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

#include <CompleteOrthogonalDecomposition.h>

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

Public Types
enum	{ MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime , MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef MatrixType_	MatrixType

typedef SolverBase< CompleteOrthogonalDecomposition >	Base

typedef PermutationIndex_	PermutationIndex

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

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex >	PermutationType

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

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

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

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

typedef MatrixType::PlainObject	PlainObject

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

typedef EigenBase< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >	Base

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

typedef Scalar	CoeffReturnType

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

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

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
	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...

HouseholderSequenceType	householderQ (void) const

HouseholderSequenceType	matrixQ (void) const

MatrixType	matrixZ () const

const MatrixType &	matrixQTZ () const

const MatrixType &	matrixT () const

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

const PermutationType &	colsPermutation () const

MatrixType::Scalar	determinant () const

MatrixType::RealScalar	absDeterminant () const

MatrixType::RealScalar	logAbsDeterminant () const

MatrixType::Scalar	signDeterminant () const

Index	rank () const

Index	dimensionOfKernel () const

bool	isInjective () const

bool	isSurjective () const

bool	isInvertible () const

const Inverse< CompleteOrthogonalDecomposition >	pseudoInverse () const

Index	rows () const

Index	cols () const

const HCoeffsType &	hCoeffs () const

const HCoeffsType &	zCoeffs () const

CompleteOrthogonalDecomposition &	setThreshold (const RealScalar &threshold)

CompleteOrthogonalDecomposition &	setThreshold (Default_t)

RealScalar	threshold () const

Index	nonzeroPivots () const

RealScalar	maxPivot () const

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

template<typename RhsType , typename DstType >
void	_solve_impl (const RhsType &rhs, DstType &dst) const

template<bool Conjugate, typename RhsType , typename DstType >
void	_solve_impl_transposed (const RhsType &rhs, DstType &dst) const

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

	~SolverBase ()

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

const ConstTransposeReturnType	transpose () const

const AdjointReturnType	adjoint () const

constexpr EIGEN_DEVICE_FUNC CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > &	derived ()

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

Public Member Functions inherited from Eigen::EigenBase< Derived >
constexpr EIGEN_DEVICE_FUNC Derived &	derived ()

constexpr EIGEN_DEVICE_FUNC const Derived &	derived () const

EIGEN_DEVICE_FUNC Derived &	const_cast_derived () const

EIGEN_DEVICE_FUNC const Derived &	const_derived () const

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index	size () const EIGEN_NOEXCEPT

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

template<typename Dest >
EIGEN_DEVICE_FUNC void	addTo (Dest &dst) const

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

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

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

template<typename Device >
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE DeviceWrapper< Derived, Device >	device (Device &device)

template<typename Device >
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE DeviceWrapper< const Derived, Device >	device (Device &device) const

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

void	computeInPlace ()

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

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

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

HCoeffsType	m_zCoeffs

RowVectorType	m_temp

Friends
template<typename Derived >
struct	internal::solve_assertion

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()

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ HCoeffsType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ HouseholderSequenceType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ IntRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ MatrixType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PermutationIndex

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PermutationType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ PlainObject

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ RealRowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

◆ RowVectorType

template<typename MatrixType_ , typename PermutationIndex_ >

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

Member Enumeration Documentation

◆ anonymous enum

template<typename MatrixType_ , typename PermutationIndex_ >

anonymous enum

Enumerator
MaxRowsAtCompileTime
MaxColsAtCompileTime

        {
     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&).

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

Eigen::CompleteOrthogonalDecomposition::m_cpqr

ColPivHouseholderQR< MatrixType, PermutationIndex > m_cpqr

Definition: CompleteOrthogonalDecomposition.h:421

Eigen::CompleteOrthogonalDecomposition::m_temp

RowVectorType m_temp

Definition: CompleteOrthogonalDecomposition.h:423

Eigen::CompleteOrthogonalDecomposition::m_zCoeffs

HCoeffsType m_zCoeffs

Definition: CompleteOrthogonalDecomposition.h:422

◆ 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()

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

Eigen::CompleteOrthogonalDecomposition::cols

Index cols() const

Definition: CompleteOrthogonalDecomposition.h:302

Eigen::CompleteOrthogonalDecomposition::rows

Index rows() const

Definition: CompleteOrthogonalDecomposition.h:301

min

#define min(a, b)

Definition: datatypes.h:22

◆ 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()

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

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::compute(), and matrix().

◆ 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&)

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

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::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

                                                   {
     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");
   }

References b, Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::cols(), Eigen::SolverBase< CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ > >::derived(), eigen_assert, EIGEN_ONLY_USED_FOR_DEBUG, Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::rows().

◆ _solve_impl()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename RhsType , typename DstType >

void Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

                                                                                                       {
   const Index rank = this->rank();
   if (rank == 0) {
     dst.setZero();
     return;
   }
  
   // Compute c = Q^* * rhs
   typename RhsType::PlainObject c(rhs);
   c.applyOnTheLeft(matrixQ().setLength(rank).adjoint());
  
   // Solve T z = c(1:rank, :)
   dst.topRows(rank) = matrixT().topLeftCorner(rank, rank).template triangularView<Upper>().solve(c.topRows(rank));
  
   const Index cols = this->cols();
   if (rank < cols) {
     // Compute y = Z^* * [ z ]
     //                   [ 0 ]
     dst.bottomRows(cols - rank).setZero();
     applyZAdjointOnTheLeftInPlace(dst);
   }
  
   // Undo permutation to get x = P^{-1} * y.
   dst = colsPermutation() * dst;
 }

References adjoint(), calibrate::c, and cols.

◆ _solve_impl_transposed()

template<typename MatrixType_ , typename PermutationIndex_ >

template<bool Conjugate, typename RhsType , typename DstType >

void Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::_solve_impl_transposed	(	const RhsType &	rhs,
		DstType &	dst
	)		const

                                                                                                                  {
   const Index rank = this->rank();
  
   if (rank == 0) {
     dst.setZero();
     return;
   }
  
   typename RhsType::PlainObject c(colsPermutation().transpose() * rhs);
  
   if (rank < cols()) {
     applyZOnTheLeftInPlace<!Conjugate>(c);
   }
  
   matrixT()
       .topLeftCorner(rank, rank)
       .template triangularView<Upper>()
       .transpose()
       .template conjugateIf<Conjugate>()
       .solveInPlace(c.topRows(rank));
  
   dst.topRows(rank) = c.topRows(rank);
   dst.bottomRows(rows() - rank).setZero();
  
   dst.applyOnTheLeft(householderQ().setLength(rank).template conjugateIf<!Conjugate>());
 }

References calibrate::c, cols, rows, and anonymous_namespace{skew_symmetric_matrix3.cpp}::transpose().

◆ 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()

                                                                                                                   {
   return m_cpqr.absDeterminant();
 }

Referenced by cod().

◆ applyZAdjointOnTheLeftInPlace()

template<typename MatrixType , typename PermutationIndex >

template<typename Rhs >

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

protected

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

                                                                                                                 {
   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));
     }
   }
 }

References adjoint(), cols, k, max, and row().

◆ 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.

                                                                                                          {
   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));
     }
   }
 }

References cols, k, max, row(), and anonymous_namespace{skew_symmetric_matrix3.cpp}::transpose().

◆ cols()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

302 { return m_cpqr.cols(); }

Eigen::ColPivHouseholderQR::cols

Index cols() const

Definition: ColPivHouseholderQR.h:327

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::cols(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

Referenced by gdb.printers._MatrixEntryIterator::__next__(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::_check_solve_assertion(), gdb.printers.EigenMatrixPrinter::children(), gdb.printers.EigenSparseMatrixPrinter::children(), gdb.printers.EigenMatrixPrinter::to_string(), and gdb.printers.EigenSparseMatrixPrinter::to_string().

◆ colsPermutation()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Returns: a const reference to the column permutation matrix

184 { return m_cpqr.colsPermutation(); }

Eigen::ColPivHouseholderQR::colsPermutation

const PermutationType & colsPermutation() const

Definition: ColPivHouseholderQR.h:192

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::colsPermutation(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

◆ compute()

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename InputType >

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

inline

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

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::compute(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::computeInPlace(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and matrix().

Referenced by cod(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::CompleteOrthogonalDecomposition().

◆ 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&)

                                                                                    {
   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));
       }
     }
   }
 }

References beta, cols, eigen_assert, k, min, and rows.

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::CompleteOrthogonalDecomposition(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::compute().

◆ 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()

                                                                                                            {
   return m_cpqr.determinant();
 }

Referenced by cod().

◆ 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&).

262 { return m_cpqr.dimensionOfKernel(); }

Eigen::ColPivHouseholderQR::dimensionOfKernel

Index dimensionOfKernel() const

Definition: ColPivHouseholderQR.h:276

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::dimensionOfKernel(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

◆ 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.

309 { return m_cpqr.hCoeffs(); }

Eigen::ColPivHouseholderQR::hCoeffs

const HCoeffsType & hCoeffs() const

Definition: ColPivHouseholderQR.h:333

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::hCoeffs(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

◆ 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

                                                                                   {
   return m_cpqr.householderQ();
 }

Referenced by cod().

◆ 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

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

References eigen_assert, Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized, and Eigen::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&).

271 { return m_cpqr.isInjective(); }

Eigen::ColPivHouseholderQR::isInjective

bool isInjective() const

Definition: ColPivHouseholderQR.h:288

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInjective(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

◆ 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&).

289 { return m_cpqr.isInvertible(); }

Eigen::ColPivHouseholderQR::isInvertible

bool isInvertible() const

Definition: ColPivHouseholderQR.h:311

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isInvertible(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

◆ 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&).

280 { return m_cpqr.isSurjective(); }

Eigen::ColPivHouseholderQR::isSurjective

bool isSurjective() const

Definition: ColPivHouseholderQR.h:300

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::isSurjective(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

◆ 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()

           {
   return m_cpqr.logAbsDeterminant();
 }

Referenced by cod().

◆ matrixQ()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

147 { return m_cpqr.householderQ(); }

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::householderQ(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr.

◆ 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

160 { return m_cpqr.matrixQR(); }

Eigen::ColPivHouseholderQR::matrixQR

const MatrixType & matrixQR() const

Definition: ColPivHouseholderQR.h:169

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixQR().

◆ 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>()

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

173 { return m_cpqr.matrixQR(); }

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::matrixQR().

◆ matrixZ()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

Returns: the matrix Z.

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

References Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::cols(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and 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.

373 { return m_cpqr.maxPivot(); }

Eigen::ColPivHouseholderQR::maxPivot

RealScalar maxPivot() const

Definition: ColPivHouseholderQR.h:398

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::maxPivot().

◆ 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()

368 { return m_cpqr.nonzeroPivots(); }

Eigen::ColPivHouseholderQR::nonzeroPivots

Index nonzeroPivots() const

Definition: ColPivHouseholderQR.h:390

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::nonzeroPivots().

◆ 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).

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

References eigen_assert, Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::m_isInitialized.

◆ 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&).

253 { return m_cpqr.rank(); }

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::rank().

◆ rows()

template<typename MatrixType_ , typename PermutationIndex_ >

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

inline

301 { return m_cpqr.rows(); }

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::rows().

Referenced by gdb.printers._MatrixEntryIterator::__next__(), Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::_check_solve_assertion(), gdb.printers.EigenMatrixPrinter::children(), gdb.printers.EigenSparseMatrixPrinter::children(), gdb.printers.EigenMatrixPrinter::to_string(), and gdb.printers.EigenSparseMatrixPrinter::to_string().

◆ 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)

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

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::setThreshold(), and Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::threshold().

◆ 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&).

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

References Eigen::Default, Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::setThreshold().

◆ signDeterminant()

template<typename MatrixType , typename PermutationIndex >

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

Returns: the sign 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(), logAbsDeterminant(), MatrixBase::determinant()

                                                                                                                {
   return m_cpqr.signDeterminant();
 }

Referenced by cod().

◆ 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&).

359 { return m_cpqr.threshold(); }

Eigen::ColPivHouseholderQR::threshold

RealScalar threshold() const

Definition: ColPivHouseholderQR.h:375

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_cpqr, and Eigen::ColPivHouseholderQR< MatrixType_, PermutationIndex_ >::threshold().

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::setThreshold().

◆ 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.

316 { return m_zCoeffs; }

References Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::m_zCoeffs.

Friends And Related Function Documentation

◆ internal::solve_assertion

template<typename MatrixType_ , typename PermutationIndex_ >

template<typename Derived >

friend struct internal::solve_assertion

friend

Member Data Documentation

◆ m_cpqr

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

◆ m_temp

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

◆ m_zCoeffs

template<typename MatrixType_ , typename PermutationIndex_ >

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

protected

Referenced by Eigen::CompleteOrthogonalDecomposition< MatrixType_, PermutationIndex_ >::zCoeffs().

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

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Friends

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()

◆ _solve_impl()

◆ _solve_impl_transposed()

◆ 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]

◆ signDeterminant()

◆ threshold()

◆ zCoeffs()

Friends And Related Function Documentation

◆ internal::solve_assertion

Member Data Documentation

◆ m_cpqr

◆ m_temp

◆ m_zCoeffs

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