Matrix¶

class Matrix(*args, **kwargs)¶

Bases: ArrayDouble2D

Implementation of a matrix and operations on matrices.

This class needs one of the following third-party to compute matrix inverse, pseudo-inverse, singular value decomposition, determinant:

If Lapack is installed and detected by ViSP, this 3rd party is used by vpMatrix . Installation instructions are provided here https://visp.inria.fr/3rd_lapack;
else if Eigen3 is installed and detected by ViSP, this 3rd party is used by vpMatrix . Installation instructions are provided here https://visp.inria.fr/3rd_eigen;
else if OpenCV is installed and detected by ViSP, this 3rd party is used, Installation instructions are provided here https://visp.inria.fr/3rd_opencv;
If none of these previous 3rd parties is installed, we use by default a Lapack built-in version.

vpMatrix class provides a data structure for the matrices as well as a set of operations on these matrices.

The vpMatrix class is derived from vpArray2D<double> .

The code below shows how to create a 2-by-3 matrix of doubles, set the element values and access them:

#include <visp3/code/vpMatrix.h

int main()
{
  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::cout << "M:" << std::endl;
  for (unsigned int i = 0; i < M.getRows(); ++i) {
    for (unsigned int j = 0; j < M.getCols(); ++j) {
      std::cout << M[i][j] << " ";
    }
    std::cout << std::endl;
  }
}

Once build, this previous code produces the following output:

M:
-1 -2 -3
4 5.5 6

If ViSP is build with c++11 enabled, you can do the same using:

#include <visp3/code/vpMatrix.h

int main()
{
  vpMatrix M( {-1, -2, -3}, {4, 5.5, 6.0f} );
  std::cout << "M:\n" << M << std::endl;
}

You can also create and initialize a matrix this way:

#include <visp3/code/vpMatrix.h

int main()
{
  vpMatrix M(2, 3, {-1, -2, -3, 4, 5.5, 6.0f} );
}

The Matrix could also be initialized using operator=(const std::initializer_list< std::initializer_list< double > > &)

int main()
{
  vpMatrix M;
  M = { {-1, -2, -3}, {4, 5.5, 6.0f} };
}

Note

See vpArray2D , vpRowVector , vpColVector , vpHomogeneousMatrix , vpRotationMatrix , vpVelocityTwistMatrix , vpForceTwistMatrix , vpHomography

Overloaded function.

__init__(self: visp._visp.core.Matrix) -> None

Basic constructor of a matrix of double. Number of columns and rows are zero.

__init__(self: visp._visp.core.Matrix, r: int, c: int) -> None

Constructor that initialize a matrix of double with 0.

Parameters:

r: Matrix number of rows.
c: Matrix number of columns.

__init__(self: visp._visp.core.Matrix, r: int, c: int, val: float) -> None

Constructor that initialize a matrix of double with val .

Parameters:

r: Matrix number of rows.
c: Matrix number of columns.
val: Each element of the matrix is set to val .

__init__(self: visp._visp.core.Matrix, M: visp._visp.core.Matrix, r: int, c: int, nrows: int, ncols: int) -> None

Construct a matrix as a sub-matrix of the input matrix M .

Note

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

__init__(self: visp._visp.core.Matrix, A: visp._visp.core.ArrayDouble2D) -> None

Create a matrix from a 2D array that could be one of the following container that inherit from vpArray2D such as vpMatrix , vpRotationMatrix , vpHomogeneousMatrix , vpPoseVector , vpColVector , vpRowVector …

The following example shows how to create a matrix from an homogeneous matrix:

vpRotationMatrix R;
vpMatrix M(R);

__init__(self: visp._visp.core.Matrix, A: visp._visp.core.Matrix) -> None
__init__(self: visp._visp.core.Matrix, list: std::initializer_list<double>) -> None

Construct a matrix from a list of double values.The following code shows how to use this constructor to initialize a 2-by-3 matrix using reshape() function:

#include <visp3/core/vpMatrix.h>

int main()
{
vpMatrix M( {-1, -2, -3, 4, 5.5, 6.0f} );
M.reshape(2, 3);
std::cout << "M:\n" << M << std::endl;
}

It produces the following output:

M:
-1  -2  -3
4  5.5  6

Parameters:

list: List of double.

__init__(self: visp._visp.core.Matrix, nrows: int, ncols: int, list: std::initializer_list<double>) -> None

Construct a matrix from a list of double values.The following code shows how to use this constructor to initialize a 2-by-3 matrix:

#include <visp3/core/vpMatrix.h>

int main()
{
vpMatrix M(2, 3, {-1, -2, -3, 4, 5.5, 6});
std::cout << "M:\n" << M << std::endl;
}

It produces the following output:

M:
-1  -2  -3
4  5.5  6

Parameters:

nrows: Matrix size.
ncols: Matrix size.
list: List of double.

__init__(self: visp._visp.core.Matrix, lists: std::initializer_list<std::initializer_list<double> >) -> None

Construct a matrix from a list of double values.

Parameters:

lists

List of double. The following code shows how to use this constructor to initialize a 2-by-3 matrix function:

#include <visp3/core/vpMatrix.h>

int main()
{
vpMatrix M( { {-1, -2, -3}, {4, 5.5, 6} } );
std::cout << "M:\n" << M << std::endl;
}

It produces the following output:

M:
-1  -2  -3
4  5.5  6

__init__(self: visp._visp.core.Matrix, np_array: numpy.ndarray[numpy.float64]) -> None

Construct a matrix by copying a 2D numpy array.

Parameters:

np_array: The numpy array to copy.

Methods

`AAt`	Overloaded function.
`AtA`	Overloaded function.
`__init__`	Overloaded function.
`add2Matrices`	Overloaded function.
`add2WeightedMatrices`	Operation C = AwA + BwB
`clear`	Removes all elements from the matrix (which are destroyed), leaving the container with a size of 0.
`computeCovarianceMatrix`	Overloaded function.
`computeCovarianceMatrixVVS`	Overloaded function.
`computeHLM`	Compute \({\bf H} + \alpha * diag({\bf H})\)
`cond`	param 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.
`createDiagonalMatrix`	Create a diagonal matrix with the element of a vector \(DA_{ii} = A_i\) .
`det`	Compute the determinant of a n-by-n matrix.
`detByLU`	Compute the determinant of a square matrix using the LU decomposition.
`detByLUEigen3`	Compute the determinant of a square matrix using the LU decomposition with Eigen3 3rd party.
`detByLULapack`	Compute the determinant of a square matrix using the LU decomposition with Lapack 3rd party.
`detByLUOpenCV`	Compute the determinant of a n-by-n matrix using the LU decomposition with OpenCV 3rd party.
`diag`	Overloaded function.
`eigenValues`	Overloaded function.
`expm`	Compute the exponential matrix of a square matrix.
`extract`	Extract a sub matrix from a matrix M .
`eye`	Overloaded function.
`frobeniusNorm`	Compute and return the Frobenius norm (also called Euclidean norm) \(\|\|A\|\| = \sqrt{ \sum{A_{ij}^2}}\) .
`getCol`	Overloaded function.
`getDiag`	Extract a diagonal vector from a matrix.
`getLapackMatrixMinSize`	Return the minimum size of rows and columns required to enable Blas/Lapack usage on matrices and vectors.
`getRow`	Overloaded function.
`hadamard`	Overloaded function.
`inducedL2Norm`	Compute and return the induced L2 norm \(\|\|A\|\| = \Sigma_{max}(A)\) which is equal to the maximum singular value of the matrix.
`infinityNorm`	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 \((m,n)\) is the matrix size.
`init`	Initialize the matrix from a part of an input matrix M .
`insert`	Overloaded function.
`insertMatrixInMatrix`	Overloaded function.
`inverseByCholesky`	Compute the inverse of a n-by-n matrix using the Cholesky decomposition.
`inverseByCholeskyLapack`	Compute the inverse of a n-by-n matrix using the Cholesky decomposition with Lapack 3rd party.
`inverseByCholeskyOpenCV`	Compute the inverse of a n-by-n matrix using the Cholesky decomposition with OpenCV 3rd party.
`inverseByLU`	Compute the inverse of a n-by-n matrix using the LU decomposition.
`inverseByLUEigen3`	Compute the inverse of a n-by-n matrix using the LU decomposition with Eigen3 3rd party.
`inverseByLULapack`	Compute the inverse of a n-by-n matrix using the LU decomposition with Lapack 3rd party.
`inverseByLUOpenCV`	Compute the inverse of a n-by-n matrix using the LU decomposition with OpenCV 3rd party.
`inverseByQR`	Compute the inverse of a n-by-n matrix using the QR decomposition.
`inverseByQRLapack`	Compute the inverse of a n-by-n matrix using the QR decomposition with Lapack 3rd party.
`inverseTriangular`	Compute the inverse of a full-rank n-by-n triangular matrix.
`juxtaposeMatrices`	Overloaded function.
`kernel`	Function to compute the null space (the kernel) of a m-by-n matrix \(\bf A\) .
`kron`	Overloaded function.
`kronStatic`	Overloaded function.
`mult2Matrices`	Overloaded function.
`multMatrixVector`	Operation w = A * v (v and w are vectors).
`negateMatrix`	Operation C = -A.
`nullSpace`	Overloaded function.
`numpy`	Numpy view of the underlying array data.
`print`	Pretty print a matrix.
`printSize`
`pseudoInverse`	Overloaded function.
`pseudoInverseEigen3`	Overloaded function.
`pseudoInverseLapack`	Overloaded function.
`pseudoInverseOpenCV`	Overloaded function.
`qr`	Compute the QR decomposition of a (m x n) matrix of rank r.
`qrPivot`	Compute the QR pivot decomposition of a (m x n) matrix of rank r.
`saveMatrix`	Save a matrix to a file.
`saveMatrixYAML`	Save a matrix in a YAML-formatted file.
`setLapackMatrixMinSize`	Modify default size used to determine if Blas/Lapack basic linear algebra operations are enabled.
`solveByQR`	Overloaded function.
`solveBySVD`	Overloaded function.
`stack`	Overloaded function.
`stackColumn`	Overloaded function.
`stackColumns`	Overloaded function.
`stackMatrices`	Overloaded function.
`stackRow`	Overloaded function.
`stackRows`	Overloaded function.
`strCppCode`	Returns a C++ code representation of this data array (see cppPrint in the C++ documentation)
`strCsv`	Returns the CSV representation of this data array (see csvPrint in the C++ documentation)
`strMaple`	Returns the CSV representation of this data array (see maplePrint in the C++ documentation)
`strMatlab`	Returns the Matlab representation of this data array (see matlabPrint in the C++ documentation)
`sub2Matrices`	Overloaded function.
`sum`	Return the sum of all the \(a_{ij}\) elements of the matrix.
`sumSquare`	Return the sum square of all the \(A_{ij}\) elements of the matrix \(A(m, n)\) .
`svd`	Matrix singular value decomposition (SVD).
`svdEigen3`	Singular value decomposition (SVD) using Eigen3 3rd party.
`svdLapack`	Singular value decomposition (SVD) using Lapack 3rd party.
`svdOpenCV`	Singular value decomposition (SVD) using OpenCV 3rd party.
`t`	Overloaded function.
`transpose`	Overloaded function.

Inherited Methods

`getCols`	Return the number of columns of the 2D array.
`insertStatic`	Insert array B in array A at the given position.
`save`	Overloaded function.
`reshape`
`getRows`	Return the number of rows of the 2D array.
`size`	Return the number of elements of the 2D array.
`getMinValue`	Return the array min value.
`getMaxValue`	Return the array max value.
`saveYAML`	Save an array in a YAML-formatted file.
`conv2`	Overloaded function.
`resize`	Set the size of the array and initialize all the values to zero.

Operators

`__add__`	Operation C = A + B (A is unchanged).
`__doc__`
`__getitem__`	Overloaded function.
`__iadd__`	Overloaded function.
`__imul__`	Multiply all the element of the matrix by x : Aij = Aij * x.
`__init__`	Overloaded function.
`__isub__`	Overloaded function.
`__itruediv__`	Divide all the element of the matrix by x : Aij = Aij / x.
`__module__`
`__mul__`	Overloaded function.
`__neg__`	Operation C = -A (A is unchanged).
`__sub__`	Operation C = A - B (A is unchanged).
`__truediv__`	Cij = Aij / x (A is unchanged)

Attributes

`LU_DECOMPOSITION`
`__annotations__`
`__hash__`

class DetMethod(self, value: int)¶

Bases: pybind11_object

Method used to compute the determinant of a square matrix.

Note

See det()

Values:

LU_DECOMPOSITION: LU decomposition method.

__and__(self, other: object) → object¶

__eq__(self, other: object) → bool¶

__ge__(self, other: object) → bool¶

__getstate__(self) → int¶

__gt__(self, other: object) → bool¶

__hash__(self) → int¶

__index__(self) → int¶

__init__(self, value: int)¶

__int__(self) → int¶

__invert__(self) → object¶

__le__(self, other: object) → bool¶

__lt__(self, other: object) → bool¶

__ne__(self, other: object) → bool¶

__or__(self, other: object) → object¶

__rand__(self, other: object) → object¶

__ror__(self, other: object) → object¶

__rxor__(self, other: object) → object¶

__setstate__(self, state: int) → None¶

__xor__(self, other: object) → object¶

property name : str¶

AAt(*args, **kwargs)¶

Overloaded function.

AAt(self: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Note

See AAt(vpMatrix &) const

Returns:: \(A*A^T\)

AAt(self: visp._visp.core.Matrix, B: visp._visp.core.Matrix) -> None

Compute the AAt operation such as \(B = A*A^T\) .

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.

Note

See AAt()

AtA(*args, **kwargs)¶

Overloaded function.

AtA(self: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Note

See AtA(vpMatrix &) const

Returns:: \(A^T*A\)

AtA(self: visp._visp.core.Matrix, B: visp._visp.core.Matrix) -> None

Compute the AtA operation such as \(B = A^T*A\) .

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.

Note

See AtA()

__add__(self, B: visp._visp.core.Matrix) → visp._visp.core.Matrix¶: Operation C = A + B (A is unchanged).

Note

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

__eq__(*args, **kwargs)¶

Overloaded function.

__eq__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Equal to comparison operator of a 2D array.

__eq__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Equal to comparison operator of a 2D array.

__eq__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Equal to comparison operator of a 2D array.

__getitem__(*args, **kwargs)¶

Overloaded function.

__getitem__(self: visp._visp.core.Matrix, arg0: tuple[int, int]) -> float
__getitem__(self: visp._visp.core.Matrix, arg0: int) -> numpy.ndarray[numpy.float64]
__getitem__(self: visp._visp.core.Matrix, arg0: slice) -> numpy.ndarray[numpy.float64]
__getitem__(self: visp._visp.core.Matrix, arg0: tuple) -> numpy.ndarray[numpy.float64]

__iadd__(*args, **kwargs)¶

Overloaded function.

__iadd__(self: visp._visp.core.Matrix, B: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Operation A = A + B.

__iadd__(self: visp._visp.core.Matrix, x: float) -> visp._visp.core.Matrix

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

__imul__(self, x: float) → visp._visp.core.Matrix¶

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.

__init__(*args, **kwargs)¶

Overloaded function.

__init__(self: visp._visp.core.Matrix) -> None

Basic constructor of a matrix of double. Number of columns and rows are zero.

__init__(self: visp._visp.core.Matrix, r: int, c: int) -> None

Constructor that initialize a matrix of double with 0.

Parameters:

r: Matrix number of rows.
c: Matrix number of columns.

__init__(self: visp._visp.core.Matrix, r: int, c: int, val: float) -> None

Constructor that initialize a matrix of double with val .

Parameters:

r: Matrix number of rows.
c: Matrix number of columns.
val: Each element of the matrix is set to val .

__init__(self: visp._visp.core.Matrix, M: visp._visp.core.Matrix, r: int, c: int, nrows: int, ncols: int) -> None

Construct a matrix as a sub-matrix of the input matrix M .

Note

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

__init__(self: visp._visp.core.Matrix, A: visp._visp.core.ArrayDouble2D) -> None

The following example shows how to create a matrix from an homogeneous matrix:

vpRotationMatrix R;
vpMatrix M(R);

__init__(self: visp._visp.core.Matrix, A: visp._visp.core.Matrix) -> None
__init__(self: visp._visp.core.Matrix, list: std::initializer_list<double>) -> None

Construct a matrix from a list of double values.The following code shows how to use this constructor to initialize a 2-by-3 matrix using reshape() function:

#include <visp3/core/vpMatrix.h>

int main()
{
vpMatrix M( {-1, -2, -3, 4, 5.5, 6.0f} );
M.reshape(2, 3);
std::cout << "M:\n" << M << std::endl;
}

It produces the following output:

M:
-1  -2  -3
4  5.5  6

Parameters:

list: List of double.

__init__(self: visp._visp.core.Matrix, nrows: int, ncols: int, list: std::initializer_list<double>) -> None

Construct a matrix from a list of double values.The following code shows how to use this constructor to initialize a 2-by-3 matrix:

#include <visp3/core/vpMatrix.h>

int main()
{
vpMatrix M(2, 3, {-1, -2, -3, 4, 5.5, 6});
std::cout << "M:\n" << M << std::endl;
}

It produces the following output:

M:
-1  -2  -3
4  5.5  6

Parameters:

nrows: Matrix size.
ncols: Matrix size.
list: List of double.

__init__(self: visp._visp.core.Matrix, lists: std::initializer_list<std::initializer_list<double> >) -> None

Construct a matrix from a list of double values.

Parameters:

lists

List of double. The following code shows how to use this constructor to initialize a 2-by-3 matrix function:

#include <visp3/core/vpMatrix.h>

int main()
{
vpMatrix M( { {-1, -2, -3}, {4, 5.5, 6} } );
std::cout << "M:\n" << M << std::endl;
}

It produces the following output:

M:
-1  -2  -3
4  5.5  6

__init__(self: visp._visp.core.Matrix, np_array: numpy.ndarray[numpy.float64]) -> None

Construct a matrix by copying a 2D numpy array.

Parameters:

np_array: The numpy array to copy.

__isub__(*args, **kwargs)¶

Overloaded function.

__isub__(self: visp._visp.core.Matrix, B: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Operation A = A - B.

__isub__(self: visp._visp.core.Matrix, x: float) -> visp._visp.core.Matrix

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

__itruediv__(self, x: float) → visp._visp.core.Matrix¶: Divide all the element of the matrix by x : Aij = Aij / x.

__mul__(*args, **kwargs)¶

Overloaded function.

__mul__(self: visp._visp.core.Matrix, B: visp._visp.core.Matrix) -> visp._visp.core.Matrix

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

Note

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

__mul__(self: visp._visp.core.Matrix, R: visp._visp.core.RotationMatrix) -> visp._visp.core.Matrix

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

__mul__(self: visp._visp.core.Matrix, R: visp._visp.core.HomogeneousMatrix) -> visp._visp.core.Matrix

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

__mul__(self: visp._visp.core.Matrix, V: visp._visp.core.VelocityTwistMatrix) -> visp._visp.core.Matrix

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

__mul__(self: visp._visp.core.Matrix, V: visp._visp.core.ForceTwistMatrix) -> visp._visp.core.Matrix

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

__mul__(self: visp._visp.core.Matrix, tv: visp._visp.core.TranslationVector) -> visp._visp.core.TranslationVector

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

__mul__(self: visp._visp.core.Matrix, v: visp._visp.core.ColVector) -> visp._visp.core.ColVector

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

Note

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

__mul__(self: visp._visp.core.Matrix, x: float) -> visp._visp.core.Matrix

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

__ne__(*args, **kwargs)¶

Overloaded function.

__ne__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Not equal to comparison operator of a 2D array.

__ne__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Not equal to comparison operator of a 2D array.

__ne__(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D) -> bool

Not equal to comparison operator of a 2D array.

__neg__(self) → visp._visp.core.Matrix¶: Operation C = -A (A is unchanged).

Note

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

__sub__(self, B: visp._visp.core.Matrix) → visp._visp.core.Matrix¶: Operation C = A - B (A is unchanged).

Note

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

__truediv__(self, x: float) → visp._visp.core.Matrix¶: Cij = Aij / x (A is unchanged)

static add2Matrices(*args, **kwargs)¶

Overloaded function.

add2Matrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.Matrix) -> None

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).

Note

See operator+()

add2Matrices(A: visp._visp.core.ColVector, B: visp._visp.core.ColVector, C: visp._visp.core.ColVector) -> None

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).

Note

See vpColVector::operator+()

static add2WeightedMatrices(A: visp._visp.core.Matrix, wA: float, B: visp._visp.core.Matrix, wB: float, C: visp._visp.core.Matrix) → None¶

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).

Note

See operator+()

clear(self) → None¶: Removes all elements from the matrix (which are destroyed), leaving the container with a size of 0.

static computeCovarianceMatrix(*args, **kwargs)¶

Overloaded function.

computeCovarianceMatrix(A: visp._visp.core.Matrix, x: visp._visp.core.ColVector, b: visp._visp.core.ColVector) -> visp._visp.core.Matrix

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.

computeCovarianceMatrix(A: visp._visp.core.Matrix, x: visp._visp.core.ColVector, b: visp._visp.core.ColVector, w: visp._visp.core.Matrix) -> visp._visp.core.Matrix

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.

static computeCovarianceMatrixVVS(*args, **kwargs)¶

Overloaded function.

computeCovarianceMatrixVVS(cMo: visp._visp.core.HomogeneousMatrix, deltaS: visp._visp.core.ColVector, Ls: visp._visp.core.Matrix, W: visp._visp.core.Matrix) -> visp._visp.core.Matrix

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.

computeCovarianceMatrixVVS(cMo: visp._visp.core.HomogeneousMatrix, deltaS: visp._visp.core.ColVector, Ls: visp._visp.core.Matrix) -> visp._visp.core.Matrix

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

static computeHLM(H: visp._visp.core.Matrix, alpha: float, HLM: visp._visp.core.Matrix) → None¶

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

Parameters:

H: visp._visp.core.Matrix¶: input Matrix \({\bf H}\) . This matrix should be square.
alpha: float¶: Scalar \(\alpha\)
HLM: visp._visp.core.Matrix¶: Resulting operation.

cond(self, svThreshold: float = 1e-6) → float¶

Parameters:

svThreshold: float = 1e-6¶: 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 condition number, the ratio of the largest singular value of the matrix to the smallest.

static conv2(*args, **kwargs)¶

Overloaded function.

conv2(M: visp._visp.core.ArrayDouble2D, kernel: visp._visp.core.ArrayDouble2D, mode: str) -> visp._visp.core.ArrayDouble2D

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

Note

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

Parameters:

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

conv2(M: visp._visp.core.ArrayDouble2D, kernel: visp._visp.core.ArrayDouble2D, res: visp._visp.core.ArrayDouble2D, mode: str) -> None

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

Note

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

Parameters:

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

static createDiagonalMatrix(A: visp._visp.core.ColVector, DA: visp._visp.core.Matrix) → None¶

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

Note

See diag()

Parameters:

A: visp._visp.core.ColVector¶: Vector which element will be put in the diagonal.
DA: visp._visp.core.Matrix¶: Diagonal matrix DA[i][i] = A[i]

det(self, method: visp._visp.core.Matrix.DetMethod) → float¶

Compute the determinant of a n-by-n 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;
}

Parameters:

method: visp._visp.core.Matrix.DetMethod¶: Method used to compute the determinant. Default LU decomposition method is faster than the method based on Gaussian elimination.

Returns:

Determinant of the matrix.

detByLU(self) → float¶

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.

#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;
}

Note

See detByLULapack() , detByLUEigen3() , detByLUOpenCV()

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

detByLUEigen3(self) → float¶

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

#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;
}

Note

See detByLU() , detByLUOpenCV() , detByLULapack()

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

detByLULapack(self) → float¶

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

#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;
}

Note

See detByLU() , detByLUEigen3() , detByLUOpenCV()

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

detByLUOpenCV(self) → float¶

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

#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;
}

Note

See detByLU() , detByLUEigen3() , detByLULapack()

Returns:: Determinant of the matrix.

diag(*args, **kwargs)¶

Overloaded function.

diag(self: visp._visp.core.Matrix, val: float = 1.0) -> None

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.

Note

See 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

Parameters:

val: Value to set.

diag(self: visp._visp.core.Matrix, A: visp._visp.core.ColVector) -> None

Create a diagonal matrix with the element of a vector.

Note

See 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

Parameters:

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

eigenValues(*args, **kwargs)¶

Overloaded function.

eigenValues(self: visp._visp.core.Matrix) -> visp._visp.core.ColVector

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

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;
}

Note

See eigenValues(vpColVector &, vpMatrix &)

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

eigenValues(self: visp._visp.core.Matrix, evalue: visp._visp.core.ColVector, evector: visp._visp.core.Matrix) -> None

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

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;
}

Note

See eigenValues()

Parameters:

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

expm(self) → visp._visp.core.Matrix¶

Compute the exponential matrix of a square matrix.

Returns:: Return the exponential matrix.

extract(self, r: int, c: int, nrows: int, ncols: int) → visp._visp.core.Matrix¶

Extract a sub matrix from a matrix M .

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

Note

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

Parameters:

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

eye(*args, **kwargs)¶

Overloaded function.

eye(self: visp._visp.core.Matrix) -> None

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

eye(self: visp._visp.core.Matrix, n: int) -> None

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

eye(self: visp._visp.core.Matrix, m: int, n: int) -> None

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

frobeniusNorm(self) → float¶

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

Note

See infinityNorm() , inducedL2Norm()

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

getCol(*args, **kwargs)¶

Overloaded function.

getCol(self: visp._visp.core.Matrix, j: int) -> visp._visp.core.ColVector

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

Parameters:

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

Returns:

The extracted column vector.

getCol(self: visp._visp.core.Matrix, j: int, i_begin: int, size: int) -> visp._visp.core.ColVector

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

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.

Returns:

The extracted column vector.

getCols(self) → int¶: Return the number of columns of the 2D array.

Note

See getRows() , size()

getDiag(self) → visp._visp.core.ColVector¶

Extract a diagonal vector from a 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

Returns:: The diagonal of the matrix.

static getLapackMatrixMinSize() → int¶

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.

Note

See setLapackMatrixMinSize()

getMaxValue(self) → float¶: Return the array max value.

getMinValue(self) → float¶: Return the array min value.

getRow(*args, **kwargs)¶

Overloaded function.

getRow(self: visp._visp.core.Matrix, i: int) -> visp._visp.core.RowVector

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

Parameters:

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

Returns:

The extracted row vector.

getRow(self: visp._visp.core.Matrix, i: int, j_begin: int, size: int) -> visp._visp.core.RowVector

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

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.

Returns:

The extracted row vector.

getRows(self) → int¶: Return the number of rows of the 2D array.

Note

See getCols() , size()

hadamard(*args, **kwargs)¶

Overloaded function.

hadamard(self: visp._visp.core.Matrix, m: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Parameters:

m: Second matrix;

Returns:

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

hadamard(self: visp._visp.core.ArrayDouble2D, m: visp._visp.core.ArrayDouble2D) -> visp._visp.core.ArrayDouble2D

Parameters:

m: Second matrix;

Returns:

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

inducedL2Norm(self) → float¶

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

Note

See infinityNorm() , frobeniusNorm()

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

infinityNorm(self) → float¶

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 \((m,n)\) is the matrix size.

Note

See frobeniusNorm() , inducedL2Norm()

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

init(self, M: visp._visp.core.Matrix, r: int, c: int, nrows: int, ncols: int) → None¶

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

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

Note

See extract()

Parameters:

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

insert(*args, **kwargs)¶

Overloaded function.

insert(self: visp._visp.core.Matrix, A: visp._visp.core.Matrix, r: int, c: int) -> None

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.

insert(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D, r: int, c: int) -> None

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.

insert(self: visp._visp.core.ArrayDouble2D, A: visp._visp.core.ArrayDouble2D, B: visp._visp.core.ArrayDouble2D, r: int, c: int) -> visp._visp.core.ArrayDouble2D

Insert array B in array A at the given position.

Warning

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

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.

static insertMatrixInMatrix(*args, **kwargs)¶

Overloaded function.

insertMatrixInMatrix(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, r: int, c: int) -> visp._visp.core.Matrix

Insert matrix B in matrix A at the given position.

Warning

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

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.

insertMatrixInMatrix(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.Matrix, r: int, c: int) -> None

Insert matrix B in matrix A at the given position.

Warning

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

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.

static insertStatic(A: visp._visp.core.ArrayDouble2D, B: visp._visp.core.ArrayDouble2D, C: visp._visp.core.ArrayDouble2D, r: int, c: int) → None¶

Insert array B in array A at the given position.

Warning

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

Parameters:

A: visp._visp.core.ArrayDouble2D¶: Main array.
B: visp._visp.core.ArrayDouble2D¶: Array to insert.
C: visp._visp.core.ArrayDouble2D¶: Result array.
r: int¶: Index of the row where to insert array B.
c: int¶: Index of the column where to insert array B.

inverseByCholesky(self) → visp._visp.core.Matrix¶

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.

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;
}

Note

See pseudoInverse()

Returns:: The inverse matrix.

inverseByCholeskyLapack(self) → visp._visp.core.Matrix¶

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.

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;
}

Note

See inverseByCholesky() , inverseByCholeskyOpenCV()

Returns:: The inverse matrix.

inverseByCholeskyOpenCV(self) → visp._visp.core.Matrix¶

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.

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;
}

Note

See inverseByCholesky() , inverseByCholeskyLapack()

Returns:: The inverse matrix.

inverseByLU(self) → visp._visp.core.Matrix¶

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.

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;
}

Note

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

Returns:: The inverse matrix.

inverseByLUEigen3(self) → visp._visp.core.Matrix¶

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

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;
}

Note

See inverseByLU() , inverseByLULapack() , inverseByLUOpenCV()

Returns:: The inverse matrix.

inverseByLULapack(self) → visp._visp.core.Matrix¶

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

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;
}

Note

See inverseByLU() , inverseByLUEigen3() , inverseByLUOpenCV()

Returns:: The inverse matrix.

inverseByLUOpenCV(self) → visp._visp.core.Matrix¶

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

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;
}

Note

See inverseByLU() , inverseByLUEigen3() , inverseByLULapack()

Returns:: The inverse matrix.

inverseByQR(self) → visp._visp.core.Matrix¶

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.

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;
}

Note

See inverseByLU() , inverseByCholesky()

Returns:: The inverse matrix.

inverseByQRLapack(self) → visp._visp.core.Matrix¶

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

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;
}

Note

See inverseByQR()

Returns:: The inverse matrix.

inverseTriangular(self, upper: bool = true) → visp._visp.core.Matrix¶

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.

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

Parameters:

upper: bool = true¶: if it is an upper triangular matrix

Returns:

The inverse matrix

static juxtaposeMatrices(*args, **kwargs)¶

Overloaded function.

juxtaposeMatrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Juxtapose to matrices C = [ A B ].

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

Warning

A and B must have the same number of rows.

Parameters:

A: Left matrix.
B: Right matrix.

Returns:

Juxtaposed matrix C = [ A B ]

juxtaposeMatrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.Matrix) -> None

Juxtapose to matrices C = [ A B ].

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

Warning

A and B must have the same number of rows.

Parameters:

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

kernel(self, kerAt: visp._visp.core.Matrix, svThreshold: float = 1e-6) → int¶

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: visp._visp.core.Matrix¶: 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: float = 1e-6¶: 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.

kron(*args, **kwargs)¶

Overloaded function.

kron(self: visp._visp.core.Matrix, m1: visp._visp.core.Matrix, out: visp._visp.core.Matrix) -> None

Compute Kronecker product matrix.

Parameters:

out: If m1.kron(m2) out contains the kronecker product’s result : \(m1 \otimes m2\) .

kron(self: visp._visp.core.Matrix, m1: visp._visp.core.Matrix) -> visp._visp.core.Matrix

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

static kronStatic(*args, **kwargs)¶

Overloaded function.

kronStatic(m1: visp._visp.core.Matrix, m2: visp._visp.core.Matrix, out: visp._visp.core.Matrix) -> None

Compute Kronecker product matrix.

Parameters:

m1: vpMatrix ;
m2: vpMatrix ;
out: The kronecker product : \(m1 \otimes m2\)

kronStatic(m1: visp._visp.core.Matrix, m2: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Parameters:

m1: vpMatrix ;
m2: vpMatrix ;

Returns:

The kronecker product : \(m1 \otimes m2\)

static mult2Matrices(*args, **kwargs)¶

Overloaded function.

mult2Matrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.Matrix) -> None

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).

Note

See operator*()

mult2Matrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.RotationMatrix) -> None

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).

mult2Matrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.HomogeneousMatrix) -> None

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).

mult2Matrices(A: visp._visp.core.Matrix, B: visp._visp.core.ColVector, C: visp._visp.core.ColVector) -> None

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).

Note

See multMatrixVector()

static multMatrixVector(A: visp._visp.core.Matrix, v: visp._visp.core.ColVector, w: visp._visp.core.ColVector) → None¶

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).

Note

See operator*(const vpColVector &v) const

static negateMatrix(A: visp._visp.core.Matrix, C: visp._visp.core.Matrix) → None¶

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).

Note

See operator-(void)

nullSpace(*args, **kwargs)¶

Overloaded function.

nullSpace(self: visp._visp.core.Matrix, kerA: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> int

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 \(n - r\) .

nullSpace(self: visp._visp.core.Matrix, kerA: visp._visp.core.Matrix, dim: int) -> int

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 \(n - r\) , by using 1e-6 as threshold for the sigular values.

numpy(self) → numpy.ndarray[numpy.float64]¶: Numpy view of the underlying array data. This numpy view can be used to directly modify the array.

print(self: visp._visp.core.Matrix, s: std::ostream, length: int, intro: str =) → int¶

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 .

Note

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

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.

printSize(self) → None¶

pseudoInverse(*args, **kwargs)¶

Overloaded function.

pseudoInverse(self: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> visp._visp.core.Matrix

Compute and return the Moore-Penros pseudo inverse \(A^+\) of a m-by-n 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 = 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

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 \(A^+\) .

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) of a m-by-n matrix \(\bf A\) and return 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

Parameters:

Ap: The Moore-Penros pseudo inverse \(A^+\) .
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.

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float = 1e-6) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) of a m-by-n matrix \(\bf A\) along with singular values and return 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;
  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

Parameters:

Ap: The Moore-Penros pseudo inverse \(A^+\) .
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.

Returns:

The rank of the matrix \(\bf A\) .

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) 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.

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

Parameters:

Ap: The Moore-Penros pseudo inverse \(A^+\) .
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: math:mbox{Im}({bf A}) that is a m-by-r matrix.
imAt: math:mbox{Im}({bf A}^T) that is n-by-r matrix.

Returns:

The rank of the matrix \(\bf A\) .

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) 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.

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

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: math:mbox { Im }({ bf A }) that is a m - by - r matrix.
imAt: math: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 \(\bf A\) .

pseudoInverse(self: visp._visp.core.Matrix, rank_in: int) -> visp._visp.core.Matrix

Compute and return the Moore-Penros pseudo inverse \(A^+\) of a m-by-n 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);

  // 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

Parameters:

rank_in: Known rank of the matrix.

Returns:

The Moore-Penros pseudo inverse \(A^+\) .

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, rank_in: int) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) of a m-by-n matrix \(\bf A\) and return 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

Parameters:

Ap: The Moore-Penros pseudo inverse \(A^+\) .
rank_in: Known rank of the matrix.

Returns:

The rank of the matrix.

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) of a m-by-n matrix \(\bf A\) along with singular values and return 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;
  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

Parameters:

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

Returns:

The rank of the matrix \(\bf A\) .

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) 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.

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

Parameters:

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

Returns:

The rank of the matrix \(\bf A\) .

pseudoInverse(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int

Compute the Moore-Penros pseudo inverse \(A^+\) 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.

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

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).
rank_in: Known rank of the matrix.
imA: math:mbox { Im }({ bf A }) that is a m - by - r matrix.
imAt: math: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 \(\bf A\) .

pseudoInverseEigen3(*args, **kwargs)¶

Overloaded function.

pseudoInverseEigen3(self: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> visp._visp.core.Matrix
pseudoInverseEigen3(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> int
pseudoInverseEigen3(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float = 1e-6) -> int
pseudoInverseEigen3(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int
pseudoInverseEigen3(self: visp._visp.core.Matrix, rank_in: int) -> visp._visp.core.Matrix
pseudoInverseEigen3(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, rank_in: int) -> int
pseudoInverseEigen3(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int) -> int
pseudoInverseEigen3(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int

pseudoInverseLapack(*args, **kwargs)¶

Overloaded function.

pseudoInverseLapack(self: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> visp._visp.core.Matrix
pseudoInverseLapack(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> int
pseudoInverseLapack(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float = 1e-6) -> int
pseudoInverseLapack(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int
pseudoInverseLapack(self: visp._visp.core.Matrix, rank_in: int) -> visp._visp.core.Matrix
pseudoInverseLapack(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, rank_in: int) -> int
pseudoInverseLapack(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int) -> int
pseudoInverseLapack(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int

pseudoInverseOpenCV(*args, **kwargs)¶

Overloaded function.

pseudoInverseOpenCV(self: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> visp._visp.core.Matrix
pseudoInverseOpenCV(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, svThreshold: float = 1e-6) -> int
pseudoInverseOpenCV(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float = 1e-6) -> int
pseudoInverseOpenCV(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, svThreshold: float, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int
pseudoInverseOpenCV(self: visp._visp.core.Matrix, rank_in: int) -> visp._visp.core.Matrix
pseudoInverseOpenCV(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, rank_in: int) -> int
pseudoInverseOpenCV(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int) -> int
pseudoInverseOpenCV(self: visp._visp.core.Matrix, Ap: visp._visp.core.Matrix, sv: visp._visp.core.ColVector, rank_in: int, imA: visp._visp.core.Matrix, imAt: visp._visp.core.Matrix, kerAt: visp._visp.core.Matrix) -> int

qr(self, Q: visp._visp.core.Matrix, R: visp._visp.core.Matrix, full: bool = false, squareR: bool = false, tol: float = 1e-6) → int¶

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.

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;
}

Note

See qrPivot()

Parameters:

Q: visp._visp.core.Matrix¶: orthogonal matrix (will be modified).
R: visp._visp.core.Matrix¶: upper-triangular matrix (will be modified).
full: bool = false¶: whether or not we want full decomposition.
squareR: bool = false¶: will return only the square (min(m,n) x min(m,n)) part of R.
tol: float = 1e-6¶: tolerance to test the rank of R.

Returns:

The rank r of the matrix.

qrPivot(self, Q: visp._visp.core.Matrix, R: visp._visp.core.Matrix, P: visp._visp.core.Matrix, full: bool = false, squareR: bool = false, tol: float = 1e-6) → int¶

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.

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;
}

Note

See qrPivot()

Parameters:

Q: visp._visp.core.Matrix¶: orthogonal matrix (will be modified).
R: visp._visp.core.Matrix¶: upper-triangular matrix (will be modified).
P: visp._visp.core.Matrix¶: the (n x n) permutation matrix.
full: bool = false¶: whether or not we want full decomposition.
squareR: bool = false¶: will return only the (r x r) part of R and the (r x n) part of P.
tol: float = 1e-6¶: tolerance to test the rank of R.

Returns:

The rank r of the matrix.

reshape(self, nrows: int, ncols: int) → None¶

resize(self, nrows: int, ncols: int, flagNullify: bool = true, recopy_: bool = true) → None¶

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

Parameters:

nrows: int¶: number of rows.
ncols: int¶: number of column.
flagNullify: bool = true¶: 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_: bool = true¶: if true, will perform an explicit recopy of the old data.

static save(*args, **kwargs)¶

Overloaded function.

save(filename: str, A: visp._visp.core.ArrayDouble2D, binary: bool = false, header: str = ) -> bool

Save a matrix to a file.

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

Note

See load()

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.

save(filename: str, A: visp._visp.core.ArrayDouble2D, binary: bool = false, header: str = ) -> bool

Save a matrix to a file.

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

Note

See load()

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.

save(filename: str, A: visp._visp.core.ArrayDouble2D, binary: bool = false, header: str = ) -> bool

Save a matrix to a file.

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

Note

See load()

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.

static saveMatrix(filename: str, M: visp._visp.core.ArrayDouble2D, binary: bool = false, header: str =) → bool¶

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

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??@@@%

Note

See loadMatrix() , saveMatrixYAML() , loadMatrixYAML()

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.

static saveMatrixYAML(filename: str, M: visp._visp.core.ArrayDouble2D, header: str =) → bool¶

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

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]

Note

See saveMatrix() , loadMatrix() , loadMatrixYAML()

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.

static saveYAML(filename: str, A: visp._visp.core.ArrayDouble2D, header: str =) → bool¶

Save an array in a YAML-formatted file.

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]

Note

See loadYAML()

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.

static setLapackMatrixMinSize(min_size: int) → None¶

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

To get more info see tutorial-basic-linear-algebra.

Note

See getLapackMatrixMinSize()

Parameters:

min_size: int¶: 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.

size(self) → int¶: Return the number of elements of the 2D array.

solveByQR(*args, **kwargs)¶

Overloaded function.

solveByQR(self: visp._visp.core.Matrix, b: visp._visp.core.ColVector, x: visp._visp.core.ColVector) -> None

Solve a linear system Ax = b using QR Decomposition.

Non destructive wrt. A and b.

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;
}

Note

See qrPivot()

Parameters:

b: Vector b
x: Vector x

solveByQR(self: visp._visp.core.Matrix, b: visp._visp.core.ColVector) -> visp._visp.core.ColVector

Solve a linear system Ax = b using QR Decomposition.

Non destructive wrt. A and B.

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;
}

Note

See qrPivot()

Parameters:

b: Vector b

Returns:

Vector x

solveBySVD(*args, **kwargs)¶

Overloaded function.

solveBySVD(self: visp._visp.core.Matrix, B: visp._visp.core.ColVector, x: visp._visp.core.ColVector) -> None

Solve a linear system \(A X = B\) using Singular Value Decomposition (SVD).

Non destructive wrt. A and B.

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;
}

Note

See solveBySVD(const vpColVector &)

Parameters:

x: Vector \(X\) .

solveBySVD(self: visp._visp.core.Matrix, B: visp._visp.core.ColVector) -> visp._visp.core.ColVector

Solve a linear system \(A X = B\) using Singular Value Decomposition (SVD).

Non destructive wrt. A and B.

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;
}

Note

See solveBySVD(const vpColVector &, vpColVector &)

Parameters:

B: Vector \(B\) .

Returns:

Vector \(X\) .

stack(*args, **kwargs)¶

Overloaded function.

stack(self: visp._visp.core.Matrix, A: visp._visp.core.Matrix) -> None

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

stack(self: visp._visp.core.Matrix, r: visp._visp.core.RowVector) -> None

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());
}

stack(self: visp._visp.core.Matrix, c: visp._visp.core.ColVector) -> None

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.

static stackColumn(*args, **kwargs)¶

Overloaded function.

stackColumn(A: visp._visp.core.Matrix, c: visp._visp.core.ColVector) -> visp._visp.core.Matrix

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

Warning

A and c must have the same number of rows.

Parameters:

A: Left matrix.
c: Right column vector.

Returns:

Stacked matrix [ A c ]

stackColumn(A: visp._visp.core.Matrix, c: visp._visp.core.ColVector, C: visp._visp.core.Matrix) -> None

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

Warning

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

Parameters:

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

stackColumns(*args, **kwargs)¶

Overloaded function.

stackColumns(self: visp._visp.core.Matrix, out: visp._visp.core.ColVector) -> None

Stacks columns of a matrix in a vector.

Parameters:

out: a vpColVector .

stackColumns(self: visp._visp.core.Matrix) -> visp._visp.core.ColVector

Returns:: a vpColVector .

static stackMatrices(*args, **kwargs)¶

Overloaded function.

stackMatrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix) -> visp._visp.core.Matrix

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

Warning

A and B must have the same number of columns.

Parameters:

A: Upper matrix.
B: Lower matrix.

Returns:

Stacked matrix [ A B ]^T

stackMatrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.Matrix) -> None

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

Warning

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

Parameters:

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

static stackRow(*args, **kwargs)¶

Overloaded function.

stackRow(A: visp._visp.core.Matrix, r: visp._visp.core.RowVector) -> visp._visp.core.Matrix

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

Warning

A and r must have the same number of columns.

Parameters:

A: Upper matrix.
r: Lower row vector.

Returns:

Stacked matrix [ A r ]^T

stackRow(A: visp._visp.core.Matrix, r: visp._visp.core.RowVector, C: visp._visp.core.Matrix) -> None

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

Warning

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

Parameters:

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

stackRows(*args, **kwargs)¶

Overloaded function.

stackRows(self: visp._visp.core.Matrix, out: visp._visp.core.RowVector) -> None

Stacks rows of a matrix in a vector

Parameters:

out: a vpRowVector .

stackRows(self: visp._visp.core.Matrix) -> visp._visp.core.RowVector

Returns:: a vpRowVector .

strCppCode(self, name: str, byte_per_byte: bool = False) → str¶

Returns a C++ code representation of this data array (see cppPrint in the C++ documentation)

Parameters:

name: str¶: variable name of the matrix.
byte_per_byte: bool = False¶: Whether to print byte per byte defaults to false.

strCsv(self) → str¶: Returns the CSV representation of this data array (see csvPrint in the C++ documentation)

strMaple(self) → str¶: Returns the CSV representation of this data array (see maplePrint in the C++ documentation)

strMatlab(self) → str¶: Returns the Matlab representation of this data array (see matlabPrint in the C++ documentation)

static sub2Matrices(*args, **kwargs)¶

Overloaded function.

sub2Matrices(A: visp._visp.core.Matrix, B: visp._visp.core.Matrix, C: visp._visp.core.Matrix) -> None

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).

Note

See operator-()

sub2Matrices(A: visp._visp.core.ColVector, B: visp._visp.core.ColVector, C: visp._visp.core.ColVector) -> None

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).

Note

See vpColVector::operator-()

sum(self) → float¶

Return the sum of all the \(a_{ij}\) elements of the matrix.

Returns:: Value of \(\sum a_{ij}\)

sumSquare(self) → float¶

Return the sum square of all the \(A_{ij}\) elements of the matrix \(A(m, n)\) .

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

svd(self, w: visp._visp.core.ColVector, V: visp._visp.core.Matrix) → None¶

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 \(M\) , this function computes it singular value decomposition such as

\[M = U \Sigma V^{\top} \]

Warning

This method is destructive wrt. to the matrix \(M\) to decompose. You should make a COPY of that matrix if needed.

The matrix object (*this) is updated with :math:`U .

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;
}

Note

See svdLapack() , svdEigen3() , svdOpenCV()

Parameters:

w: visp._visp.core.ColVector¶: Vector of singular values: \(\Sigma = diag(w)\) .
V: visp._visp.core.Matrix¶: Matrix \(V\) .

svdEigen3(self, w: visp._visp.core.ColVector, V: visp._visp.core.Matrix) → None¶

Singular value decomposition (SVD) using Eigen3 3rd party.

Given matrix \(M\) , this function computes it singular value decomposition such as

\[M = U \Sigma V^{\top} \]

Warning

This method is destructive wrt. to the matrix \(M\) to decompose. You should make a COPY of that matrix if needed.

The matrix object (*this) is updated with \(U\) .

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;
}

Note

See svd() , svdLapack() , svdOpenCV()

Parameters:

w: visp._visp.core.ColVector¶: Vector of singular values: \(\Sigma = diag(w)\) .
V: visp._visp.core.Matrix¶: Matrix \(V\) .

svdLapack(self, w: visp._visp.core.ColVector, V: visp._visp.core.Matrix) → None¶

Singular value decomposition (SVD) using Lapack 3rd party.

Given matrix \(M\) , this function computes it singular value decomposition such as

\[M = U \Sigma V^{\top} \]

Warning

This method is destructive wrt. to the matrix \(M\) to decompose. You should make a COPY of that matrix if needed.

The matrix object (*this) is updated with \(U\) .

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;
}

Note

See svd() , svdEigen3() , svdOpenCV()

Parameters:

w: visp._visp.core.ColVector¶: Vector of singular values: \(\Sigma = diag(w)\) .
V: visp._visp.core.Matrix¶: Matrix \(V\) .

svdOpenCV(self, w: visp._visp.core.ColVector, V: visp._visp.core.Matrix) → None¶

Singular value decomposition (SVD) using OpenCV 3rd party.

Given matrix \(M\) , this function computes it singular value decomposition such as

\[M = U \Sigma V^{\top} \]

Warning

This method is destructive wrt. to the matrix \(M\) to decompose. You should make a COPY of that matrix if needed.

The matrix object (*this) is updated with \(U\) .

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;
}

Note

See svd() , svdEigen3() , svdLapack()

Parameters:

w: visp._visp.core.ColVector¶: Vector of singular values: \(\Sigma = diag(w)\) .
V: visp._visp.core.Matrix¶: Matrix \(V\) .

t(*args, **kwargs)¶

Overloaded function.

t(self: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Compute and return the transpose of the matrix.

t(self: visp._visp.core.ArrayDouble2D) -> visp._visp.core.ArrayDouble2D

Compute the transpose of the array.

Returns:: vpArray2D<Type> C = A^T

transpose(*args, **kwargs)¶

Overloaded function.

transpose(self: visp._visp.core.Matrix) -> visp._visp.core.Matrix

Compute and return the transpose of the matrix.

Note

See t()

transpose(self: visp._visp.core.Matrix, At: visp._visp.core.Matrix) -> None

Compute At the transpose of the matrix.

Note

See t()

Parameters:

At: (output) : Resulting transpose matrix.

__hash__ = None¶