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Eigen::FullPivHouseholderQR< _MatrixType > Class Template Reference
[QR module]


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 51 of file QR.

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 Eigen::FullPivHouseholderQR< _MatrixType >::ColVectorType

Definition at line 72 of file QR.

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

Definition at line 67 of file QR.

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

Definition at line 65 of file QR.

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

Definition at line 70 of file QR.

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

Definition at line 68 of file QR.

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

Definition at line 66 of file QR.

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

Definition at line 55 of file QR.

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

Definition at line 69 of file QR.

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

Definition at line 64 of file QR.

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

Definition at line 71 of file QR.

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

Definition at line 63 of file QR.


Member Enumeration Documentation

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

Definition at line 56 of file QR.


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 79 of file QR.

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 94 of file QR.

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

Definition at line 103 of file QR.


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 281 of file QR.

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

Definition at line 264 of file QR.

template<typename _MatrixType>
const PermutationType& Eigen::FullPivHouseholderQR< _MatrixType >::colsPermutation (  ) const [inline]

Definition at line 152 of file QR.

template<typename MatrixType >
FullPivHouseholderQR< MatrixType > & FullPivHouseholderQR< MatrixType >::compute ( const MatrixType matrix )

Definition at line 297 of file QR.

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:
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 209 of file QR.

template<typename _MatrixType>
const HCoeffsType& Eigen::FullPivHouseholderQR< _MatrixType >::hCoeffs (  ) const [inline]

Definition at line 265 of file QR.

template<typename _MatrixType>
const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> Eigen::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 256 of file QR.

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:
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 221 of file QR.

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:
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 244 of file QR.

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:
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 233 of file QR.

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 289 of file QR.

template<typename MatrixType >
FullPivHouseholderQR< MatrixType >::MatrixQType FullPivHouseholderQR< MatrixType >::matrixQ ( void   ) const
Returns:
the matrix Q

Definition at line 429 of file QR.

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 144 of file QR.

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 is computed at the time of the construction of the QR decomposition. This method does not perform any further computation.

Definition at line 198 of file QR.

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

Definition at line 263 of file QR.

template<typename _MatrixType>
const IntColVectorType& Eigen::FullPivHouseholderQR< _MatrixType >::rowsTranspositions (  ) const [inline]

Definition at line 158 of file QR.

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:
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 134 of file QR.


Member Data Documentation

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

Definition at line 272 of file QR.

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

Definition at line 271 of file QR.

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

Definition at line 277 of file QR.

template<typename _MatrixType>
HCoeffsType Eigen::FullPivHouseholderQR< _MatrixType >::m_hCoeffs [protected]

Definition at line 269 of file QR.

template<typename _MatrixType>
bool Eigen::FullPivHouseholderQR< _MatrixType >::m_isInitialized [protected]

Definition at line 274 of file QR.

template<typename _MatrixType>
RealScalar Eigen::FullPivHouseholderQR< _MatrixType >::m_precision [protected]

Definition at line 275 of file QR.

template<typename _MatrixType>
MatrixType Eigen::FullPivHouseholderQR< _MatrixType >::m_qr [protected]

Definition at line 268 of file QR.

template<typename _MatrixType>
Index Eigen::FullPivHouseholderQR< _MatrixType >::m_rank [protected]

Definition at line 276 of file QR.

template<typename _MatrixType>
IntColVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_rows_transpositions [protected]

Definition at line 270 of file QR.

template<typename _MatrixType>
RowVectorType Eigen::FullPivHouseholderQR< _MatrixType >::m_temp [protected]

Definition at line 273 of file QR.




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