Householder rank-revealing QR decomposition of a matrix with column-pivoting. More...
#include <ColPivHouseholderQR.h>
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.
typedef internal::plain_diag_type<MatrixType>::type HCoeffsType |
typedef HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType HouseholderSequenceType |
typedef MatrixType::Index Index |
typedef internal::plain_row_type<MatrixType, Index>::type IntRowVectorType |
typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType |
typedef _MatrixType MatrixType |
typedef internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType |
typedef MatrixType::RealScalar RealScalar |
typedef internal::plain_row_type<MatrixType>::type RowVectorType |
typedef MatrixType::Scalar Scalar |
anonymous enum |
ColPivHouseholderQR | ( | ) | [inline] |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via ColPivHouseholderQR::compute(const 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.
ColPivHouseholderQR | ( | const MatrixType & | matrix | ) | [inline] |
References ColPivHouseholderQR< _MatrixType >::compute().
MatrixType::RealScalar absDeterminant | ( | ) | const |
References abs().
const PermutationType& colsPermutation | ( | ) | const [inline] |
ColPivHouseholderQR< MatrixType > & compute | ( | const MatrixType & | matrix | ) |
References abs(), abs2(), and ColPivHouseholderQR< _MatrixType >::rows().
Referenced by ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR().
Index dimensionOfKernel | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::cols(), ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::rank().
const HCoeffsType& hCoeffs | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::m_hCoeffs.
ColPivHouseholderQR< MatrixType >::HouseholderSequenceType householderQ | ( | void | ) | const |
const internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType> inverse | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.
bool isInjective | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::cols(), ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::rank().
Referenced by ColPivHouseholderQR< _MatrixType >::isInvertible().
bool isInvertible | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::isInjective(), ColPivHouseholderQR< _MatrixType >::isSurjective(), and ColPivHouseholderQR< _MatrixType >::m_isInitialized.
bool isSurjective | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::m_isInitialized, ColPivHouseholderQR< _MatrixType >::rank(), and ColPivHouseholderQR< _MatrixType >::rows().
Referenced by ColPivHouseholderQR< _MatrixType >::isInvertible().
MatrixType::RealScalar logAbsDeterminant | ( | ) | const |
const MatrixType& matrixQR | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_qr.
RealScalar maxPivot | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::m_maxpivot.
Index nonzeroPivots | ( | ) | const [inline] |
References ColPivHouseholderQR< _MatrixType >::m_isInitialized, and ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots.
References abs(), 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().
References ColPivHouseholderQR< _MatrixType >::m_qr.
Referenced by ColPivHouseholderQR< _MatrixType >::compute(), and ColPivHouseholderQR< _MatrixType >::isSurjective().
ColPivHouseholderQR& 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)
References ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold, ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold, and ColPivHouseholderQR< _MatrixType >::threshold().
ColPivHouseholderQR& 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&).
References ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.
const internal::solve_retval<ColPivHouseholderQR, Rhs> 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. |
This method just tries to find as good a solution as possible. If you want to check whether a solution exists or if it is accurate, just call this function to get a result and then compute the error of this result, or use MatrixBase::isApprox() directly, for instance like this:
bool a_solution_exists = (A*result).isApprox(b, precision);
This method avoids dividing by zero, so that the non-existence of a solution doesn't by itself mean that you'll get inf
or nan
values.
If there exists more than one solution, this method will arbitrarily choose one.
Example:
Matrix3f m = Matrix3f::Random(); Matrix3f y = Matrix3f::Random(); cout << "Here is the matrix m:" << endl << m << endl; cout << "Here is the matrix y:" << endl << y << endl; Matrix3f x; x = m.colPivHouseholderQr().solve(y); assert(y.isApprox(m*x)); cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;
Output:
Here is the matrix m: 0.68 0.597 -0.33 -0.211 0.823 0.536 0.566 -0.605 -0.444 Here is the matrix y: 0.108 -0.27 0.832 -0.0452 0.0268 0.271 0.258 0.904 0.435 Here is a solution x to the equation mx=y: 0.609 2.68 1.67 -0.231 -1.57 0.0713 0.51 3.51 1.05
References ColPivHouseholderQR< _MatrixType >::m_isInitialized.
RealScalar threshold | ( | ) | const [inline] |
Returns the threshold that will be used by certain methods such as rank().
See the documentation of setThreshold(const RealScalar&).
References ColPivHouseholderQR< _MatrixType >::m_isInitialized, ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold, ColPivHouseholderQR< _MatrixType >::m_qr, and ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.
Referenced by ColPivHouseholderQR< _MatrixType >::rank(), and ColPivHouseholderQR< _MatrixType >::setThreshold().
PermutationType m_colsPermutation [protected] |
Referenced by ColPivHouseholderQR< _MatrixType >::colsPermutation().
RealRowVectorType m_colSqNorms [protected] |
IntRowVectorType m_colsTranspositions [protected] |
HCoeffsType m_hCoeffs [protected] |
Referenced by ColPivHouseholderQR< _MatrixType >::hCoeffs().
bool m_isInitialized [protected] |
Referenced by ColPivHouseholderQR< _MatrixType >::colsPermutation(), ColPivHouseholderQR< _MatrixType >::dimensionOfKernel(), ColPivHouseholderQR< _MatrixType >::inverse(), ColPivHouseholderQR< _MatrixType >::isInjective(), ColPivHouseholderQR< _MatrixType >::isInvertible(), ColPivHouseholderQR< _MatrixType >::isSurjective(), ColPivHouseholderQR< _MatrixType >::matrixQR(), ColPivHouseholderQR< _MatrixType >::nonzeroPivots(), ColPivHouseholderQR< _MatrixType >::rank(), ColPivHouseholderQR< _MatrixType >::solve(), and ColPivHouseholderQR< _MatrixType >::threshold().
RealScalar m_maxpivot [protected] |
Index m_nonzero_pivots [protected] |
RealScalar m_prescribedThreshold [protected] |
MatrixType m_qr [protected] |
RowVectorType m_temp [protected] |
bool m_usePrescribedThreshold [protected] |