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

#include <ColPivHouseholderQR.h>

Collaboration diagram for Eigen::ColPivHouseholderQR< _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 Matrix< Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime >	MatrixQType

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

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

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

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

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

HouseholderSequenceType	householderQ (void) const

HouseholderSequenceType	matrixQ (void) const

const MatrixType &	matrixQR () const

const MatrixType &	matrixR () const

ColPivHouseholderQR &	compute (const MatrixType &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 internal::solve_retval< ColPivHouseholderQR, typename MatrixType::IdentityReturnType >	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. More...

Protected Attributes
MatrixType	m_qr

HCoeffsType	m_hCoeffs

PermutationType	m_colsPermutation

IntRowVectorType	m_colsTranspositions

RowVectorType	m_temp

RealRowVectorType	m_colSqNorms

bool	m_isInitialized

bool	m_usePrescribedThreshold

RealScalar	m_prescribedThreshold

RealScalar	m_maxpivot

Index	m_nonzero_pivots

Index	m_det_pq

Detailed Description

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

Householder rank-revealing QR decomposition of a matrix with column-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 column pivoting in order to be rank-revealing and improve numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.

See also: MatrixBase::colPivHouseholderQr()

Definition at line 222 of file ForwardDeclarations.h.

Constructor & Destructor Documentation

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&).

Definition at line 72 of file ColPivHouseholderQR.h.

       : m_qr(),
         m_hCoeffs(),
         m_colsPermutation(),
         m_colsTranspositions(),
         m_temp(),
         m_colSqNorms(),
         m_isInitialized(false) {}

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()

Definition at line 87 of file ColPivHouseholderQR.h.

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

template<typename _MatrixType>

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

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

qr.compute(matrix);

See also: compute()

Definition at line 109 of file ColPivHouseholderQR.h.

       : 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_colSqNorms(matrix.cols()),
         m_isInitialized(false),
         m_usePrescribedThreshold(false)
     {
       compute(matrix);
     }

Member Function Documentation

template<typename MatrixType >

MatrixType::RealScalar Eigen::ColPivHouseholderQR< 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 399 of file ColPivHouseholderQR.h.

 {
   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());
 }

template<typename _MatrixType>

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

inline

Returns: a const reference to the column permutation matrix

Definition at line 179 of file ColPivHouseholderQR.h.

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

template<typename MatrixType >

ColPivHouseholderQR< MatrixType > & Eigen::ColPivHouseholderQR< 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 ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)

Definition at line 422 of file ColPivHouseholderQR.h.

 {
   using std::abs;
   Index rows = matrix.rows();
   Index cols = matrix.cols();
   Index size = matrix.diagonalSize();
   
   // the column permutation is stored as int indices, so just to be sure:
   eigen_assert(cols<=NumTraits<int>::highest());
 
   m_qr = matrix;
   m_hCoeffs.resize(size);
 
   m_temp.resize(cols);
 
   m_colsTranspositions.resize(matrix.cols());
   Index number_of_transpositions = 0;
 
   m_colSqNorms.resize(cols);
   for(Index k = 0; k < cols; ++k)
     m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();
 
   RealScalar threshold_helper = m_colSqNorms.maxCoeff() * numext::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);
 
   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_colSqNorms which column has the biggest squared norm
     Index biggest_col_index;
     RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
     biggest_col_index += k;
 
     // since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
     // the actual squared norm of the selected column.
     // Note that not doing so does result in solve() sometimes returning inf/nan values
     // when running the unit test with 1000 repetitions.
     biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();
 
     // we store that back into our table: it can't hurt to correct our table.
     m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;
 
     // if the current biggest column is smaller than epsilon times the initial biggest column,
     // terminate to avoid generating nan/inf values.
     // Note that here, if we test instead for "biggest == 0", we get a failure every 1000 (or so)
     // repetitions of the unit test, with the result of solve() filled with large values of the order
     // of 1/(size*epsilon).
     if(biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
     {
       m_nonzero_pivots = k;
       m_hCoeffs.tail(size-k).setZero();
       m_qr.bottomRightCorner(rows-k,cols-k)
           .template triangularView<StrictlyLower>()
           .setZero();
       break;
     }
 
     // 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_colSqNorms.coeffRef(k), m_colSqNorms.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 squared norms of the columns
     m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
   }
 
   m_colsPermutation.setIdentity(PermIndexType(cols));
   for(PermIndexType k = 0; k < 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;
 
   return *this;
 }

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&).

Definition at line 237 of file ColPivHouseholderQR.h.

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

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.

Definition at line 302 of file ColPivHouseholderQR.h.

302 { return m_hCoeffs; }

template<typename MatrixType >

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

Returns: the matrix Q as a sequence of householder transformations

Definition at line 560 of file ColPivHouseholderQR.h.

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

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

Definition at line 379 of file ColPivHouseholderQR.h.

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

template<typename _MatrixType>

const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> Eigen::ColPivHouseholderQR< _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 288 of file ColPivHouseholderQR.h.

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

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&).

Definition at line 250 of file ColPivHouseholderQR.h.

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

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&).

Definition at line 275 of file ColPivHouseholderQR.h.

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

Definition at line 263 of file ColPivHouseholderQR.h.

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

template<typename MatrixType >

MatrixType::RealScalar Eigen::ColPivHouseholderQR< 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 408 of file ColPivHouseholderQR.h.

 {
   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();
 }

template<typename _MatrixType>

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

inline

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

Definition at line 155 of file ColPivHouseholderQR.h.

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

template<typename _MatrixType>

const MatrixType& Eigen::ColPivHouseholderQR< _MatrixType >::matrixR ( void ) 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>()
For rank-deficient matrices, use
matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Definition at line 170 of file ColPivHouseholderQR.h.

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

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.

Definition at line 371 of file ColPivHouseholderQR.h.

371 { return m_maxpivot; }

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()

Definition at line 362 of file ColPivHouseholderQR.h.

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

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&).

Definition at line 220 of file ColPivHouseholderQR.h.

     {
       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;
     }

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)

Definition at line 321 of file ColPivHouseholderQR.h.

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

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);

See the documentation of setThreshold(const RealScalar&).

Definition at line 336 of file ColPivHouseholderQR.h.

     {
       m_usePrescribedThreshold = false;
       return *this;
     }

template<typename _MatrixType>

template<typename Rhs >

const internal::solve_retval<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: The case where b is a matrix is not yet implemented. Also, this code is space inefficient.

Example:

Output:

Definition at line 141 of file ColPivHouseholderQR.h.

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

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&).

Definition at line 346 of file ColPivHouseholderQR.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/ColPivHouseholderQR.h

Public Types

Public Member Functions

Protected Attributes

Detailed Description

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

Constructor & Destructor Documentation

Member Function Documentation

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