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FullPivHouseholderQR< _MatrixType > Class Template Reference
Public Member Functions

FullPivHouseholderQR< _MatrixType > Class Template Reference
[QR module]

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

#include <FullPivHouseholderQR.h>

Public Member Functions

 FullPivHouseholderQR ()
 Default Constructor.
 FullPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 FullPivHouseholderQR (const MatrixType &matrix)
 Constructs a QR factorization from a given matrix.
template<typename Rhs >
const internal::solve_retval
< FullPivHouseholderQR, Rhs > 		solve (const MatrixBase< Rhs > &b) const
 This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition.
MatrixQReturnType matrixQ (void) const
const MatrixType & matrixQR () const
FullPivHouseholderQR & compute (const MatrixType &matrix)
 Performs the QR factorization of the given matrix 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 internal::solve_retval
< FullPivHouseholderQR,
typename
MatrixType::IdentityReturnType > 		inverse () const
const HCoeffsType & hCoeffs () const
FullPivHouseholderQR & setThreshold (const RealScalar &threshold)
 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.
FullPivHouseholderQR & setThreshold (Default_t)
 Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.
RealScalar threshold () const
 Returns the threshold that will be used by certain methods such as rank().
Index nonzeroPivots () const
RealScalar maxPivot () const

Detailed Description

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

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

Parameters:
MatrixTypethe 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 49 of file FullPivHouseholderQR.h.

Constructor & Destructor Documentation

FullPivHouseholderQR (  )

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 78 of file FullPivHouseholderQR.h.

FullPivHouseholderQR ( Index  rows,
Index  cols 
)

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 94 of file FullPivHouseholderQR.h.

FullPivHouseholderQR ( const MatrixType &  matrix )

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 116 of file FullPivHouseholderQR.h.

Member Function Documentation

MatrixType::RealScalar 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 391 of file FullPivHouseholderQR.h.

const PermutationType& colsPermutation (  ) const
Returns:
a const reference to the column permutation matrix

Definition at line 170 of file FullPivHouseholderQR.h.

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 414 of file FullPivHouseholderQR.h.

Index dimensionOfKernel (  ) const
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 235 of file FullPivHouseholderQR.h.

const HCoeffsType& hCoeffs (  ) const
Returns:
a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

Definition at line 299 of file FullPivHouseholderQR.h.

const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> inverse (  ) const
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 285 of file FullPivHouseholderQR.h.

bool isInjective (  ) const
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 248 of file FullPivHouseholderQR.h.

bool isInvertible (  ) const
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 273 of file FullPivHouseholderQR.h.

bool isSurjective (  ) const
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 261 of file FullPivHouseholderQR.h.

MatrixType::RealScalar 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 400 of file FullPivHouseholderQR.h.

FullPivHouseholderQR< MatrixType >::MatrixQReturnType matrixQ ( void   ) const
Returns:
Expression object representing the matrix Q

Definition at line 603 of file FullPivHouseholderQR.h.

const MatrixType& matrixQR (  ) const
Returns:
a reference to the matrix where the Householder QR decomposition is stored

Definition at line 161 of file FullPivHouseholderQR.h.

RealScalar maxPivot (  ) const
Returns:
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U.

Definition at line 368 of file FullPivHouseholderQR.h.

Index nonzeroPivots (  ) const
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 359 of file FullPivHouseholderQR.h.

Index rank (  ) const
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 218 of file FullPivHouseholderQR.h.

const IntDiagSizeVectorType& rowsTranspositions (  ) const
Returns:
a const reference to the vector of indices representing the rows transpositions

Definition at line 177 of file FullPivHouseholderQR.h.

FullPivHouseholderQR& setThreshold ( Default_t   )

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 333 of file FullPivHouseholderQR.h.

FullPivHouseholderQR& setThreshold ( const RealScalar &  threshold )

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:
thresholdThe 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 318 of file FullPivHouseholderQR.h.

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

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

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

Example: 

 Output: 


 

Definition at line 149 of file FullPivHouseholderQR.h.











      

        

          RealScalar threshold 
          (
           )
           const
        

      







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


See the documentation of setThreshold(const RealScalar&). 



Definition at line 343 of file FullPivHouseholderQR.h.









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