atlas::lattice Namespace Reference

Functions
template<typename I , typename O >
void	baseChange (I first, I last, O out, I firstb, I lastb)
	Functions for working with lattices. More...
template<typename I , typename O >
void	inverseBaseChange (I first, I last, O out, I firstb, I lastb)
CoweightList	perp (const WeightList &b, size_t r)
	Returns a basis of the orthogonal of the sublattice generated by b in Z^r. More...
LatticeMatrix	kernel (const LatticeMatrix &M)
LatticeMatrix	eigen_lattice (LatticeMatrix M, LatticeCoeff lambda)
LatticeMatrix	row_saturate (const LatticeMatrix &M)
template void	baseChange (WeightList::iterator, WeightList::iterator, std::back_insert_iterator< WeightList >, WeightList::iterator, WeightList::iterator)

Function Documentation

template<typename I , typename O >

void atlas::lattice::baseChange	(	I	first,
		I	last,
		O	out,
		I	firstb,
		I	lastb
	)

Functions for working with lattices.

This namespace defines some general lattice functions. It includes change of basis functions, and calculating the orthogonal of a sublattice.

In this template, we assume that |I|, and |O| are respectively random access input and output iterator types for type |Weight|, and that |[firstb,lastb[| holds a new $$-basis for the lattice, in particular that |lastb-firstb| is equal to the size of the |Weight|s.

As we iterate from |first| to |last|, we write the vectors in the new basis (this is supposed to be possible) and output the result to |O|.

Doing the base change amounts to applying the inverse of |b|'s matrix.

NOTE: we don't assume that |[firstb, lastb[| is necessarily a $$-basis of the current lattice, only that it is a basis of a full rank sublattice containing the vectors in the input range; the new coordinates will then be integers. Users should be aware of the "full rank" condition; without it the specification still makes sense, but the implementation will fail.

template void atlas::lattice::baseChange	(	WeightList::iterator	,
		WeightList::iterator	,
		std::back_insert_iterator< WeightList >	,
		WeightList::iterator	,
		WeightList::iterator
	)

LatticeMatrix atlas::lattice::eigen_lattice	(	LatticeMatrix	M,
		LatticeCoeff	lambda
	)

template<typename I , typename O >

void atlas::lattice::inverseBaseChange	(	I	first,
		I	last,
		O	out,
		I	firstb,
		I	lastb
	)

This (unsused) template function is like |baseChange|, but goes from weights expressed in terms of |[firstb, lastb[| to ones expressed in terms of the original basis. This is easier, as we don't have to invert the matrix!

LatticeMatrix atlas::lattice::kernel

(

const LatticeMatrix &

M

)

CoweightList atlas::lattice::perp	(	const WeightList &	b,
		size_t	r
	)

Returns a basis of the orthogonal of the sublattice generated by b in Z^r.

Algorithm: diagonalise the matrix |M| with columns |b| using row operations |R| (forgetting column operations |C|) as for Smith form; then image |R*M| is span of original images of canonical basis vectors (as many as nonzero factors), and orthogonal sublattice is spanned by the remaining rows of |R|.

Precondition: the (possibly dependent) vectors in |b| all have the size |r|.

LatticeMatrix atlas::lattice::row_saturate

(

const LatticeMatrix &

M

)