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HouseholderQR< _MatrixType > Class Template Reference
Public Member Functions

HouseholderQR< _MatrixType > Class Template Reference
[QR module]

Householder QR decomposition of a matrix. More...

#include <HouseholderQR.h>

Public Member Functions

 HouseholderQR ()
 Default Constructor.
 HouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 HouseholderQR (const MatrixType &matrix)
 Constructs a QR factorization from a given matrix.
template<typename Rhs >
const internal::solve_retval
< HouseholderQR, 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, if any exists.
HouseholderSequenceType householderQ () const
 This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
const MatrixType & matrixQR () const
HouseholderQR & compute (const MatrixType &matrix)
 Performs the QR factorization of the given matrix matrix.
MatrixType::RealScalar absDeterminant () const
MatrixType::RealScalar logAbsDeterminant () const
const HCoeffsType & hCoeffs () const

Detailed Description

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

Householder QR decomposition of a matrix.

Parameters:
MatrixTypethe type of the matrix of which we are computing the QR decomposition

This class performs a QR decomposition of a matrix A into matrices Q and R such that

\[ \mathbf{A} = \mathbf{Q} \, \mathbf{R} \]

by using Householder transformations. Here, Q a unitary matrix and R an upper triangular matrix. The result is stored in a compact way compatible with LAPACK.

Note that no pivoting is performed. This is not a rank-revealing decomposition. If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.

This Householder QR decomposition is faster, but less numerically stable and less feature-full than FullPivHouseholderQR or ColPivHouseholderQR.

See also:
MatrixBase::householderQr()

Definition at line 42 of file HouseholderQR.h.

Constructor & Destructor Documentation

HouseholderQR (  )

Default Constructor.

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

Definition at line 68 of file HouseholderQR.h.

HouseholderQR ( 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:
HouseholderQR()

Definition at line 76 of file HouseholderQR.h.

HouseholderQR ( 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:

 HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
 qr.compute(matrix);
See also:
compute()

Definition at line 94 of file HouseholderQR.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 205 of file HouseholderQR.h.

HouseholderQR< 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 HouseholderQR, HouseholderQR(const MatrixType&)

Definition at line 356 of file HouseholderQR.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 189 of file HouseholderQR.h.

HouseholderSequenceType householderQ ( void   ) const

This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.

The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object. Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:

Example: 

 Output: 


 

Definition at line 136 of file HouseholderQR.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 214 of file HouseholderQR.h.











      

        

          const MatrixType& matrixQR 
          (
           )
           const
        

      






Returns:
a reference to the matrix where the Householder QR decomposition is stored in a LAPACK-compatible way. 



Definition at line 145 of file HouseholderQR.h.











      

        

          const internal::solve_retval<HouseholderQR, 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, if any exists. 


Parameters:

  

    
bthe 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 122 of file HouseholderQR.h.









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