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Eigen::CompleteOrthogonalDecomposition< MatrixType > Class Template Reference

Detailed Description

template<typename MatrixType>
class Eigen::CompleteOrthogonalDecomposition< MatrixType >

Complete orthogonal decomposition (COD) of a matrix.

Parameters
MatrixTypethe type of the matrix of which we are computing the COD.

This class performs a rank-revealing complete orthogonal decomposition of a matrix A into matrices P, Q, T, and Z such that

\[ \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \begin{bmatrix} \mathbf{T} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{bmatrix} \, \mathbf{Z} \]

by using Householder transformations. Here, P is a permutation matrix, Q and Z are unitary matrices and T an upper triangular matrix of size rank-by-rank. A may be rank deficient.

This class supports the inplace decomposition mechanism.

See Also
MatrixBase::completeOrthogonalDecomposition()

Public Member Functions

MatrixType::RealScalar absDeterminant () const
 
const PermutationTypecolsPermutation () const
 
 CompleteOrthogonalDecomposition ()
 Default Constructor. More...
 
 CompleteOrthogonalDecomposition (Index rows, Index cols)
 Default Constructor with memory preallocation. More...
 
template<typename InputType >
 CompleteOrthogonalDecomposition (const EigenBase< InputType > &matrix)
 Constructs a complete orthogonal decomposition from a given matrix. More...
 
template<typename InputType >
 CompleteOrthogonalDecomposition (EigenBase< InputType > &matrix)
 Constructs a complete orthogonal decomposition from a given matrix. More...
 
Index dimensionOfKernel () const
 
const HCoeffsType & hCoeffs () const
 
HouseholderSequenceType householderQ (void) const
 
ComputationInfo info () const
 Reports whether the complete orthogonal decomposition was succesful. More...
 
bool isInjective () const
 
bool isInvertible () const
 
bool isSurjective () const
 
MatrixType::RealScalar logAbsDeterminant () const
 
const MatrixType & matrixQTZ () const
 
const MatrixType & matrixT () const
 
MatrixType matrixZ () const
 
RealScalar maxPivot () const
 
Index nonzeroPivots () const
 
const Inverse
< CompleteOrthogonalDecomposition
pseudoInverse () const
 
Index rank () const
 
CompleteOrthogonalDecompositionsetThreshold (const RealScalar &threshold)
 
CompleteOrthogonalDecompositionsetThreshold (Default_t)
 
template<typename Rhs >
const Solve
< CompleteOrthogonalDecomposition,
Rhs > 
solve (const MatrixBase< Rhs > &b) const
 
RealScalar threshold () const
 
const HCoeffsType & zCoeffs () const
 

Protected Member Functions

template<typename Rhs >
void applyZAdjointOnTheLeftInPlace (Rhs &rhs) const
 
void computeInPlace ()
 

Constructor & Destructor Documentation

template<typename MatrixType>
Eigen::CompleteOrthogonalDecomposition< MatrixType >::CompleteOrthogonalDecomposition ( )
inline

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via CompleteOrthogonalDecomposition::compute(const* MatrixType&).

template<typename MatrixType>
Eigen::CompleteOrthogonalDecomposition< MatrixType >::CompleteOrthogonalDecomposition ( Index  rows,
Index  cols 
)
inline

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See Also
CompleteOrthogonalDecomposition()
template<typename MatrixType>
template<typename InputType >
Eigen::CompleteOrthogonalDecomposition< MatrixType >::CompleteOrthogonalDecomposition ( const EigenBase< InputType > &  matrix)
inlineexplicit

Constructs a complete orthogonal decomposition from a given matrix.

This constructor computes the complete orthogonal decomposition of the matrix matrix by calling the method compute(). The default threshold for rank determination will be used. It is a short cut for:

* CompleteOrthogonalDecomposition<MatrixType> cod(matrix.rows(),
* matrix.cols());
* cod.setThreshold(Default);
* cod.compute(matrix);
*
See Also
compute()
template<typename MatrixType>
template<typename InputType >
Eigen::CompleteOrthogonalDecomposition< MatrixType >::CompleteOrthogonalDecomposition ( EigenBase< InputType > &  matrix)
inlineexplicit

Constructs a complete orthogonal decomposition from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See Also
CompleteOrthogonalDecomposition(const EigenBase&)

Member Function Documentation

template<typename MatrixType >
MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::absDeterminant ( ) const
Returns
the absolute value of the determinant of the matrix of which *this is the complete orthogonal decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the complete orthogonal decomposition has already been computed.
Note
This is only for square matrices.
Warning
a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.
See Also
logAbsDeterminant(), MatrixBase::determinant()
template<typename MatrixType >
template<typename Rhs >
void Eigen::CompleteOrthogonalDecomposition< MatrixType >::applyZAdjointOnTheLeftInPlace ( Rhs &  rhs) const
protected

Overwrites rhs with $ \mathbf{Z}^* * \mathbf{rhs} $.

template<typename MatrixType>
const PermutationType& Eigen::CompleteOrthogonalDecomposition< MatrixType >::colsPermutation ( ) const
inline
Returns
a const reference to the column permutation matrix
template<typename MatrixType >
void Eigen::CompleteOrthogonalDecomposition< MatrixType >::computeInPlace ( )
protected

Performs the complete orthogonal decomposition of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

See Also
class CompleteOrthogonalDecomposition, CompleteOrthogonalDecomposition(const MatrixType&)
template<typename MatrixType>
Index Eigen::CompleteOrthogonalDecomposition< MatrixType >::dimensionOfKernel ( ) const
inline
Returns
the dimension of the kernel of the matrix of which *this is the complete orthogonal decomposition.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename MatrixType>
const HCoeffsType& Eigen::CompleteOrthogonalDecomposition< MatrixType >::hCoeffs ( ) const
inline
Returns
a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

template<typename MatrixType >
CompleteOrthogonalDecomposition< MatrixType >::HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition< MatrixType >::householderQ ( void  ) const
Returns
the matrix Q as a sequence of householder transformations
template<typename MatrixType>
ComputationInfo Eigen::CompleteOrthogonalDecomposition< MatrixType >::info ( ) const
inline

Reports whether the complete orthogonal decomposition was succesful.

Note
This function always returns Success. It is provided for compatibility with other factorization routines.
Returns
Success
template<typename MatrixType>
bool Eigen::CompleteOrthogonalDecomposition< MatrixType >::isInjective ( ) const
inline
Returns
true if the matrix of which *this is the decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename MatrixType>
bool Eigen::CompleteOrthogonalDecomposition< MatrixType >::isInvertible ( ) const
inline
Returns
true if the matrix of which *this is the complete orthogonal decomposition is invertible.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename MatrixType>
bool Eigen::CompleteOrthogonalDecomposition< MatrixType >::isSurjective ( ) const
inline
Returns
true if the matrix of which *this is the decomposition represents a surjective linear map; false otherwise.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename MatrixType >
MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::logAbsDeterminant ( ) const
Returns
the natural log of the absolute value of the determinant of the matrix of which *this is the complete orthogonal decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the complete orthogonal decomposition has already been computed.
Note
This is only for square matrices.
This method is useful to work around the risk of overflow/underflow that's inherent to determinant computation.
See Also
absDeterminant(), MatrixBase::determinant()
template<typename MatrixType>
const MatrixType& Eigen::CompleteOrthogonalDecomposition< MatrixType >::matrixQTZ ( ) const
inline
Returns
a reference to the matrix where the complete orthogonal decomposition is stored
template<typename MatrixType>
const MatrixType& Eigen::CompleteOrthogonalDecomposition< MatrixType >::matrixT ( ) const
inline
Returns
a reference to the matrix where the complete orthogonal decomposition is stored.
Warning
The strict lower part and
cols() - rank()
right columns of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use
matrixT().template triangularView<Upper>()
For rank-deficient matrices, use
* matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
*
template<typename MatrixType>
MatrixType Eigen::CompleteOrthogonalDecomposition< MatrixType >::matrixZ ( ) const
inline
Returns
the matrix Z.
template<typename MatrixType>
RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::maxPivot ( ) const
inline
Returns
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of R.
template<typename MatrixType>
Index Eigen::CompleteOrthogonalDecomposition< MatrixType >::nonzeroPivots ( ) const
inline
Returns
the number of nonzero pivots in the complete orthogonal decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.
See Also
rank()
template<typename MatrixType>
const Inverse<CompleteOrthogonalDecomposition> Eigen::CompleteOrthogonalDecomposition< MatrixType >::pseudoInverse ( ) const
inline
Returns
the pseudo-inverse of the matrix of which *this is the complete orthogonal decomposition.
Warning
: Do not compute this->pseudoInverse()*rhs to solve a linear systems. It is more efficient and numerically stable to call this->solve(rhs).
template<typename MatrixType>
Index Eigen::CompleteOrthogonalDecomposition< MatrixType >::rank ( ) const
inline
Returns
the rank of the matrix of which *this is the complete orthogonal decomposition.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).
template<typename MatrixType>
CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition< MatrixType >::setThreshold ( const RealScalar &  threshold)
inline

Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero. Most be called before calling compute().

When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.

Parameters
thresholdThe new value to use as the threshold.

A pivot will be considered nonzero if its absolute value is strictly greater than $ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert $ where maxpivot is the biggest pivot.

If you want to come back to the default behavior, call setThreshold(Default_t)

template<typename MatrixType>
CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition< MatrixType >::setThreshold ( Default_t  )
inline

Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

You should pass the special object Eigen::Default as parameter here.

qr.setThreshold(Eigen::Default);

See the documentation of setThreshold(const RealScalar&).

template<typename MatrixType>
template<typename Rhs >
const Solve<CompleteOrthogonalDecomposition, Rhs> Eigen::CompleteOrthogonalDecomposition< MatrixType >::solve ( const MatrixBase< Rhs > &  b) const
inline

This method computes the minimum-norm solution X to a least squares problem

\[\mathrm{minimize} \|A X - B\|, \]

where A is the matrix of which *this is the complete orthogonal decomposition.

Parameters
bthe right-hand sides of the problem to solve.
Returns
a solution.
template<typename MatrixType>
RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::threshold ( ) const
inline

Returns the threshold that will be used by certain methods such as rank().

See the documentation of setThreshold(const RealScalar&).

template<typename MatrixType>
const HCoeffsType& Eigen::CompleteOrthogonalDecomposition< MatrixType >::zCoeffs ( ) const
inline
Returns
a const reference to the vector of Householder coefficients used to represent the factor Z.

For advanced uses only.


The documentation for this class was generated from the following files: