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

#include <src/eigen/Eigen/src/QR/CompleteOrthogonalDecomposition.h>

Collaboration diagram for Eigen::CompleteOrthogonalDecomposition< _MatrixType >:

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

typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::RealScalar	RealScalar

typedef MatrixType::StorageIndex	StorageIndex

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

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	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, typename internal::remove_all< typename HCoeffsType::ConjugateReturnType >::type >	HouseholderSequenceType

typedef MatrixType::PlainObject	PlainObject

Public Member Functions
	CompleteOrthogonalDecomposition ()
	Default Constructor.

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

template<typename InputType >
	CompleteOrthogonalDecomposition (const EigenBase< InputType > &matrix)
	Constructs a complete orthogonal decomposition from a given matrix.

template<typename InputType >
	CompleteOrthogonalDecomposition (EigenBase< InputType > &matrix)
	Constructs a complete orthogonal decomposition from a given matrix.

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

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::RealScalar	absDeterminant () const

MatrixType::RealScalar	logAbsDeterminant () 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 succesful.

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

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

Protected Member Functions
void	computeInPlace ()

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

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
ColPivHouseholderQR< MatrixType >	m_cpqr

HCoeffsType	m_zCoeffs

RowVectorType	m_temp

Private Types
typedef PermutationType::Index	PermIndexType

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.

See also: MatrixBase::completeOrthogonalDecomposition()

Member Typedef Documentation

◆ HCoeffsType

template<typename _MatrixType >

typedef internal::plain_diag_type<MatrixType>::type Eigen::CompleteOrthogonalDecomposition< _MatrixType >::HCoeffsType

◆ HouseholderSequenceType

template<typename _MatrixType >

typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename HCoeffsType::ConjugateReturnType>::type> Eigen::CompleteOrthogonalDecomposition< _MatrixType >::HouseholderSequenceType

◆ IntRowVectorType

template<typename _MatrixType >

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

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::MatrixType

◆ PermIndexType

template<typename _MatrixType >

typedef PermutationType::Index Eigen::CompleteOrthogonalDecomposition< _MatrixType >::PermIndexType

private

◆ PermutationType

template<typename _MatrixType >

typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::CompleteOrthogonalDecomposition< _MatrixType >::PermutationType

◆ PlainObject

template<typename _MatrixType >

typedef MatrixType::PlainObject Eigen::CompleteOrthogonalDecomposition< _MatrixType >::PlainObject

◆ RealRowVectorType

template<typename _MatrixType >

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

◆ RealScalar

template<typename _MatrixType >

typedef MatrixType::RealScalar Eigen::CompleteOrthogonalDecomposition< _MatrixType >::RealScalar

◆ RowVectorType

template<typename _MatrixType >

typedef internal::plain_row_type<MatrixType>::type Eigen::CompleteOrthogonalDecomposition< _MatrixType >::RowVectorType

◆ Scalar

template<typename _MatrixType >

typedef MatrixType::Scalar Eigen::CompleteOrthogonalDecomposition< _MatrixType >::Scalar

◆ StorageIndex

template<typename _MatrixType >

typedef MatrixType::StorageIndex Eigen::CompleteOrthogonalDecomposition< _MatrixType >::StorageIndex

Member Enumeration Documentation

◆ anonymous enum

template<typename _MatrixType >

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
MaxRowsAtCompileTime
MaxColsAtCompileTime

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

Constructor & Destructor Documentation

◆ 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&).

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

Eigen::CompleteOrthogonalDecomposition::m_cpqr

ColPivHouseholderQR< MatrixType > m_cpqr

Definition CompleteOrthogonalDecomposition.h:385

Eigen::CompleteOrthogonalDecomposition::m_zCoeffs

HCoeffsType m_zCoeffs

Definition CompleteOrthogonalDecomposition.h:386

Eigen::CompleteOrthogonalDecomposition::m_temp

RowVectorType m_temp

Definition CompleteOrthogonalDecomposition.h:387

◆ 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.

See also: CompleteOrthogonalDecomposition()

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

Eigen::CompleteOrthogonalDecomposition::cols

Index cols() const

Definition CompleteOrthogonalDecomposition.h:282

Eigen::CompleteOrthogonalDecomposition::rows

Index rows() const

Definition CompleteOrthogonalDecomposition.h:281

◆ 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);

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 >::compute(), and Eigen::EigenBase< Derived >::derived().

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◆ 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.

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 >::computeInPlace().

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Member Function Documentation

◆ _solve_impl() [1/2]

template<typename _MatrixType >

template<typename RhsType , typename DstType >

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

◆ _solve_impl() [2/2]

template<typename _MatrixType >

template<typename RhsType , typename DstType >

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

                                            {
  eigen_assert(rhs.rows() == this->rows());
 
  const Index rank = this->rank();
  if (rank == 0) {
    dst.setZero();
    return;
  }
 
  // Compute c = Q^* * rhs
  // Note that the matrix Q = H_0^* H_1^*... so its inverse is
  // Q^* = (H_0 H_1 ...)^T
  typename RhsType::PlainObject c(rhs);
  c.applyOnTheLeft(
      householderSequence(matrixQTZ(), hCoeffs()).setLength(rank).transpose());
 
  // 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 eigen_assert, and Eigen::householderSequence().

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◆ 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.

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

                                                                  {
  return m_cpqr.absDeterminant();
}

◆ applyZAdjointOnTheLeftInPlace()

template<typename MatrixType >

template<typename Rhs >

void Eigen::CompleteOrthogonalDecomposition< MatrixType >::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 MatrixType::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 row(), and tail().

Referenced by Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixZ().

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◆ check_template_parameters()

template<typename _MatrixType >

static void Eigen::CompleteOrthogonalDecomposition< _MatrixType >::check_template_parameters ( )

inlinestaticprotected

                                          {
    EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
  }

References EIGEN_STATIC_ASSERT_NON_INTEGER.

◆ cols()

template<typename _MatrixType >

Index Eigen::CompleteOrthogonalDecomposition< _MatrixType >::cols ( ) const

inline

282{ return m_cpqr.cols(); }

Eigen::ColPivHouseholderQR::cols

Index cols() const

Definition ColPivHouseholderQR.h:328

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

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◆ colsPermutation()

template<typename _MatrixType >

const PermutationType & Eigen::CompleteOrthogonalDecomposition< _MatrixType >::colsPermutation ( ) const

inline

Returns: a const reference to the column permutation matrix

                                                 {
    return m_cpqr.colsPermutation();
  }

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

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◆ compute()

template<typename _MatrixType >

template<typename InputType >

CompleteOrthogonalDecomposition & Eigen::CompleteOrthogonalDecomposition< _MatrixType >::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 >::compute(), Eigen::CompleteOrthogonalDecomposition< _MatrixType >::computeInPlace(), and Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_cpqr.

Referenced by Eigen::CompleteOrthogonalDecomposition< _MatrixType >::CompleteOrthogonalDecomposition().

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◆ 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.

See also: class CompleteOrthogonalDecomposition, CompleteOrthogonalDecomposition(const MatrixType&)

{
  check_template_parameters();
 
  // the column permutation is stored as int indices, so just to be sure:
  eigen_assert(m_cpqr.cols() <= NumTraits<int>::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).transpose(), 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 eigen_assert.

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

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◆ 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&).

242{ return m_cpqr.dimensionOfKernel(); }

Eigen::ColPivHouseholderQR::dimensionOfKernel

Index dimensionOfKernel() const

Definition ColPivHouseholderQR.h:272

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

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◆ 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.

For advanced uses only.

289{ return m_cpqr.hCoeffs(); }

Eigen::ColPivHouseholderQR::hCoeffs

const HCoeffsType & hCoeffs() const

Definition ColPivHouseholderQR.h:334

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

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◆ householderQ()

template<typename MatrixType >

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

Returns: the matrix Q as a sequence of householder transformations

                                                                {
  return m_cpqr.householderQ();
}

◆ 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

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

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

251{ return m_cpqr.isInjective(); }

Eigen::ColPivHouseholderQR::isInjective

bool isInjective() const

Definition ColPivHouseholderQR.h:285

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

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◆ 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&).

269{ return m_cpqr.isInvertible(); }

Eigen::ColPivHouseholderQR::isInvertible

bool isInvertible() const

Definition ColPivHouseholderQR.h:310

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

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◆ 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&).

260{ return m_cpqr.isSurjective(); }

Eigen::ColPivHouseholderQR::isSurjective

bool isSurjective() const

Definition ColPivHouseholderQR.h:298

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

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◆ 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.

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

                                                                     {
  return m_cpqr.logAbsDeterminant();
}

◆ matrixQ()

template<typename _MatrixType >

HouseholderSequenceType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixQ ( void ) const

inline

155{ return m_cpqr.householderQ(); }

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

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◆ 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

168{ return m_cpqr.matrixQR(); }

Eigen::ColPivHouseholderQR::matrixQR

const MatrixType & matrixQR() const

Definition ColPivHouseholderQR.h:189

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

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◆ 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>()

181{ return m_cpqr.matrixQR(); }

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

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◆ matrixZ()

template<typename _MatrixType >

MatrixType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixZ ( ) const

inline

Returns: the matrix Z.

                             {
    MatrixType Z = MatrixType::Identity(m_cpqr.cols(), m_cpqr.cols());
    applyZAdjointOnTheLeftInPlace(Z);
    return Z.adjoint();
  }

References Eigen::CompleteOrthogonalDecomposition< _MatrixType >::applyZAdjointOnTheLeftInPlace(), Eigen::ColPivHouseholderQR< _MatrixType >::cols(), and Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_cpqr.

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◆ 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.

353{ return m_cpqr.maxPivot(); }

Eigen::ColPivHouseholderQR::maxPivot

RealScalar maxPivot() const

Definition ColPivHouseholderQR.h:403

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

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◆ 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.

See also: rank()

348{ return m_cpqr.nonzeroPivots(); }

Eigen::ColPivHouseholderQR::nonzeroPivots

Index nonzeroPivots() const

Definition ColPivHouseholderQR.h:394

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

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◆ pseudoInverse()

template<typename _MatrixType >

const Inverse< CompleteOrthogonalDecomposition > Eigen::CompleteOrthogonalDecomposition< _MatrixType >::pseudoInverse ( ) const

inline

Returns: the pseudo-inverse of the matrix of which *this is the complete orthogonal decomposition.

Warning: : Do not compute this->pseudoInverse()*rhs to solve a linear systems. It is more efficient and numerically stable to call this->solve(rhs).

  {
    return Inverse<CompleteOrthogonalDecomposition>(*this);
  }

◆ 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&).

233{ return m_cpqr.rank(); }

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

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◆ rows()

template<typename _MatrixType >

Index Eigen::CompleteOrthogonalDecomposition< _MatrixType >::rows ( ) const

inline

281{ return m_cpqr.rows(); }

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

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◆ 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)

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

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

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◆ 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&).

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

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

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◆ solve()

template<typename _MatrixType >

template<typename Rhs >

const Solve< CompleteOrthogonalDecomposition, Rhs > Eigen::CompleteOrthogonalDecomposition< _MatrixType >::solve ( const MatrixBase< Rhs > & b ) const

inline

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

$\mathrm{minimize} \|A X - B\|,$

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

Parameters

b	the right-hand sides of the problem to solve.

Returns: a solution.

                                      {
    eigen_assert(m_cpqr.m_isInitialized &&
                 "CompleteOrthogonalDecomposition is not initialized.");
    return Solve<CompleteOrthogonalDecomposition, Rhs>(*this, b.derived());
  }

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

◆ 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&).

339{ return m_cpqr.threshold(); }

Eigen::ColPivHouseholderQR::threshold

RealScalar threshold() const

Definition ColPivHouseholderQR.h:378

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

Referenced by Eigen::CompleteOrthogonalDecomposition< _MatrixType >::setThreshold().

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◆ 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.

For advanced uses only.

296{ return m_zCoeffs; }

References Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_zCoeffs.

Member Data Documentation

◆ m_cpqr

template<typename _MatrixType >

ColPivHouseholderQR<MatrixType> Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_cpqr

protected

◆ m_temp

template<typename _MatrixType >

RowVectorType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_temp

protected

◆ m_zCoeffs

template<typename _MatrixType >

HCoeffsType Eigen::CompleteOrthogonalDecomposition< _MatrixType >::m_zCoeffs

protected

Referenced by Eigen::CompleteOrthogonalDecomposition< _MatrixType >::zCoeffs().

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

src/eigen/Eigen/src/Core/util/ForwardDeclarations.h
src/eigen/Eigen/src/QR/CompleteOrthogonalDecomposition.h

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Private Types

Detailed Description

Member Typedef Documentation

◆ HCoeffsType

◆ HouseholderSequenceType

◆ IntRowVectorType

◆ MatrixType

◆ PermIndexType

◆ PermutationType

◆ PlainObject

◆ RealRowVectorType

◆ RealScalar

◆ RowVectorType

◆ Scalar

◆ StorageIndex

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ CompleteOrthogonalDecomposition() [1/4]

◆ CompleteOrthogonalDecomposition() [2/4]

◆ CompleteOrthogonalDecomposition() [3/4]

◆ CompleteOrthogonalDecomposition() [4/4]

Member Function Documentation

◆ _solve_impl() [1/2]

◆ _solve_impl() [2/2]

◆ absDeterminant()

◆ applyZAdjointOnTheLeftInPlace()

◆ check_template_parameters()

◆ cols()

◆ colsPermutation()

◆ compute()

◆ computeInPlace()

◆ 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