Householder rank-revealing QR decomposition of a matrix with column-pivoting. More...
#include <ForwardDeclarations.h>
Public Member Functions | |
| template<typename RhsType , typename DstType > | |
| void | _solve_impl (const RhsType &rhs, DstType &dst) const |
| template<bool Conjugate, typename RhsType , typename DstType > | |
| void | _solve_impl_transposed (const RhsType &rhs, DstType &dst) const |
| MatrixType::RealScalar | absDeterminant () const |
| ColPivHouseholderQR () | |
| Default Constructor. More... | |
| ColPivHouseholderQR (Index rows, Index cols) | |
| Default Constructor with memory preallocation. More... | |
| template<typename InputType > | |
| ColPivHouseholderQR (const EigenBase< InputType > &matrix) | |
| Constructs a QR factorization from a given matrix. More... | |
| template<typename InputType > | |
| ColPivHouseholderQR (EigenBase< InputType > &matrix) | |
| Constructs a QR factorization from a given matrix. More... | |
| Index | cols () const |
| const PermutationType & | colsPermutation () const |
| template<typename InputType > | |
| ColPivHouseholderQR & | compute (const EigenBase< InputType > &matrix) |
| template<typename InputType > | |
| ColPivHouseholderQR< MatrixType > & | compute (const EigenBase< InputType > &matrix) |
| Index | dimensionOfKernel () const |
| const HCoeffsType & | hCoeffs () const |
| HouseholderSequenceType | householderQ () const |
| ComputationInfo | info () const |
| Reports whether the QR factorization was successful. More... | |
| const Inverse< ColPivHouseholderQR > | inverse () const |
| bool | isInjective () const |
| bool | isInvertible () const |
| bool | isSurjective () const |
| MatrixType::RealScalar | logAbsDeterminant () const |
| HouseholderSequenceType | matrixQ () const |
| const MatrixType & | matrixQR () const |
| const MatrixType & | matrixR () const |
| RealScalar | maxPivot () const |
| Index | nonzeroPivots () const |
| Index | rank () const |
| Index | rows () const |
| ColPivHouseholderQR & | setThreshold (const RealScalar &threshold) |
| ColPivHouseholderQR & | setThreshold (Default_t) |
| RealScalar | threshold () const |
 Public Member Functions inherited from Eigen::SolverBase< ColPivHouseholderQR< _MatrixType > > | |
| AdjointReturnType | adjoint () const |
| const Solve< ColPivHouseholderQR< _MatrixType >, Rhs > | solve (const MatrixBase< Rhs > &b) const |
| SolverBase () | |
| ConstTransposeReturnType | transpose () const |
| ~SolverBase () | |
| Public Member Functions inherited from Eigen::EigenBase< Derived > | |
| template<typename Dest > | |
| EIGEN_DEVICE_FUNC void | addTo (Dest &dst) const |
| template<typename Dest > | |
| EIGEN_DEVICE_FUNC void | applyThisOnTheLeft (Dest &dst) const |
| template<typename Dest > | |
| EIGEN_DEVICE_FUNC void | applyThisOnTheRight (Dest &dst) const |
| EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index | cols () const EIGEN_NOEXCEPT |
| EIGEN_DEVICE_FUNC Derived & | const_cast_derived () const |
| EIGEN_DEVICE_FUNC const Derived & | const_derived () const |
| EIGEN_DEVICE_FUNC Derived & | derived () |
| EIGEN_DEVICE_FUNC const Derived & | derived () const |
| template<typename Dest > | |
| EIGEN_DEVICE_FUNC void | evalTo (Dest &dst) const |
| EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index | rows () const EIGEN_NOEXCEPT |
| EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR Index | size () const EIGEN_NOEXCEPT |
| template<typename Dest > | |
| EIGEN_DEVICE_FUNC void | subTo (Dest &dst) const |
Protected Member Functions | |
| void | computeInPlace () |
| Protected Member Functions inherited from Eigen::SolverBase< ColPivHouseholderQR< _MatrixType > > | |
| void | _check_solve_assertion (const Rhs &b) const |
Static Protected Member Functions | |
| static void | check_template_parameters () |
Private Types | |
| typedef PermutationType::StorageIndex | PermIndexType |
Friends | |
| class | CompleteOrthogonalDecomposition< MatrixType > |
| class | SolverBase< ColPivHouseholderQR > |
Detailed Description
template<typename _MatrixType>
class Eigen::ColPivHouseholderQR< _MatrixType >
Householder rank-revealing QR decomposition of a matrix with column-pivoting.
- Template 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
![\[ \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R} \]](form_225.png)
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.
This class supports the inplace decomposition mechanism.
Definition at line 274 of file ForwardDeclarations.h.
Member Typedef Documentation
◆ Base
| typedef SolverBase<ColPivHouseholderQR> Eigen::ColPivHouseholderQR< _MatrixType >::Base |
Definition at line 57 of file ColPivHouseholderQR.h.
◆ HCoeffsType
| typedef internal::plain_diag_type<MatrixType>::type Eigen::ColPivHouseholderQR< _MatrixType >::HCoeffsType |
Definition at line 65 of file ColPivHouseholderQR.h.
◆ HouseholderSequenceType
| typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> Eigen::ColPivHouseholderQR< _MatrixType >::HouseholderSequenceType |
Definition at line 70 of file ColPivHouseholderQR.h.
◆ IntRowVectorType
| typedef internal::plain_row_type<MatrixType, Index>::type Eigen::ColPivHouseholderQR< _MatrixType >::IntRowVectorType |
Definition at line 67 of file ColPivHouseholderQR.h.
◆ MatrixType
| typedef _MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::MatrixType |
Definition at line 56 of file ColPivHouseholderQR.h.
◆ PermIndexType
|
private |
Definition at line 75 of file ColPivHouseholderQR.h.
◆ PermutationType
| typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::ColPivHouseholderQR< _MatrixType >::PermutationType |
Definition at line 66 of file ColPivHouseholderQR.h.
◆ PlainObject
| typedef MatrixType::PlainObject Eigen::ColPivHouseholderQR< _MatrixType >::PlainObject |
Definition at line 71 of file ColPivHouseholderQR.h.
◆ RealRowVectorType
| typedef internal::plain_row_type<MatrixType, RealScalar>::type Eigen::ColPivHouseholderQR< _MatrixType >::RealRowVectorType |
Definition at line 69 of file ColPivHouseholderQR.h.
◆ RowVectorType
| typedef internal::plain_row_type<MatrixType>::type Eigen::ColPivHouseholderQR< _MatrixType >::RowVectorType |
Definition at line 68 of file ColPivHouseholderQR.h.
Member Enumeration Documentation
◆ anonymous enum
| anonymous enum |
| Enumerator | |
|---|---|
| MaxRowsAtCompileTime | |
| MaxColsAtCompileTime | |
Definition at line 61 of file ColPivHouseholderQR.h.
Constructor & Destructor Documentation
◆ ColPivHouseholderQR() [1/4]
|
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 85 of file ColPivHouseholderQR.h.
◆ ColPivHouseholderQR() [2/4]
|
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 102 of file ColPivHouseholderQR.h.
◆ ColPivHouseholderQR() [3/4]
|
inlineexplicit |
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:
- See also
- compute()
Definition at line 126 of file ColPivHouseholderQR.h.
◆ ColPivHouseholderQR() [4/4]
|
inlineexplicit |
Constructs a QR factorization from a given matrix.
This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.
- See also
- ColPivHouseholderQR(const EigenBase&)
Definition at line 147 of file ColPivHouseholderQR.h.
Member Function Documentation
◆ _solve_impl()
| void Eigen::ColPivHouseholderQR< _MatrixType >::_solve_impl | ( | const RhsType & | rhs, |
| DstType & | dst | ||
| ) | const |
Definition at line 587 of file ColPivHouseholderQR.h.
◆ _solve_impl_transposed()
| void Eigen::ColPivHouseholderQR< _MatrixType >::_solve_impl_transposed | ( | const RhsType & | rhs, |
| DstType & | dst | ||
| ) | const |
Definition at line 611 of file ColPivHouseholderQR.h.
◆ absDeterminant()
| 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.
Definition at line 450 of file ColPivHouseholderQR.h.
◆ check_template_parameters()
|
inlinestaticprotected |
Definition at line 429 of file ColPivHouseholderQR.h.
◆ cols()
|
inline |
Definition at line 328 of file ColPivHouseholderQR.h.
◆ colsPermutation()
|
inline |
- Returns
- a const reference to the column permutation matrix
Definition at line 214 of file ColPivHouseholderQR.h.
◆ compute() [1/2]
| ColPivHouseholderQR& Eigen::ColPivHouseholderQR< _MatrixType >::compute | ( | const EigenBase< InputType > & | matrix | ) |
◆ compute() [2/2]
| ColPivHouseholderQR<MatrixType>& Eigen::ColPivHouseholderQR< _MatrixType >::compute | ( | const EigenBase< InputType > & | 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 474 of file ColPivHouseholderQR.h.
◆ computeInPlace()
|
protected |
Definition at line 482 of file ColPivHouseholderQR.h.
◆ dimensionOfKernel()
|
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 272 of file ColPivHouseholderQR.h.
◆ hCoeffs()
|
inline |
- Returns
- a const reference to the vector of Householder coefficients used to represent the factor
Q.
For advanced uses only.
Definition at line 334 of file ColPivHouseholderQR.h.
◆ householderQ()
| ColPivHouseholderQR< MatrixType >::HouseholderSequenceType Eigen::ColPivHouseholderQR< MatrixType >::householderQ | ( | ) | const |
- Returns
- the matrix Q as a sequence of householder transformations. You can extract the meaningful part only by using:
Definition at line 655 of file ColPivHouseholderQR.h.
◆ info()
|
inline |
Reports whether the QR factorization was successful.
- Note
- This function always returns
Success. It is provided for compatibility with other factorization routines.
- Returns
Success
Definition at line 411 of file ColPivHouseholderQR.h.
◆ inverse()
|
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 321 of file ColPivHouseholderQR.h.
◆ isInjective()
|
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 285 of file ColPivHouseholderQR.h.
◆ isInvertible()
|
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 310 of file ColPivHouseholderQR.h.
◆ isSurjective()
|
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 298 of file ColPivHouseholderQR.h.
◆ logAbsDeterminant()
| 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.
Definition at line 459 of file ColPivHouseholderQR.h.
◆ matrixQ()
|
inline |
Definition at line 182 of file ColPivHouseholderQR.h.
◆ matrixQR()
|
inline |
- Returns
- a reference to the matrix where the Householder QR decomposition is stored
Definition at line 189 of file ColPivHouseholderQR.h.
◆ matrixR()
|
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 For rank-deficient matrices, usematrixR().template triangularView<Upper>()
Definition at line 204 of file ColPivHouseholderQR.h.
◆ maxPivot()
|
inline |
- Returns
- the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of R.
Definition at line 403 of file ColPivHouseholderQR.h.
◆ nonzeroPivots()
|
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 394 of file ColPivHouseholderQR.h.
◆ rank()
|
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 255 of file ColPivHouseholderQR.h.
◆ rows()
|
inline |
Definition at line 327 of file ColPivHouseholderQR.h.
◆ setThreshold() [1/2]
|
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 
where maxpivot is the biggest pivot.
If you want to come back to the default behavior, call setThreshold(Default_t)
Definition at line 353 of file ColPivHouseholderQR.h.
◆ setThreshold() [2/2]
|
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.
See the documentation of setThreshold(const RealScalar&).
Definition at line 368 of file ColPivHouseholderQR.h.
◆ threshold()
|
inline |
Returns the threshold that will be used by certain methods such as rank().
See the documentation of setThreshold(const RealScalar&).
Definition at line 378 of file ColPivHouseholderQR.h.
Friends And Related Function Documentation
◆ CompleteOrthogonalDecomposition< MatrixType >
|
friend |
Definition at line 427 of file ColPivHouseholderQR.h.
◆ SolverBase< ColPivHouseholderQR >
|
friend |
Definition at line 58 of file ColPivHouseholderQR.h.
Member Data Documentation
◆ m_colNormsDirect
|
protected |
Definition at line 442 of file ColPivHouseholderQR.h.
◆ m_colNormsUpdated
|
protected |
Definition at line 441 of file ColPivHouseholderQR.h.
◆ m_colsPermutation
|
protected |
Definition at line 438 of file ColPivHouseholderQR.h.
◆ m_colsTranspositions
|
protected |
Definition at line 439 of file ColPivHouseholderQR.h.
◆ m_det_pq
|
protected |
Definition at line 446 of file ColPivHouseholderQR.h.
◆ m_hCoeffs
|
protected |
Definition at line 437 of file ColPivHouseholderQR.h.
◆ m_isInitialized
|
protected |
Definition at line 443 of file ColPivHouseholderQR.h.
◆ m_maxpivot
|
protected |
Definition at line 444 of file ColPivHouseholderQR.h.
◆ m_nonzero_pivots
|
protected |
Definition at line 445 of file ColPivHouseholderQR.h.
◆ m_prescribedThreshold
|
protected |
Definition at line 444 of file ColPivHouseholderQR.h.
◆ m_qr
|
protected |
Definition at line 436 of file ColPivHouseholderQR.h.
◆ m_temp
|
protected |
Definition at line 440 of file ColPivHouseholderQR.h.
◆ m_usePrescribedThreshold
|
protected |
Definition at line 443 of file ColPivHouseholderQR.h.
The documentation for this class was generated from the following files: