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FullPivHouseholderQR< _MatrixType > Class Template Reference
[QR module]


Detailed Description

template<typename _MatrixType>
class 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 50 of file FullPivHouseholderQR.h.

#include <src/QR/FullPivHouseholderQR.h>

List of all members.

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 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.
 FullPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation.
 FullPivHouseholderQR (const MatrixType &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, if any exists.
MatrixQType matrixQ (void) const
const MatrixTypematrixQR () const
FullPivHouseholderQRcompute (const MatrixType &matrix)
const PermutationTypecolsPermutation () const
const IntColVectorTyperowsTranspositions () 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 HCoeffsTypehCoeffs () 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
RealScalar m_precision
Index m_rank
Index m_det_pq

Member Typedef Documentation

template<typename _MatrixType>
typedef internal::plain_col_type<MatrixType>::type FullPivHouseholderQR< _MatrixType >::ColVectorType

Definition at line 71 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef internal::plain_diag_type<MatrixType>::type FullPivHouseholderQR< _MatrixType >::HCoeffsType

Definition at line 66 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef MatrixType::Index FullPivHouseholderQR< _MatrixType >::Index

Definition at line 64 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef internal::plain_col_type<MatrixType, Index>::type FullPivHouseholderQR< _MatrixType >::IntColVectorType

Definition at line 69 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef Matrix<Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime> FullPivHouseholderQR< _MatrixType >::IntRowVectorType

Definition at line 67 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> FullPivHouseholderQR< _MatrixType >::MatrixQType

Definition at line 65 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef _MatrixType FullPivHouseholderQR< _MatrixType >::MatrixType

Definition at line 54 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> FullPivHouseholderQR< _MatrixType >::PermutationType

Definition at line 68 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef MatrixType::RealScalar FullPivHouseholderQR< _MatrixType >::RealScalar

Definition at line 63 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType>::type FullPivHouseholderQR< _MatrixType >::RowVectorType

Definition at line 70 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
typedef MatrixType::Scalar FullPivHouseholderQR< _MatrixType >::Scalar

Definition at line 62 of file FullPivHouseholderQR.h.


Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Definition at line 55 of file FullPivHouseholderQR.h.


Constructor & Destructor Documentation

template<typename _MatrixType>
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 78 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
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.

template<typename _MatrixType>
FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR ( const MatrixType matrix ) [inline]

Member Function Documentation

template<typename MatrixType >
MatrixType::RealScalar 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 280 of file FullPivHouseholderQR.h.

References abs(), and eigen_assert.

template<typename _MatrixType>
Index FullPivHouseholderQR< _MatrixType >::cols ( void   ) const [inline]

Definition at line 263 of file FullPivHouseholderQR.h.

References FullPivHouseholderQR< _MatrixType >::m_qr.

template<typename _MatrixType>
const PermutationType& FullPivHouseholderQR< _MatrixType >::colsPermutation (  ) const [inline]
template<typename MatrixType >
FullPivHouseholderQR< MatrixType > & FullPivHouseholderQR< MatrixType >::compute ( const MatrixType matrix )
template<typename _MatrixType>
Index FullPivHouseholderQR< _MatrixType >::dimensionOfKernel (  ) const [inline]
Returns:
the dimension of the kernel of the matrix of which *this is the QR decomposition.
Note:
Since the rank is computed at the time of the construction of the QR decomposition, this method almost does not perform any further computation.

Definition at line 208 of file FullPivHouseholderQR.h.

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

template<typename _MatrixType>
const HCoeffsType& FullPivHouseholderQR< _MatrixType >::hCoeffs (  ) const [inline]
template<typename _MatrixType>
const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> 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.

Definition at line 255 of file FullPivHouseholderQR.h.

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

template<typename _MatrixType>
bool 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:
Since the rank is computed at the time of the construction of the QR decomposition, this method almost does not perform any further computation.

Definition at line 220 of file FullPivHouseholderQR.h.

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

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

template<typename _MatrixType>
bool FullPivHouseholderQR< _MatrixType >::isInvertible (  ) const [inline]
Returns:
true if the matrix of which *this is the QR decomposition is invertible.
Note:
Since the rank is computed at the time of the construction of the QR decomposition, this method almost does not perform any further computation.

Definition at line 243 of file FullPivHouseholderQR.h.

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

template<typename _MatrixType>
bool 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:
Since the rank is computed at the time of the construction of the QR decomposition, this method almost does not perform any further computation.

Definition at line 232 of file FullPivHouseholderQR.h.

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

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

template<typename MatrixType >
MatrixType::RealScalar 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 288 of file FullPivHouseholderQR.h.

References eigen_assert.

template<typename MatrixType >
FullPivHouseholderQR< MatrixType >::MatrixQType FullPivHouseholderQR< MatrixType >::matrixQ ( void   ) const
template<typename _MatrixType>
const MatrixType& FullPivHouseholderQR< _MatrixType >::matrixQR (  ) const [inline]
template<typename _MatrixType>
Index FullPivHouseholderQR< _MatrixType >::rank (  ) const [inline]
Returns:
the rank of the matrix of which *this is the QR decomposition.
Note:
This is computed at the time of the construction of the QR decomposition. This method does not perform any further computation.

Definition at line 197 of file FullPivHouseholderQR.h.

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

template<typename _MatrixType>
Index FullPivHouseholderQR< _MatrixType >::rows ( void   ) const [inline]
template<typename _MatrixType>
const IntColVectorType& FullPivHouseholderQR< _MatrixType >::rowsTranspositions (  ) const [inline]
template<typename _MatrixType>
template<typename Rhs >
const internal::solve_retval<FullPivHouseholderQR, Rhs> 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:
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 133 of file FullPivHouseholderQR.h.

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


Member Data Documentation

template<typename _MatrixType>
PermutationType FullPivHouseholderQR< _MatrixType >::m_cols_permutation [protected]
template<typename _MatrixType>
IntRowVectorType FullPivHouseholderQR< _MatrixType >::m_cols_transpositions [protected]

Definition at line 270 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
Index FullPivHouseholderQR< _MatrixType >::m_det_pq [protected]

Definition at line 276 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
HCoeffsType FullPivHouseholderQR< _MatrixType >::m_hCoeffs [protected]
template<typename _MatrixType>
bool FullPivHouseholderQR< _MatrixType >::m_isInitialized [protected]
template<typename _MatrixType>
RealScalar FullPivHouseholderQR< _MatrixType >::m_precision [protected]

Definition at line 274 of file FullPivHouseholderQR.h.

template<typename _MatrixType>
MatrixType FullPivHouseholderQR< _MatrixType >::m_qr [protected]
template<typename _MatrixType>
Index FullPivHouseholderQR< _MatrixType >::m_rank [protected]
template<typename _MatrixType>
IntColVectorType FullPivHouseholderQR< _MatrixType >::m_rows_transpositions [protected]
template<typename _MatrixType>
RowVectorType FullPivHouseholderQR< _MatrixType >::m_temp [protected]

Definition at line 272 of file FullPivHouseholderQR.h.




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