Householder rank-revealing QR decomposition of a matrix with full pivoting. More...
#include <FullPivHouseholderQR.h>
Householder rank-revealing QR decomposition of a matrix with full 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 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.
typedef internal::plain_col_type<MatrixType>::type ColVectorType |
typedef internal::plain_diag_type<MatrixType>::type HCoeffsType |
typedef MatrixType::Index Index |
typedef internal::plain_col_type<MatrixType, Index>::type IntColVectorType |
typedef Matrix<Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime> IntRowVectorType |
typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> MatrixQReturnType |
typedef _MatrixType MatrixType |
typedef MatrixType::RealScalar RealScalar |
typedef internal::plain_row_type<MatrixType>::type RowVectorType |
typedef MatrixType::Scalar Scalar |
anonymous enum |
FullPivHouseholderQR | ( | ) | [inline] |
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via FullPivHouseholderQR::compute(const 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.
FullPivHouseholderQR | ( | const MatrixType & | matrix | ) | [inline] |
References FullPivHouseholderQR< _MatrixType >::compute().
MatrixType::RealScalar absDeterminant | ( | ) | const |
References abs().
const PermutationType& colsPermutation | ( | ) | const [inline] |
FullPivHouseholderQR< MatrixType > & compute | ( | const MatrixType & | matrix | ) |
References abs(), Eigen::internal::isMuchSmallerThan(), and FullPivHouseholderQR< _MatrixType >::rows().
Referenced by FullPivHouseholderQR< _MatrixType >::FullPivHouseholderQR().
Index dimensionOfKernel | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::cols(), FullPivHouseholderQR< _MatrixType >::m_isInitialized, and FullPivHouseholderQR< _MatrixType >::rank().
const HCoeffsType& hCoeffs | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::m_hCoeffs.
const internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType> inverse | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::m_isInitialized, and FullPivHouseholderQR< _MatrixType >::m_qr.
bool isInjective | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::cols(), FullPivHouseholderQR< _MatrixType >::m_isInitialized, and FullPivHouseholderQR< _MatrixType >::rank().
Referenced by FullPivHouseholderQR< _MatrixType >::isInvertible().
bool isInvertible | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::isInjective(), FullPivHouseholderQR< _MatrixType >::isSurjective(), and FullPivHouseholderQR< _MatrixType >::m_isInitialized.
bool isSurjective | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::m_isInitialized, FullPivHouseholderQR< _MatrixType >::rank(), and FullPivHouseholderQR< _MatrixType >::rows().
Referenced by FullPivHouseholderQR< _MatrixType >::isInvertible().
MatrixType::RealScalar logAbsDeterminant | ( | ) | const |
FullPivHouseholderQR< MatrixType >::MatrixQReturnType matrixQ | ( | void | ) | const [inline] |
const MatrixType& matrixQR | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::m_isInitialized, and FullPivHouseholderQR< _MatrixType >::m_qr.
RealScalar maxPivot | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::m_maxpivot.
Index nonzeroPivots | ( | ) | const [inline] |
References FullPivHouseholderQR< _MatrixType >::m_isInitialized, and FullPivHouseholderQR< _MatrixType >::m_nonzero_pivots.
References abs(), FullPivHouseholderQR< _MatrixType >::m_isInitialized, FullPivHouseholderQR< _MatrixType >::m_maxpivot, FullPivHouseholderQR< _MatrixType >::m_nonzero_pivots, FullPivHouseholderQR< _MatrixType >::m_qr, and FullPivHouseholderQR< _MatrixType >::threshold().
Referenced by FullPivHouseholderQR< _MatrixType >::dimensionOfKernel(), FullPivHouseholderQR< _MatrixType >::isInjective(), and FullPivHouseholderQR< _MatrixType >::isSurjective().
const IntColVectorType& rowsTranspositions | ( | ) | const [inline] |
FullPivHouseholderQR& 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 FullPivHouseholderQR< _MatrixType >::m_prescribedThreshold, FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold, and FullPivHouseholderQR< _MatrixType >::threshold().
FullPivHouseholderQR& 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 FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.
const internal::solve_retval<FullPivHouseholderQR, 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.fullPivHouseholderQr().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 FullPivHouseholderQR< _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 FullPivHouseholderQR< _MatrixType >::m_isInitialized, FullPivHouseholderQR< _MatrixType >::m_prescribedThreshold, FullPivHouseholderQR< _MatrixType >::m_qr, and FullPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold.
Referenced by FullPivHouseholderQR< _MatrixType >::rank(), and FullPivHouseholderQR< _MatrixType >::setThreshold().
PermutationType m_cols_permutation [protected] |
Referenced by FullPivHouseholderQR< _MatrixType >::colsPermutation().
IntRowVectorType m_cols_transpositions [protected] |
HCoeffsType m_hCoeffs [protected] |
Referenced by FullPivHouseholderQR< _MatrixType >::hCoeffs().
bool m_isInitialized [protected] |
Referenced by FullPivHouseholderQR< _MatrixType >::colsPermutation(), FullPivHouseholderQR< _MatrixType >::dimensionOfKernel(), FullPivHouseholderQR< _MatrixType >::inverse(), FullPivHouseholderQR< _MatrixType >::isInjective(), FullPivHouseholderQR< _MatrixType >::isInvertible(), FullPivHouseholderQR< _MatrixType >::isSurjective(), FullPivHouseholderQR< _MatrixType >::matrixQR(), FullPivHouseholderQR< _MatrixType >::nonzeroPivots(), FullPivHouseholderQR< _MatrixType >::rank(), FullPivHouseholderQR< _MatrixType >::rowsTranspositions(), FullPivHouseholderQR< _MatrixType >::solve(), and FullPivHouseholderQR< _MatrixType >::threshold().
RealScalar m_maxpivot [protected] |
Index m_nonzero_pivots [protected] |
RealScalar m_precision [protected] |
RealScalar m_prescribedThreshold [protected] |
MatrixType m_qr [protected] |
IntColVectorType m_rows_transpositions [protected] |
Referenced by FullPivHouseholderQR< _MatrixType >::rowsTranspositions().
RowVectorType m_temp [protected] |
bool m_usePrescribedThreshold [protected] |