Sparse vectors. More...

#include <svectorbase.h>

Public Types
typedef Nonzero< R >	Element

Public Member Functions
SVectorBase< Real > &	operator= (const VectorBase< S > &vec)
	Assignment operator (specialization for Real). More...

Real	operator* (const SVectorBase< S > &w) const
	specialization for inner product for sparse vectors More...

Access
int	size () const
	Number of used indices. More...

int	max () const
	Maximal number of indices. More...

int	dim () const
	Dimension of the vector defined as maximal index + 1. More...

int	pos (int i) const
	Position of index `i`. More...

R	operator[] (int i) const
	Value to index `i`. More...

Nonzero< R > &	element (int n)
	Reference to the `n` 'th nonzero element. More...

const Nonzero< R > &	element (int n) const
	The `n` 'th nonzero element. More...

int &	index (int n)
	Reference to index of `n` 'th nonzero. More...

int	index (int n) const
	Index of `n` 'th nonzero. More...

R &	value (int n)
	Reference to value of `n` 'th nonzero. More...

const R &	value (int n) const
	Value of `n` 'th nonzero. More...

void	add (int i, const R &v)
	Append one nonzero (i,v). More...

void	add (int i)
	Append one uninitialized nonzero. More...

void	add (const SVectorBase &sv)
	Append nonzeros of `sv`. More...

void	add (int n, const int i[], const R v[])
	Append `n` nonzeros. More...

template<class S >
void	add (int n, const int i[], const S v[])
	Append `n` nonzeros. More...

void	add (int n, const Nonzero< R > e[])
	Append `n` nonzeros. More...

void	remove (int n, int m)
	Remove nonzeros `n` thru `m`. More...

void	remove (int n)
	Remove `n` 'th nonzero. More...

void	clear ()
	Remove all indices. More...

void	sort ()
	Sort nonzeros to increasing indices. More...

Arithmetic operations
R	maxAbs () const
	Maximum absolute value, i.e., infinity norm. More...

R	minAbs () const
	Minimum absolute value. More...

R	length () const
	Floating point approximation of euclidian norm (without any approximation guarantee). More...

R	length2 () const
	Squared norm. More...

SVectorBase< R > &	operator*= (const R &x)
	Scaling. More...

R	operator* (const VectorBase< R > &w) const
	Inner product. More...

template<class S >
R	operator* (const SVectorBase< S > &w) const
	inner product for sparse vectors More...

Constructions, destruction, and assignment
	SVectorBase (int n=0, Nonzero< R > *p_mem=nullptr)
	Default constructor. More...

	SVectorBase (const SVectorBase< R > &sv)=default

template<class S >
SVectorBase< R > &	operator= (const VectorBase< S > &vec)
	Assignment operator. More...

SVectorBase< R > &	operator= (const SVectorBase< R > &sv)
	Assignment operator. More...

SVectorBase< R > &	operator= (const SVectorBase< R > &&sv)
	move assignement operator. More...

template<class S >
SVectorBase< R > &	operator= (const SVectorBase< S > &sv)
	Assignment operator. More...

SVectorBase< Real > &	scaleAssign (int scaleExp, const SVectorBase< Real > &sv)
	scale and assign More...

SVectorBase< Real > &	scaleAssign (const int *scaleExp, const SVectorBase< Real > &sv, bool negateExp=false)
	scale and assign More...

template<class S >
SVectorBase< R > &	assignArray (const S rowValues, const int rowIndices, int rowSize)
	Assignment operator. More...

template<class S >
SVectorBase< R > &	operator= (const SSVectorBase< S > &sv)
	Assignment operator. More...

Memory
Nonzero< R > *	mem () const
	get pointer to internal memory. More...

void	set_size (int s)
	Set size of the vector. More...

void	set_max (int m)
	Set the maximum number of nonzeros in the vector. More...

void	setMem (int n, Nonzero< R > *elmem)
	Set the memory area where the nonzeros will be stored. More...

Utilities
bool	isConsistent () const
	Consistency check. More...

Private Attributes
Data
Nonzero< R > *	m_elem

int	memsize

int	memused

Friends
template<class S >
class	SVectorBase

Detailed Description

template<class R>
class soplex::SVectorBase< R >

Sparse vectors.

Class SVectorBase provides packed sparse vectors. Such are a sparse vectors, with a storage scheme that keeps all data in one contiguous block of memory. This is best suited for using them for parallel computing on a distributed memory multiprocessor.

SVectorBase does not provide any memory management (this will be done by class DSVectorBase). This means, that the constructor of SVectorBase expects memory where to save the nonzeros. Further, adding nonzeros to an SVectorBase may fail if no more memory is available for saving them (see also DSVectorBase).

When nonzeros are added to an SVectorBase, they are appended to the set of nonzeros, i.e., they recieve numbers size(), size()+1 ... . An SVectorBase can hold atmost max() nonzeros, where max() is given in the constructor. When removing nonzeros, the remaining nonzeros are renumbered. However, only the numbers greater than the number of the first removed nonzero are affected.

The following mathematical operations are provided by class SVectorBase (SVectorBase a, b, c; R x):

Operation	Description
`-=`	subtraction	`a` `-=` `b`
`+=`	addition	`a` `+=` `b`
`*`	skalar product	`x` = `a` `*` `b`
`*=`	scaling	`a` `*=` `x`
maxAbs()	infinity norm	`a.maxAbs()` == \(\\|a\\|_{\infty}\)
length()	eucledian norm	`a.length()` == \(\sqrt{a^2}\)
length2()	square norm	`a.length2()` == \(a^2\)

Operators += and -= should be used with caution, since no efficient implementation is available. One should think of assigning the left handside vector to a dense VectorBase first and perform the addition on it. The same applies to the scalar product *.

There are two numberings of the nonzeros of an SVectorBase. First, an SVectorBase is supposed to act like a linear algebra VectorBase. An index refers to this view of an SVectorBase: operator[]() is provided which returns the value at the given index of the vector, i.e., 0 for all indices which are not in the set of nonzeros. The other view of SVectorBases is that of a set of nonzeros. The nonzeros are numbered from 0 to size()-1. The methods index(int n) and value(int n) allow to access the index and value of the n 'th nonzero. n is referred to as the number of a nonzero.

Todo:: SVectorBase should get a new implementation. There maybe a lot of memory lost due to padding the Nonzero structure. A better idea seems to be class SVectorBase { int size; int used; int* idx; R* val; }; which for several reasons could be faster or slower. If SVectorBase is changed, also DSVectorBase and SVSet have to be modified.

Definition at line 139 of file svectorbase.h.

Member Typedef Documentation

◆ Element

typedef Nonzero<R> Element

Definition at line 157 of file svectorbase.h.

Constructor & Destructor Documentation

◆ SVectorBase() [1/2]

SVectorBase	(	int	n = `0`,
		Nonzero< R > *	p_mem = `nullptr`
	)

explicit

Default constructor.

The constructor expects one memory block where to store the nonzero elements. This must be passed to the constructor, where the number of Nonzeros needs that fit into the memory must be given and a pointer to the beginning of the memory block. Once this memory has been passed, it shall not be modified until the SVectorBase is no longer used.

Definition at line 625 of file svectorbase.h.

References SVectorBase< R >::setMem().

◆ SVectorBase() [2/2]

SVectorBase ( const SVectorBase< R > & sv )

default

Member Function Documentation

◆ add() [1/6]

void add ( const SVectorBase< R > & sv )

Append nonzeros of sv.

Definition at line 314 of file svectorbase.h.

References SVectorBase< R >::add(), SVectorBase< R >::m_elem, and SVectorBase< R >::size().

◆ add() [2/6]

void add ( int i )

Append one uninitialized nonzero.

Definition at line 300 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::set_size(), and SVectorBase< R >::size().

◆ add() [3/6]

void add	(	int	i,
		const R &	v
	)

Append one nonzero (i,v).

Definition at line 282 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::set_size(), and SVectorBase< R >::size().

Referenced by SVectorBase< R >::add(), DSVectorBase< R >::add(), SVSetBase< R >::add2(), and UnitVectorBase< R >::UnitVectorBase().

◆ add() [4/6]

void add	(	int	n,
		const int	i[],
		const R	v[]
	)

Append n nonzeros.

Definition at line 320 of file svectorbase.h.

References Nonzero< R >::idx, SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::set_size(), SVectorBase< R >::size(), and Nonzero< R >::val.

◆ add() [5/6]

void add	(	int	n,
		const int	i[],
		const S	v[]
	)

Append n nonzeros.

Definition at line 351 of file svectorbase.h.

References Nonzero< R >::idx, SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::set_size(), SVectorBase< R >::size(), and Nonzero< R >::val.

◆ add() [6/6]

void add	(	int	n,
		const Nonzero< R >	e[]
	)

Append n nonzeros.

Definition at line 380 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::set_size(), SVectorBase< R >::size(), and Nonzero< R >::val.

◆ assignArray()

SVectorBase< R > & assignArray	(	const S *	rowValues,
		const int *	rowIndices,
		int	rowSize
	)

Assignment operator.

Definition at line 765 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::max(), and SVectorBase< R >::set_size().

Referenced by SVSetBase< R >::add().

◆ clear()

void clear ( )

Remove all indices.

Definition at line 443 of file svectorbase.h.

References SVectorBase< R >::set_size().

Referenced by DSVectorBase< R >::add(), SPxLPBase< R >::changeCol(), SPxLPBase< R >::changeRow(), and DSVectorBase< R >::operator=().

◆ dim()

int dim ( ) const

Dimension of the vector defined as maximal index + 1.

Definition at line 178 of file svectorbase.h.

References Nonzero< R >::idx, SVectorBase< R >::m_elem, and SVectorBase< R >::size().

Referenced by SSVectorBase< R >::assign(), and VectorBase< R >::operator*().

◆ element() [1/2]

Nonzero< R > & element ( int n )

Reference to the n 'th nonzero element.

Definition at line 228 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::max().

Referenced by SSVectorBase< Rational >::assign(), SSVectorBase< R >::assign2product1(), SSVectorBase< R >::assign2productAndSetup(), SSVectorBase< R >::assign2productShort(), and SVectorBase< R >::operator*().

◆ element() [2/2]

const Nonzero< R > & element ( int n ) const

The n 'th nonzero element.

Definition at line 237 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::size().

◆ index() [1/2]

int & index ( int n )

Reference to index of n 'th nonzero.

Definition at line 246 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::size().

◆ index() [2/2]

int index ( int n ) const

Index of n 'th nonzero.

Definition at line 255 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::size().

◆ isConsistent()

bool isConsistent ( ) const

Consistency check.

Definition at line 830 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::size(), and SPX_MSG_INCONSISTENT.

Referenced by SVSetBase< R >::isConsistent(), UnitVectorBase< R >::isConsistent(), and SVectorBase< R >::scaleAssign().

◆ length()

R length ( ) const

Floating point approximation of euclidian norm (without any approximation guarantee).

Definition at line 519 of file svectorbase.h.

References SVectorBase< R >::length2().

◆ length2()

R length2 ( ) const

Squared norm.

Definition at line 525 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::size(), and Nonzero< R >::val.

Referenced by SVectorBase< R >::length().

◆ max()

int max ( ) const

Maximal number of indices.

Definition at line 171 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::memsize, and SVectorBase< R >::memused.

Referenced by SVectorBase< R >::add(), SVectorBase< R >::assignArray(), SVSetBase< R >::deleteVec(), SVectorBase< R >::element(), SVSetBase< R >::ensureMem(), SVectorBase< R >::isConsistent(), SVSetBase< R >::isConsistent(), UnitVectorBase< R >::isConsistent(), SVectorBase< R >::operator=(), SVSetBase< R >::operator=(), SVectorBase< R >::scaleAssign(), and SVSetBase< R >::xtend().

◆ maxAbs()

R maxAbs ( ) const

Maximum absolute value, i.e., infinity norm.

Definition at line 487 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::size(), and soplex::spxAbs().

◆ mem()

Nonzero< R > * mem ( ) const

get pointer to internal memory.

Definition at line 793 of file svectorbase.h.

References SVectorBase< R >::m_elem.

Referenced by SVSetBase< R >::deleteVec(), SVSetBase< R >::isConsistent(), UnitVectorBase< R >::isConsistent(), SVSetBase< R >::memPack(), SVSetBase< R >::operator=(), and SVSetBase< R >::xtend().

◆ minAbs()

R minAbs ( ) const

Minimum absolute value.

Definition at line 503 of file svectorbase.h.

References soplex::infinity, SVectorBase< R >::m_elem, SVectorBase< R >::size(), and soplex::spxAbs().

◆ operator*() [1/3]

R operator* ( const SVectorBase< S > & w ) const

inner product for sparse vectors

Definition at line 562 of file svectorbase.h.

References SVectorBase< R >::element(), Nonzero< R >::idx, SVectorBase< R >::m_elem, SVectorBase< R >::size(), and Nonzero< R >::val.

◆ operator*() [2/3]

Real operator* ( const SVectorBase< S > & w ) const

specialization for inner product for sparse vectors

Definition at line 866 of file svectorbase.h.

References SVectorBase< R >::element(), Nonzero< R >::idx, SVectorBase< R >::size(), and Nonzero< R >::val.

◆ operator*() [3/3]

R operator* ( const VectorBase< R > & w ) const

Inner product.

Definition at line 1046 of file basevectors.h.

References Nonzero< R >::idx, and Nonzero< R >::val.

◆ operator*=()

SVectorBase< R > & operator*= ( const R & x )

Scaling.

Definition at line 541 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::size(), and Nonzero< R >::val.

◆ operator=() [1/6]

SVectorBase< R > & operator= ( const SSVectorBase< S > & sv )

Assignment operator.

Definition at line 1013 of file basevectors.h.

References Nonzero< R >::idx, SSVectorBase< R >::index(), SSVectorBase< R >::isSetup(), SSVectorBase< R >::size(), Nonzero< R >::val, and SSVectorBase< R >::value().

◆ operator=() [2/6]

SVectorBase< R > & operator= ( const SVectorBase< R > && sv )

move assignement operator.

Definition at line 668 of file svectorbase.h.

◆ operator=() [3/6]

SVectorBase< R > & operator= ( const SVectorBase< R > & sv )

Assignment operator.

Definition at line 637 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::set_size(), SVectorBase< R >::size(), and Nonzero< R >::val.

◆ operator=() [4/6]

SVectorBase< R > & operator= ( const SVectorBase< S > & sv )

Assignment operator.

Definition at line 682 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::set_size(), SVectorBase< R >::size(), and Nonzero< R >::val.

◆ operator=() [5/6]

SVectorBase< Real > & operator= ( const VectorBase< S > & vec )

Assignment operator (specialization for Real).

Definition at line 982 of file basevectors.h.

References VectorBase< R >::dim(), Nonzero< R >::idx, and Nonzero< R >::val.

◆ operator=() [6/6]

SVectorBase< R > & operator= ( const VectorBase< S > & vec )

Assignment operator.

Definition at line 951 of file basevectors.h.

References VectorBase< R >::dim(), Nonzero< R >::idx, and Nonzero< R >::val.

Referenced by DSVectorBase< R >::DSVectorBase(), DSVectorBase< R >::operator=(), and SVSetBase< R >::DLPSV::operator=().

◆ operator[]()

R operator[] ( int i ) const

Value to index i.

Definition at line 217 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::pos().

◆ pos()

int pos ( int i ) const

Position of index i.

Returns: Finds the position of the first index i in the index set. If no such index i is found, -1 is returned. Otherwise, index(pos(i)) == i holds.

Definition at line 197 of file svectorbase.h.

References SVectorBase< R >::index(), SVectorBase< R >::m_elem, and SVectorBase< R >::size().

Referenced by SPxLPBase< R >::changeCol(), SPxLPBase< R >::changeElement(), SPxLPBase< R >::changeRow(), SPxLPBase< R >::doRemoveCol(), SPxLPBase< R >::doRemoveRow(), SPxLPBase< R >::isConsistent(), and SVectorBase< R >::operator[]().

◆ remove() [1/2]

void remove ( int n )

Remove n 'th nonzero.

Definition at line 430 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::set_size(), and SVectorBase< R >::size().

◆ remove() [2/2]

void remove	(	int	n,
		int	m
	)

Remove nonzeros n thru m.

Definition at line 406 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::set_size(), and SVectorBase< R >::size().

Referenced by SPxLPBase< R >::changeCol(), SPxLPBase< R >::changeElement(), SPxLPBase< R >::changeRow(), SPxLPBase< R >::doRemoveCol(), SPxLPBase< R >::doRemoveCols(), SPxLPBase< R >::doRemoveRow(), and SPxLPBase< R >::doRemoveRows().

◆ scaleAssign() [1/2]

SVectorBase< Real > & scaleAssign	(	const int *	scaleExp,
		const SVectorBase< Real > &	sv,
		bool	negateExp = `false`
	)

scale and assign

Definition at line 732 of file svectorbase.h.

References SVectorBase< R >::index(), SVectorBase< R >::isConsistent(), SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::size(), soplex::spxLdexp(), and SVectorBase< R >::value().

◆ scaleAssign() [2/2]

SVectorBase< Real > & scaleAssign	(	int	scaleExp,
		const SVectorBase< Real > &	sv
	)

scale and assign

Definition at line 713 of file svectorbase.h.

References SVectorBase< R >::index(), SVectorBase< R >::isConsistent(), SVectorBase< R >::m_elem, SVectorBase< R >::max(), SVectorBase< R >::size(), soplex::spxLdexp(), and SVectorBase< R >::value().

◆ set_max()

void set_max ( int m )

Set the maximum number of nonzeros in the vector.

Definition at line 806 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::memsize.

Referenced by SVSetBase< R >::ensureMem(), SVSetBase< R >::memPack(), and SVectorBase< R >::setMem().

◆ set_size()

void set_size ( int s )

Set size of the vector.

Definition at line 799 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::memused.

Referenced by SVectorBase< R >::add(), SVectorBase< R >::assignArray(), SVectorBase< R >::clear(), SVSetBase< R >::deleteVec(), SVSetBase< R >::memPack(), SVectorBase< R >::operator=(), SVSetBase< R >::operator=(), UnitVectorBase< R >::operator=(), SVectorBase< R >::remove(), DSVectorBase< R >::setMax(), SVectorBase< R >::setMem(), UnitVectorBase< R >::UnitVectorBase(), and SVSetBase< R >::xtend().

◆ setMem()

void setMem	(	int	n,
		Nonzero< R > *	elmem
	)

Set the memory area where the nonzeros will be stored.

Definition at line 813 of file svectorbase.h.

References SVectorBase< R >::m_elem, SVectorBase< R >::set_max(), and SVectorBase< R >::set_size().

Referenced by DSVectorBase< R >::allocMem(), SVSetBase< R >::create(), SVSetBase< R >::deleteVec(), SVSetBase< R >::memPack(), SVSetBase< R >::operator=(), DSVectorBase< R >::setMax(), SVectorBase< R >::SVectorBase(), and SVSetBase< R >::xtend().

◆ size()

int size ( ) const

Number of used indices.

Definition at line 164 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::memused.

◆ sort()

void sort ( )

Sort nonzeros to increasing indices.

Definition at line 449 of file svectorbase.h.

References Nonzero< R >::idx, SVectorBase< R >::m_elem, and SVectorBase< R >::size().

◆ value() [1/2]

R & value ( int n )

Reference to value of n 'th nonzero.

Definition at line 264 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::size().

Referenced by SPxLPBase< R >::addCols(), SPxLPBase< R >::addDualActivity(), SPxLPBase< R >::added2Set(), SPxLPBase< R >::addPrimalActivity(), SPxLPBase< R >::addRows(), SSVectorBase< R >::assign(), VectorBase< R >::assign(), SPxLPBase< R >::changeCol(), SPxLPBase< R >::changeElement(), SPxLPBase< R >::changeRow(), SPxLPBase< R >::doAddCol(), SPxLPBase< R >::doAddCols(), SPxLPBase< R >::doAddRow(), SPxLPBase< R >::doAddRows(), SPxLPBase< R >::isConsistent(), VectorBase< R >::multAdd(), VectorBase< R >::multSub(), soplex::operator*(), VectorBase< R >::operator*(), VectorBase< R >::operator-=(), soplex::operator<<(), VectorBase< R >::operator=(), SVectorBase< R >::scaleAssign(), SoPlex_getRowVectorRational(), and SoPlex_getRowVectorReal().

◆ value() [2/2]

const R & value ( int n ) const

Value of n 'th nonzero.

Definition at line 273 of file svectorbase.h.

References SVectorBase< R >::m_elem, and SVectorBase< R >::size().

Friends And Related Function Documentation

◆ SVectorBase

friend class SVectorBase

friend

Definition at line 141 of file svectorbase.h.

Member Data Documentation

◆ m_elem

Nonzero<R>* m_elem

private

◆ memsize

int memsize

private

Definition at line 150 of file svectorbase.h.

Referenced by SVectorBase< R >::max(), and SVectorBase< R >::set_max().

◆ memused

int memused

private

Definition at line 151 of file svectorbase.h.

Referenced by SVectorBase< R >::max(), SVectorBase< R >::set_size(), and SVectorBase< R >::size().

SoPlex

Public Types

Public Member Functions

Private Attributes

Friends

Detailed Description

Member Typedef Documentation

◆ Element

Constructor & Destructor Documentation

◆ SVectorBase() [1/2]

◆ SVectorBase() [2/2]

Member Function Documentation

◆ add() [1/6]

◆ add() [2/6]

◆ add() [3/6]

◆ add() [4/6]

◆ add() [5/6]

◆ add() [6/6]

◆ assignArray()

◆ clear()

◆ dim()

◆ element() [1/2]

◆ element() [2/2]

◆ index() [1/2]

◆ index() [2/2]

◆ isConsistent()

◆ length()

◆ length2()

◆ max()

◆ maxAbs()

◆ mem()

◆ minAbs()

◆ operator*() [1/3]

◆ operator*() [2/3]

◆ operator*() [3/3]

◆ operator*=()

◆ operator=() [1/6]

◆ operator=() [2/6]

◆ operator=() [3/6]

◆ operator=() [4/6]

◆ operator=() [5/6]

◆ operator=() [6/6]

◆ operator[]()

◆ pos()

◆ remove() [1/2]

◆ remove() [2/2]

◆ scaleAssign() [1/2]

◆ scaleAssign() [2/2]

◆ set_max()

◆ set_size()

◆ setMem()

◆ size()

◆ sort()

◆ value() [1/2]

◆ value() [2/2]

Friends And Related Function Documentation

◆ SVectorBase

Member Data Documentation

◆ m_elem

◆ memsize

◆ memused