Householder rank-revealing QR decomposition of a matrix with column-pivoting.
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
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.
Definition at line 50 of file ColPivHouseholderQR.h.
#include <src/QR/ColPivHouseholderQR.h>
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 PermutationMatrix < ColsAtCompileTime, MaxColsAtCompileTime > | PermutationType |
typedef internal::plain_row_type < MatrixType, Index >::type | IntRowVectorType |
typedef internal::plain_row_type < MatrixType >::type | RowVectorType |
typedef internal::plain_row_type < MatrixType, RealScalar > ::type | RealRowVectorType |
typedef HouseholderSequence < MatrixType, HCoeffsType > ::ConjugateReturnType | HouseholderSequenceType |
Public Member Functions | |
ColPivHouseholderQR () | |
Default Constructor. | |
ColPivHouseholderQR (Index rows, Index cols) | |
Default Constructor with memory preallocation. | |
ColPivHouseholderQR (const MatrixType &matrix) | |
template<typename Rhs > | |
const internal::solve_retval < ColPivHouseholderQR, 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 (void) const |
const MatrixType & | matrixQR () const |
ColPivHouseholderQR & | compute (const MatrixType &matrix) |
const PermutationType & | colsPermutation () 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 < ColPivHouseholderQR, typename MatrixType::IdentityReturnType > | inverse () const |
Index | rows () const |
Index | cols () const |
const HCoeffsType & | hCoeffs () const |
ColPivHouseholderQR & | 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. | |
ColPivHouseholderQR & | 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 |
Protected Attributes | |
MatrixType | m_qr |
HCoeffsType | m_hCoeffs |
PermutationType | m_colsPermutation |
IntRowVectorType | m_colsTranspositions |
RowVectorType | m_temp |
RealRowVectorType | m_colSqNorms |
bool | m_isInitialized |
bool | m_usePrescribedThreshold |
RealScalar | m_prescribedThreshold |
RealScalar | m_maxpivot |
Index | m_nonzero_pivots |
Index | m_det_pq |
typedef internal::plain_diag_type<MatrixType>::type ColPivHouseholderQR< _MatrixType >::HCoeffsType |
Definition at line 66 of file ColPivHouseholderQR.h.
typedef HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType ColPivHouseholderQR< _MatrixType >::HouseholderSequenceType |
Definition at line 71 of file ColPivHouseholderQR.h.
typedef MatrixType::Index ColPivHouseholderQR< _MatrixType >::Index |
Definition at line 64 of file ColPivHouseholderQR.h.
typedef internal::plain_row_type<MatrixType, Index>::type ColPivHouseholderQR< _MatrixType >::IntRowVectorType |
Definition at line 68 of file ColPivHouseholderQR.h.
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> ColPivHouseholderQR< _MatrixType >::MatrixQType |
Definition at line 65 of file ColPivHouseholderQR.h.
typedef _MatrixType ColPivHouseholderQR< _MatrixType >::MatrixType |
Definition at line 54 of file ColPivHouseholderQR.h.
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> ColPivHouseholderQR< _MatrixType >::PermutationType |
Definition at line 67 of file ColPivHouseholderQR.h.
typedef internal::plain_row_type<MatrixType, RealScalar>::type ColPivHouseholderQR< _MatrixType >::RealRowVectorType |
Definition at line 70 of file ColPivHouseholderQR.h.
typedef MatrixType::RealScalar ColPivHouseholderQR< _MatrixType >::RealScalar |
Definition at line 63 of file ColPivHouseholderQR.h.
typedef internal::plain_row_type<MatrixType>::type ColPivHouseholderQR< _MatrixType >::RowVectorType |
Definition at line 69 of file ColPivHouseholderQR.h.
typedef MatrixType::Scalar ColPivHouseholderQR< _MatrixType >::Scalar |
Definition at line 62 of file ColPivHouseholderQR.h.
anonymous enum |
Definition at line 55 of file ColPivHouseholderQR.h.
ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR | ( | ) | [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 79 of file ColPivHouseholderQR.h.
ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR | ( | 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.
Definition at line 94 of file ColPivHouseholderQR.h.
ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR | ( | const MatrixType & | matrix | ) | [inline] |
Definition at line 104 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::compute().
MatrixType::RealScalar ColPivHouseholderQR< MatrixType >::absDeterminant | ( | ) | const |
Definition at line 356 of file ColPivHouseholderQR.h.
References abs(), eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.
Index ColPivHouseholderQR< _MatrixType >::cols | ( | void | ) | const [inline] |
Definition at line 270 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::m_qr.
Referenced by ColPivHouseholderQR< _MatrixType >::compute(), ColPivHouseholderQR< _MatrixType >::dimensionOfKernel(), and ColPivHouseholderQR< _MatrixType >::isInjective().
const PermutationType& ColPivHouseholderQR< _MatrixType >::colsPermutation | ( | ) | const [inline] |
Definition at line 154 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::m_colsPermutation, and ColPivHouseholderQR< _MatrixType >::m_isInitialized.
Referenced by internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run().
ColPivHouseholderQR< MatrixType > & ColPivHouseholderQR< MatrixType >::compute | ( | const MatrixType & | matrix | ) |
Definition at line 372 of file ColPivHouseholderQR.h.
References abs(), abs2(), PermutationMatrix< SizeAtCompileTime, MaxSizeAtCompileTime >::applyTranspositionOnTheRight(), ColPivHouseholderQR< _MatrixType >::cols(), ColPivHouseholderQR< _MatrixType >::m_colsPermutation, ColPivHouseholderQR< _MatrixType >::m_colSqNorms, ColPivHouseholderQR< _MatrixType >::m_colsTranspositions, ColPivHouseholderQR< _MatrixType >::m_det_pq, ColPivHouseholderQR< _MatrixType >::m_hCoeffs, ColPivHouseholderQR< _MatrixType >::m_isInitialized, ColPivHouseholderQR< _MatrixType >::m_maxpivot, ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots, ColPivHouseholderQR< _MatrixType >::m_qr, ColPivHouseholderQR< _MatrixType >::m_temp, ColPivHouseholderQR< _MatrixType >::rows(), PermutationMatrix< SizeAtCompileTime, MaxSizeAtCompileTime >::setIdentity(), and mrpt::math::size().
Referenced by ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR().
Index ColPivHouseholderQR< _MatrixType >::dimensionOfKernel | ( | ) | const [inline] |
Definition at line 211 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::cols(), eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::rank().
const HCoeffsType& ColPivHouseholderQR< _MatrixType >::hCoeffs | ( | ) | const [inline] |
Definition at line 271 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::m_hCoeffs.
ColPivHouseholderQR< MatrixType >::HouseholderSequenceType ColPivHouseholderQR< MatrixType >::householderQ | ( | void | ) | const |
Definition at line 514 of file ColPivHouseholderQR.h.
References eigen_assert.
Referenced by internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run().
const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> ColPivHouseholderQR< _MatrixType >::inverse | ( | ) | const [inline] |
Definition at line 262 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.
bool ColPivHouseholderQR< _MatrixType >::isInjective | ( | ) | const [inline] |
Definition at line 224 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::cols(), eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::rank().
Referenced by ColPivHouseholderQR< _MatrixType >::isInvertible().
bool ColPivHouseholderQR< _MatrixType >::isInvertible | ( | ) | const [inline] |
Definition at line 249 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::isInjective(), ColPivHouseholderQR< _MatrixType >::isSurjective(), and ColPivHouseholderQR< _MatrixType >::m_isInitialized.
bool ColPivHouseholderQR< _MatrixType >::isSurjective | ( | ) | const [inline] |
Definition at line 237 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, ColPivHouseholderQR< _MatrixType >::rank(), and ColPivHouseholderQR< _MatrixType >::rows().
Referenced by ColPivHouseholderQR< _MatrixType >::isInvertible().
MatrixType::RealScalar ColPivHouseholderQR< MatrixType >::logAbsDeterminant | ( | ) | const |
Definition at line 364 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.
const MatrixType& ColPivHouseholderQR< _MatrixType >::matrixQR | ( | ) | const [inline] |
Definition at line 146 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.
Referenced by internal::qr_preconditioner_impl< MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true >::run().
RealScalar ColPivHouseholderQR< _MatrixType >::maxPivot | ( | ) | const [inline] |
Definition at line 340 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::m_maxpivot.
Index ColPivHouseholderQR< _MatrixType >::nonzeroPivots | ( | ) | const [inline] |
Definition at line 331 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots.
Index ColPivHouseholderQR< _MatrixType >::rank | ( | ) | const [inline] |
Definition at line 195 of file ColPivHouseholderQR.h.
References abs(), eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, ColPivHouseholderQR< _MatrixType >::m_maxpivot, ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots, ColPivHouseholderQR< _MatrixType >::m_qr, and ColPivHouseholderQR< _MatrixType >::threshold().
Referenced by ColPivHouseholderQR< _MatrixType >::dimensionOfKernel(), ColPivHouseholderQR< _MatrixType >::isInjective(), and ColPivHouseholderQR< _MatrixType >::isSurjective().
Index ColPivHouseholderQR< _MatrixType >::rows | ( | void | ) | const [inline] |
Definition at line 269 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::m_qr.
Referenced by ColPivHouseholderQR< _MatrixType >::compute(), and ColPivHouseholderQR< _MatrixType >::isSurjective().
ColPivHouseholderQR& ColPivHouseholderQR< _MatrixType >::setThreshold | ( | Default_t | ) | [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.
qr.setThreshold(Eigen::Default);
See the documentation of setThreshold(const RealScalar&).
Definition at line 305 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.
ColPivHouseholderQR& ColPivHouseholderQR< _MatrixType >::setThreshold | ( | const RealScalar & | threshold | ) | [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.
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 290 of file ColPivHouseholderQR.h.
References ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold, and ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.
const internal::solve_retval<ColPivHouseholderQR, Rhs> ColPivHouseholderQR< _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.
b | the right-hand-side of the equation to solve. |
Example:
Output:
Definition at line 136 of file ColPivHouseholderQR.h.
References eigen_assert, and ColPivHouseholderQR< _MatrixType >::m_isInitialized.
RealScalar ColPivHouseholderQR< _MatrixType >::threshold | ( | ) | const [inline] |
Returns the threshold that will be used by certain methods such as rank().
See the documentation of setThreshold(const RealScalar&).
Definition at line 315 of file ColPivHouseholderQR.h.
References eigen_assert, ColPivHouseholderQR< _MatrixType >::m_isInitialized, ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold, ColPivHouseholderQR< _MatrixType >::m_qr, and ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.
Referenced by ColPivHouseholderQR< _MatrixType >::rank().
PermutationType ColPivHouseholderQR< _MatrixType >::m_colsPermutation [protected] |
Definition at line 345 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::colsPermutation(), and ColPivHouseholderQR< _MatrixType >::compute().
RealRowVectorType ColPivHouseholderQR< _MatrixType >::m_colSqNorms [protected] |
Definition at line 348 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::compute().
IntRowVectorType ColPivHouseholderQR< _MatrixType >::m_colsTranspositions [protected] |
Definition at line 346 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::compute().
Index ColPivHouseholderQR< _MatrixType >::m_det_pq [protected] |
Definition at line 352 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::compute().
HCoeffsType ColPivHouseholderQR< _MatrixType >::m_hCoeffs [protected] |
Definition at line 344 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::compute(), and ColPivHouseholderQR< _MatrixType >::hCoeffs().
bool ColPivHouseholderQR< _MatrixType >::m_isInitialized [protected] |
Definition at line 349 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::absDeterminant(), ColPivHouseholderQR< _MatrixType >::colsPermutation(), ColPivHouseholderQR< _MatrixType >::compute(), ColPivHouseholderQR< _MatrixType >::dimensionOfKernel(), ColPivHouseholderQR< _MatrixType >::inverse(), ColPivHouseholderQR< _MatrixType >::isInjective(), ColPivHouseholderQR< _MatrixType >::isInvertible(), ColPivHouseholderQR< _MatrixType >::isSurjective(), ColPivHouseholderQR< _MatrixType >::logAbsDeterminant(), ColPivHouseholderQR< _MatrixType >::matrixQR(), ColPivHouseholderQR< _MatrixType >::nonzeroPivots(), ColPivHouseholderQR< _MatrixType >::rank(), ColPivHouseholderQR< _MatrixType >::solve(), and ColPivHouseholderQR< _MatrixType >::threshold().
RealScalar ColPivHouseholderQR< _MatrixType >::m_maxpivot [protected] |
Definition at line 350 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::compute(), ColPivHouseholderQR< _MatrixType >::maxPivot(), and ColPivHouseholderQR< _MatrixType >::rank().
Index ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots [protected] |
Definition at line 351 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::compute(), ColPivHouseholderQR< _MatrixType >::nonzeroPivots(), and ColPivHouseholderQR< _MatrixType >::rank().
RealScalar ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold [protected] |
Definition at line 350 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::setThreshold(), and ColPivHouseholderQR< _MatrixType >::threshold().
MatrixType ColPivHouseholderQR< _MatrixType >::m_qr [protected] |
Definition at line 343 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::absDeterminant(), ColPivHouseholderQR< _MatrixType >::cols(), ColPivHouseholderQR< _MatrixType >::compute(), ColPivHouseholderQR< _MatrixType >::inverse(), ColPivHouseholderQR< _MatrixType >::logAbsDeterminant(), ColPivHouseholderQR< _MatrixType >::matrixQR(), ColPivHouseholderQR< _MatrixType >::rank(), ColPivHouseholderQR< _MatrixType >::rows(), and ColPivHouseholderQR< _MatrixType >::threshold().
RowVectorType ColPivHouseholderQR< _MatrixType >::m_temp [protected] |
Definition at line 347 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::compute().
bool ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold [protected] |
Definition at line 349 of file ColPivHouseholderQR.h.
Referenced by ColPivHouseholderQR< _MatrixType >::setThreshold(), and ColPivHouseholderQR< _MatrixType >::threshold().
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