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

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

Inheritance diagram for Eigen::FullPivHouseholderQR< _MatrixType >:

Collaboration diagram for Eigen::FullPivHouseholderQR< _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::FullPivHouseholderQRMatrixQReturnType< MatrixType >	MatrixQReturnType

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

typedef Matrix< StorageIndex, 1, EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime, RowsAtCompileTime), RowMajor, 1, EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime, MaxRowsAtCompileTime)>	IntDiagSizeVectorType

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType

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

typedef internal::plain_col_type< MatrixType >::type	ColVectorType

typedef MatrixType::PlainObject	PlainObject

Public Member Functions
	FullPivHouseholderQR ()
	Default Constructor.

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

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

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

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

MatrixQReturnType	matrixQ (void) const

const MatrixType &	matrixQR () const

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

const PermutationType &	colsPermutation () const

const IntDiagSizeVectorType &	rowsTranspositions () 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< FullPivHouseholderQR >	inverse () const

Index	rows () const

Index	cols () const

const HCoeffsType &	hCoeffs () const

FullPivHouseholderQR &	setThreshold (const RealScalar &threshold)

FullPivHouseholderQR &	setThreshold (Default_t)

RealScalar	threshold () const

Index	nonzeroPivots () const

RealScalar	maxPivot () const

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

template<typename InputType >
FullPivHouseholderQR< 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 ()

Detailed Description

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

Householder rank-revealing QR decomposition of a matrix with full 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, P', Q and R such that

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

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

This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also: MatrixBase::fullPivHouseholderQr()

Member Typedef Documentation

◆ ColVectorType

template<typename _MatrixType >

typedef internal::plain_col_type<MatrixType>::type Eigen::FullPivHouseholderQR< _MatrixType >::ColVectorType

◆ HCoeffsType

template<typename _MatrixType >

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

◆ IntDiagSizeVectorType

template<typename _MatrixType >

typedef Matrix<StorageIndex, 1, EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1, EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime,MaxRowsAtCompileTime)> Eigen::FullPivHouseholderQR< _MatrixType >::IntDiagSizeVectorType

◆ MatrixQReturnType

template<typename _MatrixType >

typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> Eigen::FullPivHouseholderQR< _MatrixType >::MatrixQReturnType

◆ MatrixType

template<typename _MatrixType >

typedef _MatrixType Eigen::FullPivHouseholderQR< _MatrixType >::MatrixType

◆ PermutationType

template<typename _MatrixType >

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

◆ PlainObject

template<typename _MatrixType >

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

◆ RealScalar

template<typename _MatrixType >

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

◆ RowVectorType

template<typename _MatrixType >

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

◆ Scalar

template<typename _MatrixType >

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

◆ StorageIndex

template<typename _MatrixType >

typedef MatrixType::StorageIndex Eigen::FullPivHouseholderQR< _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

◆ FullPivHouseholderQR() [1/4]

template<typename _MatrixType >

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( )

inline

Default Constructor.

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

      : m_qr(),
        m_hCoeffs(),
        m_rows_transpositions(),
        m_cols_transpositions(),
        m_cols_permutation(),
        m_temp(),
        m_isInitialized(false),
        m_usePrescribedThreshold(false) {}

◆ FullPivHouseholderQR() [2/4]

template<typename _MatrixType >

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR	(	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: FullPivHouseholderQR()

      : m_qr(rows, cols),
        m_hCoeffs((std::min)(rows,cols)),
        m_rows_transpositions((std::min)(rows,cols)),
        m_cols_transpositions((std::min)(rows,cols)),
        m_cols_permutation(cols),
        m_temp(cols),
        m_isInitialized(false),
        m_usePrescribedThreshold(false) {}

◆ FullPivHouseholderQR() [3/4]

template<typename _MatrixType >

template<typename InputType >

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( 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:

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

qr.compute(matrix);

Eigen::FullPivHouseholderQR

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

Definition FullPivHouseholderQR.h:58

See also: compute()

      : m_qr(matrix.rows(), matrix.cols()),
        m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
        m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_permutation(matrix.cols()),
        m_temp(matrix.cols()),
        m_isInitialized(false),
        m_usePrescribedThreshold(false)
    {
      compute(matrix.derived());
    }

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

Here is the call graph for this function:

◆ FullPivHouseholderQR() [4/4]

template<typename _MatrixType >

template<typename InputType >

Eigen::FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( 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: FullPivHouseholderQR(const EigenBase&)

      : m_qr(matrix.derived()),
        m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
        m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
        m_cols_permutation(matrix.cols()),
        m_temp(matrix.cols()),
        m_isInitialized(false),
        m_usePrescribedThreshold(false)
    {
      computeInPlace();
    }

References Eigen::FullPivHouseholderQR< _MatrixType >::computeInPlace().

Here is the call graph for this function:

Member Function Documentation

◆ _solve_impl() [1/2]

template<typename _MatrixType >

template<typename RhsType , typename DstType >

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

◆ _solve_impl() [2/2]

template<typename _MatrixType >

template<typename RhsType , typename DstType >

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

{
  eigen_assert(rhs.rows() == rows());
  const Index l_rank = rank();
 
  // FIXME introduce nonzeroPivots() and use it here. and more generally,
  // make the same improvements in this dec as in FullPivLU.
  if(l_rank==0)
  {
    dst.setZero();
    return;
  }
 
  typename RhsType::PlainObject c(rhs);
 
  Matrix<Scalar,1,RhsType::ColsAtCompileTime> temp(rhs.cols());
  for (Index k = 0; k < l_rank; ++k)
  {
    Index remainingSize = rows()-k;
    c.row(k).swap(c.row(m_rows_transpositions.coeff(k)));
    c.bottomRightCorner(remainingSize, rhs.cols())
      .applyHouseholderOnTheLeft(m_qr.col(k).tail(remainingSize-1),
                               m_hCoeffs.coeff(k), &temp.coeffRef(0));
  }
 
  m_qr.topLeftCorner(l_rank, l_rank)
      .template triangularView<Upper>()
      .solveInPlace(c.topRows(l_rank));
 
  for(Index i = 0; i < l_rank; ++i) dst.row(m_cols_permutation.indices().coeff(i)) = c.row(i);
  for(Index i = l_rank; i < cols(); ++i) dst.row(m_cols_permutation.indices().coeff(i)).setZero();
}

References Eigen::Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols >::coeffRef(), and eigen_assert.

Here is the call graph for this function:

◆ absDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< 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 && "FullPivHouseholderQR 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::FullPivHouseholderQR< _MatrixType >::check_template_parameters ( )

inlinestaticprotected

    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

References EIGEN_STATIC_ASSERT_NON_INTEGER.

◆ cols()

template<typename _MatrixType >

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

inline

319{ return m_qr.cols(); }

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

Referenced by Eigen::FullPivHouseholderQR< _MatrixType >::dimensionOfKernel(), and Eigen::FullPivHouseholderQR< _MatrixType >::isInjective().

Here is the caller graph for this function:

◆ colsPermutation()

template<typename _MatrixType >

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

inline

Returns: a const reference to the column permutation matrix

    {
      eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
      return m_cols_permutation;
    }

References eigen_assert, Eigen::FullPivHouseholderQR< _MatrixType >::m_cols_permutation, and Eigen::FullPivHouseholderQR< _MatrixType >::m_isInitialized.

◆ compute() [1/2]

template<typename _MatrixType >

template<typename InputType >

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

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

Here is the caller graph for this function:

◆ compute() [2/2]

template<typename _MatrixType >

template<typename InputType >

FullPivHouseholderQR< MatrixType > & Eigen::FullPivHouseholderQR< _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 FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)

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

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

Here is the call graph for this function:

◆ computeInPlace()

template<typename MatrixType >

void Eigen::FullPivHouseholderQR< MatrixType >::computeInPlace

protected

{
  check_template_parameters();
 
  using std::abs;
  Index rows = m_qr.rows();
  Index cols = m_qr.cols();
  Index size = (std::min)(rows,cols);
 
  
  m_hCoeffs.resize(size);
 
  m_temp.resize(cols);
 
  m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);
 
  m_rows_transpositions.resize(size);
  m_cols_transpositions.resize(size);
  Index number_of_transpositions = 0;
 
  RealScalar biggest(0);
 
  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)
  {
    Index row_of_biggest_in_corner, col_of_biggest_in_corner;
    typedef internal::scalar_score_coeff_op<Scalar> Scoring;
    typedef typename Scoring::result_type Score;
 
    Score score = m_qr.bottomRightCorner(rows-k, cols-k)
                      .unaryExpr(Scoring())
                      .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
    row_of_biggest_in_corner += k;
    col_of_biggest_in_corner += k;
    RealScalar biggest_in_corner = internal::abs_knowing_score<Scalar>()(m_qr(row_of_biggest_in_corner, col_of_biggest_in_corner), score);
    if(k==0) biggest = biggest_in_corner;
 
    // if the corner is negligible, then we have less than full rank, and we can finish early
    if(internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
    {
      m_nonzero_pivots = k;
      for(Index i = k; i < size; i++)
      {
        m_rows_transpositions.coeffRef(i) = i;
        m_cols_transpositions.coeffRef(i) = i;
        m_hCoeffs.coeffRef(i) = Scalar(0);
      }
      break;
    }
 
    m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
    m_cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
    if(k != row_of_biggest_in_corner) {
      m_qr.row(k).tail(cols-k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols-k));
      ++number_of_transpositions;
    }
    if(k != col_of_biggest_in_corner) {
      m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
      ++number_of_transpositions;
    }
 
    RealScalar beta;
    m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
    m_qr.coeffRef(k,k) = beta;
 
    // remember the maximum absolute value of diagonal coefficients
    if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
 
    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));
  }
 
  m_cols_permutation.setIdentity(cols);
  for(Index k = 0; k < size; ++k)
    m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));
 
  m_det_pq = (number_of_transpositions%2) ? -1 : 1;
  m_isInitialized = true;
}

References Eigen::internal::isMuchSmallerThan().

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

Here is the call graph for this function:

Here is the caller graph for this function:

◆ dimensionOfKernel()

template<typename _MatrixType >

Index Eigen::FullPivHouseholderQR< _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 && "FullPivHouseholderQR is not initialized.");
      return cols() - rank();
    }

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

Here is the call graph for this function:

◆ hCoeffs()

template<typename _MatrixType >

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

inline

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

For advanced uses only.

325{ return m_hCoeffs; }

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

◆ inverse()

template<typename _MatrixType >

const Inverse< FullPivHouseholderQR > Eigen::FullPivHouseholderQR< _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 && "FullPivHouseholderQR is not initialized.");
      return Inverse<FullPivHouseholderQR>(*this);
    }

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

◆ isInjective()

template<typename _MatrixType >

bool Eigen::FullPivHouseholderQR< _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 && "FullPivHouseholderQR is not initialized.");
      return rank() == cols();
    }

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

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

Here is the call graph for this function:

Here is the caller graph for this function:

◆ isInvertible()

template<typename _MatrixType >

bool Eigen::FullPivHouseholderQR< _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 && "FullPivHouseholderQR is not initialized.");
      return isInjective() && isSurjective();
    }

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

Here is the call graph for this function:

◆ isSurjective()

template<typename _MatrixType >

bool Eigen::FullPivHouseholderQR< _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 && "FullPivHouseholderQR is not initialized.");
      return rank() == rows();
    }

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

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

Here is the call graph for this function:

Here is the caller graph for this function:

◆ logAbsDeterminant()

template<typename MatrixType >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< 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 && "FullPivHouseholderQR 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 >

FullPivHouseholderQR< MatrixType >::MatrixQReturnType Eigen::FullPivHouseholderQR< MatrixType >::matrixQ ( void ) const

inline

Returns: Expression object representing the matrix Q

{
  eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
  return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
}

References eigen_assert.

◆ matrixQR()

template<typename _MatrixType >

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

inline

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

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

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

◆ maxPivot()

template<typename _MatrixType >

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

inline

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

394{ return m_maxpivot; }

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

◆ nonzeroPivots()

template<typename _MatrixType >

Index Eigen::FullPivHouseholderQR< _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 && "LU is not initialized.");
      return m_nonzero_pivots;
    }

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

◆ rank()

template<typename _MatrixType >

Index Eigen::FullPivHouseholderQR< _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 && "FullPivHouseholderQR 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::FullPivHouseholderQR< _MatrixType >::m_isInitialized, Eigen::FullPivHouseholderQR< _MatrixType >::m_maxpivot, Eigen::FullPivHouseholderQR< _MatrixType >::m_nonzero_pivots, Eigen::FullPivHouseholderQR< _MatrixType >::m_qr, and Eigen::FullPivHouseholderQR< _MatrixType >::threshold().

Referenced by Eigen::FullPivHouseholderQR< _MatrixType >::dimensionOfKernel(), Eigen::FullPivHouseholderQR< _MatrixType >::isInjective(), and Eigen::FullPivHouseholderQR< _MatrixType >::isSurjective().

Here is the call graph for this function:

Here is the caller graph for this function:

◆ rows()

template<typename _MatrixType >

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

inline

318{ return m_qr.rows(); }

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

Referenced by Eigen::FullPivHouseholderQR< _MatrixType >::isSurjective().

Here is the caller graph for this function:

◆ rowsTranspositions()

template<typename _MatrixType >

const IntDiagSizeVectorType & Eigen::FullPivHouseholderQR< _MatrixType >::rowsTranspositions ( ) const

inline

Returns: a const reference to the vector of indices representing the rows transpositions

    {
      eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
      return m_rows_transpositions;
    }

References eigen_assert, Eigen::FullPivHouseholderQR< _MatrixType >::m_isInitialized, and Eigen::FullPivHouseholderQR< _MatrixType >::m_rows_transpositions.

◆ setThreshold() [1/2]

template<typename _MatrixType >

FullPivHouseholderQR & Eigen::FullPivHouseholderQR< _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;
    }

References Eigen::FullPivHouseholderQR< _MatrixType >::m_prescribedThreshold, Eigen::FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold, and Eigen::FullPivHouseholderQR< _MatrixType >::threshold().

Here is the call graph for this function:

◆ setThreshold() [2/2]

template<typename _MatrixType >

FullPivHouseholderQR & Eigen::FullPivHouseholderQR< _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::FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.

◆ solve()

template<typename _MatrixType >

template<typename Rhs >

const Solve< FullPivHouseholderQR, Rhs > Eigen::FullPivHouseholderQR< _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.

Parameters

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

Returns: the exact or least-square solution if the rank is greater or equal to the number of columns of A, and an arbitrary solution otherwise.

\note_about_checking_solutions

\note_about_arbitrary_choice_of_solution

Example:

Output:

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

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

◆ threshold()

template<typename _MatrixType >

RealScalar Eigen::FullPivHouseholderQR< _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::FullPivHouseholderQR< _MatrixType >::m_isInitialized, Eigen::FullPivHouseholderQR< _MatrixType >::m_prescribedThreshold, Eigen::FullPivHouseholderQR< _MatrixType >::m_qr, and Eigen::FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.

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

Here is the caller graph for this function:

Member Data Documentation

◆ m_cols_permutation

template<typename _MatrixType >

PermutationType Eigen::FullPivHouseholderQR< _MatrixType >::m_cols_permutation

protected

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

◆ m_cols_transpositions

template<typename _MatrixType >

IntDiagSizeVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_cols_transpositions

protected

◆ m_det_pq

template<typename _MatrixType >

Index Eigen::FullPivHouseholderQR< _MatrixType >::m_det_pq

protected

◆ m_hCoeffs

template<typename _MatrixType >

HCoeffsType Eigen::FullPivHouseholderQR< _MatrixType >::m_hCoeffs

protected

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

◆ m_isInitialized

template<typename _MatrixType >

bool Eigen::FullPivHouseholderQR< _MatrixType >::m_isInitialized

protected

◆ m_maxpivot

template<typename _MatrixType >

RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::m_maxpivot

protected

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

◆ m_nonzero_pivots

template<typename _MatrixType >

Index Eigen::FullPivHouseholderQR< _MatrixType >::m_nonzero_pivots

protected

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

◆ m_precision

template<typename _MatrixType >

RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::m_precision

protected

◆ m_prescribedThreshold

template<typename _MatrixType >

RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::m_prescribedThreshold

protected

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

◆ m_qr

template<typename _MatrixType >

MatrixType Eigen::FullPivHouseholderQR< _MatrixType >::m_qr

protected

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

◆ m_rows_transpositions

template<typename _MatrixType >

IntDiagSizeVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_rows_transpositions

protected

Referenced by Eigen::FullPivHouseholderQR< _MatrixType >::rowsTranspositions().

◆ m_temp

template<typename _MatrixType >

RowVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_temp

protected

◆ m_usePrescribedThreshold

template<typename _MatrixType >

bool Eigen::FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold

protected

Referenced by Eigen::FullPivHouseholderQR< _MatrixType >::setThreshold(), Eigen::FullPivHouseholderQR< _MatrixType >::setThreshold(), and Eigen::FullPivHouseholderQR< _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/FullPivHouseholderQR.h

Protected Attributes
MatrixType	m_qr

HCoeffsType	m_hCoeffs

IntDiagSizeVectorType	m_rows_transpositions

IntDiagSizeVectorType	m_cols_transpositions

PermutationType	m_cols_permutation

RowVectorType	m_temp

bool	m_isInitialized

bool	m_usePrescribedThreshold

RealScalar	m_prescribedThreshold

RealScalar	m_maxpivot

Index	m_nonzero_pivots

RealScalar	m_precision

Index	m_det_pq

Public Types

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Detailed Description

Member Typedef Documentation

◆ ColVectorType

◆ HCoeffsType

◆ IntDiagSizeVectorType

◆ MatrixQReturnType

◆ MatrixType

◆ PermutationType

◆ PlainObject

◆ RealScalar

◆ RowVectorType

◆ Scalar

◆ StorageIndex

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ FullPivHouseholderQR() [1/4]

◆ FullPivHouseholderQR() [2/4]

◆ FullPivHouseholderQR() [3/4]

◆ FullPivHouseholderQR() [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()

◆ inverse()

◆ isInjective()

◆ isInvertible()

◆ isSurjective()

◆ logAbsDeterminant()

◆ matrixQ()

◆ matrixQR()

◆ maxPivot()

◆ nonzeroPivots()

◆ rank()

◆ rows()

◆ rowsTranspositions()

◆ setThreshold() [1/2]

◆ setThreshold() [2/2]

◆ solve()

◆ threshold()

Member Data Documentation

◆ m_cols_permutation

◆ m_cols_transpositions

◆ m_det_pq

◆ m_hCoeffs

◆ m_isInitialized

◆ m_maxpivot

◆ m_nonzero_pivots

◆ m_precision

◆ m_prescribedThreshold

◆ m_qr

◆ m_rows_transpositions

◆ m_temp

◆ m_usePrescribedThreshold