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

#include <FullPivHouseholderQR.h>

Collaboration diagram for Eigen::FullPivHouseholderQR< _MatrixType >:

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

typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::RealScalar	RealScalar

typedef MatrixType::Index	Index

typedef internal::FullPivHouseholderQRMatrixQReturnType< MatrixType >	MatrixQReturnType

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

typedef Matrix< Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime >	IntRowVectorType

typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime >	PermutationType

typedef internal::plain_col_type< MatrixType, Index >::type	IntColVectorType

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

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

Public Member Functions
	FullPivHouseholderQR ()
	Default Constructor. More...

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

	FullPivHouseholderQR (const MatrixType &matrix)
	Constructs a QR factorization from a given matrix. More...

template<typename Rhs >
const internal::solve_retval< FullPivHouseholderQR, Rhs >	solve (const MatrixBase< Rhs > &b) const

MatrixQReturnType	matrixQ (void) const

const MatrixType &	matrixQR () const

FullPivHouseholderQR &	compute (const MatrixType &matrix)

const PermutationType &	colsPermutation () const

const IntColVectorType &	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 internal::solve_retval< FullPivHouseholderQR, typename MatrixType::IdentityReturnType >	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

Protected Attributes
MatrixType	m_qr

HCoeffsType	m_hCoeffs

IntColVectorType	m_rows_transpositions

IntRowVectorType	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

Detailed Description

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

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

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 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.

See also: MatrixBase::fullPivHouseholderQr()

Definition at line 223 of file ForwardDeclarations.h.

Constructor & Destructor Documentation

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&).

Definition at line 77 of file FullPivHouseholderQR.h.

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

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()

Definition at line 93 of file FullPivHouseholderQR.h.

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

template<typename _MatrixType>

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

inline

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);

See also: compute()

Definition at line 115 of file FullPivHouseholderQR.h.

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

Member Function Documentation

template<typename MatrixType >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType >::absDeterminant ( ) const

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()

Definition at line 383 of file FullPivHouseholderQR.h.

 {
   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());
 }

template<typename _MatrixType>

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

inline

Returns: a const reference to the column permutation matrix

Definition at line 168 of file FullPivHouseholderQR.h.

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

template<typename MatrixType >

FullPivHouseholderQR< MatrixType > & Eigen::FullPivHouseholderQR< MatrixType >::compute ( const MatrixType & 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&)

Definition at line 406 of file FullPivHouseholderQR.h.

 {
   using std::abs;
   Index rows = matrix.rows();
   Index cols = matrix.cols();
   Index size = (std::min)(rows,cols);
 
   m_qr = matrix;
   m_hCoeffs.resize(size);
 
   m_temp.resize(cols);
 
   m_precision = NumTraits<Scalar>::epsilon() * size;
 
   m_rows_transpositions.resize(matrix.rows());
   m_cols_transpositions.resize(matrix.cols());
   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;
     RealScalar biggest_in_corner;
 
     biggest_in_corner = m_qr.bottomRightCorner(rows-k, cols-k)
                             .cwiseAbs()
                             .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
     row_of_biggest_in_corner += k;
     col_of_biggest_in_corner += k;
     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;
 
   return *this;
 }

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&).

Definition at line 233 of file FullPivHouseholderQR.h.

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

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.

Definition at line 297 of file FullPivHouseholderQR.h.

297 { return m_hCoeffs; }

template<typename _MatrixType>

const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> Eigen::FullPivHouseholderQR< _MatrixType >::inverse ( void ) 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.

Definition at line 283 of file FullPivHouseholderQR.h.

     {
       eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
       return internal::solve_retval<FullPivHouseholderQR,typename MatrixType::IdentityReturnType>
                (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
     }

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&).

Definition at line 246 of file FullPivHouseholderQR.h.

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

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&).

Definition at line 271 of file FullPivHouseholderQR.h.

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

Definition at line 259 of file FullPivHouseholderQR.h.

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

template<typename MatrixType >

MatrixType::RealScalar Eigen::FullPivHouseholderQR< MatrixType >::logAbsDeterminant ( ) const

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()

Definition at line 392 of file FullPivHouseholderQR.h.

 {
   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();
 }

template<typename MatrixType >

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

inline

Returns: Expression object representing the matrix Q

Definition at line 604 of file FullPivHouseholderQR.h.

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

template<typename _MatrixType>

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

inline

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

Definition at line 159 of file FullPivHouseholderQR.h.

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

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.

Definition at line 366 of file FullPivHouseholderQR.h.

366 { return m_maxpivot; }

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()

Definition at line 357 of file FullPivHouseholderQR.h.

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

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&).

Definition at line 216 of file FullPivHouseholderQR.h.

     {
       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;
     }

template<typename _MatrixType>

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

inline

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

Definition at line 175 of file FullPivHouseholderQR.h.

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

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)

Definition at line 316 of file FullPivHouseholderQR.h.

     {
       m_usePrescribedThreshold = true;
       m_prescribedThreshold = threshold;
       return *this;
     }

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);

See the documentation of setThreshold(const RealScalar&).

Definition at line 331 of file FullPivHouseholderQR.h.

     {
       m_usePrescribedThreshold = false;
       return *this;
     }

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<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, if any exists.

Parameters

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

Returns: a solution.

Note: The case where b is a matrix is not yet implemented. Also, this code is space inefficient.

Example:

Output:

Definition at line 147 of file FullPivHouseholderQR.h.

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

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&).

Definition at line 341 of file FullPivHouseholderQR.h.

     {
       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() * m_qr.diagonalSize();
     }

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

C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/Core/util/ForwardDeclarations.h
C:/Users/Brig/Documents/ShapeWorksStudio/src/Surfworks/Eigen/src/QR/FullPivHouseholderQR.h

Public Types

Public Member Functions

Protected Attributes

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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