LinBox Namespace Reference

Namespace in which all linbox code resides. More...

Namespaces
	Exceptions
	Exception class for invalid matrix input.
	IndexedTags
	limited doc so far
	IteratorCategories
	Information about the type of Prime Iterator.
	MatrixHom
	Limited doc so far. Used in RationalSolver.
	Protected
	This is the namespace all LinBox internal code is in.
	Rank
	Rank of the system, if known.
	RingCategories
	some basic information about each field or ring.
	Shape
	Flags decribing the shape of the matrix.
	SparseFileFormat
	Sparse matrix format (file storage)
	SparseMatrixFormat
	Sparse matrix format (memory storage)
	Tag
	Structure for tags.
	VectorWrapper
	limited doc so far.

Data Structures
class	algoException
	Algorithmic exception. More...
class	AlgorithmMetaData
	Algorithm metadata;. More...
class	BadInputException
	The input is not as expected. More...
class	BenchmarkMetaData
	Benchmark metadata;. More...
class	BitVector
	Binary constant defined both for 32 and 64 bits. More...
class	BlackboxArchetype
	showing the member functions provided by all blackbox matrix classes. More...
class	BlackboxBlockContainerBase
	A base class for BlackboxBlockContainer. More...
class	BlackboxBlockContainerRecord
	no doc. More...
class	BlackboxContainer
	Limited doc so far. More...
class	BlackboxContainerBase
	A base class for BlackboxContainer. More...
class	BlackboxContainerSymmetric
	See base class for doc. More...
class	BlackboxContainerSymmetrize
	Symmetrizing iterator (for rank computations). More...
class	BlackboxFactory
	A tool for computations with integer and rational matrices. More...
class	BlasMatrix
	Dense matrix representation. More...
class	BlasMatrix< MultiModDouble >
	No Doc. More...
class	BlasMatrixDomain
	Interface for all functionnalities provided for BlasMatrix. More...
class	BlasMatrixDomainAddin
	C += A. More...
class	BlasMatrixDomainMulAdd< BlasVector< Field >, BlasMatrix< Field, _Rep >, BlasVector< Field > >
	what about subvector/submatrices ? More...
class	BlasMatrixDomainSubin
	C -= A. More...
class	BlasPermutation
	Lapack-style permutation. More...
class	BlasSubmatrix
	Dense Submatrix representation. More...
class	BlockBB
	converts a black box into a block black box More...
class	BlockCompose
	Blackbox of a product: , i.e . More...
class	BlockCoppersmithDomain
	Compute the linear generator of a sequence of matrices. More...
class	BlockHankelLiftingContainer
	Block Hankel LiftingContianer. More...
class	BlockLanczosSolver
	Block Lanczos iteration. More...
class	BlockMasseyDomain
	Compute the linear generator of a sequence of matrices. More...
class	BlockWiedemannLiftingContainer
	Block Wiedemann LiftingContianer. More...
class	BooleanSwitch
	Boolean switch object. More...
class	Butterfly
	Switching Network based BlackBox Matrix. More...
class	CekstvSwitch
	The default butterfly switch object. More...
struct	ChineseRemainder
	No doc. More...
struct	ChineseRemainderSequential
	No doc. More...
struct	ClassifyRing
	Default ring category. More...
class	Commentator
	Give information to user during runtime. More...
struct	Companion
	Companion matrix of a monic polynomial. More...
class	Compose
	Blackbox of a product: , i.e . More...
class	Compose< _Blackbox, _Blackbox >
	specialization for _Blackbox1 = _Blackbox2 More...
class	ComposeOwner
	Blackbox of a product: , i.e . More...
class	ComposeTraits
	used in ..., for example More...
class	ComposeTraits< BlasMatrix< Field, Rep > >
	used in smith-binary, for example More...
class	ConstantVectorStream
	Constant vector factory. More...
struct	ContainerCategories
	used to separate BLAS2 and BLAS3 operations More...
struct	ContainerTraits
	Trait for the Category. More...
struct	ContainerTraits< std::vector< _Rep > >
struct	CRABuilderEarlyMultip
	NO DOC. More...
struct	CRABuilderEarlySingle
	Heuristic Chinese Remaindering with early termination. More...
struct	CRABuilderFullMultip
	Chinese remaindering of a vector of elements without early termination. More...
struct	CRABuilderFullMultipFixed
	Chinese Remaindering Algorithm for multiple residues. More...
struct	CRABuilderFullMultipMatrix
	NO DOC. More...
struct	CRABuilderFullSingle
	Chinese Remaindering with full precision and no chance of failure. More...
struct	CRABuilderProbSingle
	Chinese Remaindering with guaranteed probability bound and early termination. More...
struct	CRABuilderSingleBase
	Abstract base class for CRA builders. More...
struct	CRAResidue
	Type information for the residue in a CRA iteration. More...
struct	CRAResidue< Integer, Function >
	Type information for the residue in a CRA iteration. More...
struct	CRAResidue< std::vector< Integer >, Function >
	Type information for the residue in a CRA iteration. More...
class	CSF
	Space efficient representation of sparse matrices. More...
struct	DataSeries
	this structure holds a bunch of timings. More...
class	DenseContainer
	Limited doc so far. More...
class	DenseMat
	to be used in standard matrix domain More...
class	DensePolynomial
	Dense Polynomial representation using Givaro. More...
class	Diagonal
	Random diagonal matrices are used heavily as preconditioners. More...
class	Diagonal< _Field, VectorCategories::DenseVectorTag >
	Specialization of Diagonal for application to dense vectors. More...
class	Diagonal< _Field, VectorCategories::SparseAssociativeVectorTag >
	Specialization of Diagonal for application to sparse associative vectors. More...
class	Diagonal< _Field, VectorCategories::SparseSequenceVectorTag >
	Specialization of Diagonal for application to sparse sequence vectors. More...
class	Dif
	Blackbox of a difference: C := A - B, i.e Cx = Ax - Bx. More...
class	DiophantineSolver
	DiophantineSolver<QSolver> creates a diophantine solver using a QSolver to generate rational solutions. More...
class	DirectSum
	If C = DirectSum(A, B) and y = xA and z = wB, then (y,z) = (x,w)C. More...
class	DixonLiftingContainer
	Dixon Lifting Container. More...
class	DotProductDomain< Givaro::Modular< uint16_t, Compute_t > >
	Specialization of DotProductDomain for unsigned short modular field. More...
class	DotProductDomain< Givaro::Modular< uint32_t, Compute_t > >
	Specialization of DotProductDomain for uint32_t modular field. More...
class	DotProductDomain< Givaro::Modular< uint64_t, Compute_t > >
	Specialization of DotProductDomain for uint64_t modular field. More...
class	DotProductDomain< Givaro::Modular< uint8_t, Compute_t > >
	Specialization of DotProductDomain for unsigned short modular field. More...
class	DotProductDomain< Givaro::ModularBalanced< double > >
	Specialization of DotProductDomain. More...
class	ElementAbstract
	Abstract element base class, a technicality. More...
class	ElementArchetype
	Field and Ring element interface specification and archetypical instance class. More...
class	ElementEnvelope
	Adaptor from archetypical interface to abstract interface, a technicality. More...
class	Eliminator
	Elimination system. More...
class	EnvironmentMetaData
	Environment metadata;. More...
class	Exception
	This is the exception class in LinBox. More...
class	FieldAbstract
	field base class. More...
class	FieldArchetype
	field specification and archetypical instance. More...
class	FieldAXPY
	FieldAXPY object. More...
class	FieldAXPY< Givaro::Modular< uint16_t, Compute_t > >
	Specialization of FieldAXPY for uint16_t modular field. More...
class	FieldAXPY< Givaro::Modular< uint32_t, Compute_t > >
	Specialization of FieldAXPY for unsigned short modular field. More...
class	FieldAXPY< Givaro::Modular< uint64_t, Compute_t > >
	Specialization of FieldAXPY for unsigned short modular field. More...
class	FieldAXPY< Givaro::Modular< uint8_t, Compute_t > >
	Specialization of FieldAXPY for uint8_t modular field. More...
class	FieldAXPY< Givaro::ModularBalanced< double > >
	Specialization of FieldAXPY. More...
class	FieldDocumentation
	This field base class exists solely to aid documentation organization. More...
class	FieldEnvelope
	Derived class used to implement the field archetypeHelps to minimize code bloat. More...
class	FieldMetaData
	Field metadata. More...
struct	FieldTraits
	FieldTrait. More...
class	GaussDomain
	Repository of functions for rank by elimination on sparse matrices. More...
class	GeneratorMetaData
	Generator metadata;. More...
class	GenericRandIter
	Random field base element generator. More...
struct	GetEntryCategory
	GetEntryCategory is specialized for BB classes that offer a local getEntry. More...
struct	GivaroRnsFixedCRA
	NO DOC... More...
class	GmpRandomPrime
	generating random prime integers, using the gmp library. More...
class	GMPRationalElement
	elements of GMP_Rationals. More...
class	Hilbert
	Example of a blackbox that is space efficient, though not time efficient. More...
class	Hilbert_JIT_Entry
	The object needed to build a Hilbert matrix as a JIT matrix. More...
class	Hom
	map element of source ring(field) to target ringAn instance of Hom is a homomorphism from a ring of type Source to a ring (usually field) of type Target. More...
class	InconsistentSystem
	Exception thrown when the system to be solved is inconsistent. More...
class	indexDomain
	Class used for permuting indices. More...
struct	IndexedCategory
	Trait to show whether or not the BB class has a Indexed iterator. More...
struct	IndexedCategory< BlasMatrix< Field, _Rep > >
class	Inverse
	A Blackbox for the inverse. More...
class	InvertTextbookDomain
	Assumes that Field is a field, not a ring. More...
class	IrrecuperableException
	Something bad an unexpected happened. More...
class	JIT_Matrix
	example of a blackbox that is space efficient, though not time efficient. More...
class	LABlockLanczosSolver
	Biorthogonalising block Lanczos iteration. More...
class	LanczosSolver
	Solve a linear system using the conjugate Lanczos iteration. More...
class	LargeDouble
	NO DOC. More...
class	LastInvariantFactor
	This is used in a Smith Form algorithm. More...
class	latticeMethod
	NTL methods. More...
class	LinboxError
	base class for execption handling in LinBox More...
struct	Local2_32
	Fast arithmetic mod 2^32, including gcd. More...
class	MaskedPrimeIterator
	Masked Prime Iterator. More...
class	MasseyDomain
	Berlekamp/Massey algorithm. More...
class	MatrixArchetype
	Directly-represented matrix archetype. More...
class	MatrixBlackbox
	Matrix black box. More...
struct	MatrixCategories
	For specializing matrix arithmetic. More...
class	MatrixContainerTrait
	NODOC. More...
class	MatrixDomain
	Class of matrix arithmetic functions. More...
class	MatrixDomain< GF2 >
	Specialization of MatrixDomain for GF2. More...
struct	MatrixHomTrait
	try to map a blackbox over a homorphic ring The most suitable type More...
class	MatrixMetaData
	Matrix metadata. More...
class	MatrixPermutation
	Permutation classique. More...
class	MatrixRank
	Compute the rank of an integer matrix in place over a finite field by Gaussian elimination. More...
class	MatrixStream
	MatrixStream. More...
class	MatrixStreamReader
	An abstract base class to represent readers for specific formats. More...
struct	MatrixTraits
	NO DOC. More...
class	MetaData
	This is the general metadata class. More...
struct	Method
	Define which method to use when working on a system. More...
struct	MethodBase
	Holds everything a method needs to know about the problem. More...
class	MGBlockLanczosSolver
	Block Lanczos iteration. More...
class	ModularCrookedRandIter
	Random field base element generator. More...
class	MoorePenrose
	Generalized inverse of a blackbox. More...
class	MVProductDomain
	Helper class to allow specializations of certain matrix-vector products. More...
class	MVProductDomain< Givaro::Modular< uint16_t, Compute_t > >
	Specialization of MVProductDomain for uint16_t modular field. More...
class	MVProductDomain< Givaro::Modular< uint32_t, Compute_t > >
	Specialization of MVProductDomain for uint32_t modular field. More...
class	MVProductDomain< Givaro::Modular< uint64_t, Compute_t > >
	Specialization of MVProductDomain for uint64_t modular field. More...
class	MVProductDomain< Givaro::Modular< uint8_t, Compute_t > >
	Specialization of MVProductDomain for uint8_t modular field. More...
class	NoHomError
	Error object for attempt to establish a Hom that cannot exist. More...
class	NotImplementedYetException
	Not implemented yet. More...
struct	NTL_PID_zz_p
	extend Wrapper of zz_p from NTL. More...
class	NTL_ZZ
	the integer ring. More...
struct	NTL_ZZ_p
	Wrapper of zz_p from NTL. More...
struct	NTL_zz_p
	long ints modulo a positive integer. More...
class	NTL_ZZ_pE
	Wrapper of ZZ_pE from NTL Define a parameterized class to handle easily Givaro::ZRing<NTL::ZZ_pE> field. More...
class	NTL_zz_pE
	zz_pE Define a parameterized class to easily handle Givaro::ZRing<NTL::zz_pE> field More...
class	NTL_zz_pE_Initialiser
	use ZZ_pEBak mechanism too ? More...
class	NTL_zz_pEX
	Ring (in fact, a unique factorization domain) of polynomial with coefficients in class NTL_zz_p (integers mod a wordsize prime). More...
class	NTL_zz_pX
	Ring (in fact, a unique factorization domain) of polynomial with coefficients in class NTL_zz_p (integers mod a wordsize prime). More...
class	NTL_ZZ_pX
	Ring (in fact, a unique factorization domain) of polynomial with coefficients in class NTL_ZZ_p (integers mod a wordsize prime). More...
class	NullMatrix
	This is a representation of the 0 by 0 empty matrix which does not occupy memory. More...
class	OneInvariantFactor
	Limited doc so far. More...
class	OpenCLEnviron
	Container for all pertenant information needed to use an OpenCL device, compile kernels for the device, track resource usage, and gain exclusive access to the device. More...
class	OpenCLMatrixDomain
	Interface for all functionnalities provided for BlasMatrix using GPUs. More...
class	ParamFuzzy
	Abstract parameterized field of "fuzzy" doubles. More...
class	PIR_ntl_ZZ_p
	extend Wrapper of ZZ_p from NTL. More...
class	PlainSubmatrix
	to be used in reference matrix domain (PlainDomain). More...
class	PlotData
	The raw data to plot. More...
class	PlotGraph
	The graph (2D). More...
class	PlotStyle
	Represents a table of values to plot (2D). More...
class	PLUQMatrix
	PLUQ factorisation. More...
class	PolynomialBB
	represent the matrix P(A) where A is a blackbox and P a polynomial More...
class	PolynomialBBOwner
	represent the matrix P(A) where A is a blackbox and P a polynomial More...
class	PolynomialRing
	Polynomials. More...
class	PowerGaussDomain
	Repository of functions for rank modulo a prime power by elimination on sparse matrices. More...
class	PowerGaussDomainPowerOfTwo
	Repository of functions for rank modulo a prime power by elimination on sparse matrices. More...
class	PreconditionFailed
	A precondition failed. More...
class	PrimeIterator
	Prime Iterator. More...
class	PrimeSequence
	Adaptor class to make a fixed-length sequence behave like a PrimeIterator. More...
class	PrimeStream
	Prime number stream. More...
class	RandIterAbstract
	Random field element generator. More...
class	RandIterArchetype
	Random field element generator archetype. More...
class	RandIterEnvelope
	Random field base element generator. More...
class	RandomDenseMatrix
	Random Dense Matrix builder. More...
class	RandomDenseStream
	Random dense vector stream. More...
class	RandomDenseStream< Field, _Vector, RandIter, VectorCategories::DenseVectorTag >
	Specialization of random dense stream for dense vectors. More...
class	RandomSparseStream
	Random sparse vector stream. More...
class	RandomSparseStream< Field, _Vector, RandIter, VectorCategories::DenseVectorTag >
	Specialization of RandomSparseStream for dense vectors. More...
class	RandomSparseStream< Field, _Vector, RandIter, VectorCategories::SparseAssociativeVectorTag >
	Specialization of RandomSparseStream for sparse associative vectors. More...
class	RandomSparseStream< Field, _Vector, RandIter, VectorCategories::SparseParallelVectorTag >
	Specialization of RandomSparseStream for sparse parallel vectors. More...
class	RandomSparseStream< Field, _Vector, RandIter, VectorCategories::SparseSequenceVectorTag >
	Specialization of RandomSparseStream for sparse sequence vectors. More...
struct	RankBuilder
	random method for constructing rank More...
struct	RationalChineseRemainder
	Chinese remainder of rationals. More...
struct	RationalChineseRemainderVarPrec
	Chinese remainder of vector of rationals. More...
class	RationalReconstruction
	Limited doc so far. More...
class	RationalSolver
	Interface for the different specialization of p-adic lifting based solvers. More...
class	RationalSolver< Ring, Field, RandomPrime, Method::BlockHankel >
	Block Hankel. More...
class	RationalSolver< Ring, Field, RandomPrime, Method::BlockWiedemann >
	partial specialization of p-adic based solver with block Wiedemann algorithm. More...
class	RationalSolver< Ring, Field, RandomPrime, Method::Dixon >
	partial specialization of p-adic based solver with Dixon algorithm. More...
class	RationalSolver< Ring, Field, RandomPrime, Method::SparseElimination >
	Sparse LU. More...
class	RationalSolver< Ring, Field, RandomPrime, Method::SymbolicNumericNorm >
	solver using a hybrid Numeric/Symbolic computation. More...
class	RationalSolver< Ring, Field, RandomPrime, Method::Wiedemann >
	Partial specialization of p-adic based solver with Wiedemann algorithm. More...
struct	RawVector
	Canonical vector types. More...
struct	Rebind
	used in support of Hom, MatrixHom More...
struct	Rebind< std::vector< T >, U >
	Rebind. More...
class	ReverseVector
	Reverse vector class This class wraps an existing vector type and reverses its direction. More...
class	RingAbstract
	Abstract ring base class. More...
class	RingArchetype
	specification and archetypic instance for the ring interfaceThe RingArchetype and its encapsulated element class contain pointers to the RingAbstract and its encapsulated ring element, respectively. More...
class	RingEnvelope
	implement the ring archetype to minimize code bloat. More...
class	RingInterface
	This ring base class exists solely to aid documentation organization. More...
class	RNS
	RNS. More...
class	ScalarMatrix
	Blackbox for aI. More...
class	SemiDIteration
	CRA iteration to get a diagonal with the same signature. More...
class	showProgression
	Show progression on the terminal (helper) More...
class	SigmaBasis
	implementation of -basis (minimal basis). More...
class	Sliced
	The Sliced Matrix class _Domain must be a GF(3) rep, BaseT must be an unsigned int type. More...
class	SlicedPolynomialMatrixAddin
	C += A. More...
class	SlicedPolynomialMatrixSubin
	C -= A. More...
class	SlicedPolynomialVectorAddin
	C += A. More...
class	SlicedPolynomialVectorSubin
	C -= A. More...
class	SmithFormBinary
	Compute Smith form. More...
class	SmithFormIliopoulos
	This is Iliopoulos' algorithm to diagonalize. More...
class	SmithFormLocal
	Smith normal form (invariant factors) of a matrix over a local ring. More...
class	Sparse_Vector
	vector< Pair<T,I> > and actualsize More...
class	SparseLULiftingContainer
	SparseLULiftingContainer. More...
class	SparseMatrix< _Field, SparseMatrixFormat::COO >
	Sparse matrix, Coordinate storage. More...
class	SparseMatrix< _Field, SparseMatrixFormat::COO::implicit >
	Sparse matrix, Coordinate storage. More...
class	SparseMatrix< _Field, SparseMatrixFormat::CSR >
	Sparse matrix, Coordinate storage. More...
class	SparseMatrix< _Field, SparseMatrixFormat::ELL >
	Sparse matrix, Coordinate storage. More...
class	SparseMatrix< _Field, SparseMatrixFormat::ELL_R >
	Sparse matrix, Coordinate storage. More...
class	SparseMatrix< _Field, SparseMatrixFormat::HYB >
	Sparse matrix, Coordinate storage. More...
class	SparseMatrix< Field_, SparseMatrixFormat::TPL >
	Sparse Matrix in Triples storage. More...
class	SparseMatrix< Field_, SparseMatrixFormat::TPL_omp >
	Sparse matrix representation which stores nonzero entries by i,j,value triples. More...
class	SparseMatrixReadHelper
	Read helper. More...
class	SparseMatrixWriteHelper
	Write helper. More...
class	Squarize
	transpose matrix without copying. More...
class	StandardBasisStream
	Stream for . More...
class	StandardBasisStream< Field, _Vector, VectorCategories::DenseVectorTag >
	Specialization of standard basis stream for dense vectors. More...
class	StandardBasisStream< Field, _Vector, VectorCategories::SparseAssociativeVectorTag >
	Specialization of standard basis stream for sparse associative vectors. More...
class	StandardBasisStream< Field, _Vector, VectorCategories::SparseParallelVectorTag >
	Specialization of standard basis stream for sparse parallel vectors. More...
class	StandardBasisStream< Field, _Vector, VectorCategories::SparseSequenceVectorTag >
	Specialization of standard basis stream for sparse sequence vectors. More...
class	StorageMetaData
	Storage metadata;. More...
class	Subiterator
	Subvector iterator class provides striding iterators. More...
class	Submatrix
	leading principal minor of existing matrix without copying. More...
class	Submatrix< Blackbox, VectorCategories::DenseVectorTag >
	Specialization for dense vectors. More...
class	Submatrix< Blackbox, VectorCategories::DenseZeroOneVectorTag >
	Specialization for dense ZeroOne vectors. More...
class	Submatrix< BlasMatrix< _Field >, VectorCategories::DenseVectorTag >
	Specialization for BlasMatrix. More...
class	SubmatrixAdapter
	Generic submatrix view adapter used internally in the OpenCLMatrixDomain. More...
class	SubmatrixOwner< Blackbox, VectorCategories::DenseVectorTag >
	Specialization for dense vectors. More...
class	Subvector
	Dense subvectorThis class provides a statically sized subvector of a random access container (such as std::vector, deque). More...
class	Sum
	blackbox of a matrix sum without copying. More...
class	SumOwner
	blackbox of a matrix sum without copying. More...
class	Sylvester
	This is a representation of the Sylvester matrix of two polynomials. More...
class	TernaryLattice
	NO DOC. More...
class	TimeWatcher
	Helper. More...
class	Toeplitz
	This is the blackbox representation of a Toeplitz matrix. More...
class	Toeplitz< typename _PRing::CoeffField, _PRing >
	Specialization for when the field of matrix elements is the same as the coefficient field of the polynomial field. More...
struct	TraceCategory
	Trait to show whether or not the BB class has a local trace function. More...
class	Transpose
	transpose matrix without copying. More...
class	TransposedBlasMatrix
	TransposedBlasMatrix. More...
class	TransposedBlasMatrix< TransposedBlasMatrix< Matrix > >
	TransposedBlasMatrix. More...
class	TransposeMatrix
	Matrix transpose. More...
class	TransposeOwner
	transpose matrix without copying. More...
class	TriangularBlasMatrix
	Triangular BLAS matrix. More...
struct	UniqueSamplingTrait
	Whether a prime generator generates a sequence with non repeating numbers. More...
class	UnparametricRandIter< NTL::ZZ_p >
	Constructor for random field element generator. More...
struct	Vector
	Vector ?? More...
struct	VectorCategories
	List of vector categories. More...
class	VectorFraction
	VectorFraction<Domain> is a vector of rational elements with common reduced denominator. More...
class	VectorStream
	Vector factory. More...
struct	VectorTraits
	Vector traits template structure. More...
class	WiedemannLiftingContainer
	Wiedemann LiftingContianer. More...
class	WiedemannSolver
	Linear system solvers based on Wiedemann's method. More...
class	ZeroOne
	Time and space efficient representation of sparse {0,1}-matrices. More...
class	ZeroOne< GF2 >
	Time and space efficient representation of sparse matrices over GF2. More...
class	ZOQuad
	A class of striped or block-decomposed zero-one matrices. More...

Typedefs
template<class CRABase >
using	ChineseRemainder = ChineseRemainderSequential< CRABase >
	Wrapper around OMP/SEQ version of ChineseRemainderXXX<CRABase>. More...
typedef Givaro::Integer	integer
	Integers in LinBox. More...
typedef std::vector< double >	dvector_t
	vector of double
typedef std::vector< dvector_t >	dmatrix_t
	matrix of double

Enumerations
enum	IterationResult { CONTINUE, SKIP, RESTART }
	Return type for CRA iteration. More...
enum	SolverReturnStatus
	define the different return status of the p-adic based solver's computation.
enum	SolverLevel
	Define the different strategy which can be used in the p-adic based solver. More...
enum	Singularity { Unknown, Singular, NonSingular }
	Singularity of the system. More...
enum	Dispatch {   Auto, Sequential, SMP, Distributed,   Combined }
	For integer-based methods that evaluate multiple times the system at different moduli, decides how to dispatch each sub-computations. More...
enum	SingularSolutionType { Deterministic, Random, Diophantine }
	For Dixon method, which solution type to get when the system is singular. More...
enum	Preconditioner {   None, Butterfly, Sparse, Toeplitz,   Symmetrize, PartialDiagonal, PartialDiagonalSymmetrize, FullDiagonal,   Dense }
	Preconditioner to ensure generic rank profile. More...
enum	PivotStrategy
	Pivoting strategy for elimination-based methods.

Functions
template<class Polynomial , class Blackbox >
Polynomial &	cia (Polynomial &P, const Blackbox &A, const Method::DenseElimination &M)
	Algorithm computing the integer characteristic polynomial of a dense matrix. More...
uint64_t	primes_count (size_t pbits)
	Lower bound on number of b-bit primes.
template<class Field >
size_t &	NullSpaceBasisIn (const Tag::Side Side, BlasMatrix< Field > &A, BlasMatrix< Field > &Ker, size_t &kerdim)
	Nullspace of a dense matrix on a finite field. More...
template<class DenseMat >
size_t &	NullSpaceBasisIn (const Tag::Side Side, BlasSubmatrix< DenseMat > &A, BlasMatrix< typename DenseMat::Field > &Ker, size_t &kerdim)
template<class Field >
size_t &	NullSpaceBasis (const Tag::Side Side, const BlasMatrix< Field > &A, BlasMatrix< Field > &Ker, size_t &kerdim)
	Nullspace of a dense matrix on a finite field. More...
template<class Field >
size_t	NullSpaceBasisIn (const Field &F, const Tag::Side Side, const size_t &m, const size_t &n, typename Field::Element *A, const size_t &lda, typename Field::Element *&Ker, size_t &ldk, size_t &kerdim)
	Computes the kernel of a dense matrix using LQUP. More...
template<class Ring >
bool	partial_hegcd (Ring &Z, typename Ring::Element &e, typename Ring::Element &b, const typename Ring::Element &n, const typename Ring::Element &d, const typename Ring::Element &denBound)
	partial_hegcd() sets e, b from the remainder sequence of n,d. More...
template<class Ring >
int	dyadicToRational (const Ring &Z, typename Ring::Element &a, typename Ring::Element &b, const typename Ring::Element &n, const typename Ring::Element &d, const typename Ring::Element &B)
	Rational reconstruction of a/b from n/d with denominator bound B. More...
template<class Blackbox , class MyMethod >
Blackbox::Field::Element &	lif_cra_det (typename Blackbox::Field::Element &d, const Blackbox &A, const RingCategories::IntegerTag &tag, const MyMethod &M)
	Compute the determinant of A over the integers. More...
std::vector< cl_platform_id >	enumPlatforms ()
	Enumerate all of the platforms currently available on the system.
std::string	getPlatformName (cl_platform_id platform)
	Get the platform name associated with the platform.
double	getPlatformVersion (cl_platform_id platform)
	Get the platform version associated with the platform.
std::vector< std::string >	getPlatformExtensions (cl_platform_id platform)
	Get the platform extensions associated with the platform.
std::vector< cl_device_id >	enumDevices (cl_platform_id platform)
	Enumerate all of the devices currently available on the platform.
cl_context	createContext (cl_platform_id platform, cl_device_id device)
	Create an OpenCL context from a platfrom and device.
template<class Prime >
bool	checkBlasPrime (const Prime p)
	NO DOC ! More...
int	large_double_division (integer &x, const integer &y, const integer &z)
	NO DOC.
template<class Domain >
void	reduceIn (Domain &D, std::pair< typename Domain::Element, typename Domain::Element > &frac)
	utility function to reduce a rational pair to lowest form
template<class Domain , class Vector >
void	vectorGcdIn (typename Domain::Element &result, Domain &D, Vector &v)
	utility function to gcd-in a vector of elements over a domain
template<class Domain , class Vector >
Domain::Element	vectorGcd (Domain &D, Vector &v)
	utility function, returns gcd of a vector of elements over a domain
template<class Domain , class IMatrix >
void	create_MatrixQadic (const Domain &D, const IMatrix &Mat, double *chunks, size_t num_chunks, const integer shift)
	split an integer matrix into a padic chunk representation More...
template<class Domain , class Vector >
void	create_VectorQadic (const Domain &D, const Vector &V, double *chunks, size_t num_chunks)
	split an integer vector into a padic chunk representation More...
template<class Domain , class Vector >
void	create_VectorQadic_32 (const Domain &D, const Vector &V, double *chunks, size_t num_chunks)
	split an integer vector into a padic chunk representation More...
template<class Field >
DenseMatrix< Field > &	genericNullspaceRandomRight (DenseMatrix< Field > &N, const FIBB< Field > &A)
	N: AN = 0, each col random.
template<class Field >
DenseMatrix< Field > &	genericNullspaceRandomLeft (DenseMatrix< Field > &N, const FIBB< Field > &A)
	N: NA = 0, each row random.
double	naturallog (const Givaro::Integer &a)
	Natural logarithm (ln). More...
template<class _Field , class _Storage >
std::ostream &	operator<< (std::ostream &os, const BlasMatrix< _Field, _Storage > &Mat)
	Write a matrix to a stream. More...
void	RandomBlasPermutation (BlasPermutation< size_t > &P)
template<class T >
std::ostream &	operator<< (std::ostream &o, const DenseMat< T > &Mat)
	Write a matrix to a stream. More...
template<class Field , class Vector >
Vector &	prepare (const Field &F, Vector &y, const typename Field::Element &a)
	y <- ay. More...
template<typename Container >
PrimeSequence< typename Container::const_iterator >	create_prime_sequence (const Container &cont)
	convenience factory to create a PrimeSequence from an STL-like container. More...
template<class Blackbox , class Polynomial , class MyMethod >
Polynomial &	charpoly (Polynomial &P, const Blackbox &A, const MyMethod &M)
	...using an optional Method parameter More...
template<class Blackbox , class Polynomial >
Polynomial &	charpoly (Polynomial &P, const Blackbox &A)
	...using default method
template<class Blackbox , class Polynomial >
Polynomial &	charpoly (Polynomial &P, const Blackbox &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	Compute the characteristic polynomial over . More...
template<class Blackbox , class Polynomial >
Polynomial &	charpoly (Polynomial &P, const Blackbox &A, const RingCategories::ModularTag &tag, const Method::Blackbox &M)
	Compute the characteristic polynomial over . More...
template<class Blackbox , class DetMethod , class DomainCategory >
Blackbox::Field::Element &	det (typename Blackbox::Field::Element &d, const Blackbox &A, const DomainCategory &tag, const DetMethod &Meth)
	Compute the determinant of A. More...
template<class Field >
Field::Element &	detInPlace (typename Field::Element &d, BlasMatrix< Field > &A)
	Rank of Blackbox A. More...
template<class Matrix , class CategoryTag >
size_t	rowEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	rowEchelon specialisation for Auto.
template<class Field >
size_t	rowEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	rowEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	rowEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	rowEchelon specialisation for Auto.
template<class Field >
size_t	rowEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	rowEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	rowEchelonize (Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	rowEchelonize specialisation for Auto.
template<class Field >
size_t	rowEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	rowEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	rowEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const Method::Auto &m)
	rowEchelonize specialisation for Auto.
template<class Field >
size_t	rowEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::Auto &m)
	rowEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedRowEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	reducedRowEchelon specialisation for Auto.
template<class Field >
size_t	reducedRowEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedRowEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedRowEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	reducedRowEchelon specialisation for Auto.
template<class Field >
size_t	reducedRowEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedRowEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedRowEchelonize (Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	reducedRowEchelonize specialisation for Auto.
template<class Field >
size_t	reducedRowEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedRowEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedRowEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const Method::Auto &m)
	reducedRowEchelonize specialisation for Auto.
template<class Field >
size_t	reducedRowEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedRowEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	colEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	colEchelon specialisation for Auto.
template<class Field >
size_t	colEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	colEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	colEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	colEchelon specialisation for Auto.
template<class Field >
size_t	colEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	colEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	colEchelonize (Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	colEchelonize specialisation for Auto.
template<class Field >
size_t	colEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	colEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	colEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const Method::Auto &m)
	colEchelonize specialisation for Auto.
template<class Field >
size_t	colEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::Auto &m)
	colEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedColEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	reducedColEchelon specialisation for Auto.
template<class Field >
size_t	reducedColEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedColEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedColEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	reducedColEchelon specialisation for Auto.
template<class Field >
size_t	reducedColEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedColEchelon specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedColEchelonize (Matrix &A, const CategoryTag &tag, const Method::Auto &m)
	reducedColEchelonize specialisation for Auto.
template<class Field >
size_t	reducedColEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedColEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag >
size_t	reducedColEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const Method::Auto &m)
	reducedColEchelonize specialisation for Auto.
template<class Field >
size_t	reducedColEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::Auto &m)
	reducedColEchelonize specialisation for Auto with DenseMatrix and ModularTag.
template<class Field >
size_t	rowEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	rowEchelon specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	rowEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	rowEchelon with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	rowEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	rowEchelonize specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	rowEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	rowEchelonize with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedRowEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedRowEchelon specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedRowEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedRowEchelon with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedRowEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedRowEchelonize specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedRowEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedRowEchelonize with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	colEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	colEchelon specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	colEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	colEchelon with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	colEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	colEchelonize specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	colEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	colEchelonize with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedColEchelon (DenseMatrix< Field > &E, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedColEchelon specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedColEchelon (DenseMatrix< Field > &E, DenseMatrix< Field > &T, const DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedColEchelon with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedColEchelonize (DenseMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedColEchelonize specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Field >
size_t	reducedColEchelonize (DenseMatrix< Field > &A, DenseMatrix< Field > &T, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	reducedColEchelonize with transformation specialisation for DenseElimination with DenseMatrix and ModularTag.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	rowEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the row echelon form of a matrix, not reduced. More...
template<class Matrix , class EchelonMethod >
size_t	rowEchelon (Matrix &E, const Matrix &A, const EchelonMethod &m)
	rowEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	rowEchelon (Matrix &E, const Matrix &A)
	rowEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	rowEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the row echelon form of a matrix, not reduced, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	rowEchelon (Matrix &E, Matrix &T, const Matrix &A, const EchelonMethod &m)
	rowEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	rowEchelon (Matrix &E, Matrix &T, const Matrix &A)
	rowEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	rowEchelonize (Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Replace the input matrix by its row echelon form, not reduced. More...
template<class Matrix , class EchelonMethod >
size_t	rowEchelonize (Matrix &A, const EchelonMethod &m)
	rowEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	rowEchelonize (Matrix &A)
	rowEchelonize dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	rowEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const EchelonMethod &m)
	Compute the row echelon form of a matrix, not reduced, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	rowEchelonize (Matrix &A, Matrix &T, const EchelonMethod &m)
	rowEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	rowEchelonize (Matrix &A, Matrix &T)
	rowEchelonize dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedRowEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the reduced row echelon form of a matrix. More...
template<class Matrix , class EchelonMethod >
size_t	reducedRowEchelon (Matrix &E, const Matrix &A, const EchelonMethod &m)
	reducedRowEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	reducedRowEchelon (Matrix &E, const Matrix &A)
	reducedRowEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedRowEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the reduced row echelon form of a matrix, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	reducedRowEchelon (Matrix &E, Matrix &T, const Matrix &A, const EchelonMethod &m)
	reducedRowEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	reducedRowEchelon (Matrix &E, Matrix &T, const Matrix &A)
	reducedRowEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedRowEchelonize (Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Replace the input matrix by its reduced row echelon form. More...
template<class Matrix , class EchelonMethod >
size_t	reducedRowEchelonize (Matrix &A, const EchelonMethod &m)
	reducedRowEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	reducedRowEchelonize (Matrix &A)
	reducedRowEchelonize dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedRowEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const EchelonMethod &m)
	Compute the reduced row echelon form of a matrix, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	reducedRowEchelonize (Matrix &A, Matrix &T, const EchelonMethod &m)
	reducedRowEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	reducedRowEchelonize (Matrix &A, Matrix &T)
	reducedRowEchelonize dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	colEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the column echelon form of a matrix, not reduced. More...
template<class Matrix , class EchelonMethod >
size_t	colEchelon (Matrix &E, const Matrix &A, const EchelonMethod &m)
	colEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	colEchelon (Matrix &E, const Matrix &A)
	colEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	colEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the column echelon form of a matrix, not reduced, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	colEchelon (Matrix &E, Matrix &T, const Matrix &A, const EchelonMethod &m)
	colEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	colEchelon (Matrix &E, Matrix &T, const Matrix &A)
	colEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	colEchelonize (Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Replace the input matrix by its column echelon form, not reduced. More...
template<class Matrix , class EchelonMethod >
size_t	colEchelonize (Matrix &A, const EchelonMethod &m)
	colEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	colEchelonize (Matrix &A)
	colEchelonize dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	colEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const EchelonMethod &m)
	Compute the column echelon form of a matrix, not reduced, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	colEchelonize (Matrix &A, Matrix &T, const EchelonMethod &m)
	colEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	colEchelonize (Matrix &A, Matrix &T)
	colEchelonize dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedColEchelon (Matrix &E, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the reduced column echelon form of a matrix. More...
template<class Matrix , class EchelonMethod >
size_t	reducedColEchelon (Matrix &E, const Matrix &A, const EchelonMethod &m)
	reducedColEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	reducedColEchelon (Matrix &E, const Matrix &A)
	reducedColEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedColEchelon (Matrix &E, Matrix &T, const Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Compute the reduced column echelon form of a matrix, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	reducedColEchelon (Matrix &E, Matrix &T, const Matrix &A, const EchelonMethod &m)
	reducedColEchelon dispatcher for automated category tag.
template<class Matrix >
size_t	reducedColEchelon (Matrix &E, Matrix &T, const Matrix &A)
	reducedColEchelon dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedColEchelonize (Matrix &A, const CategoryTag &tag, const EchelonMethod &m)
	Replace the input matrix by its reduced column echelon form. More...
template<class Matrix , class EchelonMethod >
size_t	reducedColEchelonize (Matrix &A, const EchelonMethod &m)
	reducedColEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	reducedColEchelonize (Matrix &A)
	reducedColEchelonize dispatcher for automated method.
template<class Matrix , class CategoryTag , class EchelonMethod >
size_t	reducedColEchelonize (Matrix &A, Matrix &T, const CategoryTag &tag, const EchelonMethod &m)
	Compute the reduced column echelon form of a matrix, and the related transformation matrix. More...
template<class Matrix , class EchelonMethod >
size_t	reducedColEchelonize (Matrix &A, Matrix &T, const EchelonMethod &m)
	reducedColEchelonize dispatcher for automated category tag.
template<class Matrix >
size_t	reducedColEchelonize (Matrix &A, Matrix &T)
	reducedColEchelonize dispatcher for automated method.
template<class BB >
BB::Field::Element &	getEntry (typename BB::Field::Element &x, const BB &A, const size_t i, const size_t j)
	Getting the i,j entry of the blackbox.
template<class BB , class Method >
BB::Field::Element &	getEntry (typename BB::Field::Element &x, const BB &A, const size_t i, const size_t j, Method &m)
	To ignore methods.
template<class IMatrix >
void	HadamardRowLogBound (double &logBound, double &minLogNorm, const IMatrix &A)
	Precise Hadamard bound (bound on determinant) by taking the row-wise euclidean norm. More...
template<class IMatrix >
void	HadamardColLogBound (double &logBound, double &minLogNorm, const IMatrix &A)
	Precise Hadamard bound (bound on determinant) by taking the column-wise euclidean norm. More...
template<class IMatrix >
HadamardLogBoundDetails	DetailedHadamardBound (const IMatrix &A)
	Precise Hadamard bound (bound on determinant) by taking the minimum of the column-wise and the row-wise euclidean norm. More...
template<class IMatrix >
double	HadamardBound (const IMatrix &A)
	Precise Hadamard bound (bound on determinant) by taking the minimum of the column-wise and the row-wise euclidean norm. More...
template<class IMatrix >
Integer &	InfinityNorm (Integer &max, const IMatrix &A, const MatrixCategories::RowColMatrixTag &tag)
	Returns the bit size of the Hadamard bound. More...
template<class IMatrix >
double	FastCharPolyDumasPernetWanBound (const IMatrix &A, const Integer &infnorm)
	Bound on the coefficients of the characteristic polynomial. More...
template<class IMatrix >
double	FastCharPolyGoldsteinGrahamBound (const IMatrix &A, const Integer &infnorm)
	A.J. More...
template<class Matrix , class Vector >
std::enable_if< std::is_same< typename FieldTraits< typename Matrix::Field >::categoryTag, RingCategories::IntegerTag >::value, RationalSolveHadamardBoundData >::type	RationalSolveHadamardBound (const Matrix &A, const Vector &b)
	Bound on the rational solution of a linear system Ax = b. More...
template<class Matrix , class Vector >
std::enable_if< std::is_same< typename FieldTraits< typename Matrix::Field >::categoryTag, RingCategories::RationalTag >::value, RationalSolveHadamardBoundData >::type	RationalSolveHadamardBound (const Matrix &A, const Vector &b)
	Needed to solve-cra.h, but can't be used yet.
template<class IMatrix , class MTag >
Integer &	InfinityNorm (Integer &max, const IMatrix &A, const MTag &tag)
	Returns the maximal absolute value.
template<class IMatrix >
double	FastHadamardBound (const IMatrix &A, const Integer &infnorm)
	Returns the bit size of the Hadamard bound. More...
template<class Blackbox , class MyMethod >
bool	isPositiveDefinite (const Blackbox &A, const MyMethod &M)
	Compute the isPositiveDefinite of A. More...
template<class Blackbox >
bool	isPositiveDefinite (const Blackbox &A, const RingCategories::IntegerTag &tag, const Method::Auto &M)
template<class Blackbox >
bool	isPositiveDefinite (const Blackbox &A, const RingCategories::IntegerTag &tag, const Method::DenseElimination &M)
template<class Blackbox , class MyMethod >
bool	isPositiveSemiDefinite (const Blackbox &A, const MyMethod &M)
	Determine if A is positive semidefinite. More...
template<class Blackbox >
bool	isPositiveSemiDefinite (const Blackbox &A, const RingCategories::IntegerTag &tag, const Method::DenseElimination &M)
template<class Polynomial , class Blackbox >
Polynomial &	minpoly (Polynomial &P, const Blackbox &A)
	...using default Method
template<class Blackbox , class Method , class DomainCategory >
size_t &	rank (size_t &r, const Blackbox &A, const DomainCategory &tag, const Method &M)
	Compute the rank of a linear transform A over a field by selected method. More...
template<class Blackbox >
size_t &	rank (size_t &r, const Blackbox &A)
	Compute the rank of a linear transform A over a field. More...
template<class Blackbox , class Method >
size_t &	rank (size_t &r, const Blackbox &A, const Method &M)
	Compute the rank of a linear transform A over a field. More...
template<class Blackbox >
size_t &	rankInPlace (size_t &r, Blackbox &A)
	Rank of A. More...
template<class Blackbox >
size_t &	rank (size_t &r, const Blackbox &A, const RingCategories::ModularTag &tag, const Method::Auto &m)
template<class Blackbox >
size_t &	rank (size_t &res, const Blackbox &A, const RingCategories::ModularTag &tag, const Method::Wiedemann &M)
	M may be Method::Wiedemann(). More...
template<class Field >
size_t &	rank (size_t &r, const SparseMatrix< Field, SparseMatrixFormat::SparseSeq > &A, const RingCategories::ModularTag &tag, const Method::SparseElimination &M)
	M may be Method::SparseElimination().
size_t &	rankInPlace (size_t &r, GaussDomain< GF2 >::Matrix &A, const Method::SparseElimination &)
	specialization to
size_t &	rankInPlace (size_t &r, GaussDomain< GF2 >::Matrix &A, const RingCategories::ModularTag &, const Method::SparseElimination &M)
	specialization to
template<class Field >
size_t &	rankInPlace (size_t &r, BlasMatrix< Field > &A, const RingCategories::ModularTag &tag, const Method::DenseElimination &M)
	A is modified.
template<class Blackbox , class Method >
SmithList< typename Blackbox::Field > &	smithForm (SmithList< typename Blackbox::Field > &S, const Blackbox &A, const Method &M)
	Compute the Smith form of A. More...
template<class ResultVector , class Matrix , class Vector , class CategoryTag >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Auto &m)
	Solve specialisation for Auto.
template<class ResultVector , class Field , class Vector >
ResultVector &	solve (ResultVector &x, const DenseMatrix< Field > &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::Auto &m)
	Solve specialisation for Auto with DenseMatrix and ModularTag.
template<class ResultVector , class... MatrixArgs, class Vector >
ResultVector &	solve (ResultVector &x, const SparseMatrix< MatrixArgs... > &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::Auto &m)
	Solve specialisation for Auto with SparseMatrix and ModularTag.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Auto &m)
	Solve specialisation for Auto and IntegerTag.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const RingCategories::RationalTag &tag, const Method::Auto &m)
	Solve specialisation for Auto and RationalTag.
template<class IntVector , class Matrix , class Vector >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Auto &m)
	Solve specialization for Auto and IntegerTag.
template<class ResultVector , class Matrix , class Vector , class CategoryTag >
ResultVector &	solveInPlace (ResultVector &x, Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Auto &m)
	Solve in place specialisation for Auto.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solveInPlace (ResultVector &x, Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Auto &m)
	Solve in place specialisation for Auto and IntegerTag.
template<class ResultVector , class Field , class Vector , class CategoryTag >
std::enable_if<!std::is_same< CategoryTag, RingCategories::IntegerTag >::value, ResultVector & >::type	solveInPlace (ResultVector &x, DenseMatrix< Field > &A, const Vector &b, const CategoryTag &tag, const Method::Auto &m)
	Solve in place specialisation for Auto with DenseMatrix and non-IntegerTag.
template<class ResultVector , class... MatrixArgs, class Vector , class CategoryTag >
std::enable_if<!std::is_same< CategoryTag, RingCategories::IntegerTag >::value, ResultVector & >::type	solveInPlace (ResultVector &x, SparseMatrix< MatrixArgs... > &A, const Vector &b, const CategoryTag &tag, const Method::Auto &m)
	Solve in place specialisation for Auto with SparseMatrix and non-IntegerTag.
template<class Matrix , class Vector >
void	solveInPlace (Vector &xNum, typename Vector::Element &xDen, Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Auto &m)
	Solve in place specialization for Auto and IntegerTag.
template<class ResultVector , class Matrix , class Vector , class CategoryTag >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Blackbox &m)
	Solve specialisation for Blackbox.
template<class ResultVector , class Matrix , class Vector , class CategoryTag >
ResultVector &	solveInPlace (ResultVector &x, Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Blackbox &m)
	Solve in place specialisation for Blackbox.
template<class IntVector , class Matrix , class Vector , class IterationMethod >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::CRA< IterationMethod > &m)
	Solve specialization with Chinese Remainder Algorithm method for an Integer or Rational tags. More...
template<class RatVector , class RatMatrix , class IterationMethod >
RatVector &	solve (RatVector &x, const RatMatrix &A, const RatVector &b, const RingCategories::RationalTag &tag, const Method::CRA< IterationMethod > &m)
	Solve specialization with Chinese Remainder Algorithm method for a Rational matrix.
template<class Matrix , class Vector >
Vector &	solve (Vector &x, const Matrix &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::DenseElimination &m)
	Solve specialisation for DenseElimination.
template<class Field , class Vector >
Vector &	solve (Vector &x, const DenseMatrix< Field > &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::DenseElimination &m)
	Solve specialisation for DenseElimination on dense matrices with ModularTag.
template<class IntVector , class Blackbox , class Vector >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const Blackbox &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Dixon &m)
	Solve specialisation for Dixon on blackboxes matrices.
template<class IntVector , class Ring , class Vector >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const DenseMatrix< Ring > &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Dixon &m)
	Solve specialisation for Dixon on dense matrices.
template<class IntVector , class... MatrixArgs, class Vector >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const SparseMatrix< MatrixArgs... > &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Dixon &m)
	Solve specialisation for Dixon on sparse matrices.
template<class Matrix , class Vector , class CategoryTag >
Vector &	solve (Vector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Elimination &m)
	Solve specialisation for Elimination.
template<class MatrixField , class Vector , class CategoryTag >
Vector &	solve (Vector &x, const DenseMatrix< MatrixField > &A, const Vector &b, const CategoryTag &tag, const Method::Elimination &m)
	Solve specialisation for Elimination with DenseMatrix.
template<class MatrixField , class Vector , class CategoryTag >
Vector &	solve (Vector &x, const SparseMatrix< MatrixField > &A, const Vector &b, const CategoryTag &tag, const Method::Elimination &m)
	Solve specialisation for Elimination with SparseMatrix.
template<class Matrix , class Vector , class CategoryTag >
Vector &	solveInPlace (Vector &x, Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Elimination &m)
	Solve in place specialisation for Elimination.
template<class MatrixField , class Vector , class CategoryTag >
Vector &	solveInPlace (Vector &x, DenseMatrix< MatrixField > &A, const Vector &b, const CategoryTag &tag, const Method::Elimination &m)
	Solve in place specialisation for Elimination with DenseMatrix.
template<class MatrixField , class Vector , class CategoryTag >
Vector &	solveInPlace (Vector &x, SparseMatrix< MatrixField > &A, const Vector &b, const CategoryTag &tag, const Method::Elimination &m)
	Solve in place specialisation for Elimination with SparseMatrix.
template<class Matrix , class Vector , class CategoryTag >
Vector &	solve (Vector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Lanczos &m)
	Solve specialisation for Lanczos.
template<class Matrix , class Vector >
Vector &	solve (Vector &x, const Matrix &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::Lanczos &m)
	Solve specialisation for Lanczos with ModularTag.
template<class Matrix , class Vector , class CategoryTag >
Vector &	solve (Vector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::BlockLanczos &m)
	Solve specialisation for BlockLanczos.
template<class Matrix , class Vector >
Vector &	solve (Vector &x, const Matrix &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::BlockLanczos &m)
	Solve specialisation for BlockLanczos with ModularTag.
template<class IntVector , class Ring , class Vector >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const DenseMatrix< Ring > &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::SymbolicNumericNorm &m)
	Solve specialisation for SymbolicNumericNorm with IntegerTag on DenseMatrix.
template<class Matrix , class Vector >
Vector &	solve (Vector &x, const Matrix &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::SparseElimination &m)
	Solve specialisation for SparseElimination.
template<class... MatrixArgs, class Vector >
Vector &	solve (Vector &x, const SparseMatrix< MatrixArgs... > &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::SparseElimination &m)
	Solve specialisation for SparseElimination with SparseMatrix.
template<class... MatrixArgs, class Vector >
Vector &	solveInPlace (Vector &x, SparseMatrix< MatrixArgs... > &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::SparseElimination &m)
	Solve in place specialisation for SparseElimination with SparseMatrix.
template<class ResultVector , class Matrix , class Vector , class CategoryTag >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Wiedemann &m)
	Solve specialisation for Wiedemann.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const Method::Wiedemann &m)
	Solve specialisation for Wiedemann with IntegerTag.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::Wiedemann &m)
	Solve specialisation for Wiedemann with ModularTag.
template<class ResultVector , class Matrix , class Vector , class CategoryTag >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::BlockWiedemann &m)
	Solve specialisation for BlockWiedemann.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::BlockWiedemann &m)
	Solve specialisation for BlockWiedemann with ModularTag.
template<class ResultVector , class Matrix , class Vector , class CategoryTag >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const Method::Coppersmith &m)
	Solve specialisation for Coppersmith.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const RingCategories::ModularTag &tag, const Method::Coppersmith &m)
	Solve specialisation for Coppersmith on ModularTag.
template<class ResultVector , class Matrix , class Vector , class CategoryTag , class SolveMethod >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const CategoryTag &tag, const SolveMethod &m)
	Solve Ax = b, for x. More...
template<class ResultVector , class Matrix , class Vector , class SolveMethod >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b, const SolveMethod &m)
	Solve dispatcher for automated category tag.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solve (ResultVector &x, const Matrix &A, const Vector &b)
	Solve dispatcher for automated solve method.
template<class RatVector , class RatMatrix , class Vector , class SolveMethod >
RatVector &	solve (RatVector &x, const RatMatrix &A, const Vector &b, const RingCategories::RationalTag &tag, const SolveMethod &m)
	Solve specialisation on RationalTag, with a generic method.
template<class RatVector , class Matrix , class Vector , class SolveMethod >
std::enable_if< std::is_same< typename SolveMethod::CategoryTag, RingCategories::IntegerTag >::value &&std::is_same< typename FieldTraits< typename RatVector::Field >::categoryTag, RingCategories::RationalTag >::value, RatVector & >::type	solve (RatVector &x, const Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const SolveMethod &m)
	Solve specialisation on IntegerTag with Vector<QField> as result. More...
template<class Matrix , class Vector , class SolveMethod >
std::enable_if< std::is_same< typename SolveMethod::CategoryTag, RingCategories::IntegerTag >::value, VectorFraction< typename Matrix::Field > & >::type	solve (VectorFraction< typename Matrix::Field > &x, const Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const SolveMethod &m)
	Solve specialisation on IntegerTag with VectorFraction as result. More...
template<class IntVector , class Matrix , class Vector , class CategoryTag , class SolveMethod >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const Matrix &A, const Vector &b, const CategoryTag &tag, const SolveMethod &m)
	Rational solve Ax = b, for x expressed as xNum/xDen. More...
template<class IntVector , class Matrix , class Vector , class SolveMethod >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const Matrix &A, const Vector &b, const RingCategories::IntegerTag &tag, const SolveMethod &m)
	Rational solve dispatcher for unimplemented methods.
template<class IntVector , class Matrix , class Vector , class SolveMethod >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const Matrix &A, const Vector &b, const SolveMethod &m)
	Rational solve dispatcher for automated category tag.
template<class IntVector , class Matrix , class Vector >
void	solve (IntVector &xNum, typename IntVector::Element &xDen, const Matrix &A, const Vector &b)
	Rational solve dispatcher for automated solve method.
template<class ResultVector , class Matrix , class Vector , class CategoryTag , class SolveMethod >
ResultVector &	solveInPlace (ResultVector &x, Matrix &A, const Vector &b, const CategoryTag &tag, const SolveMethod &m)
	Solve Ax = b, for x. More...
template<class ResultVector , class Matrix , class Vector , class SolveMethod >
ResultVector &	solveInPlace (ResultVector &x, Matrix &A, const Vector &b, const SolveMethod &m)
	Solve in place dispatcher for automated category tag.
template<class ResultVector , class Matrix , class Vector >
ResultVector &	solveInPlace (ResultVector &x, Matrix &A, const Vector &b)
	Solve in place dispatcher for automated solve method.
template<class IntVector , class Matrix , class Vector , class SolveMethod , class CategoryTag >
void	solveInPlace (IntVector &xNum, typename IntVector::Element &xDen, Matrix &A, const Vector &b, const CategoryTag &tag, const SolveMethod &m)
	Rational solve in place Ax = b, for x expressed as xNum/xDen. More...
template<class IntVector , class Matrix , class Vector , class SolveMethod >
void	solveInPlace (IntVector &xNum, typename IntVector::Element &xDen, Matrix &A, const Vector &b, const SolveMethod &m)
	Rational solve in place dispatcher for automated category tag.
template<class IntVector , class Matrix , class Vector >
void	solveInPlace (IntVector &xNum, typename IntVector::Element &xDen, Matrix &A, const Vector &b)
	Rational solve in place dispatcher for automated solve method.
template<class BB >
BB::Field::Element &	trace (typename BB::Field::Element &t, const BB &A)
	Sum of the eigenvalues. More...
template<class Blackbox , class MyMethod >
Blackbox::Field::Element &	valence (typename Blackbox::Field::Element &v, const Blackbox &A, const MyMethod &M)
	Compute the valence of A. More...
uint64_t	serialize (std::vector< uint8_t > &bytes, const Integer &integer)
	Serializes an Integer with its underlying __mpz_struct. More...
uint64_t	unserialize (Integer &integer, const std::vector< uint8_t > &bytes, uint64_t offset=0u)
	Unserializes an Integer.
template<class Field >
uint64_t	serialize (std::vector< uint8_t > &bytes, const BlasMatrix< Field > &M)
	Serializes a BlasMatrix. More...
template<class Field >
uint64_t	unserialize (BlasMatrix< Field > &M, const std::vector< uint8_t > &bytes, uint64_t offset=0u)
	Unserializes a BlasMatrix. More...
template<class Field >
uint64_t	serialize (std::vector< uint8_t > &bytes, const SparseMatrix< Field > &M)
	Serializes a SparseMatrix. More...
template<class Field >
uint64_t	unserialize (SparseMatrix< Field > &M, const std::vector< uint8_t > &bytes, uint64_t offset=0u)
	Unserializes a SparseMatrix. More...
template<class Field >
uint64_t	serialize (std::vector< uint8_t > &bytes, const BlasVector< Field > &V)
	Serializes a BlasVector. More...
template<class Field >
uint64_t	unserialize (BlasVector< Field > &V, const std::vector< uint8_t > &bytes, uint64_t offset=0u)
	Unserializes a BlasVector. More...
template<class Field >
std::ostream &	writeMMComment (std::ostream &os, Field &F, std::string name, std::string comment)
	Write second line and comment part of matrix market header.
template<class BB >
std::ostream &	writeMMCoordHeader (std::ostream &os, BB &A, size_t nnz, std::string name, std::string comment="")
	Write matrix market header (up to the i,j,val lines) for a sparse or structured matrix.
template<class BB >
std::ostream &	writeMMPatternHeader (std::ostream &os, BB &A, size_t nnz, std::string name, std::string comment="")
	Write matrix market header (up to the i,j lines) for a {0,1} sparse or structured matrix.
template<class BB >
std::ostream &	writeMMArrayHeader (std::ostream &os, BB &A, std::string name, std::string comment="")
	Write matrix market header (up to the entry lines) for a dense matrix.
template<class Mat >
std::ostream &	writeMMArray (std::ostream &os, Mat &A, std::string name, std::string comment="")
	Generic dense matrix writer to matrix market array format (col major).
std::string	eltype (float x)
	eltype(x) returns a string containing the name of the type of field or ring element x.
template<class Field , class Vector >
Vector	randomVector (Field &F, size_t n, typename Field::RandIter &r)
	Random vector generator This templated function takes a field and a random field element generator and returns a vector of random field elements. More...
template<class Ring , class Matrix >
Matrix &	randomAns (const Ring &R, Matrix &Mat, size_t n, size_t epr)
size_t &	RandIntInInt (const size_t &s, size_t &RIII, const int &seed=0)
	gives a random number such that . More...
void	RandomPermutation (size_t *P, const size_t &len)
	Creates a random Lapack style Permutation P of size len.
template<class Field >
bool	CheckRank (const Field &F, const typename Field ::Element *A, const size_t &m, const size_t &n, const size_t &lda, const size_t &alledged_rank)
	Checks we got the right rank. More...
template<class Field >
void	RandomMatrixWithRank (const Field &F, typename Field ::Element *A, const size_t &m, const size_t &n, const size_t &lda, const size_t &rank)
	Builds a m x n random matrix of rank rank over field F.
template<class Field >
void	RandomMatrixWithDet (const Field &F, typename Field ::Element *A, const size_t &m, const size_t &lda, const typename Field ::Element &det)
	Builds a m x m random matrix of determinant det over field F. More...
void	showAdvanceLinear (size_t curr, size_t min, size_t max)
	show the advancement (on the terminal) suppose linear advancement More...
void	showFinish (size_t curr, size_t all)
	tells the current series of measure has completed (on the terminal) More...
void	showSkip (size_t curr, size_t all)
	tells the current series of measure was skipped (on the terminal) More...
double	computeMFLOPS (const double &tim, const double mflo, const size_t rpt=1)
	computes the number of megaflops. More...
double	computeMFLOPS (const dvector_t &tim, const double mflo, Tag::TimeSelect ts=Tag::TimeSelect::bestThree)
	computes the number of megaflops. More...
bool	isDigit (const std::string &s)
	Check if a string is actually a double. More...
bool	fortifiedString (const std::string &s)
	Tells is a string has double quotes around. More...
std::string	unfortifyString (const std::string &s)
	removes the surrounding quotes. More...
std::string	fortifyString (const std::string &s)
	adds surrounding quotes. More...
std::string	getDateTime (const std::string &sep)
	get ISO time and date More...
smatrix_t	getMachineInformation ()
	get some machine information (not cpu yet)
template<class T >
std::string	toString (T &nam)
	Converts anything to a string. More...
bool	findKeyword (size_t &i, const svector_t::const_iterator &begin, const svector_t::const_iterator &end, const std::string &keyword)
	finds keyword betwen begin and end, return true if found and i is the index where it is (possibly correspondig to end)
double	fit2 (const dvector_t &X, const dvector_t &Y, int n, double x)
	fit X[n-1,n],Y[n-1,n] and return evaluation at x.
double	fit3 (const dvector_t &X, const dvector_t &Y, int n, double x)
	fit X[n-2,n],Y[n-2,n] and return evaluation at x.
Butterfly
Butterfly preconditioner and supporting function
std::vector< bool >	setButterfly (const std::vector< bool > &x, size_t j=0)
	A function used with Butterfly Blackbox Matrices. More...
template<class T >
std::enable_if<!std::is_unsigned< T >::value, bool >::value	isPositive (const T &t)
	Positiveness of an integer. More...
template<class T >
std::enable_if< std::is_unsigned< T >::value, bool >::value	isPositive (const T &t)
	Positiveness of an integer. More...
template<class T >
std::enable_if<!std::is_unsigned< T >::value, bool >::value	isNegative (const T &t)
	Positiveness of an integer. More...
template<class T >
std::enable_if< std::is_unsigned< T >::value, bool >::value	isNegative (const T &t)
	Positiveness of an integer. More...
template<typename IntType >
bool	isOdd (const IntType &value)
	Positiveness of an integer. More...
template<typename IntType >
bool	isEven (const IntType &p)
	Positiveness of an integer. More...

Detailed Description

Namespace in which all linbox code resides.

Provides way to serialize any kind of data, like matrices, vectors and Integer in a platform-independent way.

The matrix class Sliced is defined.

Bug:

it is dangerous to include matrices defs that include hom for their rebind...

The subdirectories of linbox/ contain the template source code for all the LinBox functionality. See the Modules list for the documentation of the main parts of linbox.

Warning

The timer comes from Givaro and lives in its anonymous namespace.

Bug:

those are not just traits:

Bug:

this does not belong here.

It adheres to the LinBox dense matrix interface.

It depends on SlicedBase, also defined here, which packs GF(3) elements into a pair of ints. The int type is a template parameter.

Serialize functions add data to a prexisting vector of bytes, they return the number of bytes written. Unserialize ones read a vector of bytes starting at a specific offset, the number of bytes read.

As a convention, all numbers are written little-endian.

Todo:

GMP Integers can be configured with limbs of different sizes (32 or 64 bits), depending on the machine. We do not handle that right now, but storing info about their dimension might be a good idea, to at least emit a warning.

Enumeration Type Documentation

IterationResult

enum IterationResult

strong

Return type for CRA iteration.

The function object passed to the ChineseRemainder operators should return one of these values on each iteration.

Enumerator
CONTINUE	successful iteration; add to the result and keep going
SKIP	"bad prime"; ignore this result and keep going
RESTART	all previous iterations were bad; start over with this residue

Singularity

enum Singularity

strong

Singularity of the system.

Only meaningful if the matrix is square.

Enumerator
Unknown	We don't know yet, or the matrix is not square.
Singular	The matrix is non-invertible.
NonSingular	The matrix is invertible.

Dispatch

enum Dispatch

strong

For integer-based methods that evaluate multiple times the system at different moduli, decides how to dispatch each sub-computations.

Enumerator
Auto	Let implementation decide what to use.
Sequential	All sub-computations are done sequentially.
SMP	Use symmetric multiprocessing (Paladin) to do sub-computations.
Distributed	Use MPI to distribute sub-computations accross nodes.
Combined	Use MPI then Paladin on each node.

SingularSolutionType

enum SingularSolutionType

strong

For Dixon method, which solution type to get when the system is singular.

Enumerator
Deterministic	The solution should be the easiest to compute and always the same.
Random	The solution should be random and different at each call.
Diophantine	The solution is given over the integers.

Preconditioner

enum Preconditioner

strong

Preconditioner to ensure generic rank profile.

Enumerator
None	Do not use any preconditioner.
Butterfly	Use a butterfly network, see Butterfly.
Sparse	Use a sparse preconditioner, c.f. (Mulders 2000).
Toeplitz	Use a Toeplitz preconditioner, c.f. (Kaltofen and Saunders 1991).
Symmetrize	Use At A (Eberly and Kaltofen 1997).
PartialDiagonal	Use A D, where D is a random non-singular diagonal matrix (Eberly and Kaltofen 1997).
PartialDiagonalSymmetrize	Use At D A (Eberly and Kaltofen 1997).
FullDiagonal	Use D1 At D2 A D1 (Eberly and Kaltofen 1997).
Dense	Multiply ( or add?) by a random dense matrix (used by Dixon).

Function Documentation

NullSpaceBasisIn() [1/3]

size_t & NullSpaceBasisIn	(	const Tag::Side	Side,
		BlasMatrix< Field > &	A,
		BlasMatrix< Field > &	Ker,
		size_t &	kerdim
	)

Nullspace of a dense matrix on a finite field.

A is modified.

Parameters

	F	field
	Side	SideTag::Left or SideTag::Right nullspace.
[in,out]	A	Input matrix
[out]	Ker	Nullspace of the matrix (Allocated in the routine)
[out]	kerdim	rank of the kernel

Returns

kerdim

Todo:

make it work for BlasSubmatrix too

NullSpaceBasisIn() [2/3]

size_t & NullSpaceBasisIn	(	const Tag::Side	Side,
		BlasSubmatrix< DenseMat > &	A,
		BlasMatrix< typename DenseMat::Field > &	Ker,
		size_t &	kerdim
	)

Todo:

uses too much memory

Todo:

use copy

Todo:

use copy

NullSpaceBasis()

size_t & NullSpaceBasis	(	const Tag::Side	Side,
		const BlasMatrix< Field > &	A,
		BlasMatrix< Field > &	Ker,
		size_t &	kerdim
	)

Nullspace of a dense matrix on a finite field.

A is preserved.

Parameters

	F	field
	Side	SideTag::Left or SideTag::Right nullspace.
[in]	A	Input matrix
[out]	Ker	Nullspace of the matrix (Allocated in the routine)
[out]	kerdim	rank of the kernel

Returns

kerdim

Todo:

make it work for BlasSubmatrix too

NullSpaceBasisIn() [3/3]

size_t LinBox::NullSpaceBasisIn	(	const Field &	F,
		const Tag::Side	Side,
		const size_t &	m,
		const size_t &	n,
		typename Field::Element *	A,
		const size_t &	lda,
		typename Field::Element *&	Ker,
		size_t &	ldk,
		size_t &	kerdim
	)

Computes the kernel of a dense matrix using LQUP.

Acccording to the dimensions of the input matrix, we chose different methods.

Warning

timings may vary and these choices were made on an experimental basis.

Parameters

F	Field
Side	left or right from LinBox::SideTag
m	rows
n	cols
A	input matrix
lda	leading dimension of A
Ker	Kernel. NULL if kerdim==0
ldk	leading dimension of the kernel.
kerdim	dimension of the kernel.

Returns

dimension of the kernel.

Warning

A is modified.

partial_hegcd()

bool partial_hegcd	(	Ring &	Z,
		typename Ring::Element &	e,
		typename Ring::Element &	b,
		const typename Ring::Element &	n,
		const typename Ring::Element &	d,
		const typename Ring::Element &	denBound
	)

partial_hegcd() sets e, b from the remainder sequence of n,d.

It requires positive n and d. It sets e to the first r_i (remainder) and b to the corresponding q_i (coefficient of n) such that 2r_i < |q_i| and |q_i| <= B (the given denominator bound). True is returned iff such e, b exist.

If not, b is the largest q_i such that |q_i| <= B, and e is the corresponding remainder. In this case b is the denominator of a plausibly approximated but not well approximated rational. It can be used speculatively.

dyadicToRational()

int LinBox::dyadicToRational	(	const Ring &	Z,
		typename Ring::Element &	a,
		typename Ring::Element &	b,
		const typename Ring::Element &	n,
		const typename Ring::Element &	d,
		const typename Ring::Element &	B
	)

Rational reconstruction of a/b from n/d with denominator bound B.

We give a/b, the continued fraction approximant of n/d that satisfies |a/b - n/d| < 1/2d (well approximated) and 0 < b <= B. Return value is 0, if no such approximant exists. Return value is 1, if either (i) a second well approximated rational with denominator bounded by B may exist, or (ii) the well approximated condition is not met for a/b. In these cases, a/b may be used speculatively. Return value is 2, if the approximant is guaranteed (because bB <= d).

If no fraction is well approximated the last b <= B in the remainder sequence of n,d is given.

If d = 2^k and n = sum_i=l to k n_i 2^i, then * n/d = sum_{i=l down to 0} n_i/2^{k-i} is a {dyadic rational}. Numbers of this form are produced for example by numeric-symbolic iterations.

If it is known that n/d is the most accurate approximation with denominator d to a/b, and that the denominator b is bounded by B, i.e. b <= B, then such a/b is uniquely determined, provided d >= bB. ...in that case, such a/b is returned by dyadicToRational(). This follows from two facts: First, by definition, n/d is an accurate approximation to a/b with b <= d when |n/d - a/b| < 1/2d. Otherwise (n-1)/d or (n+1)/d would be a better approximation. Second, if a/b and a'/b' are distinct rationals, then |a/b - a'/b'| >= 1/bb'. Thus if a'/b' is another rational accurately approximated by n/d, we have 1/bb' <= |a/b - a'/b'| <= |a/b - n/d| + |n/d - a'/b'| <= 1/2d + 1/2d = 1/d. So bb' > d >= bB, thus b' > B.

In summary: If it exists, the unique a/b is given such that n/d approximates a/b to within 1/2d and b <= B. Otherwise a plausible a/b is given or failure is signaled.

"Symbolic-Numeric Exact Rational Linear System Solver" by Saunders, Wood, Youse. describes the construction.

checkBlasPrime()

bool LinBox::checkBlasPrime ( const Prime  p )

inline

NO DOC !

Bug:

why is this hard coded ?

create_MatrixQadic()

void create_MatrixQadic	(	const Domain &	D,
		const IMatrix &	Mat,
		double *	chunks,
		size_t	num_chunks,
		const integer	shift
	)

split an integer matrix into a padic chunk representation

create_VectorQadic()

void LinBox::create_VectorQadic	(	const Domain &	D,
		const Vector &	V,
		double *	chunks,
		size_t	num_chunks
	)

split an integer vector into a padic chunk representation

create_VectorQadic_32()

void LinBox::create_VectorQadic_32	(	const Domain &	D,
		const Vector &	V,
		double *	chunks,
		size_t	num_chunks
	)

split an integer vector into a padic chunk representation

setButterfly()

std::vector< bool > setButterfly ( const std::vector< bool > &  x, size_t  j = 0  )

inline

A function used with Butterfly Blackbox Matrices.

This function takes an STL vector x of booleans, and returns a vector y of booleans such that setting the switches marked by true flags in y to be on (or to swap elements) the true elements x will be switched to a given contiguous block through the use of a Butterfly switching network. The integer parameter j marks where this block is to begin. If x has r true elements, the Butterfly switching network will place these elements in a contiguous block starting at j and ending at j + r - 1. Wrap around shall be considered to preserve contiguity. The value of j is defaulted to be zero, and it is only allowed to be non-zero is the size of x is a power of 2.

Returns

vector of booleans for setting switches

Parameters

x	vector of booleans marking elements to switch into contiguous block
j	offset of contiguous block

naturallog()

double LinBox::naturallog ( const Givaro::Integer &  a )

inline

Natural logarithm (ln).

log(2) being close to 0.69314718055994531

Parameters

a

integer.

Returns

ln(a).

isPositive() [1/2]

std::enable_if<!std::is_unsigned<T>::value, bool>::value LinBox::isPositive ( const T &  t )

inline

Positiveness of an integer.

Essentially usefull in debug mode to avoid compiler warnings about comparison always true for some unsigned type.

Parameters

x

integer

Returns

true iff x>=0.

isPositive() [2/2]

std::enable_if<std::is_unsigned<T>::value, bool>::value LinBox::isPositive ( const T &  t )

inline

Positiveness of an integer.

Essentially usefull in debug mode to avoid compiler warnings about comparison always true for some unsigned type.

Parameters

x

integer

Returns

true iff x>=0.

isNegative() [1/2]

std::enable_if<!std::is_unsigned<T>::value, bool>::value LinBox::isNegative ( const T &  t )

inline

Positiveness of an integer.

Essentially usefull in debug mode to avoid compiler warnings about comparison always true for some unsigned type.

Parameters

x

integer

Returns

true iff x>=0.

isNegative() [2/2]

std::enable_if<std::is_unsigned<T>::value, bool>::value LinBox::isNegative ( const T &  t )

inline

Positiveness of an integer.

Essentially usefull in debug mode to avoid compiler warnings about comparison always true for some unsigned type.

Parameters

x

integer

Returns

true iff x>=0.

isOdd()

bool LinBox::isOdd ( const IntType &  value )

inline

Positiveness of an integer.

Essentially usefull in debug mode to avoid compiler warnings about comparison always true for some unsigned type.

Parameters

x

integer

Returns

true iff x>=0.

isEven()

bool LinBox::isEven ( const IntType &  p )

inline

Positiveness of an integer.

Essentially usefull in debug mode to avoid compiler warnings about comparison always true for some unsigned type.

Parameters

x

integer

Returns

true iff x>=0.

operator<<() [1/2]

std::ostream& LinBox::operator<<	(	std::ostream &	os,
		const BlasMatrix< _Field, _Storage > &	Mat
	)

Write a matrix to a stream.

The C++ way using operator<<

Parameters

o	output stream
Mat	matrix to write.

RandomBlasPermutation()

void RandomBlasPermutation

(

BlasPermutation< size_t > &

P

)

Todo:

To be factorized.

operator<<() [2/2]

std::ostream& LinBox::operator<<	(	std::ostream &	o,
		const DenseMat< T > &	Mat
	)

Write a matrix to a stream.

The C++ way using operator<<

Parameters

o	output stream
Mat	matrix to write.

prepare()

Vector& LinBox::prepare	(	const Field &	F,
		Vector &	y,
		const typename Field::Element &	a
	)

y <- ay.

Todo:

Vector knows Field

create_prime_sequence()

PrimeSequence<typename Container::const_iterator> LinBox::create_prime_sequence

(

const Container &

cont

)

convenience factory to create a PrimeSequence from an STL-like container.

charpoly() [1/3]

Polynomial& LinBox::charpoly	(	Polynomial &	P,
		const Blackbox &	A,
		const MyMethod &	M
	)

...using an optional Method parameter

Parameters

P	- the output characteristic polynomial. If the polynomial is of degree d, this random access container has size d+1, the 0-th entry is the constant coefficient and the d-th is 1 since the charpoly is monic.
A	- a blackbox matrix Optional
M	- the method object. Generally, the default object suffices and the algorithm used is determined by the class of M. Basic methods are Method::Blackbox, Method::Elimination, and Method::Auto (the default). See methods.h for more options.

Returns

a reference to P.

charpoly() [2/3]

Polynomial& LinBox::charpoly	(	Polynomial &	P,
		const Blackbox &	A,
		const RingCategories::ModularTag &	tag,
		const Method::DenseElimination &	M
	)

Compute the characteristic polynomial over .

Compute the characteristic polynomial of a matrix using dense elimination methods

Parameters

P	Polynomial where to store the result
A	Blackbox representing the matrix
tag
M

charpoly() [3/3]

Polynomial& LinBox::charpoly	(	Polynomial &	P,
		const Blackbox &	A,
		const RingCategories::ModularTag &	tag,
		const Method::Blackbox &	M
	)

Compute the characteristic polynomial over .

Compute the characteristic polynomial of a matrix, represented via a blackBox.

Parameters

P	Polynomial where to store the result
A	Blackbox representing the matrix
tag
M

rowEchelon() [1/2]

size_t LinBox::rowEchelon ( Matrix &  E, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the row echelon form of a matrix, not reduced.

Returns the row echelon form. The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

[out]	E	the row echelon form
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

rowEchelon() [2/2]

size_t LinBox::rowEchelon ( Matrix &  E, Matrix &  T, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the row echelon form of a matrix, not reduced, and the related transformation matrix.

Returns the row echelon form E and a transformation matrix T such that T . A = E. The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

[out]	E	the row echelon form
[out]	T	the transformation matrix
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

rowEchelonize() [1/2]

size_t LinBox::rowEchelonize ( Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Replace the input matrix by its row echelon form, not reduced.

The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

	[in/out]	A matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

rowEchelonize() [2/2]

size_t LinBox::rowEchelonize ( Matrix &  A, Matrix &  T, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the row echelon form of a matrix, not reduced, and the related transformation matrix.

Returns the row echelon form E and a transformation matrix T such that T . A = E. The input matrix is replaced by the row echelon form. The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

	[in/out]	A matrix
[out]	T	the transformation matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedRowEchelon() [1/2]

size_t LinBox::reducedRowEchelon ( Matrix &  E, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the reduced row echelon form of a matrix.

Returns the reduced row echelon form. The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

[out]	E	the reduced row echelon form
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedRowEchelon() [2/2]

size_t LinBox::reducedRowEchelon ( Matrix &  E, Matrix &  T, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the reduced row echelon form of a matrix, and the related transformation matrix.

Returns the reduced row echelon form E and a transformation matrix T such that T . A = E. The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

[out]	E	the reduced row echelon form
[out]	T	the transformation matrix
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedRowEchelonize() [1/2]

size_t LinBox::reducedRowEchelonize ( Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Replace the input matrix by its reduced row echelon form.

The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

	[in/out]	A matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedRowEchelonize() [2/2]

size_t LinBox::reducedRowEchelonize ( Matrix &  A, Matrix &  T, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the reduced row echelon form of a matrix, and the related transformation matrix.

Returns the reduced row echelon form E and a transformation matrix T such that T . A = E. The input matrix is replaced by the reduced row echelon form. The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

	[in/out]	A matrix
[out]	T	the transformation matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

colEchelon() [1/2]

size_t LinBox::colEchelon ( Matrix &  E, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the column echelon form of a matrix, not reduced.

Returns the column echelon form. The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

[out]	E	the column echelon form
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

colEchelon() [2/2]

size_t LinBox::colEchelon ( Matrix &  E, Matrix &  T, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the column echelon form of a matrix, not reduced, and the related transformation matrix.

Returns the column echelon form E and a transformation matrix T such that T . A = E. The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

[out]	E	the column echelon form
[out]	T	the transformation matrix
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

colEchelonize() [1/2]

size_t LinBox::colEchelonize ( Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Replace the input matrix by its column echelon form, not reduced.

The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

	[in/out]	A matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

colEchelonize() [2/2]

size_t LinBox::colEchelonize ( Matrix &  A, Matrix &  T, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the column echelon form of a matrix, not reduced, and the related transformation matrix.

Returns the column echelon form E and a transformation matrix T such that T . A = E. The input matrix is replaced by the column echelon form. The pivots are nonzero but not necessarily ones. The form is not reduced, which means that coefficients above each pivot are not necessarily zeros.

Parameters

	[in/out]	A matrix
[out]	T	the transformation matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedColEchelon() [1/2]

size_t LinBox::reducedColEchelon ( Matrix &  E, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the reduced column echelon form of a matrix.

Returns the reduced column echelon form. The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

[out]	E	the reduced column echelon form
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedColEchelon() [2/2]

size_t LinBox::reducedColEchelon ( Matrix &  E, Matrix &  T, const Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the reduced column echelon form of a matrix, and the related transformation matrix.

Returns the reduced column echelon form E and a transformation matrix T such that T . A = E. The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

[out]	E	the reduced column echelon form
[out]	T	the transformation matrix
[in]	A	matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedColEchelonize() [1/2]

size_t LinBox::reducedColEchelonize ( Matrix &  A, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Replace the input matrix by its reduced column echelon form.

The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

	[in/out]	A matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

reducedColEchelonize() [2/2]

size_t LinBox::reducedColEchelonize ( Matrix &  A, Matrix &  T, const CategoryTag &  tag, const EchelonMethod &  m  )

inline

Compute the reduced column echelon form of a matrix, and the related transformation matrix.

Returns the reduced column echelon form E and a transformation matrix T such that T . A = E. The input matrix is replaced by the reduced column echelon form. The pivots are ones. The form is reduced, which means that coefficients above each pivot are zeros.

Parameters

	[in/out]	A matrix
[out]	T	the transformation matrix
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

the rank of the matrix

HadamardRowLogBound()

void HadamardRowLogBound	(	double &	logBound,
		double &	minLogNorm,
		const IMatrix &	A
	)

Precise Hadamard bound (bound on determinant) by taking the row-wise euclidean norm.

The result is expressed as bit size.

HadamardColLogBound()

void HadamardColLogBound	(	double &	logBound,
		double &	minLogNorm,
		const IMatrix &	A
	)

Precise Hadamard bound (bound on determinant) by taking the column-wise euclidean norm.

The result is expressed as bit size.

DetailedHadamardBound()

HadamardLogBoundDetails DetailedHadamardBound

(

const IMatrix &

A

)

Precise Hadamard bound (bound on determinant) by taking the minimum of the column-wise and the row-wise euclidean norm.

The results are expressed as bit size.

HadamardBound()

double HadamardBound

(

const IMatrix &

A

)

Precise Hadamard bound (bound on determinant) by taking the minimum of the column-wise and the row-wise euclidean norm.

The result is expressed as bit size.

InfinityNorm()

Integer & InfinityNorm ( Integer &  max, const IMatrix &  A, const MatrixCategories::RowColMatrixTag &  tag  )

inline

Returns the bit size of the Hadamard bound.

This is a larger estimation but faster to compute.

FastCharPolyDumasPernetWanBound()

double FastCharPolyDumasPernetWanBound ( const IMatrix &  A, const Integer &  infnorm  )

inline

Bound on the coefficients of the characteristic polynomial.

Bibliography:

"Efficient Computation of the Characteristic Polynomial".

Dumas Pernet Wan ISSAC'05.

Bibliography:

"Efficient Computation of the Characteristic Polynomial".

Dumas Pernet Wan ISSAC'05.

FastCharPolyGoldsteinGrahamBound()

double FastCharPolyGoldsteinGrahamBound ( const IMatrix &  A, const Integer &  infnorm  )

inline

A.J.

Goldstein et R.L. Graham. A Hadamard-type bound on the coefficients of a determinant of polynomials. SIAM Review, volume 15, 1973, pages 657?658.

Goldstein et R.L. Graham. A Hadamard-type bound on the coefficients of a determinant of polynomials. SIAM Review, volume 15, 1973, pages 657-658.

RationalSolveHadamardBound()

std::enable_if< std::is_same< typename FieldTraits< typename Matrix::Field >::categoryTag, RingCategories::IntegerTag >::value, RationalSolveHadamardBoundData >::type RationalSolveHadamardBound	(	const Matrix &	A,
		const Vector &	b
	)

Bound on the rational solution of a linear system Ax = b.

Return bounds on the bit sizes of both denominator and numerator of the solution x.

Note

Matrix and Vector should be over Integer.

FastHadamardBound()

double LinBox::FastHadamardBound ( const IMatrix &  A, const Integer &  infnorm  )

inline

Returns the bit size of the Hadamard bound.

This is a larger estimation but faster to compute.

isPositiveDefinite() [1/3]

bool LinBox::isPositiveDefinite	(	const Blackbox &	A,
		const MyMethod &	M
	)

Compute the isPositiveDefinite of A.

The isPositiveDefinite of a linear operator A, represented as a black box, is computed over the ring or field of A.

Parameters

A	Black box of which to compute the isPositiveDefinite
M	may be a Method::Auto (default), Method::Blackbox, Method::Elimination, or of other method type.

isPositiveDefinite() [2/3]

bool LinBox::isPositiveDefinite	(	const Blackbox &	A,
		const RingCategories::IntegerTag &	tag,
		const Method::Auto &	M
	)

Bug:

should try a modular minpoly and decide on the degree of that...

Bug:

this crude size check can be refined

isPositiveDefinite() [3/3]

bool LinBox::isPositiveDefinite	(	const Blackbox &	A,
		const RingCategories::IntegerTag &	tag,
		const Method::DenseElimination &	M
	)

Bug:

why map (same field)? This is a copy.

isPositiveSemiDefinite() [1/2]

bool LinBox::isPositiveSemiDefinite	(	const Blackbox &	A,
		const MyMethod &	M
	)

Determine if A is positive semidefinite.

The positive semidefiniteness of a linear operator A, represented as a black box, is computed over the ring or field (characteristic 0) of A.

Parameters

A	Black box of which to compute the isPositiveSemiDefinite
M	may be a Method::Auto (SemiDefault), Method::Blackbox, Method::Elimination, or of other method type.

isPositiveSemiDefinite() [2/2]

bool LinBox::isPositiveSemiDefinite	(	const Blackbox &	A,
		const RingCategories::IntegerTag &	tag,
		const Method::DenseElimination &	M
	)

Warning

this is a copy

rankInPlace()

size_t& LinBox::rankInPlace ( size_t &  r, Blackbox &  A  )

inline

Rank of A.

A may be modified

Parameters

A	matrix
r	rank

Bug:

there is no Elimination() method there.

rank() [1/2]

size_t& LinBox::rank ( size_t &  r, const Blackbox &  A, const RingCategories::ModularTag &  tag, const Method::Auto &  m  )

inline

Bug:

choose (benchmark) better cuttoff (size, nbnz, sparse rep)

rank() [2/2]

size_t& LinBox::rank ( size_t &  res, const Blackbox &  A, const RingCategories::ModularTag &  tag, const Method::Wiedemann &  M  )

inline

M may be Method::Wiedemann().

Bug:

This is too much for solutions. It belongs in algorithms

solve() [1/5]

void LinBox::solve ( IntVector &  xNum, typename IntVector::Element &  xDen, const Matrix &  A, const Vector &  b, const RingCategories::IntegerTag &  tag, const Method::CRA< IterationMethod > &  m  )

inline

Solve specialization with Chinese Remainder Algorithm method for an Integer or Rational tags.

If a Dispatch::Distributed is used, please note that the result will only be set on the master node.

solve() [2/5]

ResultVector& LinBox::solve ( ResultVector &  x, const Matrix &  A, const Vector &  b, const CategoryTag &  tag, const SolveMethod &  m  )

inline

Solve Ax = b, for x.

Returns a vector x such that Ax = b.

Specifically:

• A non singular: the unique solution is returned.
• A singular:
  • Consistent system: a random solution is returned. The method parameter can contain an hint that an arbitrary element of the solution space is acceptable instead, which can be faster to compute if one doesn't expect a result in that case.
  • Inconsistent system: LinboxMathInconsistentSystem is thrown.
  • Internal failure: LinboxError is thrown.

CategoryTag is defaulted to FieldTraits<Matrix::Field>::categoryTag() when omitted.

• Method::Auto
  • DenseMatrix > Method::DenseElimination
  • SparseMatrix > Method::SparseElimination
  • Otherwise | - Row or column dimension < LINBOX_USE_BLACKBOX_THRESHOLD > Method::Elimination | - Otherwise > Method::Blackbox
• Method::Elimination
  • DenseMatrix > Method::DenseElimination
  • SparseMatrix > Method::SparseElimination
  • Otherwise > Method::DenseElimination or Method::SparseElimination given matrix sparsity
• Method::DenseElimination
  • DenseMatrix | - ModularTag > LQUPMatrix<Field>::left_solve | - IntegerTag | | - RatVector > Method::Dixon | | - Otherwise > Error | - Otherwise > Error (Not implemented)
  • Otherwise > Method::DenseElimination but copy to a DenseMatrix first
• Method::SparseElimination
  • SparseMatrix | - IntegerTag > Method::Dixon | - Otherwise > GaussDomain<Field>solveInPlace
  • Otherwise > Method::SparseElimination but copy to SparseMatrix first
• Method::CRA
  • IntegerTag | - Dispatch::Distributed > ChineseRemainderDistributed | - Otherwise > RationalChineseRemainder
  • Otherwise > Error
• Method::Dixon
  • IntegerTag | - DenseMatrix > RationalSolver<..., Method::Dixon> | - SparseMatrix > RationalSolver<..., Method::SparseElimination> | - Otherwise > Error
  • Otherwise > Error
• Method::Blackbox > Method::Wiedemann
• Method::Wiedemann
  • ModularTag > WiedemannSolver
  • IntegerTag > Method::Dixon
  • Otherwise > Error
• Method::BlockWiedemann [

  Deprecated:

  , not tested]

  • ModularTag > BlockWiedemannSolver
  • Otherwise > Error

Parameters

[out]	x	solution, can be a rational solution (vector of numerators and one denominator)
[in]	A	matrix
[in]	b	target
[in]	tag	domain of computation
[in]	m	method to use (

See also

solutions/method.h)

Returns

reference to x

solve() [3/5]

std::enable_if<std::is_same<typename SolveMethod::CategoryTag, RingCategories::IntegerTag>::value && std::is_same<typename FieldTraits<typename RatVector::Field>::categoryTag, RingCategories::RationalTag>::value, RatVector&>::type LinBox::solve	(	RatVector &	x,
		const Matrix &	A,
		const Vector &	b,
		const RingCategories::IntegerTag &	tag,
		const SolveMethod &	m
	)

Solve specialisation on IntegerTag with Vector<QField> as result.

This forward to the rational interface (num, den). But will only work if the ResultVector if a vector of some Rational type.

solve() [4/5]

std::enable_if<std::is_same<typename SolveMethod::CategoryTag, RingCategories::IntegerTag>::value, VectorFraction<typename Matrix::Field>&>::type LinBox::solve	(	VectorFraction< typename Matrix::Field > &	x,
		const Matrix &	A,
		const Vector &	b,
		const RingCategories::IntegerTag &	tag,
		const SolveMethod &	m
	)

Solve specialisation on IntegerTag with VectorFraction as result.

This forward to the rational interface (num, den).

solve() [5/5]

void LinBox::solve ( IntVector &  xNum, typename IntVector::Element &  xDen, const Matrix &  A, const Vector &  b, const CategoryTag &  tag, const SolveMethod &  m  )

inline

Rational solve Ax = b, for x expressed as xNum/xDen.

Second interface for solving, only valid for RingCategories::IntegerTag.

Solve with this interface will usually go for CRA or Dixon lifting, as non-modular elimination would snowball elements to very big values.

solveInPlace() [1/2]

ResultVector& LinBox::solveInPlace ( ResultVector &  x, Matrix &  A, const Vector &  b, const CategoryTag &  tag, const SolveMethod &  m  )

inline

Solve Ax = b, for x.

Returns a vector x such that Ax = b. A can be modified.

See documentation for solve.

solveInPlace() [2/2]

void LinBox::solveInPlace ( IntVector &  xNum, typename IntVector::Element &  xDen, Matrix &  A, const Vector &  b, const CategoryTag &  tag, const SolveMethod &  m  )

inline

Rational solve in place Ax = b, for x expressed as xNum/xDen.

The matrix A might be modified.

Second interface for solving in place, only valid for RingCategories::IntegerTag.

trace()

BB::Field::Element & trace	(	typename BB::Field::Element &	t,
		const BB &	A
	)

Sum of the eigenvalues.

Also it is the sum of the diagonal entries.

Runtime on n by n matrix is n times the cost of getEntry(). This is linear in n for those classes where getEntry is constant time (eg DenseMatrix and SparseMatrix). Trace is constant time when the diagonal is necessarily constant, eg. for ScalarMatrix and Toeplitz. Worst case time is cost of n blackbox applies (matrix vector products), and apply cost typically ranges between O(n) and O(n^2).

serialize() [1/4]

uint64_t serialize ( std::vector< uint8_t > &  bytes, const Integer &  integer  )

inline

Serializes an Integer with its underlying __mpz_struct.

Returns the number of bytes written.

Format is (by bytes count): 0-3 _mp_size Number of mp_limb_t in _mp_d, can be negative to indicate negative number. 4-.. _mp_d The limbs of length (abs(_mp_size) * sizeof(mp_limb_t))

Note

As said, we're not sure of how many bytes we will write here, because mp_limb_t can be either 32 or 64 bytes-long depending on configuration. We force it to be 64 bit-long so that it becomes platform-independent.

As said, we're not sure of how many bytes we will write here, because mp_limb_t can be either 32 or 64 bytes-long depending on configuration. We force it to be 64 bit-long so that it becomes platform-independent.

serialize() [2/4]

uint64_t serialize ( std::vector< uint8_t > &  bytes, const BlasMatrix< Field > &  M  )

inline

Serializes a BlasMatrix.

Format is (by bytes count): 0-7 n Row dimension of matrix 8-15 m Column dimension of matrix 16-.. Entries of the matrix, (n * m) row-majored

unserialize() [1/3]

uint64_t unserialize ( BlasMatrix< Field > &  M, const std::vector< uint8_t > &  bytes, uint64_t  offset = 0u  )

inline

Unserializes a BlasMatrix.

The matrix will be resized if necessary.

serialize() [3/4]

uint64_t serialize ( std::vector< uint8_t > &  bytes, const SparseMatrix< Field > &  M  )

inline

Serializes a SparseMatrix.

Format is (by bytes count): 0-7 n Row dimension of matrix 8-15 m Column dimension of matrix 23-.. Entries of the matrix, only non-zero, stored as: 0-7 i Row index 8-15 j Column index 16-.. Entry value (8 bytes) End of sparse entries, with a value of 0xFFFFFFFF'FFFFFFFF

unserialize() [2/3]

uint64_t unserialize ( SparseMatrix< Field > &  M, const std::vector< uint8_t > &  bytes, uint64_t  offset = 0u  )

inline

Unserializes a SparseMatrix.

The matrix will be resized if necessary.

serialize() [4/4]

uint64_t serialize ( std::vector< uint8_t > &  bytes, const BlasVector< Field > &  V  )

inline

Serializes a BlasVector.

Format is (by bytes count): 0-7 l Length of the vector 16-.. Entries of the vector

unserialize() [3/3]

uint64_t unserialize ( BlasVector< Field > &  V, const std::vector< uint8_t > &  bytes, uint64_t  offset = 0u  )

inline

Unserializes a BlasVector.

The vector will be resized if necessary.

randomVector()

Vector LinBox::randomVector ( Field &  F, size_t  n, typename Field::RandIter &  r  )

inline

Random vector generator This templated function takes a field and a random field element generator and returns a vector of random field elements.

The vector is dense in the field elements, even if the vector is a sparse LinBox vector. The funtion is templatized by the field and the vector types being used. This function calls another function by the same name with an additional parameter of the vector category of the vector it is called with. This mechanism is used because functions cannot have partial template specializations like classes can. This new, extended function can be specialized for specific fields and vectors to allow for better performance.

Returns

v vector of random field elements

Parameters

F	Field in which arithmetic is done
n	integer number of elements in vector
r	Random field element generator

randomAns()

Matrix& LinBox::randomAns	(	const Ring &	R,
		Matrix &	Mat,
		size_t	n,
		size_t	epr
	)

Bug:

use BlasVector.

RandIntInInt()

size_t& LinBox::RandIntInInt	(	const size_t &	s,
		size_t &	RIII,
		const int &	seed = 0
	)

gives a random number such that .

basic..

Parameters

[in]	s	sup
[in]	seed	seed. If 0 (default) we create a new one.
[out]	RIII	random integer in the interval .

Returns

a reference to RIII

CheckRank()

bool LinBox::CheckRank	(	const Field &	F,
		const typename Field ::Element *	A,
		const size_t &	m,
		const size_t &	n,
		const size_t &	lda,
		const size_t &	alledged_rank
	)

Checks we got the right rank.

Parameters

F	field
A	matrix
m	rows
n	cols
lda	leadin dimmension
alledged_rank	supposedly correct rank.

Returns

alledged_rank==rank(A)

Parameters

alledged_rank

Bug:

not used

RandomMatrixWithDet()

void LinBox::RandomMatrixWithDet	(	const Field &	F,
		typename Field ::Element *	A,
		const size_t &	m,
		const size_t &	lda,
		const typename Field ::Element &	det
	)

Builds a m x m random matrix of determinant det over field F.

Parameters

det

Bug:

not used

showAdvanceLinear()

void showAdvanceLinear	(	size_t	curr,
		size_t	min,
		size_t	max
	)

show the advancement (on the terminal) suppose linear advancement

Parameters

curr	current iteration
min	starting iteration
max	terminal iteration

showFinish()

void showFinish	(	size_t	curr,
		size_t	all
	)

tells the current series of measure has completed (on the terminal)

Parameters

curr	current iteration
all	number of iterations

showSkip()

void showSkip	(	size_t	curr,
		size_t	all
	)

tells the current series of measure was skipped (on the terminal)

Parameters

curr	current iteration
all	number of iterations

computeMFLOPS() [1/2]

double computeMFLOPS	(	const double &	tim,
		const double	mflo,
		const size_t	rpt = 1
	)

computes the number of megaflops.

Parameters

tim	timer (seconds)
mflo	number of operations (1e6 operations)
rpt	number of experiences

Returns

mflo/(tim*rpt)

computeMFLOPS() [2/2]

double computeMFLOPS	(	const dvector_t &	tim,
		const double	mflo,
		Tag::TimeSelect	ts = Tag::TimeSelect::bestThree
	)

computes the number of megaflops.

Parameters

tim	timer (seconds)
mflo	number of operations (1e6 operations)
ts	number of experiences to select.

See also

TimeSelect. Default to the best three

Returns

mflo/(tim*rpt)

isDigit()

bool isDigit

(

const std::string &

s

)

Check if a string is actually a double.

Parameters

s	string to check

Returns

true/false

fortifiedString()

bool fortifiedString

(

const std::string &

s

)

Tells is a string has double quotes around.

Parameters

s	string to test

Returns

true if s[0] == s[last] == '"'

unfortifyString()

std::string unfortifyString

(

const std::string &

s

)

removes the surrounding quotes.

Parameters

s	removes quotes around if necessary

Returns

s without surrounding double quotes if necessary

fortifyString()

std::string fortifyString

(

const std::string &

s

)

adds surrounding quotes.

Parameters

s	add quotes around if necessary

Returns

s with surrounding double quotes if necessary

getDateTime()

std::string getDateTime

(

const std::string &

sep = " "

)

get ISO time and date

get ISO time and date year-time YYYY-MM-DD 'sep' HH:MM:SS 'sep' (GMT)

Parameters

sep	separation between

toString()

std::string LinBox::toString

(

T &

nam

)

Converts anything to a string.

Parameters

nam	to be put in a string.