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 Eigen  3.3.9
Eigen::CompleteOrthogonalDecomposition< _MatrixType > Class Template Reference

## Detailed Description

### template<typename _MatrixType> class Eigen::CompleteOrthogonalDecomposition< _MatrixType >

Complete orthogonal decomposition (COD) of a matrix.

Parameters
 MatrixType the 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.

MatrixBase::completeOrthogonalDecomposition()

## Public Member Functions

MatrixType::RealScalar absDeterminant () const

const PermutationTypecolsPermutation () const

CompleteOrthogonalDecomposition ()
Default Constructor. 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...

CompleteOrthogonalDecomposition (Index rows, Index cols)
Default Constructor with memory preallocation. 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< CompleteOrthogonalDecompositionpseudoInverse () 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 computeInPlace ()

## ◆ CompleteOrthogonalDecomposition() [1/4]

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&).

## ◆ CompleteOrthogonalDecomposition() [2/4]

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.

CompleteOrthogonalDecomposition()

## ◆ CompleteOrthogonalDecomposition() [3/4]

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);
compute()

## ◆ CompleteOrthogonalDecomposition() [4/4]

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.

CompleteOrthogonalDecomposition(const EigenBase&)

## ◆ absDeterminant()

template<typename MatrixType >
 MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::absDeterminant
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.
logAbsDeterminant(), MatrixBase::determinant()

template<typename MatrixType >
template<typename Rhs >
 void Eigen::CompleteOrthogonalDecomposition< MatrixType >::applyZAdjointOnTheLeftInPlace ( Rhs & rhs ) const
protected

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

## ◆ colsPermutation()

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

## ◆ computeInPlace()

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.

class CompleteOrthogonalDecomposition, CompleteOrthogonalDecomposition(const MatrixType&)

## ◆ dimensionOfKernel()

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&).

## ◆ hCoeffs()

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.

## ◆ householderQ()

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

## ◆ info()

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

## ◆ isInjective()

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&).

## ◆ isInvertible()

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&).

## ◆ isSurjective()

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&).

## ◆ logAbsDeterminant()

template<typename MatrixType >
 MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType >::logAbsDeterminant
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.
absDeterminant(), MatrixBase::determinant()

## ◆ matrixQTZ()

template<typename _MatrixType >
 const MatrixType& Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixQTZ ( ) const
inline
Returns
a reference to the matrix where the complete orthogonal decomposition is stored

## ◆ matrixT()

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>()

## ◆ matrixZ()

template<typename _MatrixType >
 MatrixType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixZ ( ) const
inline
Returns
the matrix Z.

## ◆ maxPivot()

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.

## ◆ nonzeroPivots()

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.
rank()

## ◆ pseudoInverse()

template<typename _MatrixType >
 const Inverse 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).

## ◆ rank()

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&).

## ◆ setThreshold() [1/2]

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
 threshold The 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)

## ◆ setThreshold() [2/2]

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&).

## ◆ solve()

template<typename _MatrixType >
template<typename Rhs >
 const Solve 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
 b the right-hand sides of the problem to solve.
Returns
a solution.

## ◆ threshold()

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&).

## ◆ zCoeffs()

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.