Complete orthogonal decomposition (COD) of a matrix. More...

Inheritance diagram for Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >:

Public Types
enum	{ MaxRowsAtCompileTime , MaxColsAtCompileTime }

typedef SolverBase< CompleteOrthogonalDecomposition >	Base

typedef internal::plain_diag_type< MatrixType >::type	HCoeffsType

typedef HouseholderSequence< MatrixType, internal::remove_all_t< typename HCoeffsType::ConjugateReturnType > >	HouseholderSequenceType

typedef internal::plain_row_type< MatrixType, Index >::type	IntRowVectorType

typedef MatrixType_	MatrixType

typedef PermutationIndex_	PermutationIndex

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex >	PermutationType

typedef MatrixType::PlainObject	PlainObject

typedef internal::plain_row_type< MatrixType, RealScalar >::type	RealRowVectorType

typedef internal::plain_row_type< MatrixType >::type	RowVectorType

Public Types inherited from Eigen::SolverBase< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >
enum

typedef std::conditional_t< NumTraits< Scalar >::IsComplex, CwiseUnaryOp< internal::scalar_conjugate_op< Scalar >, const ConstTransposeReturnType >, const ConstTransposeReturnType >	AdjointReturnType

typedef EigenBase< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >	Base

typedef Scalar	CoeffReturnType

typedef Transpose< const CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >	ConstTransposeReturnType

typedef internal::traits< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >::Scalar	Scalar

Public Types inherited from Eigen::EigenBase< Derived >
typedef Eigen::Index	Index
	The interface type of indices. More...

typedef internal::traits< Derived >::StorageKind	StorageKind

Public Member Functions
MatrixType::RealScalar	absDeterminant () const

Index	cols () const

const PermutationType &	colsPermutation () 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...

template<typename InputType >
CompleteOrthogonalDecomposition &	compute (const EigenBase< InputType > &matrix)

MatrixType::Scalar	determinant () const

Index	dimensionOfKernel () const

const HCoeffsType &	hCoeffs () const

HouseholderSequenceType	householderQ (void) const

ComputationInfo	info () const
	Reports whether the complete orthogonal decomposition was successful. More...

bool	isInjective () const

bool	isInvertible () const

bool	isSurjective () const

MatrixType::RealScalar	logAbsDeterminant () const

HouseholderSequenceType	matrixQ (void) 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

Index	rows () const

CompleteOrthogonalDecomposition &	setThreshold (const RealScalar &threshold)

CompleteOrthogonalDecomposition &	setThreshold (Default_t)

template<typename Rhs >
const Solve< CompleteOrthogonalDecomposition, Rhs >	solve (const MatrixBase< Rhs > &b) const

RealScalar	threshold () const

const HCoeffsType &	zCoeffs () const

Public Member Functions inherited from Eigen::SolverBase< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >
const AdjointReturnType	adjoint () const

CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > &	derived ()

const CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > &	derived () const

const Solve< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >, Rhs >	solve (const MatrixBase< Rhs > &b) const

	SolverBase ()

const ConstTransposeReturnType	transpose () const

	~SolverBase ()

Public Member Functions inherited from Eigen::EigenBase< Derived >
template<typename Dest >
void	addTo (Dest &dst) const

template<typename Dest >
void	applyThisOnTheLeft (Dest &dst) const

template<typename Dest >
void	applyThisOnTheRight (Dest &dst) const

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

Derived &	const_cast_derived () const

const Derived &	const_derived () const

Derived &	derived ()

const Derived &	derived () const

template<typename Dest >
void	evalTo (Dest &dst) const

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

template<typename Dest >
void	subTo (Dest &dst) const

Protected Member Functions
template<bool Transpose_, typename Rhs >
void	_check_solve_assertion (const Rhs &b) const

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

template<bool Conjugate, typename Rhs >
void	applyZOnTheLeftInPlace (Rhs &rhs) const

void	computeInPlace ()

Protected Member Functions inherited from Eigen::SolverBase< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >
void	_check_solve_assertion (const Rhs &b) const

Protected Attributes
ColPivHouseholderQR< MatrixType, PermutationIndex >	m_cpqr

RowVectorType	m_temp

HCoeffsType	m_zCoeffs

Detailed Description

template<typename MatrixType_, typename PermutationIndex_>
class Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >

Complete orthogonal decomposition (COD) of a matrix.

Template 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.

See also: MatrixBase::completeOrthogonalDecomposition()

Definition at line 52 of file CompleteOrthogonalDecomposition.h.

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

typedef SolverBase<CompleteOrthogonalDecomposition> Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::Base

Definition at line 57 of file CompleteOrthogonalDecomposition.h.

◆ HCoeffsType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_diag_type<MatrixType>::type Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::HCoeffsType

Definition at line 67 of file CompleteOrthogonalDecomposition.h.

◆ HouseholderSequenceType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef HouseholderSequence< MatrixType, internal::remove_all_t< typename HCoeffsType::ConjugateReturnType> > Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::HouseholderSequenceType

Definition at line 78 of file CompleteOrthogonalDecomposition.h.

◆ IntRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_row_type<MatrixType, Index>::type Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::IntRowVectorType

Definition at line 71 of file CompleteOrthogonalDecomposition.h.

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef MatrixType_ Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::MatrixType

Definition at line 56 of file CompleteOrthogonalDecomposition.h.

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

typedef PermutationIndex_ Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::PermutationIndex

Definition at line 61 of file CompleteOrthogonalDecomposition.h.

◆ PermutationType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex> Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::PermutationType

Definition at line 69 of file CompleteOrthogonalDecomposition.h.

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

typedef MatrixType::PlainObject Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::PlainObject

Definition at line 79 of file CompleteOrthogonalDecomposition.h.

◆ RealRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_row_type<MatrixType, RealScalar>::type Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::RealRowVectorType

Definition at line 74 of file CompleteOrthogonalDecomposition.h.

◆ RowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

typedef internal::plain_row_type<MatrixType>::type Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::RowVectorType

Definition at line 72 of file CompleteOrthogonalDecomposition.h.

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 63 of file CompleteOrthogonalDecomposition.h.

        {
     MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
     MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
   };

Constructor & Destructor Documentation

◆ CompleteOrthogonalDecomposition() [1/4]

template<typename MatrixType_ , typename PermutationIndex_ >

Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::CompleteOrthogonalDecomposition ( )

inline

Default Constructor.

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

Definition at line 89 of file CompleteOrthogonalDecomposition.h.

89 : m_cpqr(), m_zCoeffs(), m_temp() {}

Eigen::CompleteOrthogonalDecomposition::m_cpqr

ColPivHouseholderQR< MatrixType, PermutationIndex > m_cpqr

Definition: CompleteOrthogonalDecomposition.h:417

Eigen::CompleteOrthogonalDecomposition::m_temp

RowVectorType m_temp

Definition: CompleteOrthogonalDecomposition.h:419

Eigen::CompleteOrthogonalDecomposition::m_zCoeffs

HCoeffsType m_zCoeffs

Definition: CompleteOrthogonalDecomposition.h:418

◆ CompleteOrthogonalDecomposition() [2/4]

template<typename MatrixType_ , typename PermutationIndex_ >

Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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()

Definition at line 97 of file CompleteOrthogonalDecomposition.h.

98 : m_cpqr(rows, cols), m_zCoeffs((std::min)(rows, cols)), m_temp(cols) {}

Eigen::CompleteOrthogonalDecomposition::cols

Index cols() const

Definition: CompleteOrthogonalDecomposition.h:299

Eigen::CompleteOrthogonalDecomposition::rows

Index rows() const

Definition: CompleteOrthogonalDecomposition.h:298

Eigen::bfloat16_impl::min

bfloat16() min(const bfloat16 &a, const bfloat16 &b)

Definition: BFloat16.h:684

◆ CompleteOrthogonalDecomposition() [3/4]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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()

Definition at line 117 of file CompleteOrthogonalDecomposition.h.

       : m_cpqr(matrix.rows(), matrix.cols()),
         m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
         m_temp(matrix.cols())
   {
     compute(matrix.derived());
   }

◆ CompleteOrthogonalDecomposition() [4/4]

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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&)

Definition at line 132 of file CompleteOrthogonalDecomposition.h.

     : m_cpqr(matrix.derived()),
       m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
       m_temp(matrix.cols())
   {
     computeInPlace();
   } 

Member Function Documentation

◆ _check_solve_assertion()

template<typename MatrixType_ , typename PermutationIndex_ >

template<bool Transpose_, typename Rhs >

void Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::_check_solve_assertion ( const Rhs & b ) const

inlineprotected

Definition at line 397 of file CompleteOrthogonalDecomposition.h.

                                                   {
       EIGEN_ONLY_USED_FOR_DEBUG(b);
       eigen_assert(m_cpqr.m_isInitialized && "CompleteOrthogonalDecomposition is not initialized.");
       eigen_assert((Transpose_?derived().cols():derived().rows())==b.rows() && "CompleteOrthogonalDecomposition::solve(): invalid number of rows of the right hand side matrix b");
   }

◆ absDeterminant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::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.

See also: determinant(), logAbsDeterminant(), MatrixBase::determinant()

Definition at line 430 of file CompleteOrthogonalDecomposition.h.

                                                                                     {
   return m_cpqr.absDeterminant();
 }

◆ applyZAdjointOnTheLeftInPlace()

template<typename MatrixType , typename PermutationIndex >

template<typename Rhs >

void Eigen::CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::applyZAdjointOnTheLeftInPlace ( Rhs & rhs ) const

protected

Overwrites rhs with \( \mathbf{Z}^* * \mathbf{rhs} \).

Definition at line 526 of file CompleteOrthogonalDecomposition.h.

                     {
   const Index cols = this->cols();
   const Index nrhs = rhs.cols();
   const Index rank = this->rank();
   Matrix<typename Rhs::Scalar, Dynamic, 1> temp((std::max)(cols, nrhs));
   for (Index k = 0; k < rank; ++k) {
     if (k != rank - 1) {
       rhs.row(k).swap(rhs.row(rank - 1));
     }
     rhs.middleRows(rank - 1, cols - rank + 1)
         .applyHouseholderOnTheLeft(
             matrixQTZ().row(k).tail(cols - rank).adjoint(), zCoeffs()(k),
             &temp(0));
     if (k != rank - 1) {
       rhs.row(k).swap(rhs.row(rank - 1));
     }
   }
 }

◆ applyZOnTheLeftInPlace()

template<typename MatrixType , typename PermutationIndex >

template<bool Conjugate, typename Rhs >

void Eigen::CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::applyZOnTheLeftInPlace ( Rhs & rhs ) const

protected

Overwrites rhs with \( \mathbf{Z} * \mathbf{rhs} \) or \( \mathbf{\overline Z} * \mathbf{rhs} \) if Conjugate is set to true.

Definition at line 504 of file CompleteOrthogonalDecomposition.h.

                     {
   const Index cols = this->cols();
   const Index nrhs = rhs.cols();
   const Index rank = this->rank();
   Matrix<typename Rhs::Scalar, Dynamic, 1> temp((std::max)(cols, nrhs));
   for (Index k = rank-1; k >= 0; --k) {
     if (k != rank - 1) {
       rhs.row(k).swap(rhs.row(rank - 1));
     }
     rhs.middleRows(rank - 1, cols - rank + 1)
         .applyHouseholderOnTheLeft(
             matrixQTZ().row(k).tail(cols - rank).transpose().template conjugateIf<!Conjugate>(), zCoeffs().template conjugateIf<Conjugate>()(k),
             &temp(0));
     if (k != rank - 1) {
       rhs.row(k).swap(rhs.row(rank - 1));
     }
   }
 }

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::cols ( void ) const

inline

Definition at line 299 of file CompleteOrthogonalDecomposition.h.

299 { return m_cpqr.cols(); }

Eigen::ColPivHouseholderQR::cols

Index cols() const

Definition: ColPivHouseholderQR.h:328

◆ colsPermutation()

template<typename MatrixType_ , typename PermutationIndex_ >

const PermutationType& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::colsPermutation ( ) const

inline

Returns: a const reference to the column permutation matrix

Definition at line 193 of file CompleteOrthogonalDecomposition.h.

                                                  {
     return m_cpqr.colsPermutation();
   }

◆ compute()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::compute ( const EigenBase< InputType > & matrix )

inline

Definition at line 185 of file CompleteOrthogonalDecomposition.h.

                                                                                {
     // Compute the column pivoted QR factorization A P = Q R.
     m_cpqr.compute(matrix);
     computeInPlace();
     return *this;
   }

◆ computeInPlace()

template<typename MatrixType , typename PermutationIndex >

void Eigen::CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::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&)

Definition at line 448 of file CompleteOrthogonalDecomposition.h.

 {
   eigen_assert(m_cpqr.cols() <= NumTraits<PermutationIndex>::highest());
  
   const Index rank = m_cpqr.rank();
   const Index cols = m_cpqr.cols();
   const Index rows = m_cpqr.rows();
   m_zCoeffs.resize((std::min)(rows, cols));
   m_temp.resize(cols);
  
   if (rank < cols) {
     // We have reduced the (permuted) matrix to the form
     //   [R11 R12]
     //   [ 0  R22]
     // where R11 is r-by-r (r = rank) upper triangular, R12 is
     // r-by-(n-r), and R22 is empty or the norm of R22 is negligible.
     // We now compute the complete orthogonal decomposition by applying
     // Householder transformations from the right to the upper trapezoidal
     // matrix X = [R11 R12] to zero out R12 and obtain the factorization
     // [R11 R12] = [T11 0] * Z, where T11 is r-by-r upper triangular and
     // Z = Z(0) * Z(1) ... Z(r-1) is an n-by-n orthogonal matrix.
     // We store the data representing Z in R12 and m_zCoeffs.
     for (Index k = rank - 1; k >= 0; --k) {
       if (k != rank - 1) {
         // Given the API for Householder reflectors, it is more convenient if
         // we swap the leading parts of columns k and r-1 (zero-based) to form
         // the matrix X_k = [X(0:k, k), X(0:k, r:n)]
         m_cpqr.m_qr.col(k).head(k + 1).swap(
             m_cpqr.m_qr.col(rank - 1).head(k + 1));
       }
       // Construct Householder reflector Z(k) to zero out the last row of X_k,
       // i.e. choose Z(k) such that
       // [X(k, k), X(k, r:n)] * Z(k) = [beta, 0, .., 0].
       RealScalar beta;
       m_cpqr.m_qr.row(k)
           .tail(cols - rank + 1)
           .makeHouseholderInPlace(m_zCoeffs(k), beta);
       m_cpqr.m_qr(k, rank - 1) = beta;
       if (k > 0) {
         // Apply Z(k) to the first k rows of X_k
         m_cpqr.m_qr.topRightCorner(k, cols - rank + 1)
             .applyHouseholderOnTheRight(
                 m_cpqr.m_qr.row(k).tail(cols - rank).adjoint(), m_zCoeffs(k),
                 &m_temp(0));
       }
       if (k != rank - 1) {
         // Swap X(0:k,k) back to its proper location.
         m_cpqr.m_qr.col(k).head(k + 1).swap(
             m_cpqr.m_qr.col(rank - 1).head(k + 1));
       }
     }
   }
 }

◆ determinant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::Scalar Eigen::CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::determinant

Returns: 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: absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()

Definition at line 424 of file CompleteOrthogonalDecomposition.h.

                                                                                  {
   return m_cpqr.determinant();
 }

◆ dimensionOfKernel()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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&).

Definition at line 258 of file CompleteOrthogonalDecomposition.h.

258 { return m_cpqr.dimensionOfKernel(); }

Eigen::ColPivHouseholderQR::dimensionOfKernel

Index dimensionOfKernel() const

Definition: ColPivHouseholderQR.h:272

◆ hCoeffs()

template<typename MatrixType_ , typename PermutationIndex_ >

const HCoeffsType& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::hCoeffs ( ) const

inline

Returns: a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

Definition at line 306 of file CompleteOrthogonalDecomposition.h.

306 { return m_cpqr.hCoeffs(); }

Eigen::ColPivHouseholderQR::hCoeffs

const HCoeffsType & hCoeffs() const

Definition: ColPivHouseholderQR.h:334

◆ householderQ()

template<typename MatrixType , typename PermutationIndex >

CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::householderQ ( void ) const

Returns: the matrix Q as a sequence of householder transformations

Definition at line 634 of file CompleteOrthogonalDecomposition.h.

                                                                                   {
   return m_cpqr.householderQ();
 }

◆ info()

template<typename MatrixType_ , typename PermutationIndex_ >

ComputationInfo Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::info ( ) const

inline

Reports whether the complete orthogonal decomposition was successful.

Note: This function always returns Success. It is provided for compatibility with other factorization routines.

Returns: Success

Definition at line 380 of file CompleteOrthogonalDecomposition.h.

                                {
     eigen_assert(m_cpqr.m_isInitialized && "Decomposition is not initialized.");
     return Success;
   }

◆ isInjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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&).

Definition at line 267 of file CompleteOrthogonalDecomposition.h.

267 { return m_cpqr.isInjective(); }

Eigen::ColPivHouseholderQR::isInjective

bool isInjective() const

Definition: ColPivHouseholderQR.h:285

◆ isInvertible()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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&).

Definition at line 285 of file CompleteOrthogonalDecomposition.h.

285 { return m_cpqr.isInvertible(); }

Eigen::ColPivHouseholderQR::isInvertible

bool isInvertible() const

Definition: ColPivHouseholderQR.h:310

◆ isSurjective()

template<typename MatrixType_ , typename PermutationIndex_ >

bool Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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&).

Definition at line 276 of file CompleteOrthogonalDecomposition.h.

276 { return m_cpqr.isSurjective(); }

Eigen::ColPivHouseholderQR::isSurjective

bool isSurjective() const

Definition: ColPivHouseholderQR.h:298

◆ logAbsDeterminant()

template<typename MatrixType , typename PermutationIndex >

MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType, PermutationIndex >::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.

See also: determinant(), absDeterminant(), MatrixBase::determinant()

Definition at line 436 of file CompleteOrthogonalDecomposition.h.

                                                                                        {
   return m_cpqr.logAbsDeterminant();
 }

◆ matrixQ()

template<typename MatrixType_ , typename PermutationIndex_ >

HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::matrixQ ( void ) const

inline

Definition at line 156 of file CompleteOrthogonalDecomposition.h.

156 { return m_cpqr.householderQ(); }

◆ matrixQTZ()

template<typename MatrixType_ , typename PermutationIndex_ >

const MatrixType& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::matrixQTZ ( ) const

inline

Returns: a reference to the matrix where the complete orthogonal decomposition is stored

Definition at line 169 of file CompleteOrthogonalDecomposition.h.

169 { return m_cpqr.matrixQR(); }

Eigen::ColPivHouseholderQR::matrixQR

const MatrixType & matrixQR() const

Definition: ColPivHouseholderQR.h:174

◆ matrixT()

template<typename MatrixType_ , typename PermutationIndex_ >

const MatrixType& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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>()

Eigen::CompleteOrthogonalDecomposition::matrixT
const MatrixType & matrixT() const
Definition: CompleteOrthogonalDecomposition.h:182

For rank-deficient matrices, use
matrixT().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Definition at line 182 of file CompleteOrthogonalDecomposition.h.

182 { return m_cpqr.matrixQR(); }

◆ matrixZ()

template<typename MatrixType_ , typename PermutationIndex_ >

MatrixType Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::matrixZ ( ) const

inline

Returns: the matrix Z.

Definition at line 160 of file CompleteOrthogonalDecomposition.h.

                              {
     MatrixType Z = MatrixType::Identity(m_cpqr.cols(), m_cpqr.cols());
     applyZOnTheLeftInPlace<false>(Z);
     return Z;
   }

◆ maxPivot()

template<typename MatrixType_ , typename PermutationIndex_ >

RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::maxPivot ( ) const

inline

Returns: the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of R.

Definition at line 370 of file CompleteOrthogonalDecomposition.h.

370 { return m_cpqr.maxPivot(); }

Eigen::ColPivHouseholderQR::maxPivot

RealScalar maxPivot() const

Definition: ColPivHouseholderQR.h:403

◆ nonzeroPivots()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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()

Definition at line 365 of file CompleteOrthogonalDecomposition.h.

365 { return m_cpqr.nonzeroPivots(); }

Eigen::ColPivHouseholderQR::nonzeroPivots

Index nonzeroPivots() const

Definition: ColPivHouseholderQR.h:394

◆ pseudoInverse()

template<typename MatrixType_ , typename PermutationIndex_ >

const Inverse<CompleteOrthogonalDecomposition> Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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).

Definition at line 292 of file CompleteOrthogonalDecomposition.h.

   {
     eigen_assert(m_cpqr.m_isInitialized && "CompleteOrthogonalDecomposition is not initialized.");
     return Inverse<CompleteOrthogonalDecomposition>(*this);
   }

◆ rank()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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&).

Definition at line 249 of file CompleteOrthogonalDecomposition.h.

249 { return m_cpqr.rank(); }

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

Index Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::rows ( void ) const

inline

Definition at line 298 of file CompleteOrthogonalDecomposition.h.

298 { return m_cpqr.rows(); }

◆ setThreshold() [1/2]

template<typename MatrixType_ , typename PermutationIndex_ >

CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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)

Definition at line 334 of file CompleteOrthogonalDecomposition.h.

                                                                              {
     m_cpqr.setThreshold(threshold);
     return *this;
   }

◆ setThreshold() [2/2]

template<typename MatrixType_ , typename PermutationIndex_ >

CompleteOrthogonalDecomposition& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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);

qr

HouseholderQR< MatrixXf > qr(A)

See the documentation of setThreshold(const RealScalar&).

Definition at line 347 of file CompleteOrthogonalDecomposition.h.

                                                            {
     m_cpqr.setThreshold(Default);
     return *this;
   }

◆ solve()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename Rhs >

const Solve<CompleteOrthogonalDecomposition, Rhs> Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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_ , typename PermutationIndex_ >

RealScalar Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::threshold ( ) const

inline

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

See the documentation of setThreshold(const RealScalar&).

Definition at line 356 of file CompleteOrthogonalDecomposition.h.

356 { return m_cpqr.threshold(); }

Eigen::ColPivHouseholderQR::threshold

RealScalar threshold() const

Definition: ColPivHouseholderQR.h:378

◆ zCoeffs()

template<typename MatrixType_ , typename PermutationIndex_ >

const HCoeffsType& Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::zCoeffs ( ) const

inline

Returns: a const reference to the vector of Householder coefficients used to represent the factor Z.

For advanced uses only.

Definition at line 313 of file CompleteOrthogonalDecomposition.h.

313 { return m_zCoeffs; }

Member Data Documentation

◆ m_cpqr

template<typename MatrixType_ , typename PermutationIndex_ >

ColPivHouseholderQR<MatrixType, PermutationIndex> Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr

protected

Definition at line 417 of file CompleteOrthogonalDecomposition.h.

◆ m_temp

template<typename MatrixType_ , typename PermutationIndex_ >

RowVectorType Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_temp

protected

Definition at line 419 of file CompleteOrthogonalDecomposition.h.

◆ m_zCoeffs

template<typename MatrixType_ , typename PermutationIndex_ >

HCoeffsType Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_zCoeffs

protected

Definition at line 418 of file CompleteOrthogonalDecomposition.h.

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

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Detailed Description

template<typename MatrixType_, typename PermutationIndex_> class Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >

Member Typedef Documentation

◆ Base

◆ HCoeffsType

◆ HouseholderSequenceType

◆ IntRowVectorType

◆ MatrixType

◆ PermutationIndex

◆ PermutationType

◆ PlainObject

◆ RealRowVectorType

◆ RowVectorType

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ CompleteOrthogonalDecomposition() [1/4]

◆ CompleteOrthogonalDecomposition() [2/4]

◆ CompleteOrthogonalDecomposition() [3/4]

◆ CompleteOrthogonalDecomposition() [4/4]

Member Function Documentation

◆ _check_solve_assertion()

◆ absDeterminant()

◆ applyZAdjointOnTheLeftInPlace()

◆ applyZOnTheLeftInPlace()

◆ cols()

◆ colsPermutation()

◆ compute()

◆ computeInPlace()

◆ determinant()

◆ dimensionOfKernel()

◆ hCoeffs()

◆ householderQ()

◆ info()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQ()

◆ matrixQTZ()

◆ matrixT()

◆ matrixZ()

◆ maxPivot()

◆ nonzeroPivots()

◆ pseudoInverse()

◆ rank()

◆ rows()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ solve()

◆ threshold()

◆ zCoeffs()

Member Data Documentation

◆ m_cpqr

◆ m_temp

◆ m_zCoeffs

template<typename MatrixType_, typename PermutationIndex_>
class Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >