#include <visp3/core/vpSubMatrix.h>

Inheritance diagram for vpSubMatrix:

Public Types
enum	vpDetMethod { LU_DECOMPOSITION }

Public Member Functions
	vpSubMatrix ()

	vpSubMatrix (vpMatrix &m, const unsigned int &row, const unsigned int &col, const unsigned int &nrows, const unsigned int &ncols)

virtual	~vpSubMatrix () vp_override

void	init (vpMatrix &m, const unsigned int &row, const unsigned int &col, const unsigned int &nrows, const unsigned int &ncols)

void	checkParentStatus () const

vpSubMatrix &	operator= (const vpSubMatrix &B)

vpSubMatrix &	operator= (const vpMatrix &B)

vpSubMatrix &	operator= (const double &x)

void	clear ()

Setting a diagonal matrix <br>
void	diag (const double &val=1.0)

void	diag (const vpColVector &A)

void	eye ()

Assignment operators
vpMatrix &	operator<< (double *)

vpMatrix &	operator<< (double val)

vpMatrix &	operator, (double val)

Stacking <br>
void	stack (const vpMatrix &A)

void	stack (const vpRowVector &r)

void	stack (const vpColVector &c)

void	stackColumns (vpColVector &out)

vpColVector	stackColumns ()

void	stackRows (vpRowVector &out)

vpRowVector	stackRows ()

Matrix insertion
void	insert (const vpMatrix &A, unsigned int r, unsigned int c)

Inherited functionalities from vpArray2D
void	insert (const vpArray2D< double > &A, unsigned int r, unsigned int c)

vpArray2D< double >	hadamard (const vpArray2D< double > &m) const

unsigned int	getCols () const

double	getMaxValue () const

double	getMinValue () const

unsigned int	getRows () const

unsigned int	size () const

void	resize (unsigned int nrows, unsigned int ncols, bool flagNullify=true, bool recopy_=true)

void	reshape (unsigned int nrows, unsigned int ncols)

bool	operator== (const vpArray2D< double > &A) const

bool	operator!= (const vpArray2D< double > &A) const

double *	operator[] (unsigned int i)

double *	operator[] (unsigned int i) const

Columns, rows, sub-matrices extraction
vpMatrix	extract (unsigned int r, unsigned int c, unsigned int nrows, unsigned int ncols) const

vpColVector	getCol (unsigned int j) const

vpColVector	getCol (unsigned int j, unsigned int i_begin, unsigned int size) const

vpRowVector	getRow (unsigned int i) const

vpRowVector	getRow (unsigned int i, unsigned int j_begin, unsigned int size) const

vpColVector	getDiag () const

void	init (const vpMatrix &M, unsigned int r, unsigned int c, unsigned int nrows, unsigned int ncols)

Matrix operations <br>
double	det (vpDetMethod method=LU_DECOMPOSITION) const

double	detByLU () const

double	detByLUEigen3 () const

double	detByLULapack () const

double	detByLUOpenCV () const

vpMatrix	expm () const

vpMatrix &	operator+= (const vpMatrix &B)

vpMatrix &	operator+= (double x)

vpMatrix &	operator-= (const vpMatrix &B)

vpMatrix &	operator-= (double x)

vpMatrix	operator* (const vpMatrix &B) const

vpMatrix	operator* (const vpRotationMatrix &R) const

vpMatrix	operator* (const vpHomogeneousMatrix &R) const

vpMatrix	operator* (const vpVelocityTwistMatrix &V) const

vpMatrix	operator* (const vpForceTwistMatrix &V) const

vpTranslationVector	operator* (const vpTranslationVector &tv) const

vpColVector	operator* (const vpColVector &v) const

vpMatrix	operator* (double x) const

vpMatrix	operator+ (const vpMatrix &B) const

vpMatrix	operator- (const vpMatrix &B) const

vpMatrix	operator- () const

vpMatrix &	operator*= (double x)

vpMatrix &	operator/= (double x)

vpMatrix	operator/ (double x) const

double	sum () const

double	sumSquare () const

Hadamard product <br>
vpMatrix	hadamard (const vpMatrix &m) const

Kronecker product <br>
void	kron (const vpMatrix &m1, vpMatrix &out) const

vpMatrix	kron (const vpMatrix &m1) const

Transpose <br>
vpMatrix	t () const

vpMatrix	transpose () const

void	transpose (vpMatrix &At) const

vpMatrix	AAt () const

void	AAt (vpMatrix &B) const

vpMatrix	AtA () const

void	AtA (vpMatrix &B) const

Matrix inversion <br>
vpMatrix	inverseByLU () const

vpMatrix	inverseByLUEigen3 () const

vpMatrix	inverseByLULapack () const

vpMatrix	inverseByLUOpenCV () const

vpMatrix	inverseByCholesky () const

vpMatrix	inverseByCholeskyLapack () const

vpMatrix	inverseByCholeskyOpenCV () const

vpMatrix	inverseByQR () const

vpMatrix	inverseByQRLapack () const

vpMatrix	inverseTriangular (bool upper=true) const

vpMatrix	pseudoInverse (double svThreshold=1e-6) const

unsigned int	pseudoInverse (vpMatrix &Ap, double svThreshold=1e-6) const

unsigned int	pseudoInverse (vpMatrix &Ap, vpColVector &sv, double svThreshold=1e-6) const

unsigned int	pseudoInverse (vpMatrix &Ap, vpColVector &sv, double svThreshold, vpMatrix &imA, vpMatrix &imAt) const

unsigned int	pseudoInverse (vpMatrix &Ap, vpColVector &sv, double svThreshold, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

vpMatrix	pseudoInverse (int rank_in) const

int	pseudoInverse (vpMatrix &Ap, int rank_in) const

int	pseudoInverse (vpMatrix &Ap, vpColVector &sv, int rank_in) const

int	pseudoInverse (vpMatrix &Ap, vpColVector &sv, int rank_in, vpMatrix &imA, vpMatrix &imAt) const

int	pseudoInverse (vpMatrix &Ap, vpColVector &sv, int rank_in, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

vpMatrix	pseudoInverseLapack (double svThreshold=1e-6) const

unsigned int	pseudoInverseLapack (vpMatrix &Ap, double svThreshold=1e-6) const

unsigned int	pseudoInverseLapack (vpMatrix &Ap, vpColVector &sv, double svThreshold=1e-6) const

unsigned int	pseudoInverseLapack (vpMatrix &Ap, vpColVector &sv, double svThreshold, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

vpMatrix	pseudoInverseLapack (int rank_in) const

int	pseudoInverseLapack (vpMatrix &Ap, int rank_in) const

int	pseudoInverseLapack (vpMatrix &Ap, vpColVector &sv, int rank_in) const

int	pseudoInverseLapack (vpMatrix &Ap, vpColVector &sv, int rank_in, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

vpMatrix	pseudoInverseEigen3 (double svThreshold=1e-6) const

unsigned int	pseudoInverseEigen3 (vpMatrix &Ap, double svThreshold=1e-6) const

unsigned int	pseudoInverseEigen3 (vpMatrix &Ap, vpColVector &sv, double svThreshold=1e-6) const

unsigned int	pseudoInverseEigen3 (vpMatrix &Ap, vpColVector &sv, double svThreshold, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

vpMatrix	pseudoInverseEigen3 (int rank_in) const

int	pseudoInverseEigen3 (vpMatrix &Ap, int rank_in) const

int	pseudoInverseEigen3 (vpMatrix &Ap, vpColVector &sv, int rank_in) const

int	pseudoInverseEigen3 (vpMatrix &Ap, vpColVector &sv, int rank_in, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

vpMatrix	pseudoInverseOpenCV (double svThreshold=1e-6) const

unsigned int	pseudoInverseOpenCV (vpMatrix &Ap, double svThreshold=1e-6) const

unsigned int	pseudoInverseOpenCV (vpMatrix &Ap, vpColVector &sv, double svThreshold=1e-6) const

unsigned int	pseudoInverseOpenCV (vpMatrix &Ap, vpColVector &sv, double svThreshold, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

vpMatrix	pseudoInverseOpenCV (int rank_in) const

int	pseudoInverseOpenCV (vpMatrix &Ap, int rank_in) const

int	pseudoInverseOpenCV (vpMatrix &Ap, vpColVector &sv, int rank_in) const

int	pseudoInverseOpenCV (vpMatrix &Ap, vpColVector &sv, int rank_in, vpMatrix &imA, vpMatrix &imAt, vpMatrix &kerAt) const

SVD decomposition <br>
double	cond (double svThreshold=1e-6) const

unsigned int	kernel (vpMatrix &kerAt, double svThreshold=1e-6) const

unsigned int	nullSpace (vpMatrix &kerA, double svThreshold=1e-6) const

unsigned int	nullSpace (vpMatrix &kerA, int dim) const

void	solveBySVD (const vpColVector &B, vpColVector &x) const

vpColVector	solveBySVD (const vpColVector &B) const

void	svd (vpColVector &w, vpMatrix &V)

void	svdEigen3 (vpColVector &w, vpMatrix &V)

void	svdLapack (vpColVector &w, vpMatrix &V)

void	svdOpenCV (vpColVector &w, vpMatrix &V)

QR decomposition <br>
unsigned int	qr (vpMatrix &Q, vpMatrix &R, bool full=false, bool squareR=false, double tol=1e-6) const

unsigned int	qrPivot (vpMatrix &Q, vpMatrix &R, vpMatrix &P, bool full=false, bool squareR=false, double tol=1e-6) const

void	solveByQR (const vpColVector &b, vpColVector &x) const

vpColVector	solveByQR (const vpColVector &b) const

Eigen values <br>
vpColVector	eigenValues () const

void	eigenValues (vpColVector &evalue, vpMatrix &evector) const

Norms <br>
double	frobeniusNorm () const

double	inducedL2Norm () const

double	infinityNorm () const

Printing <br>
std::ostream &	cppPrint (std::ostream &os, const std::string &matrixName="A", bool octet=false) const

std::ostream &	csvPrint (std::ostream &os) const

std::ostream &	maplePrint (std::ostream &os) const

std::ostream &	matlabPrint (std::ostream &os) const

int	print (std::ostream &s, unsigned int length, const std::string &intro="") const

void	printSize () const

Static Public Member Functions
Linear algebra optimization <br>
static unsigned int	getLapackMatrixMinSize ()

static void	setLapackMatrixMinSize (unsigned int min_size)

Stacking with Static Public Member Functions <br>
static vpMatrix	stack (const vpMatrix &A, const vpMatrix &B)

static vpMatrix	stack (const vpMatrix &A, const vpRowVector &r)

static vpMatrix	stack (const vpMatrix &A, const vpColVector &c)

static void	stack (const vpMatrix &A, const vpMatrix &B, vpMatrix &C)

static void	stack (const vpMatrix &A, const vpRowVector &r, vpMatrix &C)

static void	stack (const vpMatrix &A, const vpColVector &c, vpMatrix &C)

static vpMatrix	juxtaposeMatrices (const vpMatrix &A, const vpMatrix &B)

static void	juxtaposeMatrices (const vpMatrix &A, const vpMatrix &B, vpMatrix &C)

Matrix insertion with Static Public Member Functions <br>
static vpMatrix	insert (const vpMatrix &A, const vpMatrix &B, unsigned int r, unsigned int c)

static void	insert (const vpMatrix &A, const vpMatrix &B, vpMatrix &C, unsigned int r, unsigned int c)

Kronecker product with Static Public Member Functions <br>
static void	kron (const vpMatrix &m1, const vpMatrix &m2, vpMatrix &out)

static vpMatrix	kron (const vpMatrix &m1, const vpMatrix &m2)

Setting a diagonal matrix with Static Public Member Functions <br>
static void	createDiagonalMatrix (const vpColVector &A, vpMatrix &DA)

Matrix operations with Static Public Member Functions <br>
static void	add2Matrices (const vpMatrix &A, const vpMatrix &B, vpMatrix &C)

static void	add2Matrices (const vpColVector &A, const vpColVector &B, vpColVector &C)

static void	add2WeightedMatrices (const vpMatrix &A, const double &wA, const vpMatrix &B, const double &wB, vpMatrix &C)

static void	computeHLM (const vpMatrix &H, const double &alpha, vpMatrix &HLM)

static void	mult2Matrices (const vpMatrix &A, const vpMatrix &B, vpMatrix &C)

static void	mult2Matrices (const vpMatrix &A, const vpMatrix &B, vpRotationMatrix &C)

static void	mult2Matrices (const vpMatrix &A, const vpMatrix &B, vpHomogeneousMatrix &C)

static void	mult2Matrices (const vpMatrix &A, const vpColVector &B, vpColVector &C)

static void	multMatrixVector (const vpMatrix &A, const vpColVector &v, vpColVector &w)

static void	negateMatrix (const vpMatrix &A, vpMatrix &C)

static void	sub2Matrices (const vpMatrix &A, const vpMatrix &B, vpMatrix &C)

static void	sub2Matrices (const vpColVector &A, const vpColVector &B, vpColVector &C)

Covariance computation with Static Public Member Functions <br>
static vpMatrix	computeCovarianceMatrix (const vpMatrix &A, const vpColVector &x, const vpColVector &b)

static vpMatrix	computeCovarianceMatrix (const vpMatrix &A, const vpColVector &x, const vpColVector &b, const vpMatrix &w)

static vpMatrix	computeCovarianceMatrixVVS (const vpHomogeneousMatrix &cMo, const vpColVector &deltaS, const vpMatrix &Ls, const vpMatrix &W)

static vpMatrix	computeCovarianceMatrixVVS (const vpHomogeneousMatrix &cMo, const vpColVector &deltaS, const vpMatrix &Ls)

Public Attributes
double *	data

Protected Attributes
unsigned int	pRowNum

unsigned int	pColNum

vpMatrix *	parent

unsigned int	rowNum

unsigned int	colNum

double **	rowPtrs

unsigned int	dsize

Related Functions
(Note that these are not member functions.)
void	insert (const vpMatrix &A, const vpMatrix &B, vpMatrix &C, unsigned int r, unsigned int c)

vpMatrix	operator* (const double &x, const vpMatrix &B)

enum	vpGEMMmethod

bool	operator== (const vpArray2D< double > &A) const

bool	operator== (const vpArray2D< float > &A) const

bool	operator!= (const vpArray2D< double > &A) const

void	vpGEMM (const vpArray2D< double > &A, const vpArray2D< double > &B, const double &alpha, const vpArray2D< double > &C, const double &beta, vpArray2D< double > &D, const unsigned int &ops=0)

Inherited I/O from vpArray2D with Static Public Member Functions
vpArray2D< double >	insert (const vpArray2D< double > &A, const vpArray2D< double > &B, unsigned int r, unsigned int c)

static bool	load (const std::string &filename, vpArray2D< double > &A, bool binary=false, char *header=nullptr)

static bool	loadYAML (const std::string &filename, vpArray2D< double > &A, char *header=nullptr)

static bool	save (const std::string &filename, const vpArray2D< double > &A, bool binary=false, const char *header="")

static bool	saveYAML (const std::string &filename, const vpArray2D< double > &A, const char *header="")

static vpArray2D< double >	conv2 (const vpArray2D< double > &M, const vpArray2D< double > &kernel, const std::string &mode)

static void	conv2 (const vpArray2D< double > &M, const vpArray2D< double > &kernel, vpArray2D< double > &res, const std::string &mode)

static void	insert (const vpArray2D< double > &A, const vpArray2D< double > &B, vpArray2D< double > &C, unsigned int r, unsigned int c)

Deprecated functions
vp_deprecated void	init ()

vp_deprecated void	stackMatrices (const vpMatrix &A)

vp_deprecated void	setIdentity (const double &val=1.0)

vp_deprecated vpRowVector	row (unsigned int i)

vp_deprecated vpColVector	column (unsigned int j)

static vp_deprecated vpMatrix	stackMatrices (const vpMatrix &A, const vpMatrix &B)

static vp_deprecated void	stackMatrices (const vpMatrix &A, const vpMatrix &B, vpMatrix &C)

static vp_deprecated vpMatrix	stackMatrices (const vpMatrix &A, const vpRowVector &B)

static vp_deprecated void	stackMatrices (const vpMatrix &A, const vpRowVector &B, vpMatrix &C)

static vp_deprecated vpMatrix	stackMatrices (const vpColVector &A, const vpColVector &B)

static vp_deprecated void	stackMatrices (const vpColVector &A, const vpColVector &B, vpColVector &C)

Matrix I/O with Static Public Member Functions <br>
vp_deprecated double	euclideanNorm () const

static bool	loadMatrix (const std::string &filename, vpArray2D< double > &M, bool binary=false, char *header=nullptr)

static bool	loadMatrixYAML (const std::string &filename, vpArray2D< double > &M, char *header=nullptr)

static bool	saveMatrix (const std::string &filename, const vpArray2D< double > &M, bool binary=false, const char *header="")

static bool	saveMatrixYAML (const std::string &filename, const vpArray2D< double > &M, const char *header="")

Detailed Description

Definition of the vpSubMatrix class that provides a mask on a vpMatrix. All properties of vpMatrix are available with a vpSubMatrix.

See also: vpMatrix vpColVector vpRowVector

Definition at line 54 of file vpSubMatrix.h.

Member Enumeration Documentation

◆ vpDetMethod

enum vpMatrix::vpDetMethod

inherited

Method used to compute the determinant of a square matrix.

See also: det()

Enumerator
LU_DECOMPOSITION	LU decomposition method.

Definition at line 152 of file vpMatrix.h.

Constructor & Destructor Documentation

◆ vpSubMatrix() [1/2]

vpSubMatrix::vpSubMatrix ( )

Default constructor.

Definition at line 45 of file vpSubMatrix.cpp.

◆ vpSubMatrix() [2/2]

vpSubMatrix::vpSubMatrix	(	vpMatrix &	m,
		const unsigned int &	row_offset,
		const unsigned int &	col_offset,
		const unsigned int &	nrows,
		const unsigned int &	ncols
	)

Constructor.

Parameters

m	: parent matrix
row_offset	: row offset
col_offset	: col offset
nrows	: number of rows of the sub matrix
ncols	: number of columns of the sub matrix

Definition at line 55 of file vpSubMatrix.cpp.

References vpMatrix::init().

◆ ~vpSubMatrix()

vpSubMatrix::~vpSubMatrix ( )

virtual

Destructor.

Definition at line 182 of file vpSubMatrix.cpp.

References vpArray2D< double >::data.

Member Function Documentation

◆ AAt() [1/2]

vpMatrix vpMatrix::AAt ( ) const

inherited

Computes the operation

Returns

See also: AAt(vpMatrix &) const

Definition at line 517 of file vpMatrix.cpp.

Referenced by vpServo::computeControlLaw(), and vpServo::computeProjectionOperators().

◆ AAt() [2/2]

void vpMatrix::AAt ( vpMatrix & B ) const

inherited

Compute the AAt operation such as .

The result is placed in the parameter B and not returned.

A new matrix won't be allocated for every use of the function. This results in a speed gain if used many times with the same result matrix size.

See also: AAt()

Definition at line 537 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, vpArray2D< Type >::rowNum, and vpArray2D< double >::rowPtrs.

◆ add2Matrices() [1/2]

void vpMatrix::add2Matrices	(	const vpColVector &	A,
		const vpColVector &	B,
		vpColVector &	C
	)

staticinherited

Warning: This function is provided for compat with previous releases. You should rather use the functionalities provided in vpColVector class.

Operation C = A + B.

The result is placed in the third parameter C and not returned. A new vector won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

See also: vpColVector::operator+()

Definition at line 1455 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpColVector::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

◆ add2Matrices() [2/2]

void vpMatrix::add2Matrices	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C
	)

staticinherited

Operation C = A + B.

The result is placed in the third parameter C and not returned. A new matrix won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

See also: operator+()

Definition at line 1421 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

Referenced by vpMatrix::operator+().

◆ add2WeightedMatrices()

void vpMatrix::add2WeightedMatrices	(	const vpMatrix &	A,
		const double &	wA,
		const vpMatrix &	B,
		const double &	wB,
		vpMatrix &	C
	)

staticinherited

Operation C = A*wA + B*wB

The result is placed in the third parameter C and not returned. A new matrix won't be allocated for every use of the function (Speed gain if used many times with the same result matrix size).

See also: operator+()

Definition at line 1389 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

◆ AtA() [1/2]

vpMatrix vpMatrix::AtA ( ) const

inherited

Compute the AtA operation such as

Returns

See also: AtA(vpMatrix &) const

Examples: photometricVisualServoingWithoutVpServo.cpp, and testMatrixInverse.cpp.

Definition at line 645 of file vpMatrix.cpp.

Referenced by vpServo::computeProjectionOperators(), vpMbDepthDenseTracker::computeVVS(), vpMbDepthNormalTracker::computeVVS(), vpMbGenericTracker::computeVVS(), vpMbEdgeTracker::computeVVSFirstPhasePoseEstimation(), vpTemplateTrackerWarpHomographySL3::findWarp(), and vpNurbs::globalCurveApprox().

◆ AtA() [2/2]

void vpMatrix::AtA ( vpMatrix & B ) const

inherited

Compute the AtA operation such as .

The result is placed in the parameter B and not returned.

A new matrix won't be allocated for every use of the function. This results in a speed gain if used many times with the same result matrix size.

See also: AtA()

Definition at line 593 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpArray2D< Type >::rowNum.

◆ checkParentStatus()

void vpSubMatrix::checkParentStatus ( ) const

Check is parent vpRowVector has changed since initialization.

This method can be used to detect if the parent matrix always exits or its size have not changed and throw an exception is not.

Definition at line 107 of file vpSubMatrix.cpp.

References vpArray2D< double >::data, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpMatrixException::incorrectMatrixSizeError, parent, pColNum, and pRowNum.

◆ clear()

void vpMatrix::clear ( )

inlineinherited

Removes all elements from the matrix (which are destroyed), leaving the container with a size of 0.

Definition at line 209 of file vpMatrix.h.

◆ column()

vpColVector vpMatrix::column ( unsigned int j )

inherited

Deprecated:: This method is deprecated. You should rather use getCol(). More precisely, the following code:

vpMatrix L;
unsigned int column_index = ...;
... = L.column(column_index);

should be replaced with:

... = L.getCol(column_index - 1);

Warning: Notice column(1) is the 0-th column. This function returns the j-th columns of the matrix.

Parameters

j	: Index of the column to extract noting that column index start at 1 to get the first column.

Definition at line 6680 of file vpMatrix.cpp.

References vpArray2D< double >::getRows().

◆ computeCovarianceMatrix() [1/2]

vpMatrix vpMatrix::computeCovarianceMatrix	(	const vpMatrix &	A,
		const vpColVector &	x,
		const vpColVector &	b
	)

staticinherited

Compute the covariance matrix of the parameters x from a least squares minimization defined as: Ax = b

Parameters

A	: Matrix A from Ax = b.
x	: Vector x from Ax = b corresponding to the parameters to estimate.
b	: Vector b from Ax = b.

Definition at line 59 of file vpMatrix_covariance.cpp.

References vpException::divideByZeroError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpMatrix::pseudoInverse(), and vpMatrix::t().

Referenced by vpMatrix::computeCovarianceMatrixVVS(), and vpPose::poseVirtualVSrobust().

◆ computeCovarianceMatrix() [2/2]

vpMatrix vpMatrix::computeCovarianceMatrix	(	const vpMatrix &	A,
		const vpColVector &	x,
		const vpColVector &	b,
		const vpMatrix &	W
	)

staticinherited

Compute the covariance matrix of the parameters x from a least squares minimization defined as: WAx = Wb

Parameters

A	: Matrix A from WAx = Wb.
x	: Vector x from WAx = Wb corresponding to the parameters to estimate.
b	: Vector b from WAx = Wb.
W	: Diagonal weigths matrix from WAx = Wb.

Definition at line 92 of file vpMatrix_covariance.cpp.

References vpException::divideByZeroError, vpArray2D< Type >::getCols(), vpMatrix::pseudoInverse(), and vpMatrix::t().

◆ computeCovarianceMatrixVVS() [1/2]

vpMatrix vpMatrix::computeCovarianceMatrixVVS	(	const vpHomogeneousMatrix &	cMo,
		const vpColVector &	deltaS,
		const vpMatrix &	Ls
	)

staticinherited

Compute the covariance matrix of an image-based virtual visual servoing. This assumes the optimization has been done via v = Ls.pseudoInverse() * DeltaS.

Parameters

cMo	: Pose matrix that has been computed with the v.
deltaS	: Error vector used in v = Ls.pseudoInverse() * DeltaS
Ls	: interaction matrix used in v = Ls.pseudoInverse() * DeltaS

Definition at line 127 of file vpMatrix_covariance.cpp.

References vpMatrix::computeCovarianceMatrix(), and vpMatrix::computeCovarianceMatrixVVS().

◆ computeCovarianceMatrixVVS() [2/2]

vpMatrix vpMatrix::computeCovarianceMatrixVVS	(	const vpHomogeneousMatrix &	cMo,
		const vpColVector &	deltaS,
		const vpMatrix &	Ls,
		const vpMatrix &	W
	)

staticinherited

Compute the covariance matrix of an image-based virtual visual servoing. This assumes the optimization has been done via v = (W * Ls).pseudoInverse() W * DeltaS.

Parameters

cMo	: Pose matrix that has been computed with the v.
deltaS	: Error vector used in v = (W * Ls).pseudoInverse() * W * DeltaS.
Ls	: interaction matrix used in v = (W * Ls).pseudoInverse() * W * DeltaS.
W	: Weight matrix used in v = (W * Ls).pseudoInverse() * W * DeltaS.

Definition at line 152 of file vpMatrix_covariance.cpp.

References vpMatrix::computeCovarianceMatrix().

Referenced by vpMbTracker::computeCovarianceMatrixVVS(), vpMatrix::computeCovarianceMatrixVVS(), and vpPose::poseVirtualVS().

◆ computeHLM()

void vpMatrix::computeHLM	(	const vpMatrix &	H,
		const double &	alpha,
		vpMatrix &	HLM
	)

staticinherited

Compute ${\bf H} + \alpha * diag({\bf H})$

Parameters

H	: input Matrix ${\bf H}$ . This matrix should be square.
alpha	: Scalar $\alpha$
HLM	: Resulting operation.

Definition at line 6493 of file vpMatrix.cpp.

References vpException::dimensionError, vpArray2D< Type >::getCols(), and vpArray2D< Type >::getRows().

◆ cond()

double vpMatrix::cond ( double svThreshold = 1e-6 ) const

inherited

Returns: The condition number, the ratio of the largest singular value of the matrix to the smallest.

Parameters

svThreshold Threshold used to test the singular values. If a singular value is lower than this threshold we consider that the matrix is not full rank.

Examples: testMatrixConditionNumber.cpp.

Definition at line 6430 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), vpMatrix::insert(), vpColVector::resize(), vpArray2D< Type >::resize(), and vpMatrix::svd().

Referenced by vpTemplateTrackerMIESM::trackNoPyr(), vpTemplateTrackerMIForwardAdditional::trackNoPyr(), vpTemplateTrackerMIForwardCompositional::trackNoPyr(), and vpTemplateTrackerMIInverseCompositional::trackNoPyr().

◆ conv2() [1/2]

vpArray2D< double > vpArray2D< double >::conv2	(	const vpArray2D< Type > &	M,
		const vpArray2D< Type > &	kernel,
		const std::string &	mode
	)

staticinherited

Perform a 2D convolution similar to Matlab conv2 function: $M \star kernel$ .

Parameters

M	: First matrix.
kernel	: Second matrix.
mode	: Convolution mode: "full" (default), "same", "valid".

Convolution mode: full, same, valid (image credit: Theano doc).

Note: This is a very basic implementation that does not use FFT.

Examples: testMatrixConvolution.cpp.

Definition at line 1029 of file vpArray2D.h.

◆ conv2() [2/2]

void vpArray2D< double >::conv2	(	const vpArray2D< Type > &	M,
		const vpArray2D< Type > &	kernel,
		vpArray2D< Type > &	res,
		const std::string &	mode
	)

staticinherited

Perform a 2D convolution similar to Matlab conv2 function: $M \star kernel$ .

Parameters

M	: First array.
kernel	: Second array.
res	: Result.
mode	: Convolution mode: "full" (default), "same", "valid".

Convolution mode: full, same, valid (image credit: Theano doc).

Note: This is a very basic implementation that does not use FFT.

Definition at line 1043 of file vpArray2D.h.

◆ cppPrint()

std::ostream & vpMatrix::cppPrint	(	std::ostream &	os,
		const std::string &	matrixName = `"A"`,
		bool	octet = `false`
	)		const

inherited

Print to be used as part of a C++ code later.

Parameters

os	: the stream to be printed in.
matrixName	: name of the matrix, "A" by default.
octet	: if false, print using double, if true, print byte per byte each bytes of the double array.

The following code shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(2,3);
  int cpt = 0;
  for (unsigned int i=0; i<M.getRows(); i++)
    for (unsigned int j=0; j<M.getCols(); j++)
      M[i][j] = cpt++;
 
  M.cppPrint(std::cout, "M");
}

It produces the following output that could be copy/paste in a C++ code:

vpMatrix M (2, 3);
M[0][0] = 0;
M[0][1] = 1;
M[0][2] = 2;
 
M[1][0] = 3;
M[1][1] = 4;
M[1][2] = 5;

Definition at line 5642 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), and vpArray2D< double >::getRows().

◆ createDiagonalMatrix()

void vpMatrix::createDiagonalMatrix	(	const vpColVector &	A,
		vpMatrix &	DA
	)

staticinherited

Create a diagonal matrix with the element of a vector $DA_{ii} = A_i$ .

Parameters

A	: Vector which element will be put in the diagonal.
DA	: Diagonal matrix DA[i][i] = A[i]

See also: diag()

Definition at line 929 of file vpMatrix.cpp.

References vpArray2D< Type >::getRows(), and vpArray2D< Type >::resize().

◆ csvPrint()

std::ostream & vpMatrix::csvPrint ( std::ostream & os ) const

inherited

Print/save a matrix in csv format.

The following code

#include <visp3/core/vpMatrix.h>
 
int main()
{
  std::ofstream ofs("log.csv", std::ofstream::out);
  vpMatrix M(2,3);
  int cpt = 0;
  for (unsigned int i=0; i<M.getRows(); i++)
    for (unsigned int j=0; j<M.getCols(); j++)
      M[i][j] = cpt++;
 
  M.csvPrint(ofs);
 
  ofs.close();
}

produces log.csv file that contains:

0, 1, 2

3, 4, 5

Definition at line 5591 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), and vpArray2D< double >::getRows().

◆ det()

double vpMatrix::det ( vpDetMethod method = LU_DECOMPOSITION ) const

inherited

Compute the determinant of a n-by-n matrix.

Parameters

method : Method used to compute the determinant. Default LU decomposition method is faster than the method based on Gaussian elimination.

Returns: Determinant of the matrix.

#include <iostream>
 
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3,3);
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.;
  A[1][0] = 1/3.; A[1][1] = 1/4.; A[1][2] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/7.; A[2][2] = 1/8.;
  std::cout << "Initial matrix: \n" << A << std::endl;
 
  // Compute the determinant
  std:: cout << "Determinant by default method           : " << A.det() << std::endl;
  std:: cout << "Determinant by LU decomposition         : " << A.detByLU() << std::endl;
  std:: cout << "Determinant by LU decomposition (Lapack): " << A.detByLULapack() << std::endl;
  std:: cout << "Determinant by LU decomposition (OpenCV): " << A.detByLUOpenCV() << std::endl;
  std:: cout << "Determinant by LU decomposition (Eigen3): " << A.detByLUEigen3() << std::endl;
}

Examples: testMatrixInverse.cpp.

Definition at line 6275 of file vpMatrix.cpp.

References vpMatrix::detByLU(), and vpMatrix::LU_DECOMPOSITION.

Referenced by vpHomogeneousMatrix::compute3d3dTransformation(), vpMatrix::detByLULapack(), vpMatrix::detByLUOpenCV(), vpTemplateTrackerTriangle::init(), vpMatrix::inverseByLU(), and vpRotationMatrix::orthogonalize().

◆ detByLU()

double vpMatrix::detByLU ( ) const

inherited

Compute the determinant of a square matrix using the LU decomposition.

This function calls the first following function that is available:

detByLULapack() if Lapack 3rd party is installed
detByLUEigen3() if Eigen3 3rd party is installed
detByLUOpenCV() if OpenCV 3rd party is installed.

If none of these previous 3rd parties is installed, we use by default detByLULapack() with a Lapack built-in version.

Returns: The determinant of the matrix if the matrix is square.

#include <iostream>
 
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3,3);
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.;
  A[1][0] = 1/3.; A[1][1] = 1/4.; A[1][2] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/7.; A[2][2] = 1/8.;
  std::cout << "Initial matrix: \n" << A << std::endl;
 
  // Compute the determinant
  std:: cout << "Determinant by default method           : " << A.det() << std::endl;
  std:: cout << "Determinant by LU decomposition         : " << A.detByLU() << std::endl;
}

See also: detByLULapack(), detByLUEigen3(), detByLUOpenCV()

Definition at line 224 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpMatrix::detByLUEigen3(), vpMatrix::detByLULapack(), vpMatrix::detByLUOpenCV(), vpException::fatalError, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::det().

◆ detByLUEigen3()

double vpMatrix::detByLUEigen3 ( ) const

inherited

Compute the determinant of a square matrix using the LU decomposition with Eigen3 3rd party.

Returns: The determinant of the matrix if the matrix is square.

#include <iostream>
 
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3,3);
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.;
  A[1][0] = 1/3.; A[1][1] = 1/4.; A[1][2] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/7.; A[2][2] = 1/8.;
  std::cout << "Initial matrix: \n" << A << std::endl;
 
  // Compute the determinant
  std:: cout << "Determinant by LU decomposition (Eigen3): " << A.detByLUEigen3() << std::endl;
}

See also: detByLU(), detByLUOpenCV(), detByLULapack()

Definition at line 615 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::data, vpException::fatalError, vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), and vpArray2D< double >::rowNum.

Referenced by vpMatrix::detByLU().

◆ detByLULapack()

double vpMatrix::detByLULapack ( ) const

inherited

Compute the determinant of a square matrix using the LU decomposition with Lapack 3rd party.

Returns: The determinant of the matrix if the matrix is square.

#include <iostream>
 
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3,3);
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.;
  A[1][0] = 1/3.; A[1][1] = 1/4.; A[1][2] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/7.; A[2][2] = 1/8.;
  std::cout << "Initial matrix: \n" << A << std::endl;
 
  // Compute the determinant
  std:: cout << "Determinant by LU decomposition (Lapack): " << A.detByLULapack() << std::endl;
}

See also: detByLU(), detByLUEigen3(), detByLUOpenCV()

Definition at line 379 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpMatrix::det(), vpException::fatalError, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::detByLU().

◆ detByLUOpenCV()

double vpMatrix::detByLUOpenCV ( ) const

inherited

Compute the determinant of a n-by-n matrix using the LU decomposition with OpenCV 3rd party.

Returns: Determinant of the matrix.

#include <iostream>
 
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3,3);
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.;
  A[1][0] = 1/3.; A[1][1] = 1/4.; A[1][2] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/7.; A[2][2] = 1/8.;
  std::cout << "Initial matrix: \n" << A << std::endl;
 
  // Compute the determinant
  std:: cout << "Determinant by LU decomposition (OpenCV): " << A.detByLUOpenCV() << std::endl;
}

See also: detByLU(), detByLUEigen3(), detByLULapack()

Definition at line 525 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::data, vpMatrix::det(), vpException::fatalError, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::detByLU().

◆ diag() [1/2]

void vpMatrix::diag ( const double & val = 1.0 )

inherited

Set the matrix as a diagonal matrix where each element on the diagonal is set to val. Elements that are not on the diagonal are set to 0.

Parameters

val	: Value to set.

See also: eye()

#include <iostream>
 
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3, 4);
 
  A.diag(2);
 
  std::cout << "A:\n" << A << std::endl;
}

Matrix A is now equal to:

0 0 0
2 0 0
0 2 0

Examples: testSvd.cpp.

Definition at line 910 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::getDiag().

◆ diag() [2/2]

void vpMatrix::diag ( const vpColVector & A )

inherited

Create a diagonal matrix with the element of a vector.

Parameters

A	: Vector which element will be put in the diagonal.

See also: createDiagonalMatrix()

#include <iostream>
 
#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A;
  vpColVector v(3);
 
  v[0] = 1;
  v[1] = 2;
  v[2] = 3;
 
  A.diag(v);
 
  std::cout << "A:\n" << A << std::endl;
}

Matrix A is now equal to:

0 0
2 0
0 3

Definition at line 869 of file vpMatrix.cpp.

References vpArray2D< Type >::getRows(), and vpArray2D< double >::resize().

◆ eigenValues() [1/2]

vpColVector vpMatrix::eigenValues ( ) const

inherited

Compute the eigenvalues of a n-by-n real symmetric matrix using Lapack 3rd party.

Returns: The eigenvalues of a n-by-n real symmetric matrix, sorted in ascending order.

Exceptions

vpException::dimensionError	If the matrix is not square.
vpException::fatalError	If the matrix is not symmetric.
vpException::functionNotImplementedError	If the Lapack 3rd party is not detected.

Here an example:

#include <iostream>
 
#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3,3); // A is a symmetric matrix
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.;
  A[1][0] = 1/2.; A[1][1] = 1/3.; A[1][2] = 1/4.;
  A[2][0] = 1/3.; A[2][1] = 1/4.; A[2][2] = 1/5.;
  std::cout << "Initial symmetric matrix: \n" << A << std::endl;
 
  // Compute the eigen values
  vpColVector evalue; // Eigenvalues
  evalue = A.eigenValues();
  std::cout << "Eigen values: \n" << evalue << std::endl;
}

See also: eigenValues(vpColVector &, vpMatrix &)

Definition at line 5836 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpException::dimensionError, vpException::fatalError, vpException::functionNotImplementedError, vpColVector::resize(), vpArray2D< double >::rowNum, and vpMatrix::t().

Referenced by vpQuadProg::fromCanonicalCost(), and vpMath::lineFitting().

◆ eigenValues() [2/2]

void vpMatrix::eigenValues	(	vpColVector &	evalue,
		vpMatrix &	evector
	)		const

inherited

Compute the eigenvalues of a n-by-n real symmetric matrix using Lapack 3rd party.

Parameters

evalue	: Eigenvalues of the matrix, sorted in ascending order.
evector	: Corresponding eigenvectors of the matrix.

Exceptions

vpException::dimensionError	If the matrix is not square.
vpException::fatalError	If the matrix is not symmetric.
vpException::functionNotImplementedError	If Lapack 3rd party is not detected.

Here an example:

#include <iostream>
 
#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4); // A is a symmetric matrix
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/2.; A[1][1] = 1/3.; A[1][2] = 1/4.; A[1][3] = 1/5.;
  A[2][0] = 1/3.; A[2][1] = 1/4.; A[2][2] = 1/5.; A[2][3] = 1/6.;
  A[3][0] = 1/4.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
  std::cout << "Initial symmetric matrix: \n" << A << std::endl;
 
  vpColVector d; // Eigenvalues
  vpMatrix    V; // Eigenvectors
 
  // Compute the eigenvalues and eigenvectors
  A.eigenValues(d, V);
  std::cout << "Eigen values: \n" << d << std::endl;
  std::cout << "Eigen vectors: \n" << V << std::endl;
 
  vpMatrix D;
  D.diag(d); // Eigenvalues are on the diagonal
 
  std::cout << "D: " << D << std::endl;
 
  // Verification: A * V = V * D
  std::cout << "AV-VD = 0 ? \n" << (A*V) - (V*D) << std::endl;
}

See also: eigenValues()

Definition at line 5957 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpException::dimensionError, vpException::fatalError, vpException::functionNotImplementedError, vpColVector::resize(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpMatrix::t().

◆ euclideanNorm()

double vpMatrix::euclideanNorm ( ) const

inherited

Deprecated:: This function is deprecated. You should rather use frobeniusNorm().

Compute and return the Euclidean norm (also called Frobenius norm) $||A|| = \sqrt{ \sum{A_{ij}^2}}$ .

Returns: The Euclidean norm (also called Frobenius norm) if the matrix is initialized, 0 otherwise.

See also: frobeniusNorm(), infinityNorm(), inducedL2Norm()

Definition at line 6619 of file vpMatrix.cpp.

References vpMatrix::frobeniusNorm().

◆ expm()

vpMatrix vpMatrix::expm ( ) const

inherited

Compute the exponential matrix of a square matrix.

Returns: Return the exponential matrix.

Definition at line 6293 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::dimensionError, vpMatrix::eye(), vpMatrix::inverseByLU(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpMatrix::sum().

Referenced by vpTemplateTrackerWarpHomographySL3::computeCoeff().

◆ extract()

vpMatrix vpMatrix::extract	(	unsigned int	r,
		unsigned int	c,
		unsigned int	nrows,
		unsigned int	ncols
	)		const

inherited

Extract a sub matrix from a matrix M.

Parameters

r	: row index in matrix M.
c	: column index in matrix M.
nrows	: Number of rows of the matrix that should be extracted.
ncols	: Number of columns of the matrix that should be extracted.

The following code shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(4,5);
  int val = 0;
  for(size_t i=0; i<M.getRows(); i++) {
    for(size_t j=0; j<M.getCols(); j++) {
      M[i][j] = val++;
    }
  }
  M.print(std::cout, 4, "M ");
  vpMatrix N = M.extract(0, 1, 2, 3);
  N.print(std::cout, 4, "N ");
}

It produces the following output:

M [4,5]=
1  2  3  4
6  7  8  9
11 12 13 14
16 17 18 19
N [2,3]=
2 3
7 8

See also: init(const vpMatrix &, unsigned int, unsigned int, unsigned int, unsigned int)

Definition at line 404 of file vpMatrix.cpp.

References vpException::dimensionError, vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), and vpArray2D< Type >::resize().

Referenced by vpLinProg::colReduction(), vpLinProg::rowReduction(), and vpMatrix::solveByQR().

◆ eye()

void vpMatrix::eye ( )

inherited

Set an m-by-n matrix to identity with ones on the diagonal and zeros else where.

Examples: testMatrix.cpp.

Definition at line 448 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, and vpArray2D< double >::rowNum.

Referenced by vpLinProg::colReduction(), vpServo::computeControlLaw(), vpMbDepthDenseTracker::computeVVS(), vpMbDepthNormalTracker::computeVVS(), vpMbKltTracker::computeVVS(), vpMbEdgeTracker::computeVVS(), vpMbEdgeKltTracker::computeVVS(), vpMbGenericTracker::computeVVS(), vpMbTracker::computeVVSPoseEstimation(), vpMatrix::expm(), vpMatrix::eye(), vpTemplateTrackerWarpHomographySL3::getdW0(), vpTemplateTrackerWarpHomographySL3::getdWdp0(), vpFeatureThetaU::interaction(), vpPose::poseFromRectangle(), vpRobotKinova::setCartVelocity(), vpServo::setServo(), vpLinProg::solveLP(), vpMbTracker::vpMbTracker(), and vpServo::vpServo().

◆ frobeniusNorm()

double vpMatrix::frobeniusNorm ( ) const

inherited

Compute and return the Frobenius norm (also called Euclidean norm) $||A|| = \sqrt{ \sum{A_{ij}^2}}$ .

Returns: The Frobenius norm (also called Euclidean norm) if the matrix is initialized, 0 otherwise.

See also: infinityNorm(), inducedL2Norm()

Examples: testMatrixPseudoInverse.cpp, and testSvd.cpp.

Definition at line 6514 of file vpMatrix.cpp.

References vpArray2D< double >::data, and vpArray2D< double >::dsize.

Referenced by vpMatrix::euclideanNorm().

◆ getCol() [1/2]

vpColVector vpMatrix::getCol ( unsigned int j ) const

inherited

Extract a column vector from a matrix.

Warning: All the indexes start from 0 in this function.

Parameters

j	: Index of the column to extract. If j=0, the first column is extracted.

Returns: The extracted column vector.

The following example shows how to use this function:

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  for (unsigned int i = 0; i < A.getRows(); i++)
    for (unsigned int j = 0; j < A.getCols(); j++)
      A[i][j] = i*A.getCols()+j;
 
  A.print(std::cout, 4);
 
  vpColVector cv = A.getCol(1);
  std::cout << "Column vector: \n" << cv << std::endl;
}

It produces the following output :

[4, 4] =
0  1  2  3
4  5  6  7
8  9 10 11
12 13 14 15
column vector :
1
5
9
13

Examples: testMatrix.cpp.

Definition at line 4975 of file vpMatrix.cpp.

References vpArray2D< double >::rowNum.

Referenced by vpLinProg::colReduction(), vpHomography::DLT(), vpMbtFaceDepthNormal::estimatePlaneEquationSVD(), vpMatrix::kernel(), vpPose::poseFromRectangle(), vpServo::secondaryTaskJointLimitAvoidance(), and vpLinProg::simplex().

◆ getCol() [2/2]

vpColVector vpMatrix::getCol	(	unsigned int	j,
		unsigned int	i_begin,
		unsigned int	column_size
	)		const

inherited

Extract a column vector from a matrix.

Warning: All the indexes start from 0 in this function.

Parameters

j	: Index of the column to extract. If col=0, the first column is extracted.
i_begin	: Index of the row that gives the location of the first element of the column vector to extract.
column_size	: Size of the column vector to extract.

Returns: The extracted column vector.

The following example shows how to use this function:

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  for(unsigned int i=0; i < A.getRows(); i++)
    for(unsigned int j=0; j < A.getCols(); j++)
      A[i][j] = i*A.getCols()+j;
 
  A.print(std::cout, 4);
 
  vpColVector cv = A.getCol(1, 1, 3);
  std::cout << "Column vector: \n" << cv << std::endl;
}

It produces the following output :

[4, 4] =
0  1  2  3
4  5  6  7
8  9 10 11
12 13 14 15
column vector :
5
9
13

Definition at line 4923 of file vpMatrix.cpp.

References vpException::dimensionError, vpArray2D< double >::getCols(), and vpArray2D< double >::getRows().

◆ getCols()

unsigned int vpArray2D< double >::getCols ( ) const

inlineinherited

Return the number of columns of the 2D array.

See also: getRows(), size()

Examples: servoViper850Point2DArtVelocity-jointAvoidance-basic.cpp, testAprilTag.cpp, testArray2D.cpp, testColVector.cpp, testDisplacement.cpp, testMatrix.cpp, testMatrixConditionNumber.cpp, testMatrixConvolution.cpp, testMatrixDeterminant.cpp, testMatrixInitialization.cpp, testMatrixInverse.cpp, testMatrixPseudoInverse.cpp, testPoseVector.cpp, testRotation.cpp, testRowVector.cpp, testSvd.cpp, testTranslationVector.cpp, and tutorial-matlab.cpp.

Definition at line 327 of file vpArray2D.h.

◆ getDiag()

vpColVector vpMatrix::getDiag ( ) const

inherited

Extract a diagonal vector from a matrix.

Returns: The diagonal of the matrix.

Warning: An empty vector is returned if the matrix is empty.

The following example shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(3, 4);
 
  for (unsigned int i = 0; i < A.getRows(); i++)
    for (unsigned int j = 0; j < A.getCols(); j++)
      A[i][j] = i*A.getCols()+j;
 
  A.print(std::cout, 4);
 
  vpColVector diag = A.getDiag();
  std::cout << "Diag vector: \n" << diag.t() << std::endl;
}

It produces the following output :

[3, 4] =
0  1  2  3
4  5  6  7
8  9 10 11
Diag vector :
0  5  10

Definition at line 5102 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpMatrix::diag(), and vpArray2D< double >::rowNum.

◆ getLapackMatrixMinSize()

static unsigned int vpMatrix::getLapackMatrixMinSize ( )

inlinestaticinherited

Return the minimum size of rows and columns required to enable Blas/Lapack usage on matrices and vectors.

To get more info see Tutorial: Basic linear algebra operations.

See also: setLapackMatrixMinSize()

Definition at line 236 of file vpMatrix.h.

◆ getMaxValue()

double vpArray2D< double >::getMaxValue

inherited

Return the array max value.

Examples: servoMomentImage.cpp.

Definition at line 329 of file vpArray2D.h.

◆ getMinValue()

double vpArray2D< double >::getMinValue

inherited

Return the array min value.

Examples: servoMomentImage.cpp.

Definition at line 331 of file vpArray2D.h.

◆ getRow() [1/2]

vpRowVector vpMatrix::getRow ( unsigned int i ) const

inherited

Extract a row vector from a matrix.

Warning: All the indexes start from 0 in this function.

Parameters

i	: Index of the row to extract. If i=0, the first row is extracted.

Returns: The extracted row vector.

The following example shows how to use this function:

#include <visp3/core/vpMatrix.h>
#include <visp3/core/vpRowVector.h>
 
int main()
{
  vpMatrix A(4,4);
 
  for(unsigned int i=0; i < A.getRows(); i++)
    for(unsigned int j=0; j < A.getCols(); j++)
      A[i][j] = i*A.getCols()+j;
 
  A.print(std::cout, 4);
 
  vpRowVector rv = A.getRow(1);
  std::cout << "Row vector: \n" << rv << std::endl;
}

It produces the following output:

[4,4]=
1  2  3
5  6  7
9 10 11
13 14 15
Row vector:
5  6  7

Examples: testMatrix.cpp.

Definition at line 5013 of file vpMatrix.cpp.

References vpArray2D< double >::colNum.

Referenced by vpLinProg::allClose(), vpLinProg::allLesser(), vpLinProg::solveLP(), and vpQuadProg::solveQPi().

◆ getRow() [2/2]

vpRowVector vpMatrix::getRow	(	unsigned int	i,
		unsigned int	j_begin,
		unsigned int	row_size
	)		const

inherited

Extract a row vector from a matrix.

Warning: All the indexes start from 0 in this function.

Parameters

i	: Index of the row to extract. If i=0, the first row is extracted.
j_begin	: Index of the column that gives the location of the first element of the row vector to extract.
row_size	: Size of the row vector to extract.

Returns: The extracted row vector.

The following example shows how to use this function:

#include <visp3/core/vpMatrix.h>
#include <visp3/core/vpRowVector.h>
 
int main()
{
  vpMatrix A(4,4);
 
  for(unsigned int i=0; i < A.getRows(); i++)
    for(unsigned int j=0; j < A.getCols(); j++)
      A[i][j] = i*A.getCols()+j;
 
  A.print(std::cout, 4);
 
  vpRowVector rv = A.getRow(1, 1, 3);
  std::cout << "Row vector: \n" << rv << std::endl;
}

It produces the following output :

[4, 4] =
1  2  3
5  6  7
9 10 11
13 14 15
Row vector :
6  7

Definition at line 5054 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::dimensionError, and vpArray2D< double >::rowNum.

◆ getRows()

unsigned int vpArray2D< double >::getRows ( ) const

inlineinherited

Return the number of rows of the 2D array.

See also: getCols(), size()

Definition at line 337 of file vpArray2D.h.

◆ hadamard() [1/2]

vpArray2D< double > vpArray2D< double >::hadamard ( const vpArray2D< Type > & m ) const

inherited

Compute the Hadamard product (element wise matrix multiplication).

Parameters

m	: Second matrix;

Returns: m1.hadamard(m2) The Hadamard product : $m1 \circ m2 = (m1 \circ m2)_{i,j} = (m1)_{i,j} (m2)_{i,j}$

Examples: testArray2D.cpp.

Definition at line 625 of file vpArray2D.h.

◆ hadamard() [2/2]

vpMatrix vpMatrix::hadamard ( const vpMatrix & m ) const

inherited

Compute the Hadamard product (element wise matrix multiplication).

Parameters

m	: Second matrix;

Returns: m1.hadamard(m2) The Hadamard product : $m1 \circ m2 = (m1 \circ m2)_{i,j} = (m1)_{i,j} (m2)_{i,j}$

Examples: testMatrix.cpp.

Definition at line 1863 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::data, vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< double >::dsize, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), and vpArray2D< double >::rowNum.

◆ inducedL2Norm()

double vpMatrix::inducedL2Norm ( ) const

inherited

Compute and return the induced L2 norm $||A|| = \Sigma_{max}(A)$ which is equal to the maximum singular value of the matrix.

Returns: The induced L2 norm if the matrix is initialized, 0 otherwise.

See also: infinityNorm(), frobeniusNorm()

Examples: testMatrixConditionNumber.cpp.

Definition at line 6533 of file vpMatrix.cpp.

References vpArray2D< double >::dsize, vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), vpArray2D< Type >::size(), and vpMatrix::svd().

◆ infinityNorm()

double vpMatrix::infinityNorm ( ) const

inherited

Compute and return the infinity norm ${||A||}_{\infty} = max\left(\sum_{j=0}^{n}{\mid A_{ij} \mid}\right)$ with $i \in \{0, ..., m\}$ where is the matrix size.

Returns: The infinity norm if the matrix is initialized, 0 otherwise.

See also: frobeniusNorm(), inducedL2Norm()

Definition at line 6574 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

Referenced by vpLinProg::colReduction(), and vpLinProg::rowReduction().

◆ init() [1/3]

vp_deprecated void vpMatrix::init ( )

inlineinherited

Deprecated:: Only provided for compatibility with ViSP previous releases. This function does nothing.

Definition at line 989 of file vpMatrix.h.

Referenced by vpMatrix::vpMatrix(), and vpSubMatrix().

◆ init() [2/3]

void vpMatrix::init	(	const vpMatrix &	M,
		unsigned int	r,
		unsigned int	c,
		unsigned int	nrows,
		unsigned int	ncols
	)

inherited

Initialize the matrix from a part of an input matrix M.

Parameters

M	: Input matrix used for initialization.
r	: row index in matrix M.
c	: column index in matrix M.
nrows	: Number of rows of the matrix that should be initialized.
ncols	: Number of columns of the matrix that should be initialized.

The sub-matrix starting from M[r][c] element and ending on M[r+nrows-1][c+ncols-1] element is used to initialize the matrix.

The following code shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(4,5);
  int val = 0;
  for(size_t i=0; i<M.getRows(); i++) {
    for(size_t j=0; j<M.getCols(); j++) {
      M[i][j] = val++;
    }
  }
  M.print(std::cout, 4, "M ");
 
  vpMatrix N;
  N.init(M, 0, 1, 2, 3);
  N.print(std::cout, 4, "N ");
}

It produces the following output:

M [4,5]=
1  2  3  4
6  7  8  9
11 12 13 14
16 17 18 19
N [2,3]=
2 3
7 8

See also: extract()

Examples: testMatrix.cpp.

Definition at line 340 of file vpMatrix.cpp.

References vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::resize(), and vpArray2D< double >::rowPtrs.

◆ init() [3/3]

void vpSubMatrix::init	(	vpMatrix &	m,
		const unsigned int &	row_offset,
		const unsigned int &	col_offset,
		const unsigned int &	nrows,
		const unsigned int &	ncols
	)

Initialisation of vpMatrix.

Initialisation of a sub matrix.

Parameters

m	: parent matrix
row_offset	: row offset
col_offset	: col offset
nrows	: number of rows of the sub matrix
ncols	: number of columns of the sub matrix

Definition at line 70 of file vpSubMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::data, vpArray2D< Type >::data, vpArray2D< double >::dsize, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpMatrixException::incorrectMatrixSizeError, parent, pColNum, pRowNum, vpArray2D< double >::rowNum, vpArray2D< double >::rowPtrs, and vpMatrixException::subMatrixError.

◆ insert() [1/5]

vpArray2D< double > vpArray2D< double >::insert	(	const vpArray2D< Type > &	A,
		const vpArray2D< Type > &	B,
		unsigned int	r,
		unsigned int	c
	)

inherited

Insert array B in array A at the given position.

Parameters

A	: Main array.
B	: Array to insert.
r	: Index of the row where to add the array.
c	: Index of the column where to add the array.

Returns: Array with B insert in A.

Warning: Throw exception if the sizes of the arrays do not allow the insertion.

Definition at line 1057 of file vpArray2D.h.

◆ insert() [2/5]

void vpArray2D< double >::insert	(	const vpArray2D< Type > &	A,
		unsigned int	r,
		unsigned int	c
	)

inlineinherited

Insert array A at the given position in the current array.

Warning: Throw vpException::dimensionError if the dimensions of the matrices do not allow the operation.

Parameters

A	: The array to insert.
r	: The index of the row to begin to insert data.
c	: The index of the column to begin to insert data.

Definition at line 484 of file vpArray2D.h.

◆ insert() [3/5]

vpMatrix vpMatrix::insert	(	const vpMatrix &	A,
		const vpMatrix &	B,
		unsigned int	r,
		unsigned int	c
	)

staticinherited

Insert matrix B in matrix A at the given position.

Parameters

A	: Main matrix.
B	: Matrix to insert.
r	: Index of the row where to add the matrix.
c	: Index of the column where to add the matrix.

Returns: Matrix with B insert in A.

Warning: Throw exception if the sizes of the matrices do not allow the insertion.

Definition at line 5273 of file vpMatrix.cpp.

References vpArray2D< Type >::insert(), and vpMatrix::vpMatrix().

◆ insert() [4/5]

void insert	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C,
		unsigned int	r,
		unsigned int	c
	)

staticinherited

Insert matrix B in matrix A at the given position.

Parameters

A	: Main matrix.
B	: Matrix to insert.
C	: Result matrix.
r	: Index of the row where to insert matrix B.
c	: Index of the column where to insert matrix B.

Warning: Throw exception if the sizes of the matrices do not allow the insertion.

Definition at line 5295 of file vpMatrix.cpp.

References vpArray2D< Type >::insert().

◆ insert() [5/5]

void vpMatrix::insert	(	const vpMatrix &	A,
		unsigned int	r,
		unsigned int	c
	)

inherited

Insert matrix A at the given position in the current matrix.

Warning: Throw vpException::dimensionError if the dimensions of the matrices do not allow the operation.

Parameters

A	: The matrix to insert.
r	: The index of the row to begin to insert data.
c	: The index of the column to begin to insert data.

Examples: testMatrix.cpp, and testMatrixPseudoInverse.cpp.

Definition at line 5781 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< double >::data, vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::rowNum, and vpArray2D< Type >::size().

Referenced by vpMbDepthDenseTracker::computeVVSInteractionMatrixAndResidu(), vpMbDepthNormalTracker::computeVVSInteractionMatrixAndResidu(), vpMbGenericTracker::computeVVSInteractionMatrixAndResidu(), vpMatrix::cond(), vpNurbs::curveKnotIns(), vpMatrix::juxtaposeMatrices(), vpMatrix::kernel(), vpMatrix::nullSpace(), vpRobotKinova::setCartVelocity(), and vpMatrix::stack().

◆ inverseByCholesky()

vpMatrix vpMatrix::inverseByCholesky ( ) const

inherited

Compute the inverse of a n-by-n matrix using the Cholesky decomposition. The matrix must be real symmetric positive defined.

This function calls the first following function that is available:

inverseByCholeskyLapack() if Lapack 3rd party is installed
inverseByLUOpenCV() if OpenCV 3rd party is installed.

If none of these 3rd parties is installed we use a Lapack built-in version.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  // Symmetric matrix
  A[0][0] = 1/1.; A[0][1] = 1/5.; A[0][2] = 1/6.; A[0][3] = 1/7.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/3.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Compute the inverse
  vpMatrix A_1; // A^-1
  A_1 = A.inverseByCholesky();
  std::cout << "Inverse by Cholesky: \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: pseudoInverse()

Definition at line 112 of file vpMatrix_cholesky.cpp.

References vpException::fatalError, vpMatrix::inverseByCholeskyLapack(), and vpMatrix::inverseByCholeskyOpenCV().

◆ inverseByCholeskyLapack()

vpMatrix vpMatrix::inverseByCholeskyLapack ( ) const

inherited

Compute the inverse of a n-by-n matrix using the Cholesky decomposition with Lapack 3rd party. The matrix must be real symmetric positive defined.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  unsigned int n = 4;
  vpMatrix A(n, n);
  vpMatrix I;
  I.eye(4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Make matrix symmetric positive
  A = 0.5*(A+A.t());
  A = A + n*I;
 
  // Compute the inverse
  vpMatrix A_1 = A.inverseByCholeskyLapack();
  std::cout << "Inverse by Cholesky (Lapack): \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByCholesky(), inverseByCholeskyOpenCV()

Definition at line 162 of file vpMatrix_cholesky.cpp.

References vpException::badValue, vpArray2D< double >::colNum, vpArray2D< Type >::data, vpException::fatalError, vpArray2D< Type >::getCols(), vpArray2D< double >::getRows(), vpArray2D< Type >::getRows(), vpMatrixException::matrixError, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::inverseByCholesky().

◆ inverseByCholeskyOpenCV()

vpMatrix vpMatrix::inverseByCholeskyOpenCV ( ) const

inherited

Compute the inverse of a n-by-n matrix using the Cholesky decomposition with OpenCV 3rd party. The matrix must be real symmetric positive defined.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  unsigned int n = 4;
  vpMatrix A(n, n);
  vpMatrix I;
  I.eye(4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Make matrix symmetric positive
  A = 0.5*(A+A.t());
  A = A + n*I;
 
  // Compute the inverse
  vpMatrix A_1 = A.inverseByCholeskyOpenCV();
  std::cout << "Inverse by Cholesky (OpenCV): \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByCholesky(), inverseByCholeskyLapack()

Definition at line 255 of file vpMatrix_cholesky.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::fatalError, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::inverseByCholesky().

◆ inverseByLU()

vpMatrix vpMatrix::inverseByLU ( ) const

inherited

Compute the inverse of a n-by-n matrix using the LU decomposition.

This function calls the first following function that is available:

inverseByLULapack() if Lapack 3rd party is installed
inverseByLUEigen3() if Eigen3 3rd party is installed
inverseByLUOpenCV() if OpenCV 3rd party is installed

If none of these previous 3rd parties is installed, we use by default inverseByLULapack() with a Lapack built-in version.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Compute the inverse
  vpMatrix A_1 = A.inverseByLU();
 
  std::cout << "Inverse by LU ";
#if defined(VISP_HAVE_LAPACK)
  std::cout << "(using Lapack)";
#elif defined(VISP_HAVE_EIGEN3)
  std::cout << "(using Eigen3)";
#elif defined(VISP_HAVE_OPENCV)
  std::cout << "(using OpenCV)";
#endif
  std::cout << ": \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByLULapack(), inverseByLUEigen3(), inverseByLUOpenCV(), pseudoInverse()

Examples: photometricVisualServoingWithoutVpServo.cpp, and testMatrixConditionNumber.cpp.

Definition at line 127 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpMatrix::det(), vpException::fatalError, vpMatrix::inverseByLUEigen3(), vpMatrix::inverseByLULapack(), vpMatrix::inverseByLUOpenCV(), vpArray2D< Type >::resize(), and vpArray2D< double >::rowNum.

◆ inverseByLUEigen3()

vpMatrix vpMatrix::inverseByLUEigen3 ( ) const

inherited

Compute the inverse of a n-by-n matrix using the LU decomposition with Eigen3 3rd party.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Compute the inverse
  vpMatrix A_1; // A^-1
  A_1 = A.inverseByLUEigen3();
  std::cout << "Inverse by LU (Eigen3): \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByLU(), inverseByLULapack(), inverseByLUOpenCV()

Definition at line 573 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::fatalError, vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), and vpArray2D< double >::rowNum.

Referenced by vpMatrix::inverseByLU().

◆ inverseByLULapack()

vpMatrix vpMatrix::inverseByLULapack ( ) const

inherited

Compute the inverse of a n-by-n matrix using the LU decomposition with Lapack 3rd party.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Compute the inverse
  vpMatrix A_1; // A^-1
  A_1 = A.inverseByLULapack();
  std::cout << "Inverse by LU (Lapack): \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByLU(), inverseByLUEigen3(), inverseByLUOpenCV()

Definition at line 282 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpException::fatalError, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::inverseByLU().

◆ inverseByLUOpenCV()

vpMatrix vpMatrix::inverseByLUOpenCV ( ) const

inherited

Compute the inverse of a n-by-n matrix using the LU decomposition with OpenCV 3rd party.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Compute the inverse
  vpMatrix A_1; // A^-1
  A_1 = A.inverseByLUOpenCV();
  std::cout << "Inverse by LU (OpenCV): \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByLU(), inverseByLUEigen3(), inverseByLULapack()

Definition at line 484 of file vpMatrix_lu.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::fatalError, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::inverseByLU().

◆ inverseByQR()

vpMatrix vpMatrix::inverseByQR ( ) const

inherited

Compute the inverse of a n-by-n matrix using the QR decomposition. Only available if Lapack 3rd party is installed. If Lapack is not installed we use a Lapack built-in version.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Compute the inverse
  vpMatrix A_1 = A.inverseByQR();
  std::cout << "Inverse by QR: \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByLU(), inverseByCholesky()

Definition at line 381 of file vpMatrix_qr.cpp.

References vpException::fatalError, and vpMatrix::inverseByQRLapack().

Referenced by vpLinProg::simplex().

◆ inverseByQRLapack()

vpMatrix vpMatrix::inverseByQRLapack ( ) const

inherited

Compute the inverse of a n-by-n matrix using the QR decomposition with Lapack 3rd party.

Returns: The inverse matrix.

Here an example:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(4,4);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.; A[0][3] = 1/4.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.; A[1][3] = 1/5.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.; A[2][3] = 1/6.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.; A[3][3] = 1/7.;
 
  // Compute the inverse
  vpMatrix A_1 = A.inverseByQRLapack();
  std::cout << "Inverse by QR: \n" << A_1 << std::endl;
 
  std::cout << "A*A^-1: \n" << A * A_1 << std::endl;
}

See also: inverseByQR()

Definition at line 151 of file vpMatrix_qr.cpp.

References vpException::badValue, vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::getRows(), vpMatrixException::matrixError, vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< double >::rowNum.

Referenced by vpMatrix::inverseByQR().

◆ inverseTriangular()

vpMatrix vpMatrix::inverseTriangular ( bool upper = true ) const

inherited

Compute the inverse of a full-rank n-by-n triangular matrix. Only available if Lapack 3rd party is installed. If Lapack is not installed we use a Lapack built-in version.

Parameters

upper : if it is an upper triangular matrix

The function does not check if the matrix is actually upper or lower triangular.

Returns: The inverse matrix

Definition at line 1010 of file vpMatrix_qr.cpp.

References vpException::badValue, vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< Type >::data, vpException::dimensionError, vpException::fatalError, vpMatrixException::rankDeficient, vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< double >::rowNum.

Referenced by vpLinProg::colReduction(), vpLinProg::rowReduction(), and vpMatrix::solveByQR().

◆ juxtaposeMatrices() [1/2]

vpMatrix vpMatrix::juxtaposeMatrices	(	const vpMatrix &	A,
		const vpMatrix &	B
	)

staticinherited

Juxtapose to matrices C = [ A B ].

$C = \left( \begin{array}{cc} A & B \end{array}\right)$

Parameters

A	: Left matrix.
B	: Right matrix.

Returns: Juxtaposed matrix C = [ A B ]

Warning: A and B must have the same number of rows.

Examples: testMatrix.cpp.

Definition at line 5315 of file vpMatrix.cpp.

Referenced by vpLinProg::colReduction().

◆ juxtaposeMatrices() [2/2]

void juxtaposeMatrices	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C
	)

staticinherited

Juxtapose to matrices C = [ A B ].

$C = \left( \begin{array}{cc} A & B \end{array}\right)$

Parameters

A	: Left matrix.
B	: Right matrix.
C	: Juxtaposed matrix C = [ A B ]

Warning: A and B must have the same number of rows.

Definition at line 5336 of file vpMatrix.cpp.

References vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpMatrix::insert(), and vpArray2D< Type >::resize().

◆ kernel()

unsigned int vpMatrix::kernel	(	vpMatrix &	kerAt,
		double	svThreshold = `1e-6`
	)		const

inherited

Function to compute the null space (the kernel) of a m-by-n matrix $\bf A$ .

The null space of a matrix $\bf A$ is defined as $\mbox{Ker}({\bf A}) = { {\bf X} : {\bf A}*{\bf X} = {\bf 0}}$ .

Parameters

kerAt	The matrix that contains the null space (kernel) of $\bf A$ defined by the matrix ${\bf X}^T$ . If matrix $\bf A$ is full rank, the dimension of `kerAt` is (0, n), otherwise the dimension is (n-r, n). This matrix is thus the transpose of $\mbox{Ker}({\bf A})$ .
svThreshold	Threshold used to test the singular values. If a singular value is lower than this threshold we consider that the matrix is not full rank.

Returns: The rank of the matrix.

Examples: servoViper850Point2DArtVelocity-jointAvoidance-basic.cpp, and testMatrixConditionNumber.cpp.

Definition at line 6056 of file vpMatrix.cpp.

References vpMatrix::getCol(), vpArray2D< double >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::getRows(), vpMatrix::insert(), vpColVector::resize(), vpArray2D< Type >::resize(), vpColVector::sumSquare(), and vpMatrix::svd().

◆ kron() [1/4]

vpMatrix vpMatrix::kron ( const vpMatrix & m ) const

inherited

Compute Kronecker product matrix.

Parameters

m	: vpMatrix;

Returns: m1.kron(m2) The kronecker product : $m1 \otimes m2$

Definition at line 1958 of file vpMatrix.cpp.

References vpMatrix::kron().

◆ kron() [2/4]

vpMatrix vpMatrix::kron	(	const vpMatrix &	m1,
		const vpMatrix &	m2
	)

staticinherited

Compute Kronecker product matrix.

Parameters

m1	: vpMatrix;
m2	: vpMatrix;

Returns: The kronecker product : $m1 \otimes m2$

Definition at line 1927 of file vpMatrix.cpp.

References vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), and vpArray2D< Type >::resize().

◆ kron() [3/4]

void vpMatrix::kron	(	const vpMatrix &	m1,
		const vpMatrix &	m2,
		vpMatrix &	out
	)

staticinherited

Compute Kronecker product matrix.

Parameters

m1	: vpMatrix;
m2	: vpMatrix;
out	: The kronecker product : $m1 \otimes m2$

Definition at line 1889 of file vpMatrix.cpp.

References vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), and vpArray2D< Type >::resize().

◆ kron() [4/4]

void vpMatrix::kron	(	const vpMatrix &	m,
		vpMatrix &	out
	)		const

inherited

Compute Kronecker product matrix.

Parameters

m	: vpMatrix.
out	: If m1.kron(m2) out contains the kronecker product's result : $m1 \otimes m2$ .

Definition at line 1919 of file vpMatrix.cpp.

Referenced by vpMatrix::kron().

◆ load()

static bool vpArray2D< double >::load	(	const std::string &	filename,
		vpArray2D< Type > &	A,
		bool	binary = `false`,
		char *	header = `nullptr`
	)

inlinestaticinherited

Load a matrix from a file.

Parameters

filename	: Absolute file name.
A	: Array to be loaded
binary	: If true the matrix is loaded from a binary file, else from a text file.
header	: Header of the file is loaded in this parameter.

Returns: Returns true if success.

See also: save()

Definition at line 653 of file vpArray2D.h.

◆ loadMatrix()

static bool vpMatrix::loadMatrix	(	const std::string &	filename,
		vpArray2D< double > &	M,
		bool	binary = `false`,
		char *	header = `nullptr`
	)

inlinestaticinherited

Load a matrix from a file. This function overloads vpArray2D::load().

Parameters

filename	: Absolute file name.
M	: Matrix to be loaded.
binary	: If true the matrix data are considered as binary, otherwise as human readable (text) data. Using binary data allows to keep data precision.
header	: Header of the file is loaded in this parameter.

Returns: Returns true if success, false otherwise.

The following example shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  std::string filename("matrix.bin");
  bool binary_data = true;
  {
    vpMatrix M(2, 3);
    M[0][0] = -1; M[0][1] =  -2; M[0][2] = -3;
    M[1][0] =  4; M[1][1] = 5.5; M[1][2] =  6.0f;
 
    std::string header("My header");
 
    if (vpMatrix::saveMatrix(filename, M, binary_data, header.c_str())) {
      std::cout << "Matrix saved in " << filename << std::endl;
      M.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot save matrix in " << filename << std::endl;
    }
  }
  {
    vpMatrix N;
    char header[FILENAME_MAX];
    if (vpMatrix::loadMatrix(filename, N, binary_data, header)) {
      std::cout << "Matrix loaded from " << filename << std::endl;
      N.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot load matrix from " << filename << std::endl;
    }
  }
}

The output of this example is the following:

Matrix saved in matrix.bin
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0
Matrix loaded from matrix.bin
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0

And the content of matrix.bin file where data are saved as binary data is the following:

% cat matrix.bin
My header??@@@%

See also: saveMatrix(), saveMatrixYAML(), loadMatrixYAML()

Examples: testMatrix.cpp.

Definition at line 748 of file vpMatrix.h.

References vpArray2D< Type >::load().

Referenced by vpDot2::defineDots().

◆ loadMatrixYAML()

static bool vpMatrix::loadMatrixYAML	(	const std::string &	filename,
		vpArray2D< double > &	M,
		char *	header = `nullptr`
	)

inlinestaticinherited

Load a matrix from a YAML-formatted file. This function overloads vpArray2D::loadYAML().

Parameters

filename	: Absolute YAML file name.
M	: Matrix to be loaded from the file.
header	: Header of the file is loaded in this parameter.

Returns: Returns true when success, false otherwise.

The following example shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  std::string filename("matrix.yaml");
  {
    vpMatrix M(2, 3);
    M[0][0] = -1; M[0][1] =  -2; M[0][2] = -3;
    M[1][0] =  4; M[1][1] = 5.5; M[1][2] =  6.0f;
 
    std::string header("My header");
 
    if (vpMatrix::saveMatrixYAML(filename, M, header.c_str())) {
      std::cout << "Matrix saved in " << filename << std::endl;
      M.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot save matrix in " << filename << std::endl;
    }
  }
  {
    vpMatrix N;
    char header[FILENAME_MAX];
    if (vpMatrix::loadMatrixYAML(filename, N, header)) {
      std::cout << "Matrix loaded from " << filename << std::endl;
      N.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot load matrix from " << filename << std::endl;
    }
  }
}

The output of this example is the following:

Matrix saved in matrix.yaml
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0
Matrix loaded from matrix.yaml
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0

And the content of matrix.yaml file is the following:

% cat matrix.yaml
My header
rows: 2
cols: 3
data:
  - [-1, -2, -3]
  - [4, 5.5, 6]

See also: saveMatrixYAML(), saveMatrix(), loadMatrix()

Examples: testMatrix.cpp.

Definition at line 823 of file vpMatrix.h.

References vpArray2D< Type >::loadYAML().

◆ loadYAML()

static bool vpArray2D< double >::loadYAML	(	const std::string &	filename,
		vpArray2D< Type > &	A,
		char *	header = `nullptr`
	)

inlinestaticinherited

Load an array from a YAML-formatted file.

Parameters

filename	: absolute file name.
A	: array to be loaded from the file.
header	: header of the file is loaded in this parameter.

Returns: Returns true on success.

See also: saveYAML()

Examples: servoFlirPtuIBVS.cpp, servoFrankaIBVS.cpp, servoFrankaPBVS.cpp, servoUniversalRobotsIBVS.cpp, servoUniversalRobotsPBVS.cpp, tutorial-flir-ptu-ibvs.cpp, tutorial-hsv-segmentation-pcl-viewer.cpp, tutorial-hsv-segmentation-pcl.cpp, and tutorial-hsv-segmentation.cpp.

Definition at line 767 of file vpArray2D.h.

◆ maplePrint()

std::ostream & vpMatrix::maplePrint ( std::ostream & os ) const

inherited

Print using Maple syntax, to copy/paste in Maple later.

The following code

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(2,3);
  int cpt = 0;
  for (unsigned int i=0; i<M.getRows(); i++)
    for (unsigned int j=0; j<M.getCols(); j++)
      M[i][j] = cpt++;
 
  std::cout << "M = "; M.maplePrint(std::cout);
}

produces this output:

M = ([
[0, 1, 2, ],
[3, 4, 5, ],
])

that could be copy/paste in Maple.

Definition at line 5548 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), and vpArray2D< double >::getRows().

◆ matlabPrint()

std::ostream & vpMatrix::matlabPrint ( std::ostream & os ) const

inherited

Print using Matlab syntax, to copy/paste in Matlab later.

The following code

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(2,3);
  int cpt = 0;
  for (unsigned int i=0; i<M.getRows(); i++)
    for (unsigned int j=0; j<M.getCols(); j++)
      M[i][j] = cpt++;
 
  std::cout << "M = "; M.matlabPrint(std::cout);
}

produces this output:

M = [ 0, 1, 2, ;

3, 4, 5, ]

that could be copy/paste in Matlab:

>> M = [ 0, 1, 2, ;
3, 4, 5, ]
 
M =
 
    0    1    2
    3    4    5
 
>>

Definition at line 5501 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), and vpArray2D< double >::getRows().

◆ mult2Matrices() [1/4]

void vpMatrix::mult2Matrices	(	const vpMatrix &	A,
		const vpColVector &	B,
		vpColVector &	C
	)

staticinherited

Warning: This function is provided for compat with previous releases. You should rather use multMatrixVector() that is more explicit.

Operation C = A * B.

The result is placed in the third parameter C and not returned. A new matrix won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

See also: multMatrixVector()

Definition at line 1189 of file vpMatrix.cpp.

References vpMatrix::multMatrixVector().

◆ mult2Matrices() [2/4]

void vpMatrix::mult2Matrices	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpHomogeneousMatrix &	C
	)

staticinherited

Warning: This function is provided for compat with previous releases. You should rather use the functionalities provided in vpHomogeneousMatrix class.

Operation C = A * B.

The result is placed in the third parameter C and not returned. A new matrix won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

Exceptions

vpException::dimensionError If matrices are not 4-by-4 dimension.

Definition at line 1130 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

◆ mult2Matrices() [3/4]

void vpMatrix::mult2Matrices	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C
	)

staticinherited

Operation C = A * B.

The result is placed in the third parameter C and not returned. A new matrix won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

See also: operator*()

Definition at line 1035 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

Referenced by vpMatrix::operator*().

◆ mult2Matrices() [4/4]

void vpMatrix::mult2Matrices	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpRotationMatrix &	C
	)

staticinherited

Warning: This function is provided for compat with previous releases. You should rather use the functionalities provided in vpRotationMatrix class.

Operation C = A * B.

The result is placed in the third parameter C and not returned. A new matrix won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

Exceptions

vpException::dimensionError If matrices are not 3-by-3 dimension.

Definition at line 1093 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

◆ multMatrixVector()

void vpMatrix::multMatrixVector	(	const vpMatrix &	A,
		const vpColVector &	v,
		vpColVector &	w
	)

staticinherited

Operation w = A * v (v and w are vectors).

A new matrix won't be allocated for every use of the function (Speed gain if used many times with the same result matrix size).

See also: operator*(const vpColVector &v) const

Definition at line 984 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpColVector::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

Referenced by vpMatrix::mult2Matrices(), and vpMatrix::operator*().

◆ negateMatrix()

void vpMatrix::negateMatrix	(	const vpMatrix &	A,
		vpMatrix &	C
	)

staticinherited

Operation C = -A.

The result is placed in the second parameter C and not returned. A new matrix won't be allocated for every use of the function (Speed gain if used many times with the same result matrix size).

See also: operator-(void)

Definition at line 1617 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

Referenced by vpMatrix::operator-().

◆ nullSpace() [1/2]

unsigned int vpMatrix::nullSpace	(	vpMatrix &	kerA,
		double	svThreshold = `1e-6`
	)		const

inherited

Function to compute the null space (the kernel) of a m-by-n matrix $\bf A$ .

The null space of a matrix $\bf A$ is defined as $\mbox{Ker}({\bf A}) = { {\bf X} : {\bf A}*{\bf X} = {\bf 0}}$ .

Parameters

kerA	The matrix that contains the null space (kernel) of $\bf A$ . If matrix $\bf A$ is full rank, the dimension of `kerA` is (n, 0), otherwise its dimension is (n, n-r).
svThreshold	Threshold used to test the singular values. The dimension of kerA corresponds to the number of singular values lower than this threshold

Returns: The dimension of the nullspace, that is .

Definition at line 6130 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), vpMatrix::insert(), vpColVector::resize(), vpArray2D< Type >::resize(), and vpMatrix::svd().

Referenced by vpMeEllipse::leastSquare(), and vpMeEllipse::leastSquareRobust().

◆ nullSpace() [2/2]

unsigned int vpMatrix::nullSpace	(	vpMatrix &	kerA,
		int	dim
	)		const

inherited

Function to compute the null space (the kernel) of a m-by-n matrix $\bf A$ .

The null space of a matrix $\bf A$ is defined as $\mbox{Ker}({\bf A}) = { {\bf X} : {\bf A}*{\bf X} = {\bf 0}}$ .

Parameters

kerA	The matrix that contains the null space (kernel) of $\bf A$ . If matrix $\bf A$ is full rank, the dimension of `kerA` is (n, 0), otherwise its dimension is (n, n-r).
dim	the dimension of the null space when it is known a priori

Returns: The estimated dimension of the nullspace, that is , by using 1e-6 as threshold for the sigular values.

Definition at line 6197 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), vpMatrix::insert(), vpColVector::resize(), vpArray2D< Type >::resize(), and vpMatrix::svd().

◆ operator!=()

bool operator!= ( const vpArray2D< Type > & A ) const

inherited

Not equal to comparison operator of a 2D array.

Definition at line 509 of file vpArray2D.h.

◆ operator*() [1/8]

vpColVector vpMatrix::operator* ( const vpColVector & v ) const

inherited

Operation w = A * v (matrix A is unchanged, v and w are column vectors).

See also: multMatrixVector() to avoid matrix allocation for each use.

Definition at line 969 of file vpMatrix.cpp.

References vpMatrix::multMatrixVector().

◆ operator*() [2/8]

vpMatrix vpMatrix::operator* ( const vpForceTwistMatrix & V ) const

inherited

Operator that allow to multiply a matrix by a force/torque twist matrix. The matrix should be of dimension m-by-6.

Definition at line 1327 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< double >::data, vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator*() [3/8]

vpMatrix vpMatrix::operator* ( const vpHomogeneousMatrix & M ) const

inherited

Operator that allow to multiply a matrix by a homogeneous matrix. The matrix should be of dimension m-by-4.

Definition at line 1241 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator*() [4/8]

vpMatrix vpMatrix::operator* ( const vpMatrix & B ) const

inherited

Operation C = A * B (A is unchanged).

See also: mult2Matrices() to avoid matrix allocation for each use.

Definition at line 1198 of file vpMatrix.cpp.

References vpMatrix::mult2Matrices().

◆ operator*() [5/8]

vpMatrix vpMatrix::operator* ( const vpRotationMatrix & R ) const

inherited

Operator that allow to multiply a matrix by a rotation matrix. The matrix should be of dimension m-by-3.

Definition at line 1211 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator*() [6/8]

vpTranslationVector vpMatrix::operator* ( const vpTranslationVector & tv ) const

inherited

Operator that allows to multiply a matrix by a translation vector. The matrix should be of dimension (3x3)

Definition at line 943 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator*() [7/8]

vpMatrix vpMatrix::operator* ( const vpVelocityTwistMatrix & V ) const

inherited

Operator that allow to multiply a matrix by a velocity twist matrix. The matrix should be of dimension m-by-6.

Definition at line 1271 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< double >::data, vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator*() [8/8]

vpMatrix vpMatrix::operator* ( double x ) const

inherited

Operator that allows to multiply all the elements of a matrix by a scalar.

Definition at line 1693 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator*=()

vpMatrix & vpMatrix::operator*= ( double x )

inherited

Multiply all the element of the matrix by x : Aij = Aij * x.

Operator that allows to multiply all the elements of a matrix by a scalar.

Definition at line 1764 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator+()

vpMatrix vpMatrix::operator+ ( const vpMatrix & B ) const

inherited

Operation C = A + B (A is unchanged).

See also: add2Matrices() to avoid matrix allocation for each use.

Definition at line 1481 of file vpMatrix.cpp.

References vpMatrix::add2Matrices().

◆ operator+=() [1/2]

vpMatrix & vpMatrix::operator+= ( const vpMatrix & B )

inherited

Operation A = A + B.

Definition at line 1572 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::rowNum, vpArray2D< double >::rowPtrs, and vpArray2D< Type >::rowPtrs.

◆ operator+=() [2/2]

vpMatrix & vpMatrix::operator+= ( double x )

inherited

Add x to all the element of the matrix : Aij = Aij + x.

Definition at line 1737 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator,()

vpMatrix & vpMatrix::operator, ( double val )

inherited

Definition at line 827 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::resize(), and vpArray2D< double >::rowPtrs.

◆ operator-() [1/2]

vpMatrix vpMatrix::operator- ( void ) const

inherited

Operation C = -A (A is unchanged).

See also: negateMatrix() to avoid matrix allocation for each use.

Definition at line 1637 of file vpMatrix.cpp.

References vpMatrix::negateMatrix().

◆ operator-() [2/2]

vpMatrix vpMatrix::operator- ( const vpMatrix & B ) const

inherited

Operation C = A - B (A is unchanged).

See also: sub2Matrices() to avoid matrix allocation for each use.

Definition at line 1563 of file vpMatrix.cpp.

References vpMatrix::sub2Matrices().

◆ operator-=() [1/2]

vpMatrix & vpMatrix::operator-= ( const vpMatrix & B )

inherited

Operation A = A - B.

Definition at line 1591 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::rowNum, vpArray2D< double >::rowPtrs, and vpArray2D< Type >::rowPtrs.

◆ operator-=() [2/2]

vpMatrix & vpMatrix::operator-= ( double x )

inherited

subtract x to all the element of the matrix : Aij = Aij - x

Definition at line 1749 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator/()

vpMatrix vpMatrix::operator/ ( double x ) const

inherited

Cij = Aij / x (A is unchanged)

Definition at line 1712 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::divideByZeroError, vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator/=()

vpMatrix & vpMatrix::operator/= ( double x )

inherited

Divide all the element of the matrix by x : Aij = Aij / x.

Definition at line 1780 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::divideByZeroError, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator<<() [1/2]

vpMatrix & vpMatrix::operator<< ( double * x )

inherited

Assignment from an array of double. This method has to be used carefully since the array allocated behind x pointer should have the same dimension than the matrix.

Definition at line 810 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator<<() [2/2]

vpMatrix & vpMatrix::operator<< ( double val )

inherited

Definition at line 820 of file vpMatrix.cpp.

References vpArray2D< double >::resize(), and vpArray2D< double >::rowPtrs.

◆ operator=() [1/3]

vpSubMatrix & vpSubMatrix::operator= ( const double & x )

Operation such as subA = x.

Operation A = x.

Parameters

x	: a scalar

Definition at line 171 of file vpSubMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator=() [2/3]

vpSubMatrix & vpSubMatrix::operator= ( const vpMatrix & B )

Operation such as subA = B.

Operation A = B.

Parameters

B	: a matrix

Definition at line 123 of file vpSubMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpMatrixException::incorrectMatrixSizeError, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ operator=() [3/3]

vpSubMatrix & vpSubMatrix::operator= ( const vpSubMatrix & B )

Operation such as subA = subB.

Operation A = B.

Parameters

B	: a subMatrix

Definition at line 144 of file vpSubMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpMatrixException::incorrectMatrixSizeError, parent, pColNum, pRowNum, vpArray2D< double >::rowNum, vpArray2D< double >::rowPtrs, and vpArray2D< Type >::rowPtrs.

◆ operator==()

bool vpArray2D< double >::operator== ( const vpArray2D< Type > & A ) const

inherited

Equal to comparison operator of a 2D array.

Definition at line 505 of file vpArray2D.h.

◆ operator[]() [1/2]

double * vpArray2D< double >::operator[] ( unsigned int i )

inlineinherited

Set element $A_{ij} = x$ using A[i][j] = x.

Definition at line 591 of file vpArray2D.h.

◆ operator[]() [2/2]

double * vpArray2D< double >::operator[] ( unsigned int i ) const

inlineinherited

Get element $x = A_{ij}$ using x = A[i][j].

Definition at line 593 of file vpArray2D.h.

◆ print()

int vpMatrix::print	(	std::ostream &	s,
		unsigned int	length,
		const std::string &	intro = `""`
	)		const

inherited

Pretty print a matrix. The data are tabulated. The common widths before and after the decimal point are set with respect to the parameter length.

Parameters

s	: Stream used for the printing.
length	: The suggested width of each matrix element. If needed, the used `length` grows in order to accommodate the whole integral part, and shrinks the decimal part to print only `length` digits.
intro	: The introduction which is printed before the matrix. Can be set to zero (or omitted), in which case the introduction is not printed.

Returns: Returns the common total width for all matrix elements.

See also: std::ostream &operator<<(std::ostream &s, const vpArray2D<Type> &A)

Examples: testMatrix.cpp, and testMatrixConditionNumber.cpp.

Definition at line 5382 of file vpMatrix.cpp.

References vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), vpMath::maximum(), and vpArray2D< double >::size().

Referenced by vpServo::computeControlLaw().

◆ printSize()

void vpMatrix::printSize ( ) const

inlineinherited

Definition at line 584 of file vpMatrix.h.

References vpArray2D< Type >::getCols(), and vpArray2D< Type >::getRows().

◆ pseudoInverse() [1/10]

vpMatrix vpMatrix::pseudoInverse ( double svThreshold = 1e-6 ) const

inherited

Compute and return the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ .

Note: By default, this function uses Lapack 3rd party. It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names (Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

svThreshold : Threshold used to test the singular values. If a singular value is lower than this threshold we consider that the matrix is not full rank.

Returns: The Moore-Penros pseudo inverse .

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpMatrix A_p = A.pseudoInverse();
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518

Examples: testFrankaCartVelocity-2.cpp, testMatrixConditionNumber.cpp, testRobotViper650-frames.cpp, and testRobotViper850-frames.cpp.

Definition at line 2343 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

Referenced by vpSimulatorAfma6::computeArticularVelocity(), vpSimulatorViper850::computeArticularVelocity(), vpServo::computeControlLaw(), vpMatrix::computeCovarianceMatrix(), vpMbEdgeTracker::computeVVSFirstPhasePoseEstimation(), vpMbTracker::computeVVSPoseEstimation(), vpQuadProg::fromCanonicalCost(), vpNurbs::globalCurveApprox(), vpNurbs::globalCurveInterp(), vpHomography::inverse(), vpMeLine::leastSquare(), vpHomogeneousMatrix::mean(), vpRotationMatrix::mean(), vpPose::poseDementhonNonPlan(), vpPose::poseDementhonPlan(), vpPose::poseFromRectangle(), vpMatrix::pseudoInverse(), vpHomography::robust(), vpMatrix::solveBySVD(), and vpQuadProg::solveQPi().

◆ pseudoInverse() [2/10]

vpMatrix vpMatrix::pseudoInverse ( int rank_in ) const

inherited

Compute and return the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ .

Note: By default, this function uses Lapack 3rd party. It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names (Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

[in] rank_in : Known rank of the matrix.

Returns: The Moore-Penros pseudo inverse .

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  // This matrix rank is 2
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  int rank_in = 2;
  vpMatrix A_p = A.pseudoInverseLapack(rank_in);
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518

Definition at line 2408 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

◆ pseudoInverse() [3/10]

unsigned int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		double	svThreshold = `1e-6`
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ and return the rank of the matrix.

Note: By default, this function uses Lapack 3rd party. It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names (Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

Ap	: The Moore-Penros pseudo inverse .
svThreshold	: Threshold used to test the singular values. If a singular value is lower than this threshold we consider that the matrix is not full rank.

Returns: The rank of the matrix.

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpMatrix A_p;
  unsigned int rank = A.pseudoInverse(A_p);
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
  std::cout << "Rank: " << rank << std::endl;
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518
Rank: 2

Definition at line 2201 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

◆ pseudoInverse() [4/10]

int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		int	rank_in
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ and return the rank of the matrix.

Note: By default, this function uses Lapack 3rd party. It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names (Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

	Ap	: The Moore-Penros pseudo inverse .
[in]	rank_in	: Known rank of the matrix.

Returns: The rank of the matrix.

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  // This matrix rank is 2
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpMatrix A_p;
  int rank_in = 2;
  int rank_out = A.pseudoInverse(A_p, rank_in);
  if (rank_out != rank_in) {
    std::cout << "There is a possibility that the pseudo-inverse in wrong." << std::endl;
    std::cout << "Are you sure that the matrix rank is " << rank_in << std::endl;
  }
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
  std::cout << "Rank in : " << rank_in << std::endl;
  std::cout << "Rank out: " << rank_out << std::endl;
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518
Rank in : 2
Rank out: 2

Definition at line 2277 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

◆ pseudoInverse() [5/10]

unsigned int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold,
		vpMatrix &	imA,
		vpMatrix &	imAt
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ along with singular values, $\mbox{Im}(A)$ and $\mbox{Im}(A^T)$ and return the rank of the matrix.

See pseudoInverse(vpMatrix &, vpColVector &, double, vpMatrix &, vpMatrix &, vpMatrix &) const for a complete description of this function.

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

Ap	: The Moore-Penros pseudo inverse .
sv	Vector corresponding to matrix singular values. The size of this vector is equal to min(m, n).
svThreshold	: Threshold used to test the singular values. If a singular value is lower than this threshold we consider that the matrix is not full rank.
imA	$\mbox{Im}({\bf A})$ that is a m-by-r matrix.
imAt	$\mbox{Im}({\bf A}^T)$ that is n-by-r matrix.

Returns: The rank of the matrix $\bf A$ .

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpMatrix A_p;
  vpColVector sv;
  vpMatrix imA, imAt;
  unsigned int rank = A.pseudoInverse(A_p, sv, 1e-6, imA, imAt);
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
  std::cout << "Rank: " << rank << std::endl;
  std::cout << "Singular values: " << sv.t() << std::endl;
  imA.print(std::cout, 10, "Im(A): ");
  imAt.print(std::cout, 10, "Im(A^T): ");
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518
Rank: 2
Singular values: 6.874359351  4.443330227
Im(A): [2,2]=
  0.81458 -0.58003
  0.58003  0.81458
Im(A^T): [3,2]=
  -0.100515 -0.994397
  0.524244 -0.024967
  0.845615 -0.102722

Definition at line 4543 of file vpMatrix.cpp.

References vpMatrix::pseudoInverse().

◆ pseudoInverse() [6/10]

unsigned int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ along with singular values, $\mbox{Im}(A)$ , $\mbox{Im}(A^T)$ and $\mbox{Ker}(A)$ and return the rank of the matrix.

Note: By default, this function uses Lapack 3rd party. It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names (Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n-by-n matrix, you have to use rather inverseByLU(), inverseByCholesky(), or inverseByQR() that are kwown as faster.

Parameters

Ap	: The Moore-Penros pseudo inverse ${\bf A}^+$ .
sv	: Vector corresponding to matrix A singular values. The size of this vector is equal to min(m, n).
svThreshold	: Threshold used to test the singular values.If a singular value is lower than this threshold we consider that the matrix is not full rank.
imA	: $\mbox { Im }({ \bf A })$ that is a m - by - r matrix.
imAt	: $\mbox { Im }({ \bf A } ^ T)$ that is n - by - r matrix.
kerAt	: The matrix that contains the null space(kernel) of $\bf A$ defined by the matrix ${ \bf X } ^ T$ .If matrix $\bf A$ is full rank, the dimension of `kerAt` is(0, n), otherwise the dimension is (n - r, n). This matrix is thus the transpose of $\mbox { Ker }({ \bf A })$ .

Returns: The rank of the matrix $\bf A$ .

Here an example to compute the pseudo - inverse of a 2 - by - 3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpColVector sv;
  vpMatrix A_p, imA, imAt, kerAt;
  unsigned int rank = A.pseudoInverse(A_p, sv, 1e-6, imA, imAt, kerAt);
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
  std::cout << "Rank: " << rank << std::endl;
  std::cout << "Singular values: " << sv.t() << std::endl;
  imA.print(std::cout, 10, "Im(A): ");
  imAt.print(std::cout, 10, "Im(A^T): ");
 
  if (kerAt.size()) {
    kerAt.t().print(std::cout, 10, "Ker(A): ");
  }
  else {
    std::cout << "Ker(A) empty " << std::endl;
  }
 
  // Reconstruct matrix A from ImA, ImAt, KerAt
  vpMatrix S(rank, A.getCols());
  for (unsigned int i = 0; i< rank; i++)
    S[i][i] = sv[i];
  vpMatrix Vt(A.getCols(), A.getCols());
  Vt.insert(imAt.t(), 0, 0);
  Vt.insert(kerAt, rank, 0);
  (imA *S *Vt).print(std::cout, 10, "Im(A) * S * [Im(A^T) | Ker(A)]^T:");
}

Once build, the previous example produces the following output :

A: [2,3] =
  2  3  5
  -4  2  3
A^+(pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518
Rank: 2
Singular values: 6.874359351  4.443330227
Im(A): [2,2]=
  0.81458 -0.58003
  0.58003  0.81458
Im(A^T): [3,2] =
  -0.100515 -0.994397
  0.524244 -0.024967
  0.845615 -0.102722
Ker(A): [3,1]=
  -0.032738
  -0.851202
  0.523816
Im(A) * S * [Im(A^T) | Ker(A)]^T: [2,3]=
  2  3  5
  -4  2  3

Definition at line 4734 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

◆ pseudoInverse() [7/10]

unsigned int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold = `1e-6`
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ along with singular values and return the rank of the matrix.

Note: By default, this function uses Lapack 3rd party. It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names (Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

Ap	: The Moore-Penros pseudo inverse .
sv	Vector corresponding to matrix singular values. The size of this vector is equal to min(m, n).
svThreshold	: Threshold used to test the singular values. If a singular value is lower than this threshold we consider that the matrix is not full rank.

Returns: The rank of the matrix $\bf A$ .

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpMatrix A_p;
  vpColVector sv;
  unsigned int rank = A.pseudoInverse(A_p, sv);
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
 
  std::cout << "Rank: " << rank << std::endl;
  std::cout << "Singular values: " << sv.t() << std::endl;
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518
Rank: 2
Singular values: 6.874359351  4.443330227

Definition at line 4370 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

◆ pseudoInverse() [8/10]

int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ along with singular values and return the rank of the matrix.

Note: By default, this function uses Lapack 3rd party. It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names (Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

	Ap	: The Moore-Penros pseudo inverse .
	sv	Vector corresponding to matrix singular values. The size of this vector is equal to min(m, n).
[in]	rank_in	: Known rank of the matrix.

Returns: The rank of the matrix $\bf A$ .

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpMatrix A_p;
  vpColVector sv;
  int rank_in = 2;
  int rank_out = A.pseudoInverse(A_p, sv, rank_in);
  if (rank_out != rank_in) {
    std::cout << "There is a possibility that the pseudo-inverse in wrong." << std::endl;
    std::cout << "Are you sure that the matrix rank is " << rank_in << std::endl;
  }
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
 
  std::cout << "Rank in : " << rank_in << std::endl;
  std::cout << "Rank out: " << rank_out << std::endl;
  std::cout << "Singular values: " << sv.t() << std::endl;
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518
Rank in : 2
Rank out: 2
Singular values: 6.874359351  4.443330227

Definition at line 4453 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

◆ pseudoInverse() [9/10]

int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in,
		vpMatrix &	imA,
		vpMatrix &	imAt
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ along with singular values, $\mbox{Im}(A)$ and $\mbox{Im}(A^T)$ and return the rank of the matrix.

See pseudoInverse(vpMatrix &, vpColVector &, double, vpMatrix &, vpMatrix &, vpMatrix &) const for a complete description of this function.

Warning: To inverse a square n-by-n matrix, you have to use rather one of the following functions inverseByLU(), inverseByQR(), inverseByCholesky() that are kwown as faster.

Parameters

	Ap	: The Moore-Penros pseudo inverse .
	sv	Vector corresponding to matrix singular values. The size of this vector is equal to min(m, n).
[in]	rank_in	: Known rank of the matrix.
	imA	$\mbox{Im}({\bf A})$ that is a m-by-r matrix.
	imAt	$\mbox{Im}({\bf A}^T)$ that is n-by-r matrix.

Returns: The rank of the matrix $\bf A$ .

Here an example to compute the pseudo-inverse of a 2-by-3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpMatrix A_p;
  vpColVector sv;
  vpMatrix imA, imAt;
  int rank_in = 2;
  int rank_out = A.pseudoInverse(A_p, sv, rank_in, imA, imAt);
  if (rank_out != rank_in) {
    std::cout << "There is a possibility that the pseudo-inverse in wrong." << std::endl;
    std::cout << "Are you sure that the matrix rank is " << rank_in << std::endl;
  }
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
  std::cout << "Rank in : " << rank_in << std::endl;
  std::cout << "Rank out: " << rank_in << std::endl;
  std::cout << "Singular values: " << sv.t() << std::endl;
  imA.print(std::cout, 10, "Im(A): ");
  imAt.print(std::cout, 10, "Im(A^T): ");
}

Once build, the previous example produces the following output:

A: [2,3]=
  2  3  5
  -4  2  3
A^+ (pseudo-inverse): [3,2]=
  0.117899 -0.190782
  0.065380  0.039657
  0.113612  0.052518
Rank: 2
Singular values: 6.874359351  4.443330227
Im(A): [2,2]=
  0.81458 -0.58003
  0.58003  0.81458
Im(A^T): [3,2]=
  -0.100515 -0.994397
  0.524244 -0.024967
  0.845615 -0.102722

Definition at line 4627 of file vpMatrix.cpp.

References vpMatrix::pseudoInverse().

◆ pseudoInverse() [10/10]

int vpMatrix::pseudoInverse	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

Compute the Moore-Penros pseudo inverse of a m-by-n matrix $\bf A$ along with singular values, $\mbox{Im}(A)$ , $\mbox{Im}(A^T)$ and $\mbox { Ker }(A)$ and return the rank of the matrix.

Note: By default, this function uses Lapack 3rd party.It is also possible to use a specific 3rd party suffixing this function name with one of the following 3rd party names(Lapack, Eigen3 or OpenCV).

Warning: To inverse a square n - by - n matrix, you have to use rather inverseByLU(), inverseByCholesky(), or inverseByQR() that are kwown as faster.

Parameters

	Ap	: The Moore - Penros pseudo inverse ${ \bf A } ^ +$ .
	sv	: Vector corresponding to matrix singular values.The size of this vector is equal to min(m, n).
[in]	rank_in	: Known rank of the matrix.
	imA	: $\mbox { Im }({ \bf A })$ that is a m - by - r matrix.
	imAt	: $\mbox { Im }({ \bf A } ^T)$ that is n - by - r matrix.
	kerAt	: The matrix that contains the null space(kernel) of $\bf A$ defined by the matrix ${ \bf X } ^ T$ .If matrix $\bf A$ is full rank, the dimension of `kerAt` is(0, n), otherwise the dimension is(n - r, n).This matrix is thus the transpose of $\mbox { Ker }({ \bf A })$ .

Returns: The rank of the matrix $\bf A$ .

Here an example to compute the pseudo - inverse of a 2 - by - 3 matrix that is rank 2.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix A(2, 3);
 
  A[0][0] = 2; A[0][1] = 3; A[0][2] = 5;
  A[1][0] = -4; A[1][1] = 2; A[1][2] = 3;
 
  A.print(std::cout, 10, "A: ");
 
  vpColVector sv;
  vpMatrix A_p, imA, imAt, kerAt;
  int rank_in = 2;
  int rank_out = A.pseudoInverse(A_p, sv, rank_in, imA, imAt, kerAt);
  if (rank_out != rank_in) {
    std::cout << "There is a possibility that the pseudo-inverse in wrong." << std::endl;
    std::cout << "Are you sure that the matrix rank is " << rank_in << std::endl;
  }
 
  A_p.print(std::cout, 10, "A^+ (pseudo-inverse): ");
  std::cout << "Rank in : " << rank_in << std::endl;
  std::cout << "Rank out: " << rank_out << std::endl;
  std::cout << "Singular values: " << sv.t() << std::endl;
  imA.print(std::cout, 10, "Im(A): ");
  imAt.print(std::cout, 10, "Im(A^T): ");
 
  if (kerAt.size()) {
    kerAt.t().print(std::cout, 10, "Ker(A): ");
  }
  else {
    std::cout << "Ker(A) empty " << std::endl;
  }
 
  // Reconstruct matrix A from ImA, ImAt, KerAt
  vpMatrix S(rank, A.getCols());
  for (unsigned int i = 0; i < rank_in; i++)
    S[i][i] = sv[i];
  vpMatrix Vt(A.getCols(), A.getCols());
  Vt.insert(imAt.t(), 0, 0);
  Vt.insert(kerAt, rank, 0);
  (imA * S * Vt).print(std::cout, 10, "Im(A) * S * [Im(A^T) | Ker(A)]^T:");
}

Once build, the previous example produces the following output :

A : [2, 3] =
2  3  5
- 4  2  3
A ^ +(pseudo - inverse) : [3, 2] =
0.117899 - 0.190782
0.065380  0.039657
0.113612  0.052518
Rank in : 2
Rank out : 2
Singular values : 6.874359351  4.443330227
Im(A) : [2, 2] =
0.81458 - 0.58003
0.58003  0.81458
Im(A ^ T) : [3, 2] =
-0.100515 - 0.994397
0.524244 - 0.024967
0.845615 - 0.102722
Ker(A) : [3, 1] =
-0.032738
- 0.851202
0.523816
Im(A) * S *[Im(A ^ T) | Ker(A)] ^ T : [2, 3] =
2  3  5
- 4  2  3

Definition at line 4861 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::pseudoInverseEigen3(), vpMatrix::pseudoInverseLapack(), and vpMatrix::pseudoInverseOpenCV().

◆ pseudoInverseEigen3() [1/8]

vpMatrix vpMatrix::pseudoInverseEigen3 ( double svThreshold = 1e-6 ) const

inherited

Referenced by vpMatrix::pseudoInverse().

◆ pseudoInverseEigen3() [2/8]

vpMatrix vpMatrix::pseudoInverseEigen3 ( int rank_in ) const

inherited

◆ pseudoInverseEigen3() [3/8]

unsigned int vpMatrix::pseudoInverseEigen3	(	vpMatrix &	Ap,
		double	svThreshold = `1e-6`
	)		const

inherited

◆ pseudoInverseEigen3() [4/8]

int vpMatrix::pseudoInverseEigen3	(	vpMatrix &	Ap,
		int	rank_in
	)		const

inherited

◆ pseudoInverseEigen3() [5/8]

unsigned int vpMatrix::pseudoInverseEigen3	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

◆ pseudoInverseEigen3() [6/8]

unsigned int vpMatrix::pseudoInverseEigen3	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold = `1e-6`
	)		const

inherited

◆ pseudoInverseEigen3() [7/8]

int vpMatrix::pseudoInverseEigen3	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in
	)		const

inherited

◆ pseudoInverseEigen3() [8/8]

int vpMatrix::pseudoInverseEigen3	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

◆ pseudoInverseLapack() [1/8]

vpMatrix vpMatrix::pseudoInverseLapack ( double svThreshold = 1e-6 ) const

inherited

Referenced by vpMatrix::pseudoInverse().

◆ pseudoInverseLapack() [2/8]

vpMatrix vpMatrix::pseudoInverseLapack ( int rank_in ) const

inherited

◆ pseudoInverseLapack() [3/8]

unsigned int vpMatrix::pseudoInverseLapack	(	vpMatrix &	Ap,
		double	svThreshold = `1e-6`
	)		const

inherited

◆ pseudoInverseLapack() [4/8]

int vpMatrix::pseudoInverseLapack	(	vpMatrix &	Ap,
		int	rank_in
	)		const

inherited

◆ pseudoInverseLapack() [5/8]

unsigned int vpMatrix::pseudoInverseLapack	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

◆ pseudoInverseLapack() [6/8]

unsigned int vpMatrix::pseudoInverseLapack	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold = `1e-6`
	)		const

inherited

◆ pseudoInverseLapack() [7/8]

int vpMatrix::pseudoInverseLapack	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in
	)		const

inherited

◆ pseudoInverseLapack() [8/8]

int vpMatrix::pseudoInverseLapack	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

◆ pseudoInverseOpenCV() [1/8]

vpMatrix vpMatrix::pseudoInverseOpenCV ( double svThreshold = 1e-6 ) const

inherited

Referenced by vpMatrix::pseudoInverse().

◆ pseudoInverseOpenCV() [2/8]

vpMatrix vpMatrix::pseudoInverseOpenCV ( int rank_in ) const

inherited

◆ pseudoInverseOpenCV() [3/8]

unsigned int vpMatrix::pseudoInverseOpenCV	(	vpMatrix &	Ap,
		double	svThreshold = `1e-6`
	)		const

inherited

◆ pseudoInverseOpenCV() [4/8]

int vpMatrix::pseudoInverseOpenCV	(	vpMatrix &	Ap,
		int	rank_in
	)		const

inherited

◆ pseudoInverseOpenCV() [5/8]

unsigned int vpMatrix::pseudoInverseOpenCV	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

◆ pseudoInverseOpenCV() [6/8]

unsigned int vpMatrix::pseudoInverseOpenCV	(	vpMatrix &	Ap,
		vpColVector &	sv,
		double	svThreshold = `1e-6`
	)		const

inherited

◆ pseudoInverseOpenCV() [7/8]

int vpMatrix::pseudoInverseOpenCV	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in
	)		const

inherited

◆ pseudoInverseOpenCV() [8/8]

int vpMatrix::pseudoInverseOpenCV	(	vpMatrix &	Ap,
		vpColVector &	sv,
		int	rank_in,
		vpMatrix &	imA,
		vpMatrix &	imAt,
		vpMatrix &	kerAt
	)		const

inherited

◆ qr()

unsigned int vpMatrix::qr	(	vpMatrix &	Q,
		vpMatrix &	R,
		bool	full = `false`,
		bool	squareR = `false`,
		double	tol = `1e-6`
	)		const

inherited

Compute the QR decomposition of a (m x n) matrix of rank r. Only available if Lapack 3rd party is installed. If Lapack is not installed we use a Lapack built-in version.

Parameters

Q	: orthogonal matrix (will be modified).
R	: upper-triangular matrix (will be modified).
full	: whether or not we want full decomposition.
squareR	: will return only the square (min(m,n) x min(m,n)) part of R.
tol	: tolerance to test the rank of R.

Returns: The rank r of the matrix.

If full is false (default) then Q is (m x min(n,m)) and R is (min(n,m) x n). We then have this = QR.

If full is true and m > n then Q is (m x m) and R is (n x n). In this case this = Q (R, 0)^T

If squareR is true and n > m then R is (m x m). If r = m then R is invertible.

Here an example:

#include <visp3/core/vpMatrix.h>
 
double residual(vpMatrix M1, vpMatrix M2)
{
    return (M1 - M2).frobeniusNorm();
}
 
int main()
{
  vpMatrix A(4,3);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/3.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/2.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/6.;
 
  // Economic QR (Q 4x3, R 3x3)
  vpMatrix Q, R;
  int r = A.qr(A, R);
  std::cout << "QR Residual: "
            << residual(A, Q*R) << std::endl;
 
  // Full QR (Q 4x4, R 3x3)
  r = A.qr(Q, R, true);
  std::cout << "Full QR Residual: "
            << residual(A, Q.extract(0, 0, 4, 3)*R) << std::endl;
}

See also: qrPivot()

Definition at line 444 of file vpMatrix_qr.cpp.

References vpException::badValue, vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::fatalError, vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< double >::rowNum.

Referenced by vpLinProg::colReduction().

◆ qrPivot()

unsigned int vpMatrix::qrPivot	(	vpMatrix &	Q,
		vpMatrix &	R,
		vpMatrix &	P,
		bool	full = `false`,
		bool	squareR = `false`,
		double	tol = `1e-6`
	)		const

inherited

Compute the QR pivot decomposition of a (m x n) matrix of rank r. Only available if Lapack 3rd party is installed. If Lapack is not installed we use a Lapack built-in version.

Parameters

Q	: orthogonal matrix (will be modified).
R	: upper-triangular matrix (will be modified).
P	: the (n x n) permutation matrix.
full	: whether or not we want full decomposition.
squareR	: will return only the (r x r) part of R and the (r x n) part of P.
tol	: tolerance to test the rank of R.

Returns: The rank r of the matrix.

If full is false (default) then Q is (m x min(n,m)) and R is (min(n,m) x n). We then have this.P = Q.R.

If full is true and m > n then Q is (m x m) and R is (n x n). In this case this.P = Q (R, 0)^T

If squareR is true then R is (r x r) invertible.

Here an example:

#include <visp3/core/vpMatrix.h>
 
double residual(vpMatrix M1, vpMatrix M2)
{
    return (M1 - M2).frobeniusNorm();
}
 
int main()
{
  vpMatrix A(4,3);
 
  A[0][0] = 1/1.; A[0][1] = 1/2.; A[0][2] = 1/2.;
  A[1][0] = 1/5.; A[1][1] = 1/3.; A[1][2] = 1/3.;
  A[2][0] = 1/6.; A[2][1] = 1/4.; A[2][2] = 1/4.;
  A[3][0] = 1/7.; A[3][1] = 1/5.; A[3][2] = 1/5.;
  // A is (4x3) but rank 2
 
  // Economic QR (Q 4x3, R 3x3)
  vpMatrix Q, R, P;
  int r = A.qrPivot(Q, R, P);
  std::cout << "A rank: " << r << std::endl;
  std::cout << "Residual: " << residual(A*P, Q*R) << std::endl;
 
  // Full QR (Q 4x4, R 3x3)
  r = A.qrPivot(Q, R, P, true);
  std::cout << "QRPivot Residual: " <<
  residual(A*P, Q.extract(0, 0, 4, 3)*R) << std::endl;
 
  // Using permutation matrix: keep only non-null part of R
  Q.resize(4, r, false);            // Q is 4 x 2
  R = R.extract(0, 0, r, 3)*P.t();  // R is 2 x 3
  std::cout << "Full QRPivot Residual: " <<
  residual(A, Q*R) << std::endl;
}

See also: qrPivot()

Definition at line 726 of file vpMatrix_qr.cpp.

References vpException::badValue, vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::fatalError, vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< double >::rowNum.

Referenced by vpLinProg::colReduction(), vpLinProg::rowReduction(), and vpMatrix::solveByQR().

◆ reshape()

void vpArray2D< double >::reshape	(	unsigned int	nrows,
		unsigned int	ncols
	)

inlineinherited

Examples: testMatrixInitialization.cpp.

Definition at line 443 of file vpArray2D.h.

◆ resize()

void vpArray2D< double >::resize	(	unsigned int	nrows,
		unsigned int	ncols,
		bool	flagNullify = `true`,
		bool	recopy_ = `true`
	)

inlineinherited

Set the size of the array and initialize all the values to zero.

Parameters

nrows	: number of rows.
ncols	: number of column.
flagNullify	: if true, then the array is re-initialized to 0 after resize. If false, the initial values from the common part of the array (common part between old and new version of the array) are kept. Default value is true.
recopy_	: if true, will perform an explicit recopy of the old data.

Examples: testArray2D.cpp, testMatrix.cpp, testMatrixDeterminant.cpp, testMatrixInverse.cpp, testMatrixPseudoInverse.cpp, and testSvd.cpp.

Definition at line 352 of file vpArray2D.h.

◆ row()

vpRowVector vpMatrix::row ( unsigned int i )

inherited

Deprecated:: This method is deprecated. You should rather use getRow(). More precisely, the following code:

vpMatrix L;
unsigned int row_index = ...;
... = L.row(row_index);

should be replaced with:

... = L.getRow(row_index - 1);

Warning: Notice row(1) is the 0th row. This function returns the i-th row of the matrix.

Parameters

i	: Index of the row to extract noting that row index start at 1 to get the first row.

Definition at line 6653 of file vpMatrix.cpp.

References vpArray2D< double >::getCols().

◆ save()

static bool vpArray2D< double >::save	(	const std::string &	filename,
		const vpArray2D< Type > &	A,
		bool	binary = `false`,
		const char *	header = `""`
	)

inlinestaticinherited

Save a matrix to a file.

Parameters

filename	: Absolute file name.
A	: Array to be saved.
binary	: If true the matrix is saved in a binary file, else a text file.
header	: Optional line that will be saved at the beginning of the file.

Returns: Returns true if success.

Warning : If you save the matrix as in a text file the precision is less than if you save it in a binary file.

See also: load()

Definition at line 855 of file vpArray2D.h.

◆ saveMatrix()

static bool vpMatrix::saveMatrix	(	const std::string &	filename,
		const vpArray2D< double > &	M,
		bool	binary = `false`,
		const char *	header = `""`
	)

inlinestaticinherited

Save a matrix to a file. This function overloads vpArray2D::save().

Parameters

filename	: Absolute file name.
M	: Matrix to be saved.
binary	: If true the matrix is save as a binary file, otherwise as a text file.
header	: Optional line that will be saved at the beginning of the file as a header.

Returns: Returns true if no problem appends.

Warning: If you save the matrix as a text file the precision is less than if you save it as a binary file.

The following example shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  std::string filename("matrix.bin");
  bool binary_data = true;
  {
    vpMatrix M(2, 3);
    M[0][0] = -1; M[0][1] =  -2; M[0][2] = -3;
    M[1][0] =  4; M[1][1] = 5.5; M[1][2] =  6.0f;
 
    std::string header("My header");
 
    if (vpMatrix::saveMatrix(filename, M, binary_data, header.c_str())) {
      std::cout << "Matrix saved in " << filename << std::endl;
      M.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot save matrix in " << filename << std::endl;
    }
  }
  {
    vpMatrix N;
    char header[FILENAME_MAX];
    if (vpMatrix::loadMatrix(filename, N, binary_data, header)) {
      std::cout << "Matrix loaded from " << filename << std::endl;
      N.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot load matrix from " << filename << std::endl;
    }
  }
}

The output of this example is the following:

Matrix saved in matrix.bin
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0
Matrix loaded from matrix.bin
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0

And the content of matrix.bin file where data are saved as binary data is the following:

% cat matrix.bin
My header??@@@%

See also: loadMatrix(), saveMatrixYAML(), loadMatrixYAML()

Examples: testMatrix.cpp.

Definition at line 896 of file vpMatrix.h.

References vpArray2D< Type >::save().

Referenced by vpDot2::defineDots().

◆ saveMatrixYAML()

static bool vpMatrix::saveMatrixYAML	(	const std::string &	filename,
		const vpArray2D< double > &	M,
		const char *	header = `""`
	)

inlinestaticinherited

Save a matrix in a YAML-formatted file. This function overloads vpArray2D::saveYAML().

Parameters

filename	: Absolute file name.
M	: Matrix to be saved in the file.
header	: Optional lines that will be saved at the beginning of the file as a header.

Returns: Returns true if success.

The following example shows how to use this function:

#include <visp3/core/vpMatrix.h>
 
int main()
{
  std::string filename("matrix.yaml");
  {
    vpMatrix M(2, 3);
    M[0][0] = -1; M[0][1] =  -2; M[0][2] = -3;
    M[1][0] =  4; M[1][1] = 5.5; M[1][2] =  6.0f;
 
    std::string header("My header");
 
    if (vpMatrix::saveMatrixYAML(filename, M, header.c_str())) {
      std::cout << "Matrix saved in " << filename << std::endl;
      M.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot save matrix in " << filename << std::endl;
    }
  }
  {
    vpMatrix N;
    char header[FILENAME_MAX];
    if (vpMatrix::loadMatrixYAML(filename, N, header)) {
      std::cout << "Matrix loaded from " << filename << std::endl;
      N.print(std::cout, 10, header);
    } else {
      std::cout << "Cannot load matrix from " << filename << std::endl;
    }
  }
}

The output of this example is the following:

Matrix saved in matrix.yaml
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0
Matrix loaded from matrix.yaml
My header[2,3]=
  -1.0 -2.0 -3.0
   4.0  5.5  6.0

And the content of matrix.yaml file is the following:

% cat matrix.yaml
My header
rows: 2
cols: 3
data:
  - [-1, -2, -3]
  - [4, 5.5, 6]

See also: saveMatrix(), loadMatrix(), loadMatrixYAML()

Examples: testMatrix.cpp.

Definition at line 972 of file vpMatrix.h.

References vpArray2D< Type >::saveYAML().

◆ saveYAML()

static bool vpArray2D< double >::saveYAML	(	const std::string &	filename,
		const vpArray2D< Type > &	A,
		const char *	header = `""`
	)

inlinestaticinherited

Save an array in a YAML-formatted file.

Parameters

filename	: absolute file name.
A	: array to be saved in the file.
header	: optional lines that will be saved at the beginning of the file. Should be YAML-formatted and will adapt to the indentation if any.

Returns: Returns true if success.

Here is an example of outputs.

vpArray2D<double> M(3,4);
vpArray2D::saveYAML("matrix.yml", M, "example: a YAML-formatted header");
vpArray2D::saveYAML("matrixIndent.yml", M, "example:\n    - a YAML-formatted \
header\n    - with inner indentation");

Content of matrix.yml:

example: a YAML-formatted header
rows: 3
cols: 4
data:
  - [0, 0, 0, 0]
  - [0, 0, 0, 0]
  - [0, 0, 0, 0]

Content of matrixIndent.yml:

example:
    - a YAML-formatted header
    - with inner indentation
rows: 3
cols: 4
data:
    - [0, 0, 0, 0]
    - [0, 0, 0, 0]
    - [0, 0, 0, 0]

See also: loadYAML()

Definition at line 950 of file vpArray2D.h.

◆ setIdentity()

void vpMatrix::setIdentity ( const double & val = 1.0 )

inherited

Deprecated:: You should rather use diag(const double &)

Deprecated:: You should rather use diag(const double &)

Set the matrix diagonal elements to val. More generally set M[i][i] = val.

Definition at line 6696 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, and vpArray2D< double >::rowNum.

Referenced by vpServo::secondaryTask().

◆ setLapackMatrixMinSize()

static void vpMatrix::setLapackMatrixMinSize ( unsigned int min_size )

inlinestaticinherited

Modify default size used to determine if Blas/Lapack basic linear algebra operations are enabled.

To get more info see Tutorial: Basic linear algebra operations.

Parameters

min_size : Minimum size of rows and columns required for a matrix or a vector to use Blas/Lapack third parties like MKL, OpenBLAS, Netlib or Atlas. When matrix or vector size is lower or equal to this parameter, Blas/Lapack is not used. In that case we prefer use naive code that runs faster for small matrices.

See also: getLapackMatrixMinSize()

Definition at line 250 of file vpMatrix.h.

◆ size()

unsigned int vpArray2D< double >::size ( ) const

inlineinherited

Return the number of elements of the 2D array.

Examples: testColVector.cpp, testDisplacement.cpp, testMatrix.cpp, testMatrixInitialization.cpp, testPoseVector.cpp, testRobotViper650-frames.cpp, testRobotViper850-frames.cpp, testRotation.cpp, testRowVector.cpp, testTranslationVector.cpp, testTukeyEstimator.cpp, testUniversalRobotsGetData.cpp, tutorial-ibvs-4pts-wireframe-robot-afma6.cpp, tutorial-ibvs-4pts-wireframe-robot-viper.cpp, tutorial-mb-generic-tracker-rgbd-realsense-json.cpp, tutorial-mb-generic-tracker-rgbd-realsense.cpp, and tutorial-mb-generic-tracker-rgbd-structure-core.cpp.

Definition at line 339 of file vpArray2D.h.

◆ solveByQR() [1/2]

vpColVector vpMatrix::solveByQR ( const vpColVector & b ) const

inherited

Solve a linear system Ax = b using QR Decomposition.

Non destructive wrt. A and B.

Parameters

b	: Vector b

Returns: Vector x

Warning: If Ax = b does not have a solution, this method does not return the least-square minimizer. Use solveBySVD() to get this vector.

Here an example:

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
int main()
{
  vpMatrix A(3,3);
  A[0][0] = 4.64;
  A[0][1] = 0.288;
  A[0][2] = -0.384;
  A[1][0] = 0.288;
  A[1][1] = 7.3296;
  A[1][2] = 2.2272;
  A[2][0] = -0.384;
  A[2][1] = 2.2272;
  A[2][2] = 6.0304;
  vpColVector X(3), B(3);
  B[0] = 1;
  B[1] = 2;
  B[2] = 3;
  X = A.solveByQR(B);
  // Obtained values of X
  // X[0] = 0.2468;
  // X[1] = 0.120782;
  // X[2] = 0.468587;
  std::cout << "X:\n" << X << std::endl;
}

See also: qrPivot()

Definition at line 1205 of file vpMatrix_qr.cpp.

References vpArray2D< double >::colNum, and vpMatrix::solveByQR().

◆ solveByQR() [2/2]

void vpMatrix::solveByQR	(	const vpColVector &	b,
		vpColVector &	x
	)		const

inherited

Solve a linear system Ax = b using QR Decomposition.

Non destructive wrt. A and b.

Parameters

b	: Vector b
x	: Vector x

Warning: If Ax = b does not have a solution, this method does not return the least-square minimizer. Use solveBySVD() to get this vector.

Here an example:

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
int main()
{
  vpMatrix A(3,3);
  A[0][0] = 4.64;
  A[0][1] = 0.288;
  A[0][2] = -0.384;
  A[1][0] = 0.288;
  A[1][1] = 7.3296;
  A[1][2] = 2.2272;
  A[2][0] = -0.384;
  A[2][1] = 2.2272;
  A[2][2] = 6.0304;
  vpColVector X(3), B(3);
  B[0] = 1;
  B[1] = 2;
  B[2] = 3;
  A.solveByQR(B, X);
  // Obtained values of X
  // X[0] = 0.2468;
  // X[1] = 0.120782;
  // X[2] = 0.468587;
  std::cout << "X:\n" << X << std::endl;
}

See also: qrPivot()

Definition at line 1155 of file vpMatrix_qr.cpp.

References vpArray2D< double >::colNum, vpMatrix::extract(), vpMatrix::inverseTriangular(), vpMatrix::qrPivot(), and vpMatrix::t().

Referenced by vpMatrix::solveByQR(), and vpQuadProg::solveSVDorQR().

◆ solveBySVD() [1/2]

vpColVector vpMatrix::solveBySVD ( const vpColVector & B ) const

inherited

Solve a linear system using Singular Value Decomposition (SVD).

Non destructive wrt. A and B.

Parameters

B	: Vector .

Returns: Vector .

Here an example:

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
vpMatrix A(3,3);
 
A[0][0] = 4.64;
A[0][1] = 0.288;
A[0][2] = -0.384;
 
A[1][0] = 0.288;
A[1][1] = 7.3296;
A[1][2] = 2.2272;
 
A[2][0] = -0.384;
A[2][1] = 2.2272;
A[2][2] = 6.0304;
 
vpColVector X(3), B(3);
B[0] = 1;
B[1] = 2;
B[2] = 3;
 
X = A.solveBySVD(B);
// Obtained values of X
// X[0] = 0.2468;
// X[1] = 0.120782;
// X[2] = 0.468587;
 
std::cout << "X:\n" << X << std::endl;
}

See also: solveBySVD(const vpColVector &, vpColVector &)

Definition at line 2061 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, and vpMatrix::solveBySVD().

◆ solveBySVD() [2/2]

void vpMatrix::solveBySVD	(	const vpColVector &	b,
		vpColVector &	x
	)		const

inherited

Solve a linear system using Singular Value Decomposition (SVD).

Non destructive wrt. A and B.

Parameters

b	: Vector .
x	: Vector .

Here an example:

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
vpMatrix A(3,3);
 
A[0][0] = 4.64;
A[0][1] = 0.288;
A[0][2] = -0.384;
 
A[1][0] = 0.288;
A[1][1] = 7.3296;
A[1][2] = 2.2272;
 
A[2][0] = -0.384;
A[2][1] = 2.2272;
A[2][2] = 6.0304;
 
vpColVector X(3), B(3);
B[0] = 1;
B[1] = 2;
B[2] = 3;
 
A.solveBySVD(B, X);
 
// Obtained values of X
// X[0] = 0.2468;
// X[1] = 0.120782;
// X[2] = 0.468587;
 
std::cout << "X:\n" << X << std::endl;
}

See also: solveBySVD(const vpColVector &)

Examples: quadprog.cpp, and quadprog_eq.cpp.

Definition at line 2010 of file vpMatrix.cpp.

References vpMatrix::pseudoInverse().

Referenced by vpMeEllipse::leastSquare(), vpMeEllipse::leastSquareRobust(), vpQuadProg::solveByProjection(), vpMatrix::solveBySVD(), vpQuadProg::solveQPe(), and vpQuadProg::solveSVDorQR().

◆ stack() [1/9]

void vpMatrix::stack ( const vpColVector & c )

inherited

Stack column vector c at the right of the current matrix, or copy if the matrix has no dimensions: this = [ this c ].

Here an example for a robot velocity log matrix:

vpMatrix log;
vpColVector v(6);
for(unsigned int i = 0; i<100;i++)
{
  robot.getVelocity(vpRobot::ARTICULAR_FRAME, v);
  log.stack(v);
}

Here the log matrix has size 6 rows by 100 columns.

Definition at line 5742 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::dimensionError, vpArray2D< Type >::getRows(), vpArray2D< double >::resize(), vpArray2D< double >::rowNum, vpArray2D< double >::rowPtrs, and vpArray2D< Type >::size().

◆ stack() [2/9]

void vpMatrix::stack ( const vpMatrix & A )

inherited

Stack A at the end of the current matrix, or copy if the matrix has no dimensions : this = [ this A ]^T.

Examples: testMatrix.cpp.

Definition at line 5669 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpMatrix::insert(), vpArray2D< double >::resize(), and vpArray2D< double >::rowNum.

Referenced by vpFeatureVanishingPoint::interaction(), vpFeatureEllipse::interaction(), vpFeaturePoint::interaction(), vpFeaturePoint3D::interaction(), vpFeaturePointPolar::interaction(), vpFeatureSegment::interaction(), vpFeatureThetaU::interaction(), vpFeatureTranslation::interaction(), vpGenericFeature::interaction(), vpLinProg::rowReduction(), vpMatrix::stack(), vpColVector::stackMatrices(), and vpMatrix::stackMatrices().

◆ stack() [3/9]

vpMatrix vpMatrix::stack	(	const vpMatrix &	A,
		const vpColVector &	c
	)

staticinherited

Stack column vector c to matrix A and return the resulting matrix [ A c ]

Parameters

A	: Left matrix.
c	: Right column vector.

Returns: Stacked matrix [ A c ]

Warning: A and c must have the same number of rows.

Definition at line 5231 of file vpMatrix.cpp.

References vpMatrix::stack().

◆ stack() [4/9]

void vpMatrix::stack	(	const vpMatrix &	A,
		const vpColVector &	c,
		vpMatrix &	C
	)

staticinherited

Stack column vector c to the end of matrix A and return the resulting matrix in C.

Parameters

A	: Left matrix.
c	: Right column vector.
C	: Stacked matrix C = [ A c ]

Warning: A and c must have the same number of rows. A and C must be two different objects.

Definition at line 5250 of file vpMatrix.cpp.

References vpArray2D< Type >::data, and vpMatrix::stack().

◆ stack() [5/9]

vpMatrix vpMatrix::stack	(	const vpMatrix &	A,
		const vpMatrix &	B
	)

staticinherited

Stack matrix B to the end of matrix A and return the resulting matrix [ A B ]^T

Parameters

A	: Upper matrix.
B	: Lower matrix.

Returns: Stacked matrix [ A B ]^T

Warning: A and B must have the same number of columns.

Definition at line 5128 of file vpMatrix.cpp.

References vpMatrix::stack().

◆ stack() [6/9]

void vpMatrix::stack	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C
	)

staticinherited

Stack matrix B to the end of matrix A and return the resulting matrix in C.

Parameters

A	: Upper matrix.
B	: Lower matrix.
C	: Stacked matrix C = [ A B ]^T

Warning: A and B must have the same number of columns. A and C, B and C must be two different objects.

Definition at line 5148 of file vpMatrix.cpp.

References vpArray2D< Type >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), and vpArray2D< Type >::size().

◆ stack() [7/9]

vpMatrix vpMatrix::stack	(	const vpMatrix &	A,
		const vpRowVector &	r
	)

staticinherited

Stack row vector r to matrix A and return the resulting matrix [ A r ]^T

Parameters

A	: Upper matrix.
r	: Lower row vector.

Returns: Stacked matrix [ A r ]^T

Warning: A and r must have the same number of columns.

Definition at line 5192 of file vpMatrix.cpp.

References vpMatrix::stack().

◆ stack() [8/9]

void vpMatrix::stack	(	const vpMatrix &	A,
		const vpRowVector &	r,
		vpMatrix &	C
	)

staticinherited

Stack row vector r to the end of matrix A and return the resulting matrix in C.

Parameters

A	: Upper matrix.
r	: Lower row vector.
C	: Stacked matrix C = [ A r ]^T

Warning: A and r must have the same number of columns. A and C must be two different objects.

Definition at line 5211 of file vpMatrix.cpp.

References vpArray2D< Type >::data, and vpMatrix::stack().

◆ stack() [9/9]

void vpMatrix::stack ( const vpRowVector & r )

inherited

Stack row vector r at the end of the current matrix, or copy if the matrix has no dimensions: this = [ this r ]^T.

Here an example for a robot velocity log :

vpMatrix A;
vpColVector v(6);
for(unsigned int i = 0;i<100;i++)
{
  robot.getVelocity(vpRobot::ARTICULAR_FRAME, v);
  Velocities.stack(v.t());
}

Definition at line 5701 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< double >::resize(), vpArray2D< double >::rowNum, vpArray2D< double >::size(), and vpArray2D< Type >::size().

◆ stackColumns() [1/2]

vpColVector vpMatrix::stackColumns ( )

inherited

Stacks columns of a matrix in a vector.

Returns: a vpColVector.

Definition at line 1827 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, and vpArray2D< double >::rowNum.

◆ stackColumns() [2/2]

void vpMatrix::stackColumns ( vpColVector & out )

inherited

Stacks columns of a matrix in a vector.

Parameters

out	: a vpColVector.

Definition at line 1809 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpArray2D< double >::colNum, vpArray2D< Type >::data, vpColVector::resize(), vpArray2D< double >::rowNum, vpArray2D< Type >::rowNum, and vpArray2D< double >::rowPtrs.

◆ stackMatrices() [1/7]

vpMatrix vpMatrix::stackMatrices	(	const vpColVector &	A,
		const vpColVector &	B
	)

staticinherited

Deprecated:: You should rather use vpColVector::stack(const vpColVector &A, const vpColVector &B)

Definition at line 6621 of file vpMatrix.cpp.

References vpColVector::stack().

◆ stackMatrices() [2/7]

void vpMatrix::stackMatrices	(	const vpColVector &	A,
		const vpColVector &	B,
		vpColVector &	C
	)

staticinherited

Deprecated:: You should rather use vpColVector::stack(const vpColVector &A, const vpColVector &B, vpColVector &C)

Definition at line 6626 of file vpMatrix.cpp.

References vpColVector::stack().

◆ stackMatrices() [3/7]

vp_deprecated void vpMatrix::stackMatrices ( const vpMatrix & A )

inlineinherited

Deprecated:: You should rather use stack(const vpMatrix &A)

Definition at line 994 of file vpMatrix.h.

◆ stackMatrices() [4/7]

static vp_deprecated vpMatrix vpMatrix::stackMatrices	(	const vpMatrix &	A,
		const vpMatrix &	B
	)

inlinestaticinherited

Deprecated:: You should rather use stack(const vpMatrix &A, const vpMatrix &B)

Definition at line 999 of file vpMatrix.h.

◆ stackMatrices() [5/7]

static vp_deprecated void vpMatrix::stackMatrices	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C
	)

inlinestaticinherited

Deprecated:: You should rather use stack(const vpMatrix &A, const vpMatrix &B, vpMatrix &C)

Definition at line 1004 of file vpMatrix.h.

◆ stackMatrices() [6/7]

vpMatrix vpMatrix::stackMatrices	(	const vpMatrix &	A,
		const vpRowVector &	B
	)

staticinherited

Deprecated:: You should rather use stack(const vpMatrix &A, const vpMatrix &B)

Definition at line 6631 of file vpMatrix.cpp.

References vpMatrix::stack().

◆ stackMatrices() [7/7]

void vpMatrix::stackMatrices	(	const vpMatrix &	A,
		const vpRowVector &	B,
		vpMatrix &	C
	)

staticinherited

Deprecated:: You should rather use stack(const vpMatrix &A, const vpRowVector &B, vpMatrix &C)

Definition at line 6633 of file vpMatrix.cpp.

References vpMatrix::stack().

◆ stackRows() [1/2]

vpRowVector vpMatrix::stackRows ( )

inherited

Stacks rows of a matrix in a vector.

Returns: a vpRowVector.

Definition at line 1850 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, and vpArray2D< double >::rowNum.

◆ stackRows() [2/2]

void vpMatrix::stackRows ( vpRowVector & out )

inherited

Stacks rows of a matrix in a vector

Parameters

out	: a vpRowVector.

Definition at line 1838 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::data, vpArray2D< Type >::data, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpRowVector::resize(), and vpArray2D< double >::rowNum.

◆ sub2Matrices() [1/2]

void vpMatrix::sub2Matrices	(	const vpColVector &	A,
		const vpColVector &	B,
		vpColVector &	C
	)

staticinherited

Warning: This function is provided for compat with previous releases. You should rather use the functionalities provided in vpColVector class.

Operation C = A - B on column vectors.

The result is placed in the third parameter C and not returned. A new vector won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

Exceptions

vpException::dimensionError If A and B vectors have not the same size.

See also: vpColVector::operator-()

Definition at line 1503 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpColVector::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

◆ sub2Matrices() [2/2]

void vpMatrix::sub2Matrices	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C
	)

staticinherited

Operation C = A - B.

The result is placed in the third parameter C and not returned. A new matrix won't be allocated for every use of the function (speed gain if used many times with the same result matrix size).

Exceptions

vpException::dimensionError If A and B matrices have not the same size.

See also: operator-()

Definition at line 1537 of file vpMatrix.cpp.

References vpArray2D< Type >::colNum, vpException::dimensionError, vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< Type >::resize(), vpArray2D< Type >::rowNum, and vpArray2D< Type >::rowPtrs.

Referenced by vpMatrix::operator-().

◆ sum()

double vpMatrix::sum ( ) const

inherited

Return the sum of all the $a_{ij}$ elements of the matrix.

Returns: Value of $\sum a_{ij}$

Definition at line 1644 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

Referenced by vpMatrix::expm().

◆ sumSquare()

double vpMatrix::sumSquare ( ) const

inherited

Return the sum square of all the $A_{ij}$ elements of the matrix .

Returns: The value $\sum A_{ij}^{2}$ .

Definition at line 6595 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< double >::rowNum, and vpArray2D< double >::rowPtrs.

◆ svd()

void vpMatrix::svd	(	vpColVector &	w,
		vpMatrix &	V
	)

inherited

Matrix singular value decomposition (SVD).

This function calls the first following function that is available:

svdLapack() if Lapack 3rd party is installed
svdEigen3() if Eigen3 3rd party is installed
svdOpenCV() if OpenCV 3rd party is installed.

If none of these previous 3rd parties is installed, we use by default svdLapack() with a Lapack built-in version.

Given matrix , this function computes it singular value decomposition such as

$M = U \Sigma V^{\top}$

Warning: This method is destructive wrt. to the matrix to decompose. You should make a COPY of that matrix if needed.

Parameters

w	: Vector of singular values: $\Sigma = diag(w)$ .
V	: Matrix .

The matrix object `(*this) is updated with .

Note: The singular values are ordered in decreasing fashion in w. It means that the highest singular value is in w[0].

Here an example of SVD decomposition of a non square Matrix M.

#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(3,2);
  M[0][0] = 1;   M[0][1] = 6;
  M[1][0] = 2;   M[1][1] = 8;
  M[2][0] = 0.5; M[2][1] = 9;
 
  vpColVector w;
  vpMatrix V, Sigma, U = M;
 
  U.svd(w, V);
 
  // Construct the diagonal matrix from the singular values
  Sigma.diag(w);
 
  // Reconstruct the initial matrix using the decomposition
  vpMatrix Mrec =  U * Sigma * V.t();
 
  // Here, Mrec is obtained equal to the initial value of M
  // Mrec[0][0] = 1;   Mrec[0][1] = 6;
  // Mrec[1][0] = 2;   Mrec[1][1] = 8;
  // Mrec[2][0] = 0.5; Mrec[2][1] = 9;
 
  std::cout << "Reconstructed M matrix: \n" << Mrec << std::endl;
}

See also: svdLapack(), svdEigen3(), svdOpenCV()

Examples: servoMomentImage.cpp.

Definition at line 2132 of file vpMatrix.cpp.

References vpException::fatalError, vpMatrix::svdEigen3(), vpMatrix::svdLapack(), and vpMatrix::svdOpenCV().

Referenced by vpHomogeneousMatrix::compute3d3dTransformation(), vpMatrix::cond(), vpHomography::DLT(), vpMbtFaceDepthNormal::estimatePlaneEquationSVD(), vpMatrix::inducedL2Norm(), vpMatrix::kernel(), vpMatrix::nullSpace(), vpRotationMatrix::orthogonalize(), and vpMatrix::svdEigen3().

◆ svdEigen3()

void vpMatrix::svdEigen3	(	vpColVector &	w,
		vpMatrix &	V
	)

inherited

Singular value decomposition (SVD) using Eigen3 3rd party.

Given matrix , this function computes it singular value decomposition such as

$M = U \Sigma V^{\top}$

Warning: This method is destructive wrt. to the matrix to decompose. You should make a COPY of that matrix if needed.

Parameters

w	: Vector of singular values: $\Sigma = diag(w)$ .
V	: Matrix .

The matrix object (*this) is updated with .

Note: The singular values are ordered in decreasing fashion in w. It means that the highest singular value is in w[0].

Here an example of SVD decomposition of a non square Matrix M.

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(3,2);
  M[0][0] = 1;
  M[1][0] = 2;
  M[2][0] = 0.5;
 
  M[0][1] = 6;
  M[1][1] = 8 ;
  M[2][1] = 9 ;
 
  vpMatrix V;
  vpColVector w;
  vpMatrix Mrec;
  vpMatrix Sigma;
 
  M.svdEigen3(w, V);
  // Here M is modified and is now equal to U
 
  // Construct the diagonal matrix from the singular values
  Sigma.diag(w);
 
  // Reconstruct the initial matrix M using the decomposition
  Mrec =  M * Sigma * V.t();
 
  // Here, Mrec is obtained equal to the initial value of M
  // Mrec[0][0] = 1;
  // Mrec[1][0] = 2;
  // Mrec[2][0] = 0.5;
  // Mrec[0][1] = 6;
  // Mrec[1][1] = 8 ;
  // Mrec[2][1] = 9 ;
 
  std::cout << "Reconstructed M matrix: \n" << Mrec << std::endl;
}

See also: svd(), svdLapack(), svdOpenCV()

Definition at line 439 of file vpMatrix_svd.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpArray2D< double >::getCols(), vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), vpArray2D< double >::getRows(), vpColVector::resize(), vpArray2D< double >::resize(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, vpArray2D< Type >::size(), and vpMatrix::svd().

Referenced by vpMatrix::svd().

◆ svdLapack()

void vpMatrix::svdLapack	(	vpColVector &	w,
		vpMatrix &	V
	)

inherited

Singular value decomposition (SVD) using Lapack 3rd party.

Given matrix , this function computes it singular value decomposition such as

$M = U \Sigma V^{\top}$

Warning: This method is destructive wrt. to the matrix to decompose. You should make a COPY of that matrix if needed.

Parameters

w	: Vector of singular values: $\Sigma = diag(w)$ .
V	: Matrix .

The matrix object (*this) is updated with .

Note: The singular values are ordered in decreasing fashion in w. It means that the highest singular value is in w[0].

Here an example of SVD decomposition of a non square Matrix M.

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(3,2);
  M[0][0] = 1;
  M[1][0] = 2;
  M[2][0] = 0.5;
 
  M[0][1] = 6;
  M[1][1] = 8 ;
  M[2][1] = 9 ;
 
  vpMatrix V;
  vpColVector w;
  vpMatrix Mrec;
  vpMatrix Sigma;
 
  M.svdLapack(w, V);
  // Here M is modified and is now equal to U
 
  // Construct the diagonal matrix from the singular values
  Sigma.diag(w);
 
  // Reconstruct the initial matrix M using the decomposition
  Mrec =  M * Sigma * V.t();
 
  // Here, Mrec is obtained equal to the initial value of M
  // Mrec[0][0] = 1;
  // Mrec[1][0] = 2;
  // Mrec[2][0] = 0.5;
  // Mrec[0][1] = 6;
  // Mrec[1][1] = 8 ;
  // Mrec[2][1] = 9 ;
 
  std::cout << "Reconstructed M matrix: \n" << Mrec << std::endl;
}

See also: svd(), svdEigen3(), svdOpenCV()

Definition at line 236 of file vpMatrix_svd.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpException::fatalError, vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), vpColVector::resize(), vpArray2D< Type >::resize(), vpArray2D< double >::rowNum, and vpMatrix::transpose().

Referenced by vpMatrix::svd().

◆ svdOpenCV()

void vpMatrix::svdOpenCV	(	vpColVector &	w,
		vpMatrix &	V
	)

inherited

Singular value decomposition (SVD) using OpenCV 3rd party.

Given matrix , this function computes it singular value decomposition such as

$M = U \Sigma V^{\top}$

Warning: This method is destructive wrt. to the matrix to decompose. You should make a COPY of that matrix if needed.

Parameters

w	: Vector of singular values: $\Sigma = diag(w)$ .
V	: Matrix .

The matrix object (*this) is updated with .

Note: The singular values are ordered in decreasing fashion in w. It means that the highest singular value is in w[0].

Here an example of SVD decomposition of a non square Matrix M.

#include <visp3/core/vpColVector.h>
#include <visp3/core/vpMatrix.h>
 
int main()
{
  vpMatrix M(3,2);
  M[0][0] = 1;
  M[1][0] = 2;
  M[2][0] = 0.5;
 
  M[0][1] = 6;
  M[1][1] = 8 ;
  M[2][1] = 9 ;
 
  vpMatrix V;
  vpColVector w;
  vpMatrix Mrec;
  vpMatrix Sigma;
 
  M.svdOpenCV(w, V);
  // Here M is modified and is now equal to U
 
  // Construct the diagonal matrix from the singular values
  Sigma.diag(w);
 
  // Reconstruct the initial matrix M using the decomposition
  Mrec =  M * Sigma * V.t();
 
  // Here, Mrec is obtained equal to the initial value of M
  // Mrec[0][0] = 1;
  // Mrec[1][0] = 2;
  // Mrec[2][0] = 0.5;
  // Mrec[0][1] = 6;
  // Mrec[1][1] = 8 ;
  // Mrec[2][1] = 9 ;
 
  std::cout << "Reconstructed M matrix: \n" << Mrec << std::endl;
}

See also: svd(), svdEigen3(), svdLapack()

Definition at line 149 of file vpMatrix_svd.cpp.

References vpArray2D< Type >::data, vpArray2D< double >::data, vpArray2D< double >::getCols(), vpArray2D< double >::getRows(), vpColVector::resize(), vpArray2D< Type >::resize(), vpArray2D< double >::resize(), and vpMatrix::transpose().

Referenced by vpMatrix::svd().

◆ t()

vpMatrix vpMatrix::t ( ) const

inherited

Compute and return the transpose of the matrix.

Examples: photometricVisualServoingWithoutVpServo.cpp, servoViper850Point2DArtVelocity-jointAvoidance-basic.cpp, testMatrixInverse.cpp, testMatrixPseudoInverse.cpp, and testSvd.cpp.

Definition at line 465 of file vpMatrix.cpp.

References vpMatrix::transpose().

Referenced by vpLinProg::colReduction(), vpHomogeneousMatrix::compute3d3dTransformation(), vpServo::computeControlLaw(), vpMatrix::computeCovarianceMatrix(), vpTemplateTracker::computeOptimalBrentGain(), vpMatrix::eigenValues(), vpMbtFaceDepthNormal::estimatePlaneEquationSVD(), vpKalmanFilter::filtering(), vpTemplateTrackerWarpHomographySL3::findWarp(), vpQuadProg::fromCanonicalCost(), vpHomogeneousMatrix::mean(), vpRotationMatrix::mean(), vpRotationMatrix::orthogonalize(), vpKalmanFilter::prediction(), vpLinProg::simplex(), and vpMatrix::solveByQR().

◆ transpose() [1/2]

vpMatrix vpMatrix::transpose ( ) const

inherited

Compute and return the transpose of the matrix.

See also: t()

Examples: testMatrixPseudoInverse.cpp.

Definition at line 472 of file vpMatrix.cpp.

Referenced by vpLinProg::colReduction(), vpQuadProg::fromCanonicalCost(), vpLinProg::rowReduction(), vpLinProg::solveLP(), vpQuadProg::solveQPi(), vpMatrix::svdLapack(), vpMatrix::svdOpenCV(), and vpMatrix::t().

◆ transpose() [2/2]

void vpMatrix::transpose ( vpMatrix & At ) const

inherited

Compute At the transpose of the matrix.

Parameters

At	(output) : Resulting transpose matrix.

See also: t()

Definition at line 484 of file vpMatrix.cpp.

References vpArray2D< double >::colNum, vpArray2D< Type >::data, vpArray2D< double >::data, vpArray2D< Type >::resize(), and vpArray2D< double >::rowNum.

Friends And Related Function Documentation

◆ insert() [1/2]

void vpArray2D< double >::insert	(	const vpArray2D< Type > &	A,
		const vpArray2D< Type > &	B,
		vpArray2D< Type > &	C,
		unsigned int	r,
		unsigned int	c
	)

Parameters

A	: Main array.
B	: Array to insert.
C	: Result array.
r	: Index of the row where to insert array B.
c	: Index of the column where to insert array B.

Warning: Throw exception if the sizes of the arrays do not allow the insertion.

Definition at line 1072 of file vpArray2D.h.

◆ insert() [2/2]

void insert	(	const vpMatrix &	A,
		const vpMatrix &	B,
		vpMatrix &	C,
		unsigned int	r,
		unsigned int	c
	)

Parameters

A	: Main matrix.
B	: Matrix to insert.
C	: Result matrix.
r	: Index of the row where to insert matrix B.
c	: Index of the column where to insert matrix B.

Warning: Throw exception if the sizes of the matrices do not allow the insertion.

Definition at line 5295 of file vpMatrix.cpp.

◆ operator!=()

bool operator!= ( const vpArray2D< Type > & A ) const

◆ operator*()

vpMatrix operator*	(	const double &	x,
		const vpMatrix &	B
	)

Definition at line 1668 of file vpMatrix.cpp.

References vpArray2D< Type >::getCols(), vpArray2D< Type >::getRows(), and vpArray2D< Type >::resize().

◆ operator==() [1/2]

bool operator== ( const vpArray2D< double > & A ) const

◆ operator==() [2/2]

bool operator== ( const vpArray2D< float > & A ) const

◆ vpGEMM()

void vpGEMM	(	const vpArray2D< double > &	A,
		const vpArray2D< double > &	B,
		const double &	alpha,
		const vpArray2D< double > &	C,
		const double &	beta,
		vpArray2D< double > &	D,
		const unsigned int &	ops = `0`
	)

For example, to compute D = alpha*A^T*B^T+beta*C we need to call :

vpGEMM(A, B, alpha, C, beta, D, VP_GEMM_A_T + VP_GEMM_B_T);

vpArray2D< double >::vpGEMM

void vpGEMM(const vpArray2D< double > &A, const vpArray2D< double > &B, const double &alpha, const vpArray2D< double > &C, const double &beta, vpArray2D< double > &D, const unsigned int &ops=0)

Definition: vpGEMM.h:388

If C is not used, vpGEMM must be called using an empty array null. Thus to compute D = alpha*A^T*B, we have to call:

vpGEMM(A, B, alpha, null, 0, D, VP_GEMM_B_T);

Exceptions

vpException::incorrectMatrixSizeError if the sizes of the matrices do not allow the operations.

Parameters

A	: An array that could be a vpMatrix.
B	: An array that could be a vpMatrix.
alpha	: A scalar.
C	: An array that could be a vpMatrix.
beta	: A scalar.
D	: The resulting array that could be a vpMatrix.
ops	: A scalar describing operation applied on the matrices. Possible values are the one defined in vpGEMMmethod(): VP_GEMM_A_T, VP_GEMM_B_T, VP_GEMM_C_T.

Definition at line 388 of file vpGEMM.h.

◆ vpGEMMmethod

enum vpGEMMmethod

Operations are :

VP_GEMM_A_T to use the transpose matrix of A instead of the matrix A
VP_GEMM_B_T to use the transpose matrix of B instead of the matrix B
VP_GEMM_C_T to use the transpose matrix of C instead of the matrix C

Definition at line 52 of file vpGEMM.h.

Member Data Documentation

◆ colNum

unsigned int vpArray2D< double >::colNum

protectedinherited

Number of columns in the array.

Definition at line 131 of file vpArray2D.h.

◆ data

double * vpArray2D< double >::data

inherited

Address of the first element of the data array.

Examples: testArray2D.cpp, testDisplacement.cpp, testMatrix.cpp, testQuaternion.cpp, testRotation.cpp, testTranslationVector.cpp, testUniversalRobotsGetData.cpp, tutorial-bridge-opencv-matrix.cpp, and tutorial-matlab.cpp.

Definition at line 139 of file vpArray2D.h.

◆ dsize

unsigned int vpArray2D< double >::dsize

protectedinherited

Current array size (rowNum * colNum)

Definition at line 135 of file vpArray2D.h.

◆ parent

vpMatrix* vpSubMatrix::parent

protected

Definition at line 67 of file vpSubMatrix.h.

Referenced by checkParentStatus(), init(), and operator=().

◆ pColNum

unsigned int vpSubMatrix::pColNum

protected

Definition at line 66 of file vpSubMatrix.h.

Referenced by checkParentStatus(), init(), and operator=().

◆ pRowNum

unsigned int vpSubMatrix::pRowNum

protected

Definition at line 65 of file vpSubMatrix.h.

Referenced by checkParentStatus(), init(), and operator=().

◆ rowNum

unsigned int vpArray2D< double >::rowNum

protectedinherited

Number of rows in the array.

Definition at line 129 of file vpArray2D.h.

◆ rowPtrs

double ** vpArray2D< double >::rowPtrs

protectedinherited

Address of the first element of each rows.

Definition at line 133 of file vpArray2D.h.

Public Types

Public Member Functions

Static Public Member Functions

Public Attributes

Protected Attributes

Related Functions

Inherited I/O from vpArray2D with Static Public Member Functions

Deprecated functions

Matrix I/O with Static Public Member Functions <br>

Detailed Description

Member Enumeration Documentation

◆ vpDetMethod

Constructor & Destructor Documentation

◆ vpSubMatrix() [1/2]

◆ vpSubMatrix() [2/2]

◆ ~vpSubMatrix()

Member Function Documentation

◆ AAt() [1/2]

◆ AAt() [2/2]

◆ add2Matrices() [1/2]

◆ add2Matrices() [2/2]

◆ add2WeightedMatrices()

◆ AtA() [1/2]

◆ AtA() [2/2]

◆ checkParentStatus()

◆ clear()

◆ column()

◆ computeCovarianceMatrix() [1/2]

◆ computeCovarianceMatrix() [2/2]

◆ computeCovarianceMatrixVVS() [1/2]

◆ computeCovarianceMatrixVVS() [2/2]

◆ computeHLM()

◆ cond()

◆ conv2() [1/2]

◆ conv2() [2/2]

◆ cppPrint()

◆ createDiagonalMatrix()

◆ csvPrint()

◆ det()

◆ detByLU()

◆ detByLUEigen3()

◆ detByLULapack()

◆ detByLUOpenCV()

◆ diag() [1/2]

◆ diag() [2/2]

◆ eigenValues() [1/2]

◆ eigenValues() [2/2]

◆ euclideanNorm()

◆ expm()

◆ extract()

◆ eye()

◆ frobeniusNorm()

◆ getCol() [1/2]

◆ getCol() [2/2]

◆ getCols()

◆ getDiag()

◆ getLapackMatrixMinSize()

◆ getMaxValue()

◆ getMinValue()

◆ getRow() [1/2]

◆ getRow() [2/2]

◆ getRows()

◆ hadamard() [1/2]

◆ hadamard() [2/2]

◆ inducedL2Norm()

◆ infinityNorm()

◆ init() [1/3]

◆ init() [2/3]

◆ init() [3/3]

◆ insert() [1/5]

◆ insert() [2/5]

◆ insert() [3/5]

◆ insert() [4/5]

◆ insert() [5/5]

◆ inverseByCholesky()

◆ inverseByCholeskyLapack()

◆ inverseByCholeskyOpenCV()

◆ inverseByLU()

◆ inverseByLUEigen3()

◆ inverseByLULapack()