors and their norms.
std::string	toString () const
	Returns a string that represents this lattice and its parameters.
void	write () const
	Writes on standard output the string returned by toString.

Protected Attributes
IntVec	m_a
	Vector of multipliers (generating vector) of the rank 1 lattice rule.
Protected Attributes inherited from LatticeTester::IntLattice< Int, Real >
Int	m_modulo
	The scaling factor m used for rescaling the lattice.
int64_t	m_maxDim
	The maximal dimension for this lattice.
int64_t	m_dim
	The current dimension of the primal basis, which is the number of (independent) vectors in it.
int64_t	m_dimdual
	The current dimension of the m-dual basis, which is the number of (independent) vectors in its basis.
IntMat	m_basis
	The rows of the m_dim x m_dim upper left corner of this matrix are the primal basis vectors.
IntMat	m_dualbasis
	The rows of the m_dim x m_dim upper left corner this matrix are the m-dual basis vectors.
NormType	m_norm
	The NormType used to measure the vector lengths for this lattice.
RealVec	m_vecNorm
	A vector that stores the norm of each basis vector.
RealVec	m_dualvecNorm
	Similar to vecNorm, but for the m-dual basis.

Additional Inherited Members
Protected Member Functions inherited from LatticeTester::IntLatticeExt< Int, Real >
virtual void	kill ()

Detailed Description

template<typename Int, typename Real>
class LatticeTester::Rank1Lattice< Int, Real >

This subclass of IntLatticeExt defines a general rank 1 lattice rule in \(d\) dimensions, whose points \(\mathbb{u}_i\) are defined by.

\begin{equation} \mathbf{u}_i = (i \mathbf{a} \mod m)/m, \end{equation}

where \(\mathbf{a} = (a_1,a_2,\dots,a_d) \in \mathbb{Z}_m^d\) is called the generating vector. The lattice is rescaled simply by removing the division by \(m\). Given \(\mathbf{a}\), a basis of the rescaled (integer) lattice is given by

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

where \(\mathbf{e}_i\) is the \(i^\text{th}\) unit vector.

A condition often required when building a rank 1 lattice is that \(\gcd(a_j, m) = 1\) for \(j =1, \dots,d\). When this condition is verified, each lower-dimensional projection of the lattice contains the same number of points (has the same density \(m\)) as the full lattice. When searching for lattices that satisfy this condition, one may assume without loss of generality generality that \(a_1 = 1\). Under that condition, it is straightforward to construct a basis for a projection that contains the first coordinate, and also its m-dual basis. In this class, we assume that \(a_1 = 1\) and \(\gcd(a_j, m) = 1\) for all \(j\). We exploit these conditions when building bases for the lattice, the projections, and their m-duals.

The functions buildBasis and incDimBasis always build and update the primal basis, because it is required to update the m-dual. To build only the m-dual, use buildDualBasis.

Constructor & Destructor Documentation

Rank1Lattice() [1/4]

template<typename Int , typename Real >

LatticeTester::Rank1Lattice< Int, Real >::Rank1Lattice	(	const Int &	m,
		const IntVec &	aa,
		NormType	norm = L2NORM )

This constructor takes as input the modulus m, the generating vector aa, and the norm used to measure the vector lengths.

The maximal dimension maxDim will be the length d of the vector aa. The coefficient \(a_j\) will be aa[j-1]. One must have aa[0] = 1. The variable withPrimal indicates if the primal basis will be maintained or not, and withDual indicates if the dual basis will be maintained or not. This constructor does not build the basis, to leave more flexibility in the dimension when doing so.

Rank1Lattice() [2/4]

template<typename Int , typename Real >

LatticeTester::Rank1Lattice< Int, Real >::Rank1Lattice	(	const Int &	m,
		const Int &	a,
		int64_t	maxDim,
		NormType	norm = L2NORM )

Constructor for the special case of a Korobov lattice.

Here the generating vector has the form \(\ba = (1, a, a^2 mod m, a^3 mod m, ...)\) where \(a\) is an integer such that \(1 < a < m\).

Rank1Lattice() [3/4]

template<typename Int , typename Real >

LatticeTester::Rank1Lattice< Int, Real >::Rank1Lattice	(	const Int &	m,
		int64_t	maxDim,
		NormType	norm = L2NORM )

This constructor does not specify the generating vector.

Rank1Lattice() [4/4]

template<typename Int , typename Real >

LatticeTester::Rank1Lattice< Int, Real >::Rank1Lattice

(

const Rank1Lattice< Int, Real > &

Lat

)

Copy constructor.

~Rank1Lattice()

template<typename Int , typename Real >

LatticeTester::Rank1Lattice< Int, Real >::~Rank1Lattice

(

)

Destructor.

Member Function Documentation

buildBasis()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::buildBasis ( int64_t dim )

virtual

Builds a basis in dim dimensions.

This dim must not exceed this->maxDim(). This initial primal basis will be upper triangular. This function can be called only when withPrimal is set to true. If withDual is true, it also builds an m-dual lower-triangular basis.

Reimplemented from LatticeTester::IntLatticeExt< Int, Real >.

buildDualBasis()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::buildDualBasis ( int64_t dim )

virtual

Builds only an m-dual lower triangular basis (and not the primal) directly in dim dimensions.

This dim must not exceed maxDim.

Reimplemented from LatticeTester::IntLatticeExt< Int, Real >.

buildProjection()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::buildProjection ( IntLattice< Int, Real > & projLattice, const Coordinates & proj )

overridevirtual

This method overrides its namesake in IntLattice.

The projection of this Rank1Lattice over the coordinates in proj is returned in projLattice. The implementation used here exploits the rank-1 lattice structure and it is simpler and faster than the general one. See Section 5.5 of the guide. When the first coordinate is 1 and belongs to the projection, both the primal and m-dual constructions are direct, just by selecting the rows and columns whose indices are in proj. Otherwise, those rows and columns plus the first row form a set of proj.size()+1 generating vectors for the primal. This number must not exceed the maxDim of projLattice.

Reimplemented from LatticeTester::IntLattice< Int, Real >.

buildProjectionDual()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::buildProjectionDual ( IntLattice< Int, Real > & projLattice, const Coordinates & proj )

overridevirtual

Similar to buildProjection, but builds a basis for the m-dual of the projection.

Reimplemented from LatticeTester::IntLattice< Int, Real >.

incDimBasis()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::incDimBasis ( )

virtual

Increases the current dimension of the primal basis by 1 and updates the basis.

This function can be called only when withPrimal is set to true. If withDual, it also increases the m-dual basis and makes it the m-dual of the primal basis. The new increased dimension must not exceed maxDim.

Reimplemented from LatticeTester::IntLatticeExt< Int, Real >.

incDimDualBasis()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::incDimDualBasis ( )

virtual

Increases the current dimension of only the m-dual basis by 1.

The primal basis is left unchanged (not updated). The new increased dimension must not exceed maxDim. This method uses the simplified method given in the lattice tester guide: the new m-dual basis vector is simply \(\bw_d = (-a_d, 0, ..., 0, 1)\).

Reimplemented from LatticeTester::IntLatticeExt< Int, Real >.

operator=()

template<typename Int , typename Real >

Rank1Lattice< Int, Real > & LatticeTester::Rank1Lattice< Int, Real >::operator=

(

const Rank1Lattice< Int, Real > &

Lat

)

Assigns Lat to this object.

seta()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::seta

(

const Int &

a

)

Sets this lattice to a Korobov lattice with multiplier a.

Sets this lattice to a Korobov lattice with multiplier a, with dim = maxDim.

The maximum dimension is unchanged.

setaa()

template<typename Int , typename Real >

void LatticeTester::Rank1Lattice< Int, Real >::setaa

(

const IntVec &

aa

)

Sets the generating vector to aa.

The maximum dimension will be set to its dimension.

toStringCoef()

template<typename Int , typename Real >

std::string LatticeTester::Rank1Lattice< Int, Real >::toStringCoef

(

)

const

Returns the first dim components of the generating vector \(\ba\) as a string, where dim is the current lattice dimension.

Member Data Documentation

m_a

template<typename Int , typename Real >

IntVec LatticeTester::Rank1Lattice< Int, Real >::m_a

protected

Vector of multipliers (generating vector) of the rank 1 lattice rule.

They are stored for up to maxDim() dimensions. The first coordinate has index 0.

---

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

• include/latticetester/Rank1Lattice.h