Eigen  3.3.3
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()

List of all members.

Public Member Functions

MatrixType::RealScalar absDeterminant () const
const PermutationTypecolsPermutation () const
 CompleteOrthogonalDecomposition ()
 Default Constructor.
 CompleteOrthogonalDecomposition (Index rows, Index cols)
 Default Constructor with memory preallocation.
template<typename InputType >
 CompleteOrthogonalDecomposition (const EigenBase< InputType > &matrix)
 Constructs a complete orthogonal decomposition from a given matrix.
template<typename InputType >
 CompleteOrthogonalDecomposition (EigenBase< InputType > &matrix)
 Constructs a complete orthogonal decomposition from a given matrix.
Index dimensionOfKernel () const
const HCoeffsType & hCoeffs () const
HouseholderSequenceType householderQ (void) const
ComputationInfo info () const
 Reports whether the complete orthogonal decomposition was succesful.
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) [inline, explicit]

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) [inline, explicit]

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 file:
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