template<typename _MatrixType>
FullPivHouseholderQR class
Householder rank-revealing QR decomposition of a matrix with full pivoting.
| Template parameters | |
|---|---|
| _MatrixType | the type of the matrix of which we are computing the QR decomposition |
Contents
This class performs a rank-revealing QR decomposition of a matrix A into matrices P, P', Q and R such that
by using Householder transformations. Here, P and P' are permutation matrices, 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.
This class supports the inplace decomposition mechanism.
Base classes
-
template<typename Derived>class SolverBase
- A base class for matrix decomposition and solvers.
Public types
- enum (anonymous) { MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
- using Base = SolverBase<FullPivHouseholderQR>
- using ColVectorType = internal::plain_col_type<MatrixType>::type
- using HCoeffsType = internal::plain_diag_type<MatrixType>::type
- using IntDiagSizeVectorType = Matrix<StorageIndex, 1, EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime, RowsAtCompileTime), RowMajor, 1, EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime, MaxRowsAtCompileTime)>
- using MatrixQReturnType = internal::FullPivHouseholderQRMatrixQReturnType<MatrixType>
- using MatrixType = _MatrixType
- using PermutationType = PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime>
- using PlainObject = MatrixType::PlainObject
- using RowVectorType = internal::plain_row_type<MatrixType>::type
Constructors, destructors, conversion operators
- FullPivHouseholderQR()
- Default Constructor.
- FullPivHouseholderQR(Index rows, Index cols)
- Default Constructor with memory preallocation.
-
template<typename InputType>FullPivHouseholderQR(const EigenBase<InputType>& matrix) explicit
- Constructs a QR factorization from a given matrix.
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template<typename InputType>FullPivHouseholderQR(EigenBase<InputType>& matrix) explicit
- Constructs a QR factorization from a given matrix.
Public functions
- auto absDeterminant() const -> MatrixType::RealScalar
- auto cols() const -> Index
- auto colsPermutation() const -> const PermutationType&
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template<typename InputType>auto compute(const EigenBase<InputType>& matrix) -> FullPivHouseholderQR&
-
template<typename InputType>auto compute(const EigenBase<InputType>& matrix) -> FullPivHouseholderQR<MatrixType>&
- auto dimensionOfKernel() const -> Index
- auto hCoeffs() const -> const HCoeffsType&
- auto inverse() const -> const Inverse<FullPivHouseholderQR>
- auto isInjective() const -> bool
- auto isInvertible() const -> bool
- auto isSurjective() const -> bool
- auto logAbsDeterminant() const -> MatrixType::RealScalar
- auto matrixQ(void) const -> MatrixQReturnType
- auto matrixQR() const -> const MatrixType&
- auto maxPivot() const -> RealScalar
- auto nonzeroPivots() const -> Index
- auto rank() const -> Index
- auto rows() const -> Index
- auto rowsTranspositions() const -> const IntDiagSizeVectorType&
- auto setThreshold(const RealScalar& threshold) -> FullPivHouseholderQR&
- auto setThreshold(Default_t) -> FullPivHouseholderQR&
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template<typename Rhs>auto solve(const MatrixBase<Rhs>& b) const -> const Solve<FullPivHouseholderQR, Rhs>
- auto threshold() const -> RealScalar
Protected static functions
- static void check_template_parameters()
Protected functions
- void computeInPlace()
Protected variables
- PermutationType m_cols_permutation
- IntDiagSizeVectorType m_cols_transpositions
- Index m_det_pq
- HCoeffsType m_hCoeffs
- bool m_isInitialized
- RealScalar m_maxpivot
- Index m_nonzero_pivots
- RealScalar m_precision
- RealScalar m_prescribedThreshold
- MatrixType m_qr
- IntDiagSizeVectorType m_rows_transpositions
- RowVectorType m_temp
- bool m_usePrescribedThreshold
Enum documentation
template<typename _MatrixType>
enum Eigen::
| Enumerators | |
|---|---|
| MaxRowsAtCompileTime | |
| MaxColsAtCompileTime |
Typedef documentation
template<typename _MatrixType>
typedef SolverBase<FullPivHouseholderQR> Eigen::
template<typename _MatrixType>
typedef internal::plain_col_type<MatrixType>::type Eigen::
template<typename _MatrixType>
typedef internal::plain_diag_type<MatrixType>::type Eigen::
template<typename _MatrixType>
typedef Matrix<StorageIndex, 1, EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime, RowsAtCompileTime), RowMajor, 1, EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime, MaxRowsAtCompileTime)> Eigen::
template<typename _MatrixType>
typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> Eigen::
template<typename _MatrixType>
typedef _MatrixType Eigen::
template<typename _MatrixType>
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::
template<typename _MatrixType>
typedef MatrixType::PlainObject Eigen::
template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType>::type Eigen::
Function documentation
template<typename _MatrixType>
Eigen::
Default Constructor.
The default constructor is useful in cases in which the user intends to perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).
template<typename _MatrixType>
Eigen::
Default Constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
template<typename _MatrixType>
template<typename InputType>
Eigen::
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:
FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols()); qr.compute(matrix);
template<typename _MatrixType>
template<typename InputType>
Eigen::
Constructs a QR factorization from a given matrix.
This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::
template<typename _MatrixType>
MatrixType::RealScalar Eigen::
| 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. |
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template<typename _MatrixType>
const PermutationType& Eigen::
| Returns | a const reference to the column permutation matrix |
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template<typename _MatrixType>
template<typename InputType>
FullPivHouseholderQR& Eigen::
template<typename _MatrixType>
template<typename InputType>
FullPivHouseholderQR<MatrixType>& Eigen::
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.
template<typename _MatrixType>
Index Eigen::
| Returns | the dimension of the kernel of the matrix of which *this is the QR decomposition. |
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template<typename _MatrixType>
const HCoeffsType& Eigen::
| Returns | a const reference to the vector of Householder coefficients used to represent the factor Q. |
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For advanced uses only.
template<typename _MatrixType>
const Inverse<FullPivHouseholderQR> Eigen::
| Returns | the inverse of the matrix of which *this is the QR decomposition. |
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template<typename _MatrixType>
bool Eigen::
| Returns | true if the matrix of which *this is the QR decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise. |
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template<typename _MatrixType>
bool Eigen::
| Returns | true if the matrix of which *this is the QR decomposition is invertible. |
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template<typename _MatrixType>
bool Eigen::
| Returns | true if the matrix of which *this is the QR decomposition represents a surjective linear map; false otherwise. |
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template<typename _MatrixType>
MatrixType::RealScalar Eigen::
| 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. |
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template<typename _MatrixType>
MatrixQReturnType Eigen::
| Returns | Expression object representing the matrix Q |
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template<typename _MatrixType>
const MatrixType& Eigen::
| Returns | a reference to the matrix where the Householder QR decomposition is stored |
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template<typename _MatrixType>
RealScalar Eigen::
| Returns | the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of U. |
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template<typename _MatrixType>
Index Eigen::
| 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. |
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template<typename _MatrixType>
const IntDiagSizeVectorType& Eigen::
| Returns | a const reference to the vector of indices representing the rows transpositions |
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template<typename _MatrixType>
FullPivHouseholderQR& Eigen::
| Parameters | |
|---|---|
| threshold | The new value to use as the 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. 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.
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_
template<typename _MatrixType>
FullPivHouseholderQR& Eigen::
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&).
template<typename _MatrixType>
template<typename Rhs>
const Solve<FullPivHouseholderQR, Rhs> Eigen::
| Parameters | |
|---|---|
| b | the right-hand-side of the equation to solve. |
| Returns | the exact or least-square solution if the rank is greater or equal to the number of columns of A, and an arbitrary solution otherwise. |
This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition.
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::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
template<typename _MatrixType>
RealScalar Eigen::
Returns the threshold that will be used by certain methods such as rank().
See the documentation of setThreshold(const RealScalar&).