Householder rank-revealing QR decomposition of a matrix with column-pivoting. More...

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

Inheritance diagram for Eigen::ColPivHouseholderQR< _MatrixType >:

Collaboration diagram for Eigen::ColPivHouseholderQR< _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
	ColPivHouseholderQR ()
	Default Constructor.

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

template<typename InputType >
	ColPivHouseholderQR (const EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix.

template<typename InputType >
	ColPivHouseholderQR (EigenBase< InputType > &matrix)
	Constructs a QR factorization from a given matrix.

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

HouseholderSequenceType	householderQ () const

HouseholderSequenceType	matrixQ () const

const MatrixType &	matrixQR () const

const MatrixType &	matrixR () const

template<typename InputType >
ColPivHouseholderQR &	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< ColPivHouseholderQR >	inverse () const

Index	rows () const

Index	cols () const

const HCoeffsType &	hCoeffs () const

ColPivHouseholderQR &	setThreshold (const RealScalar &threshold)

ColPivHouseholderQR &	setThreshold (Default_t)

RealScalar	threshold () const

Index	nonzeroPivots () const

RealScalar	maxPivot () const

ComputationInfo	info () const
	Reports whether the QR factorization was succesful.

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

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

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

Protected Member Functions
void	computeInPlace ()

Static Protected Member Functions
static void	check_template_parameters ()

Private Types
typedef PermutationType::StorageIndex	PermIndexType

Friends
class	CompleteOrthogonalDecomposition< MatrixType >

Detailed Description

template<typename _MatrixType>
class Eigen::ColPivHouseholderQR< _MatrixType >

Householder rank-revealing QR decomposition of a matrix with column-pivoting.

Template Parameters

_MatrixType the type of the matrix of which we are computing the QR decomposition

This class performs a rank-revealing QR decomposition of a matrix A into matrices P, Q and R such that

$\mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}$

by using Householder transformations. Here, P is a permutation matrix, Q a unitary matrix and R an upper triangular matrix.

This decomposition performs column pivoting in order to be rank-revealing and improve numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::colPivHouseholderQr()

Member Typedef Documentation

◆ HCoeffsType

template<typename _MatrixType >

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

◆ HouseholderSequenceType

template<typename _MatrixType >

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

◆ IntRowVectorType

template<typename _MatrixType >

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

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::MatrixType

◆ PermIndexType

template<typename _MatrixType >

typedef PermutationType::StorageIndex Eigen::ColPivHouseholderQR< _MatrixType >::PermIndexType

private

◆ PermutationType

template<typename _MatrixType >

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

◆ PlainObject

template<typename _MatrixType >

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

◆ RealRowVectorType

template<typename _MatrixType >

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

◆ RealScalar

template<typename _MatrixType >

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

◆ RowVectorType

template<typename _MatrixType >

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

◆ Scalar

template<typename _MatrixType >

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

◆ StorageIndex

template<typename _MatrixType >

typedef MatrixType::StorageIndex Eigen::ColPivHouseholderQR< _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

◆ ColPivHouseholderQR() [1/4]

template<typename _MatrixType >

Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( )

inline

Default Constructor.

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

      : m_qr(),
        m_hCoeffs(),
        m_colsPermutation(),
        m_colsTranspositions(),
        m_temp(),
        m_colNormsUpdated(),
        m_colNormsDirect(),
        m_isInitialized(false),
        m_usePrescribedThreshold(false) {}

◆ ColPivHouseholderQR() [2/4]

template<typename _MatrixType >

Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR	(	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: ColPivHouseholderQR()

      : m_qr(rows, cols),
        m_hCoeffs((std::min)(rows,cols)),
        m_colsPermutation(PermIndexType(cols)),
        m_colsTranspositions(cols),
        m_temp(cols),
        m_colNormsUpdated(cols),
        m_colNormsDirect(cols),
        m_isInitialized(false),
        m_usePrescribedThreshold(false) {}

◆ ColPivHouseholderQR() [3/4]

template<typename _MatrixType >

template<typename InputType >

Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( const EigenBase< InputType > & matrix )

inlineexplicit

Constructs a QR factorization from a given matrix.

This constructor computes the QR factorization of the matrix matrix by calling the method compute(). It is a short cut for:

ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());

qr.compute(matrix);

Eigen::ColPivHouseholderQR

Householder rank-revealing QR decomposition of a matrix with column-pivoting.

Definition ColPivHouseholderQR.h:49

See also: compute()

      : m_qr(matrix.rows(), matrix.cols()),
        m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
        m_colsPermutation(PermIndexType(matrix.cols())),
        m_colsTranspositions(matrix.cols()),
        m_temp(matrix.cols()),
        m_colNormsUpdated(matrix.cols()),
        m_colNormsDirect(matrix.cols()),
        m_isInitialized(false),
        m_usePrescribedThreshold(false)
    {
      compute(matrix.derived());
    }

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

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◆ ColPivHouseholderQR() [4/4]

template<typename _MatrixType >

template<typename InputType >

Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( EigenBase< InputType > & matrix )

inlineexplicit

Constructs a QR factorization from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See also: ColPivHouseholderQR(const EigenBase&)

      : m_qr(matrix.derived()),
        m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
        m_colsPermutation(PermIndexType(matrix.cols())),
        m_colsTranspositions(matrix.cols()),
        m_temp(matrix.cols()),
        m_colNormsUpdated(matrix.cols()),
        m_colNormsDirect(matrix.cols()),
        m_isInitialized(false),
        m_usePrescribedThreshold(false)
    {
      computeInPlace();
    }

References Eigen::ColPivHouseholderQR< _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::ColPivHouseholderQR< _MatrixType >::_solve_impl	(	const RhsType &	rhs,
		DstType &	dst
	)		const

◆ _solve_impl() [2/2]

template<typename _MatrixType >

template<typename RhsType , typename DstType >

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

{
  eigen_assert(rhs.rows() == rows());
 
  const Index nonzero_pivots = nonzeroPivots();
 
  if(nonzero_pivots == 0)
  {
    dst.setZero();
    return;
  }
 
  typename RhsType::PlainObject c(rhs);
 
  // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
  c.applyOnTheLeft(householderSequence(m_qr, m_hCoeffs)
                    .setLength(nonzero_pivots)
                    .transpose()
    );
 
  m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
      .template triangularView<Upper>()
      .solveInPlace(c.topRows(nonzero_pivots));
 
  for(Index i = 0; i < nonzero_pivots; ++i) dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
  for(Index i = nonzero_pivots; i < cols(); ++i) dst.row(m_colsPermutation.indices().coeff(i)).setZero();
}

References eigen_assert, and Eigen::householderSequence().

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◆ absDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::ColPivHouseholderQR< MatrixType >::absDeterminant

Returns: the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR 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()

{
  using std::abs;
  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  return abs(m_qr.diagonal().prod());
}

References eigen_assert.

◆ check_template_parameters()

template<typename _MatrixType >

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

inlinestaticprotected

    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

References EIGEN_STATIC_ASSERT_NON_INTEGER.

◆ cols()

template<typename _MatrixType >

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

inline

328{ return m_qr.cols(); }

References Eigen::ColPivHouseholderQR< _MatrixType >::m_qr.

Referenced by Eigen::CompleteOrthogonalDecomposition< _MatrixType >::cols(), Eigen::ColPivHouseholderQR< _MatrixType >::dimensionOfKernel(), Eigen::ColPivHouseholderQR< _MatrixType >::isInjective(), and Eigen::CompleteOrthogonalDecomposition< _MatrixType >::matrixZ().

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◆ colsPermutation()

template<typename _MatrixType >

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

inline

Returns: a const reference to the column permutation matrix

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_colsPermutation;
    }

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

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

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◆ compute() [1/2]

template<typename _MatrixType >

template<typename InputType >

ColPivHouseholderQR & Eigen::ColPivHouseholderQR< _MatrixType >::compute ( const EigenBase< InputType > & matrix )

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

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◆ compute() [2/2]

template<typename _MatrixType >

template<typename InputType >

ColPivHouseholderQR< MatrixType > & Eigen::ColPivHouseholderQR< _MatrixType >::compute ( const EigenBase< InputType > & matrix )

Performs the QR factorization of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

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

{
  m_qr = matrix.derived();
  computeInPlace();
  return *this;
}

References Eigen::EigenBase< Derived >::derived().

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◆ computeInPlace()

template<typename MatrixType >

void Eigen::ColPivHouseholderQR< MatrixType >::computeInPlace

protected

{
  check_template_parameters();
 
  // the column permutation is stored as int indices, so just to be sure:
  eigen_assert(m_qr.cols()<=NumTraits<int>::highest());
 
  using std::abs;
 
  Index rows = m_qr.rows();
  Index cols = m_qr.cols();
  Index size = m_qr.diagonalSize();
 
  m_hCoeffs.resize(size);
 
  m_temp.resize(cols);
 
  m_colsTranspositions.resize(m_qr.cols());
  Index number_of_transpositions = 0;
 
  m_colNormsUpdated.resize(cols);
  m_colNormsDirect.resize(cols);
  for (Index k = 0; k < cols; ++k) {
    // colNormsDirect(k) caches the most recent directly computed norm of
    // column k.
    m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
    m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
  }
 
  RealScalar threshold_helper =  numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
  RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());
 
  m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
  m_maxpivot = RealScalar(0);
 
  for(Index k = 0; k < size; ++k)
  {
    // first, we look up in our table m_colNormsUpdated which column has the biggest norm
    Index biggest_col_index;
    RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols-k).maxCoeff(&biggest_col_index));
    biggest_col_index += k;
 
    // Track the number of meaningful pivots but do not stop the decomposition to make
    // sure that the initial matrix is properly reproduced. See bug 941.
    if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
      m_nonzero_pivots = k;
 
    // apply the transposition to the columns
    m_colsTranspositions.coeffRef(k) = biggest_col_index;
    if(k != biggest_col_index) {
      m_qr.col(k).swap(m_qr.col(biggest_col_index));
      std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
      std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.coeffRef(biggest_col_index));
      ++number_of_transpositions;
    }
 
    // generate the householder vector, store it below the diagonal
    RealScalar beta;
    m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
 
    // apply the householder transformation to the diagonal coefficient
    m_qr.coeffRef(k,k) = beta;
 
    // remember the maximum absolute value of diagonal coefficients
    if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
 
    // apply the householder transformation
    m_qr.bottomRightCorner(rows-k, cols-k-1)
        .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
 
    // update our table of norms of the columns
    for (Index j = k + 1; j < cols; ++j) {
      // The following implements the stable norm downgrade step discussed in
      // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
      // and used in LAPACK routines xGEQPF and xGEQP3.
      // See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
      if (m_colNormsUpdated.coeffRef(j) != RealScalar(0)) {
        RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
        temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
        temp = temp <  RealScalar(0) ? RealScalar(0) : temp;
        RealScalar temp2 = temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) /
                                                           m_colNormsDirect.coeffRef(j));
        if (temp2 <= norm_downdate_threshold) {
          // The updated norm has become too inaccurate so re-compute the column
          // norm directly.
          m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
          m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
        } else {
          m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
        }
      }
    }
  }
 
  m_colsPermutation.setIdentity(PermIndexType(cols));
  for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
    m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
 
  m_det_pq = (number_of_transpositions%2) ? -1 : 1;
  m_isInitialized = true;
}

References eigen_assert, and Eigen::numext::sqrt().

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

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◆ dimensionOfKernel()

template<typename _MatrixType >

Index Eigen::ColPivHouseholderQR< _MatrixType >::dimensionOfKernel ( ) const

inline

Returns: the dimension of the kernel of the matrix of which *this is the QR 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&).

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return cols() - rank();
    }

References Eigen::ColPivHouseholderQR< _MatrixType >::cols(), eigen_assert, Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized, and Eigen::ColPivHouseholderQR< _MatrixType >::rank().

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

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◆ hCoeffs()

template<typename _MatrixType >

const HCoeffsType & Eigen::ColPivHouseholderQR< _MatrixType >::hCoeffs ( ) const

inline

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

For advanced uses only.

334{ return m_hCoeffs; }

References Eigen::ColPivHouseholderQR< _MatrixType >::m_hCoeffs.

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

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◆ householderQ()

template<typename MatrixType >

ColPivHouseholderQR< MatrixType >::HouseholderSequenceType Eigen::ColPivHouseholderQR< MatrixType >::householderQ

Returns: the matrix Q as a sequence of householder transformations. You can extract the meaningful part only by using:
qr.householderQ().setLength(qr.nonzeroPivots())

{
  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
  return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
}

References eigen_assert.

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

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◆ info()

template<typename _MatrixType >

ComputationInfo Eigen::ColPivHouseholderQR< _MatrixType >::info ( ) const

inline

Reports whether the QR factorization was succesful.

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

Returns: Success

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

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

◆ inverse()

template<typename _MatrixType >

const Inverse< ColPivHouseholderQR > Eigen::ColPivHouseholderQR< _MatrixType >::inverse ( ) const

inline

Returns: the inverse of the matrix of which *this is the QR decomposition.

Note: If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.

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

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

◆ isInjective()

template<typename _MatrixType >

bool Eigen::ColPivHouseholderQR< _MatrixType >::isInjective ( ) const

inline

Returns: true if the matrix of which *this is the QR 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&).

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return rank() == cols();
    }

References Eigen::ColPivHouseholderQR< _MatrixType >::cols(), eigen_assert, Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized, and Eigen::ColPivHouseholderQR< _MatrixType >::rank().

Referenced by Eigen::CompleteOrthogonalDecomposition< _MatrixType >::isInjective(), and Eigen::ColPivHouseholderQR< _MatrixType >::isInvertible().

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◆ isInvertible()

template<typename _MatrixType >

bool Eigen::ColPivHouseholderQR< _MatrixType >::isInvertible ( ) const

inline

Returns: true if the matrix of which *this is the QR 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&).

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return isInjective() && isSurjective();
    }

References eigen_assert, Eigen::ColPivHouseholderQR< _MatrixType >::isInjective(), Eigen::ColPivHouseholderQR< _MatrixType >::isSurjective(), and Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized.

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

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◆ isSurjective()

template<typename _MatrixType >

bool Eigen::ColPivHouseholderQR< _MatrixType >::isSurjective ( ) const

inline

Returns: true if the matrix of which *this is the QR 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&).

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return rank() == rows();
    }

References eigen_assert, Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized, Eigen::ColPivHouseholderQR< _MatrixType >::rank(), and Eigen::ColPivHouseholderQR< _MatrixType >::rows().

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::isInvertible(), and Eigen::CompleteOrthogonalDecomposition< _MatrixType >::isSurjective().

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◆ logAbsDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::ColPivHouseholderQR< MatrixType >::logAbsDeterminant

Returns: the natural log of the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR 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()

{
  eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
  eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
  return m_qr.diagonal().cwiseAbs().array().log().sum();
}

References eigen_assert.

◆ matrixQ()

template<typename _MatrixType >

HouseholderSequenceType Eigen::ColPivHouseholderQR< _MatrixType >::matrixQ ( ) const

inline

    {
      return householderQ();
    }

References Eigen::ColPivHouseholderQR< _MatrixType >::householderQ().

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◆ matrixQR()

template<typename _MatrixType >

const MatrixType & Eigen::ColPivHouseholderQR< _MatrixType >::matrixQR ( ) const

inline

Returns: a reference to the matrix where the Householder QR decomposition is stored

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_qr;
    }

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

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

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◆ matrixR()

template<typename _MatrixType >

const MatrixType & Eigen::ColPivHouseholderQR< _MatrixType >::matrixR ( ) const

inline

Returns: a reference to the matrix where the result Householder QR is stored

Warning: The strict lower part of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use
matrixR().template triangularView<Upper>()

Eigen::ColPivHouseholderQR::matrixR
const MatrixType & matrixR() const
Definition ColPivHouseholderQR.h:204

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

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_qr;
    }

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

◆ maxPivot()

template<typename _MatrixType >

RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::maxPivot ( ) const

inline

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

403{ return m_maxpivot; }

References Eigen::ColPivHouseholderQR< _MatrixType >::m_maxpivot.

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

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◆ nonzeroPivots()

template<typename _MatrixType >

Index Eigen::ColPivHouseholderQR< _MatrixType >::nonzeroPivots ( ) const

inline

Returns: the number of nonzero pivots in the QR 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()

    {
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      return m_nonzero_pivots;
    }

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

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

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◆ rank()

template<typename _MatrixType >

Index Eigen::ColPivHouseholderQR< _MatrixType >::rank ( ) const

inline

Returns: the rank of the matrix of which *this is the QR 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&).

    {
      using std::abs;
      eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
      RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
      Index result = 0;
      for(Index i = 0; i < m_nonzero_pivots; ++i)
        result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
      return result;
    }

References eigen_assert, Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized, Eigen::ColPivHouseholderQR< _MatrixType >::m_maxpivot, Eigen::ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots, Eigen::ColPivHouseholderQR< _MatrixType >::m_qr, and Eigen::ColPivHouseholderQR< _MatrixType >::threshold().

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::dimensionOfKernel(), Eigen::ColPivHouseholderQR< _MatrixType >::isInjective(), Eigen::ColPivHouseholderQR< _MatrixType >::isSurjective(), and Eigen::CompleteOrthogonalDecomposition< _MatrixType >::rank().

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◆ rows()

template<typename _MatrixType >

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

inline

327{ return m_qr.rows(); }

References Eigen::ColPivHouseholderQR< _MatrixType >::m_qr.

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::isSurjective(), and Eigen::CompleteOrthogonalDecomposition< _MatrixType >::rows().

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◆ setThreshold() [1/2]

template<typename _MatrixType >

ColPivHouseholderQR & Eigen::ColPivHouseholderQR< _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. This is not used for the QR decomposition itself.

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_usePrescribedThreshold = true;
      m_prescribedThreshold = threshold;
      return *this;
    }

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

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◆ setThreshold() [2/2]

template<typename _MatrixType >

ColPivHouseholderQR & Eigen::ColPivHouseholderQR< _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);

Eigen::Default

@ Default

Definition Constants.h:352

See the documentation of setThreshold(const RealScalar&).

    {
      m_usePrescribedThreshold = false;
      return *this;
    }

References Eigen::ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.

◆ solve()

template<typename _MatrixType >

template<typename Rhs >

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

inline

This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition, if any exists.

Parameters

b	the right-hand-side of the equation to solve.

Returns: a solution.

\note_about_checking_solutions

\note_about_arbitrary_choice_of_solution

Example:

Output:

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

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

Referenced by igl::mvc().

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◆ threshold()

template<typename _MatrixType >

RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::threshold ( ) const

inline

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

See the documentation of setThreshold(const RealScalar&).

    {
      eigen_assert(m_isInitialized || m_usePrescribedThreshold);
      return m_usePrescribedThreshold ? m_prescribedThreshold
      // this formula comes from experimenting (see "LU precision tuning" thread on the list)
      // and turns out to be identical to Higham's formula used already in LDLt.
                                      : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
    }

References eigen_assert, Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized, Eigen::ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold, Eigen::ColPivHouseholderQR< _MatrixType >::m_qr, and Eigen::ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.

Referenced by Eigen::MatrixBase< Homogeneous< MatrixType, _Direction > >::colPivHouseholderQr(), Eigen::ColPivHouseholderQR< _MatrixType >::rank(), and Eigen::CompleteOrthogonalDecomposition< _MatrixType >::threshold().

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Friends And Related Symbol Documentation

◆ CompleteOrthogonalDecomposition< MatrixType >

template<typename _MatrixType >

friend class CompleteOrthogonalDecomposition< MatrixType >

friend

Member Data Documentation

◆ m_colNormsDirect

template<typename _MatrixType >

RealRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colNormsDirect

protected

◆ m_colNormsUpdated

template<typename _MatrixType >

RealRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colNormsUpdated

protected

◆ m_colsPermutation

template<typename _MatrixType >

PermutationType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsPermutation

protected

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::colsPermutation().

◆ m_colsTranspositions

template<typename _MatrixType >

IntRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsTranspositions

protected

◆ m_det_pq

template<typename _MatrixType >

Index Eigen::ColPivHouseholderQR< _MatrixType >::m_det_pq

protected

◆ m_hCoeffs

template<typename _MatrixType >

HCoeffsType Eigen::ColPivHouseholderQR< _MatrixType >::m_hCoeffs

protected

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::hCoeffs().

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized

protected

◆ m_maxpivot

template<typename _MatrixType >

RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_maxpivot

protected

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::maxPivot(), and Eigen::ColPivHouseholderQR< _MatrixType >::rank().

◆ m_nonzero_pivots

template<typename _MatrixType >

Index Eigen::ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots

protected

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::nonzeroPivots(), and Eigen::ColPivHouseholderQR< _MatrixType >::rank().

◆ m_prescribedThreshold

template<typename _MatrixType >

RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold

protected

Referenced by Eigen::MatrixBase< Homogeneous< MatrixType, _Direction > >::colPivHouseholderQr(), and Eigen::ColPivHouseholderQR< _MatrixType >::threshold().

◆ m_qr

template<typename _MatrixType >

MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::m_qr

protected

Referenced by Eigen::ColPivHouseholderQR< _MatrixType >::cols(), Eigen::ColPivHouseholderQR< _MatrixType >::matrixQR(), Eigen::ColPivHouseholderQR< _MatrixType >::matrixR(), Eigen::ColPivHouseholderQR< _MatrixType >::rank(), Eigen::ColPivHouseholderQR< _MatrixType >::rows(), and Eigen::ColPivHouseholderQR< _MatrixType >::threshold().

◆ m_temp

template<typename _MatrixType >

RowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_temp

protected

◆ m_usePrescribedThreshold

template<typename _MatrixType >

bool Eigen::ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold

protected

Referenced by Eigen::MatrixBase< Homogeneous< MatrixType, _Direction > >::colPivHouseholderQr(), Eigen::ColPivHouseholderQR< _MatrixType >::setThreshold(), and Eigen::ColPivHouseholderQR< _MatrixType >::threshold().

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

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

Protected Attributes
MatrixType	m_qr

HCoeffsType	m_hCoeffs

PermutationType	m_colsPermutation

IntRowVectorType	m_colsTranspositions

RowVectorType	m_temp

RealRowVectorType	m_colNormsUpdated

RealRowVectorType	m_colNormsDirect

bool	m_isInitialized

bool	m_usePrescribedThreshold

RealScalar	m_prescribedThreshold

RealScalar	m_maxpivot

Index	m_nonzero_pivots

Index	m_det_pq

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Private Types

Friends

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

◆ ColPivHouseholderQR() [1/4]

◆ ColPivHouseholderQR() [2/4]

◆ ColPivHouseholderQR() [3/4]

◆ ColPivHouseholderQR() [4/4]

Member Function Documentation

◆ _solve_impl() [1/2]

◆ _solve_impl() [2/2]

◆ absDeterminant()

◆ check_template_parameters()

◆ cols()

◆ colsPermutation()

◆ compute() [1/2]

◆ compute() [2/2]

◆ computeInPlace()

◆ dimensionOfKernel()

◆ hCoeffs()

◆ householderQ()

◆ info()

◆ inverse()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQ()

◆ matrixQR()

◆ matrixR()

◆ maxPivot()

◆ nonzeroPivots()

◆ rank()

◆ rows()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ solve()

◆ threshold()

Friends And Related Symbol Documentation

◆ CompleteOrthogonalDecomposition< MatrixType >

Member Data Documentation

◆ m_colNormsDirect

◆ m_colNormsUpdated

◆ m_colsPermutation

◆ m_colsTranspositions

◆ m_det_pq

◆ m_hCoeffs

◆ m_isInitialized

◆ m_maxpivot

◆ m_nonzero_pivots

◆ m_prescribedThreshold

◆ m_qr

◆ m_temp

◆ m_usePrescribedThreshold