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f7a056bf04
This MR fixes a bunch of smaller issues, making the following changes: * Template parameters in the documentation are documented with `\tparam` instead of `\param` * Superfluous semicolon warnings fixed * Fixed the type of literals used to initialize float variables
634 lines
23 KiB
C++
634 lines
23 KiB
C++
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2016 Rasmus Munk Larsen <rmlarsen@google.com>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H
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#define EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H
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#include "./InternalHeaderCheck.h"
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namespace Eigen {
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namespace internal {
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template <typename MatrixType_>
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struct traits<CompleteOrthogonalDecomposition<MatrixType_> >
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: traits<MatrixType_> {
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typedef MatrixXpr XprKind;
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typedef SolverStorage StorageKind;
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typedef int StorageIndex;
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enum { Flags = 0 };
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};
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} // end namespace internal
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/** \ingroup QR_Module
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*
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* \class CompleteOrthogonalDecomposition
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*
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* \brief Complete orthogonal decomposition (COD) of a matrix.
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*
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* \tparam MatrixType_ the type of the matrix of which we are computing the COD.
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*
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* This class performs a rank-revealing complete orthogonal decomposition of a
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* matrix \b A into matrices \b P, \b Q, \b T, and \b Z such that
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* \f[
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* \mathbf{A} \, \mathbf{P} = \mathbf{Q} \,
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* \begin{bmatrix} \mathbf{T} & \mathbf{0} \\
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* \mathbf{0} & \mathbf{0} \end{bmatrix} \, \mathbf{Z}
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* \f]
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* by using Householder transformations. Here, \b P is a permutation matrix,
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* \b Q and \b Z are unitary matrices and \b T an upper triangular matrix of
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* size rank-by-rank. \b A may be rank deficient.
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*
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* This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
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*
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* \sa MatrixBase::completeOrthogonalDecomposition()
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*/
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template <typename MatrixType_> class CompleteOrthogonalDecomposition
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: public SolverBase<CompleteOrthogonalDecomposition<MatrixType_> >
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{
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public:
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typedef MatrixType_ MatrixType;
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typedef SolverBase<CompleteOrthogonalDecomposition> Base;
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template<typename Derived>
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friend struct internal::solve_assertion;
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EIGEN_GENERIC_PUBLIC_INTERFACE(CompleteOrthogonalDecomposition)
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enum {
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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};
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typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
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typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime>
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PermutationType;
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typedef typename internal::plain_row_type<MatrixType, Index>::type
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IntRowVectorType;
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typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
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typedef typename internal::plain_row_type<MatrixType, RealScalar>::type
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RealRowVectorType;
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typedef HouseholderSequence<
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MatrixType, typename internal::remove_all<
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typename HCoeffsType::ConjugateReturnType>::type>
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HouseholderSequenceType;
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typedef typename MatrixType::PlainObject PlainObject;
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private:
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typedef typename PermutationType::Index PermIndexType;
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public:
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/**
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* \brief Default Constructor.
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*
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via
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* \c CompleteOrthogonalDecomposition::compute(const* MatrixType&).
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*/
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CompleteOrthogonalDecomposition() : m_cpqr(), m_zCoeffs(), m_temp() {}
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/** \brief Default Constructor with memory preallocation
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*
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* Like the default constructor but with preallocation of the internal data
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* according to the specified problem \a size.
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* \sa CompleteOrthogonalDecomposition()
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*/
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CompleteOrthogonalDecomposition(Index rows, Index cols)
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: m_cpqr(rows, cols), m_zCoeffs((std::min)(rows, cols)), m_temp(cols) {}
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/** \brief Constructs a complete orthogonal decomposition from a given
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* matrix.
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*
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* This constructor computes the complete orthogonal decomposition of the
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* matrix \a matrix by calling the method compute(). The default
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* threshold for rank determination will be used. It is a short cut for:
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*
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* \code
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* CompleteOrthogonalDecomposition<MatrixType> cod(matrix.rows(),
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* matrix.cols());
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* cod.setThreshold(Default);
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* cod.compute(matrix);
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* \endcode
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*
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* \sa compute()
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*/
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template <typename InputType>
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explicit CompleteOrthogonalDecomposition(const EigenBase<InputType>& matrix)
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: m_cpqr(matrix.rows(), matrix.cols()),
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m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
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m_temp(matrix.cols())
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{
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compute(matrix.derived());
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}
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/** \brief Constructs a complete orthogonal decomposition from a given matrix
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*
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* This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
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*
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* \sa CompleteOrthogonalDecomposition(const EigenBase&)
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*/
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template<typename InputType>
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explicit CompleteOrthogonalDecomposition(EigenBase<InputType>& matrix)
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: m_cpqr(matrix.derived()),
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m_zCoeffs((std::min)(matrix.rows(), matrix.cols())),
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m_temp(matrix.cols())
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{
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computeInPlace();
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}
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#ifdef EIGEN_PARSED_BY_DOXYGEN
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/** This method computes the minimum-norm solution X to a least squares
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* problem \f[\mathrm{minimize} \|A X - B\|, \f] where \b A is the matrix of
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* which \c *this is the complete orthogonal decomposition.
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*
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* \param b the right-hand sides of the problem to solve.
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*
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* \returns a solution.
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*
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*/
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template <typename Rhs>
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inline const Solve<CompleteOrthogonalDecomposition, Rhs> solve(
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const MatrixBase<Rhs>& b) const;
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#endif
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HouseholderSequenceType householderQ(void) const;
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HouseholderSequenceType matrixQ(void) const { return m_cpqr.householderQ(); }
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/** \returns the matrix \b Z.
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*/
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MatrixType matrixZ() const {
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MatrixType Z = MatrixType::Identity(m_cpqr.cols(), m_cpqr.cols());
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applyZOnTheLeftInPlace<false>(Z);
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return Z;
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}
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/** \returns a reference to the matrix where the complete orthogonal
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* decomposition is stored
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*/
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const MatrixType& matrixQTZ() const { return m_cpqr.matrixQR(); }
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/** \returns a reference to the matrix where the complete orthogonal
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* decomposition is stored.
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* \warning The strict lower part and \code cols() - rank() \endcode right
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* columns of this matrix contains internal values.
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* Only the upper triangular part should be referenced. To get it, use
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* \code matrixT().template triangularView<Upper>() \endcode
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* For rank-deficient matrices, use
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* \code
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* matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
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* \endcode
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*/
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const MatrixType& matrixT() const { return m_cpqr.matrixQR(); }
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template <typename InputType>
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CompleteOrthogonalDecomposition& compute(const EigenBase<InputType>& matrix) {
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// Compute the column pivoted QR factorization A P = Q R.
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m_cpqr.compute(matrix);
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computeInPlace();
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return *this;
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}
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/** \returns a const reference to the column permutation matrix */
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const PermutationType& colsPermutation() const {
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return m_cpqr.colsPermutation();
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}
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/** \returns the absolute value of the determinant of the matrix of which
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* *this is the complete orthogonal decomposition. It has only linear
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* complexity (that is, O(n) where n is the dimension of the square matrix)
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* as the complete orthogonal decomposition has already been computed.
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*
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* \note This is only for square matrices.
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*
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* \warning a determinant can be very big or small, so for matrices
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* of large enough dimension, there is a risk of overflow/underflow.
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* One way to work around that is to use logAbsDeterminant() instead.
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*
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* \sa logAbsDeterminant(), MatrixBase::determinant()
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*/
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typename MatrixType::RealScalar absDeterminant() const;
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/** \returns the natural log of the absolute value of the determinant of the
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* matrix of which *this is the complete orthogonal decomposition. It has
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* only linear complexity (that is, O(n) where n is the dimension of the
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* square matrix) as the complete orthogonal decomposition has already been
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* computed.
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*
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* \note This is only for square matrices.
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*
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* \note This method is useful to work around the risk of overflow/underflow
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* that's inherent to determinant computation.
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*
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* \sa absDeterminant(), MatrixBase::determinant()
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*/
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typename MatrixType::RealScalar logAbsDeterminant() const;
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/** \returns the rank of the matrix of which *this is the complete orthogonal
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* decomposition.
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*
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* \note This method has to determine which pivots should be considered
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* nonzero. For that, it uses the threshold value that you can control by
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* calling setThreshold(const RealScalar&).
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*/
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inline Index rank() const { return m_cpqr.rank(); }
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/** \returns the dimension of the kernel of the matrix of which *this is the
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* complete orthogonal decomposition.
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*
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* \note This method has to determine which pivots should be considered
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* nonzero. For that, it uses the threshold value that you can control by
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* calling setThreshold(const RealScalar&).
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*/
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inline Index dimensionOfKernel() const { return m_cpqr.dimensionOfKernel(); }
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/** \returns true if the matrix of which *this is the decomposition represents
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* an injective linear map, i.e. has trivial kernel; false otherwise.
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*
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* \note This method has to determine which pivots should be considered
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* nonzero. For that, it uses the threshold value that you can control by
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* calling setThreshold(const RealScalar&).
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*/
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inline bool isInjective() const { return m_cpqr.isInjective(); }
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/** \returns true if the matrix of which *this is the decomposition represents
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* a surjective linear map; false otherwise.
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*
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* \note This method has to determine which pivots should be considered
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* nonzero. For that, it uses the threshold value that you can control by
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* calling setThreshold(const RealScalar&).
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*/
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inline bool isSurjective() const { return m_cpqr.isSurjective(); }
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/** \returns true if the matrix of which *this is the complete orthogonal
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* decomposition is invertible.
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*
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* \note This method has to determine which pivots should be considered
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* nonzero. For that, it uses the threshold value that you can control by
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* calling setThreshold(const RealScalar&).
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*/
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inline bool isInvertible() const { return m_cpqr.isInvertible(); }
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/** \returns the pseudo-inverse of the matrix of which *this is the complete
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* orthogonal decomposition.
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* \warning: Do not compute \c this->pseudoInverse()*rhs to solve a linear systems.
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* It is more efficient and numerically stable to call \c this->solve(rhs).
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*/
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inline const Inverse<CompleteOrthogonalDecomposition> pseudoInverse() const
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{
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eigen_assert(m_cpqr.m_isInitialized && "CompleteOrthogonalDecomposition is not initialized.");
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return Inverse<CompleteOrthogonalDecomposition>(*this);
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}
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inline Index rows() const { return m_cpqr.rows(); }
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inline Index cols() const { return m_cpqr.cols(); }
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/** \returns a const reference to the vector of Householder coefficients used
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* to represent the factor \c Q.
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*
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* For advanced uses only.
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*/
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inline const HCoeffsType& hCoeffs() const { return m_cpqr.hCoeffs(); }
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/** \returns a const reference to the vector of Householder coefficients
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* used to represent the factor \c Z.
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*
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* For advanced uses only.
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*/
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const HCoeffsType& zCoeffs() const { return m_zCoeffs; }
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/** Allows to prescribe a threshold to be used by certain methods, such as
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* rank(), who need to determine when pivots are to be considered nonzero.
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* Most be called before calling compute().
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*
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* When it needs to get the threshold value, Eigen calls threshold(). By
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* default, this uses a formula to automatically determine a reasonable
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* threshold. Once you have called the present method
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* setThreshold(const RealScalar&), your value is used instead.
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*
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* \param threshold The new value to use as the threshold.
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*
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* A pivot will be considered nonzero if its absolute value is strictly
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* greater than
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* \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
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* where maxpivot is the biggest pivot.
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*
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* If you want to come back to the default behavior, call
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* setThreshold(Default_t)
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*/
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CompleteOrthogonalDecomposition& setThreshold(const RealScalar& threshold) {
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m_cpqr.setThreshold(threshold);
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return *this;
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}
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/** Allows to come back to the default behavior, letting Eigen use its default
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* formula for determining the threshold.
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*
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* You should pass the special object Eigen::Default as parameter here.
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* \code qr.setThreshold(Eigen::Default); \endcode
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*
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* See the documentation of setThreshold(const RealScalar&).
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*/
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CompleteOrthogonalDecomposition& setThreshold(Default_t) {
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m_cpqr.setThreshold(Default);
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return *this;
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}
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/** Returns the threshold that will be used by certain methods such as rank().
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*
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* See the documentation of setThreshold(const RealScalar&).
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*/
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RealScalar threshold() const { return m_cpqr.threshold(); }
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/** \returns the number of nonzero pivots in the complete orthogonal
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* decomposition. Here nonzero is meant in the exact sense, not in a
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* fuzzy sense. So that notion isn't really intrinsically interesting,
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* but it is still useful when implementing algorithms.
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*
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* \sa rank()
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*/
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inline Index nonzeroPivots() const { return m_cpqr.nonzeroPivots(); }
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/** \returns the absolute value of the biggest pivot, i.e. the biggest
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* diagonal coefficient of R.
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*/
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inline RealScalar maxPivot() const { return m_cpqr.maxPivot(); }
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/** \brief Reports whether the complete orthogonal decomposition was
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* successful.
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*
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* \note This function always returns \c Success. It is provided for
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* compatibility
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* with other factorization routines.
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* \returns \c Success
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*/
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ComputationInfo info() const {
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eigen_assert(m_cpqr.m_isInitialized && "Decomposition is not initialized.");
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return Success;
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}
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#ifndef EIGEN_PARSED_BY_DOXYGEN
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template <typename RhsType, typename DstType>
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void _solve_impl(const RhsType& rhs, DstType& dst) const;
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template<bool Conjugate, typename RhsType, typename DstType>
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void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const;
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#endif
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protected:
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EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
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template<bool Transpose_, typename Rhs>
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void _check_solve_assertion(const Rhs& b) const {
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EIGEN_ONLY_USED_FOR_DEBUG(b);
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eigen_assert(m_cpqr.m_isInitialized && "CompleteOrthogonalDecomposition is not initialized.");
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eigen_assert((Transpose_?derived().cols():derived().rows())==b.rows() && "CompleteOrthogonalDecomposition::solve(): invalid number of rows of the right hand side matrix b");
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}
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void computeInPlace();
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/** Overwrites \b rhs with \f$ \mathbf{Z} * \mathbf{rhs} \f$ or
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* \f$ \mathbf{\overline Z} * \mathbf{rhs} \f$ if \c Conjugate
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* is set to \c true.
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*/
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template <bool Conjugate, typename Rhs>
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void applyZOnTheLeftInPlace(Rhs& rhs) const;
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/** Overwrites \b rhs with \f$ \mathbf{Z}^* * \mathbf{rhs} \f$.
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*/
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template <typename Rhs>
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void applyZAdjointOnTheLeftInPlace(Rhs& rhs) const;
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ColPivHouseholderQR<MatrixType> m_cpqr;
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HCoeffsType m_zCoeffs;
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RowVectorType m_temp;
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};
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template <typename MatrixType>
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typename MatrixType::RealScalar
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CompleteOrthogonalDecomposition<MatrixType>::absDeterminant() const {
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return m_cpqr.absDeterminant();
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}
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template <typename MatrixType>
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typename MatrixType::RealScalar
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CompleteOrthogonalDecomposition<MatrixType>::logAbsDeterminant() const {
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return m_cpqr.logAbsDeterminant();
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}
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/** Performs the complete orthogonal decomposition of the given matrix \a
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* matrix. The result of the factorization is stored into \c *this, and a
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* reference to \c *this is returned.
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*
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* \sa class CompleteOrthogonalDecomposition,
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* CompleteOrthogonalDecomposition(const MatrixType&)
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*/
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template <typename MatrixType>
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void CompleteOrthogonalDecomposition<MatrixType>::computeInPlace()
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{
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// the column permutation is stored as int indices, so just to be sure:
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eigen_assert(m_cpqr.cols() <= NumTraits<int>::highest());
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const Index rank = m_cpqr.rank();
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const Index cols = m_cpqr.cols();
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const Index rows = m_cpqr.rows();
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m_zCoeffs.resize((std::min)(rows, cols));
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m_temp.resize(cols);
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if (rank < cols) {
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// We have reduced the (permuted) matrix to the form
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// [R11 R12]
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// [ 0 R22]
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// where R11 is r-by-r (r = rank) upper triangular, R12 is
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// r-by-(n-r), and R22 is empty or the norm of R22 is negligible.
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// We now compute the complete orthogonal decomposition by applying
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// Householder transformations from the right to the upper trapezoidal
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// matrix X = [R11 R12] to zero out R12 and obtain the factorization
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// [R11 R12] = [T11 0] * Z, where T11 is r-by-r upper triangular and
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// Z = Z(0) * Z(1) ... Z(r-1) is an n-by-n orthogonal matrix.
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// We store the data representing Z in R12 and m_zCoeffs.
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for (Index k = rank - 1; k >= 0; --k) {
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if (k != rank - 1) {
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// Given the API for Householder reflectors, it is more convenient if
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// we swap the leading parts of columns k and r-1 (zero-based) to form
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// the matrix X_k = [X(0:k, k), X(0:k, r:n)]
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m_cpqr.m_qr.col(k).head(k + 1).swap(
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m_cpqr.m_qr.col(rank - 1).head(k + 1));
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}
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// Construct Householder reflector Z(k) to zero out the last row of X_k,
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// i.e. choose Z(k) such that
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// [X(k, k), X(k, r:n)] * Z(k) = [beta, 0, .., 0].
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RealScalar beta;
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m_cpqr.m_qr.row(k)
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.tail(cols - rank + 1)
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.makeHouseholderInPlace(m_zCoeffs(k), beta);
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m_cpqr.m_qr(k, rank - 1) = beta;
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if (k > 0) {
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// Apply Z(k) to the first k rows of X_k
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m_cpqr.m_qr.topRightCorner(k, cols - rank + 1)
|
|
.applyHouseholderOnTheRight(
|
|
m_cpqr.m_qr.row(k).tail(cols - rank).adjoint(), m_zCoeffs(k),
|
|
&m_temp(0));
|
|
}
|
|
if (k != rank - 1) {
|
|
// Swap X(0:k,k) back to its proper location.
|
|
m_cpqr.m_qr.col(k).head(k + 1).swap(
|
|
m_cpqr.m_qr.col(rank - 1).head(k + 1));
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
template <typename MatrixType>
|
|
template <bool Conjugate, typename Rhs>
|
|
void CompleteOrthogonalDecomposition<MatrixType>::applyZOnTheLeftInPlace(
|
|
Rhs& rhs) const {
|
|
const Index cols = this->cols();
|
|
const Index nrhs = rhs.cols();
|
|
const Index rank = this->rank();
|
|
Matrix<typename Rhs::Scalar, Dynamic, 1> temp((std::max)(cols, nrhs));
|
|
for (Index k = rank-1; k >= 0; --k) {
|
|
if (k != rank - 1) {
|
|
rhs.row(k).swap(rhs.row(rank - 1));
|
|
}
|
|
rhs.middleRows(rank - 1, cols - rank + 1)
|
|
.applyHouseholderOnTheLeft(
|
|
matrixQTZ().row(k).tail(cols - rank).transpose().template conjugateIf<!Conjugate>(), zCoeffs().template conjugateIf<Conjugate>()(k),
|
|
&temp(0));
|
|
if (k != rank - 1) {
|
|
rhs.row(k).swap(rhs.row(rank - 1));
|
|
}
|
|
}
|
|
}
|
|
|
|
template <typename MatrixType>
|
|
template <typename Rhs>
|
|
void CompleteOrthogonalDecomposition<MatrixType>::applyZAdjointOnTheLeftInPlace(
|
|
Rhs& rhs) const {
|
|
const Index cols = this->cols();
|
|
const Index nrhs = rhs.cols();
|
|
const Index rank = this->rank();
|
|
Matrix<typename Rhs::Scalar, Dynamic, 1> temp((std::max)(cols, nrhs));
|
|
for (Index k = 0; k < rank; ++k) {
|
|
if (k != rank - 1) {
|
|
rhs.row(k).swap(rhs.row(rank - 1));
|
|
}
|
|
rhs.middleRows(rank - 1, cols - rank + 1)
|
|
.applyHouseholderOnTheLeft(
|
|
matrixQTZ().row(k).tail(cols - rank).adjoint(), zCoeffs()(k),
|
|
&temp(0));
|
|
if (k != rank - 1) {
|
|
rhs.row(k).swap(rhs.row(rank - 1));
|
|
}
|
|
}
|
|
}
|
|
|
|
#ifndef EIGEN_PARSED_BY_DOXYGEN
|
|
template <typename MatrixType_>
|
|
template <typename RhsType, typename DstType>
|
|
void CompleteOrthogonalDecomposition<MatrixType_>::_solve_impl(
|
|
const RhsType& rhs, DstType& dst) const {
|
|
const Index rank = this->rank();
|
|
if (rank == 0) {
|
|
dst.setZero();
|
|
return;
|
|
}
|
|
|
|
// Compute c = Q^* * rhs
|
|
typename RhsType::PlainObject c(rhs);
|
|
c.applyOnTheLeft(matrixQ().setLength(rank).adjoint());
|
|
|
|
// Solve T z = c(1:rank, :)
|
|
dst.topRows(rank) = matrixT()
|
|
.topLeftCorner(rank, rank)
|
|
.template triangularView<Upper>()
|
|
.solve(c.topRows(rank));
|
|
|
|
const Index cols = this->cols();
|
|
if (rank < cols) {
|
|
// Compute y = Z^* * [ z ]
|
|
// [ 0 ]
|
|
dst.bottomRows(cols - rank).setZero();
|
|
applyZAdjointOnTheLeftInPlace(dst);
|
|
}
|
|
|
|
// Undo permutation to get x = P^{-1} * y.
|
|
dst = colsPermutation() * dst;
|
|
}
|
|
|
|
template<typename MatrixType_>
|
|
template<bool Conjugate, typename RhsType, typename DstType>
|
|
void CompleteOrthogonalDecomposition<MatrixType_>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
|
|
{
|
|
const Index rank = this->rank();
|
|
|
|
if (rank == 0) {
|
|
dst.setZero();
|
|
return;
|
|
}
|
|
|
|
typename RhsType::PlainObject c(colsPermutation().transpose()*rhs);
|
|
|
|
if (rank < cols()) {
|
|
applyZOnTheLeftInPlace<!Conjugate>(c);
|
|
}
|
|
|
|
matrixT().topLeftCorner(rank, rank)
|
|
.template triangularView<Upper>()
|
|
.transpose().template conjugateIf<Conjugate>()
|
|
.solveInPlace(c.topRows(rank));
|
|
|
|
dst.topRows(rank) = c.topRows(rank);
|
|
dst.bottomRows(rows()-rank).setZero();
|
|
|
|
dst.applyOnTheLeft(householderQ().setLength(rank).template conjugateIf<!Conjugate>() );
|
|
}
|
|
#endif
|
|
|
|
namespace internal {
|
|
|
|
template<typename MatrixType>
|
|
struct traits<Inverse<CompleteOrthogonalDecomposition<MatrixType> > >
|
|
: traits<typename Transpose<typename MatrixType::PlainObject>::PlainObject>
|
|
{
|
|
enum { Flags = 0 };
|
|
};
|
|
|
|
template<typename DstXprType, typename MatrixType>
|
|
struct Assignment<DstXprType, Inverse<CompleteOrthogonalDecomposition<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename CompleteOrthogonalDecomposition<MatrixType>::Scalar>, Dense2Dense>
|
|
{
|
|
typedef CompleteOrthogonalDecomposition<MatrixType> CodType;
|
|
typedef Inverse<CodType> SrcXprType;
|
|
static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename CodType::Scalar> &)
|
|
{
|
|
typedef Matrix<typename CodType::Scalar, CodType::RowsAtCompileTime, CodType::RowsAtCompileTime, 0, CodType::MaxRowsAtCompileTime, CodType::MaxRowsAtCompileTime> IdentityMatrixType;
|
|
dst = src.nestedExpression().solve(IdentityMatrixType::Identity(src.cols(), src.cols()));
|
|
}
|
|
};
|
|
|
|
} // end namespace internal
|
|
|
|
/** \returns the matrix Q as a sequence of householder transformations */
|
|
template <typename MatrixType>
|
|
typename CompleteOrthogonalDecomposition<MatrixType>::HouseholderSequenceType
|
|
CompleteOrthogonalDecomposition<MatrixType>::householderQ() const {
|
|
return m_cpqr.householderQ();
|
|
}
|
|
|
|
/** \return the complete orthogonal decomposition of \c *this.
|
|
*
|
|
* \sa class CompleteOrthogonalDecomposition
|
|
*/
|
|
template <typename Derived>
|
|
const CompleteOrthogonalDecomposition<typename MatrixBase<Derived>::PlainObject>
|
|
MatrixBase<Derived>::completeOrthogonalDecomposition() const {
|
|
return CompleteOrthogonalDecomposition<PlainObject>(eval());
|
|
}
|
|
|
|
} // end namespace Eigen
|
|
|
|
#endif // EIGEN_COMPLETEORTHOGONALDECOMPOSITION_H
|