578 lines
22 KiB
C++
578 lines
22 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) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
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// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.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_COLPIVOTINGHOUSEHOLDERQR_H
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#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
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namespace Eigen {
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/** \ingroup QR_Module
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*
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* \class ColPivHouseholderQR
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*
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* \brief Householder rank-revealing QR decomposition of a matrix with column-pivoting
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*
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* \param MatrixType the type of the matrix of which we are computing the QR decomposition
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*
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* This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b Q and \b R
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* such that
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* \f[
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* \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
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* \f]
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* by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
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* upper triangular matrix.
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*
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* This decomposition performs column pivoting in order to be rank-revealing and improve
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* numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
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*
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* \sa MatrixBase::colPivHouseholderQr()
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*/
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template<typename _MatrixType> class ColPivHouseholderQR
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{
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public:
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typedef _MatrixType MatrixType;
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enum {
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RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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Options = MatrixType::Options,
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MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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};
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef typename MatrixType::Index Index;
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typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
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typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
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typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
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typedef typename internal::plain_row_type<MatrixType, Index>::type 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 RealRowVectorType;
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typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
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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 ColPivHouseholderQR::compute(const MatrixType&).
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*/
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ColPivHouseholderQR()
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: m_qr(),
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m_hCoeffs(),
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m_colsPermutation(),
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m_colsTranspositions(),
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m_temp(),
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m_colSqNorms(),
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m_isInitialized(false),
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m_usePrescribedThreshold(false) {}
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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 ColPivHouseholderQR()
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*/
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ColPivHouseholderQR(Index rows, Index cols)
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: m_qr(rows, cols),
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m_hCoeffs((std::min)(rows,cols)),
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m_colsPermutation(PermIndexType(cols)),
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m_colsTranspositions(cols),
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m_temp(cols),
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m_colSqNorms(cols),
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m_isInitialized(false),
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m_usePrescribedThreshold(false) {}
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/** \brief Constructs a QR factorization from a given matrix
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*
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* This constructor computes the QR factorization of the matrix \a matrix by calling
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* the method compute(). It is a short cut for:
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*
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* \code
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* ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
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* qr.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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ColPivHouseholderQR(const MatrixType& matrix)
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: m_qr(matrix.rows(), matrix.cols()),
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m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
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m_colsPermutation(PermIndexType(matrix.cols())),
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m_colsTranspositions(matrix.cols()),
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m_temp(matrix.cols()),
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m_colSqNorms(matrix.cols()),
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m_isInitialized(false),
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m_usePrescribedThreshold(false)
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{
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compute(matrix);
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}
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/** This method finds a solution x to the equation Ax=b, where A is the matrix of which
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* *this is the QR decomposition, if any exists.
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*
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* \param b the right-hand-side of the equation to solve.
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*
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* \returns a solution.
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*
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* \note_about_checking_solutions
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*
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* \note_about_arbitrary_choice_of_solution
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*
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* Example: \include ColPivHouseholderQR_solve.cpp
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* Output: \verbinclude ColPivHouseholderQR_solve.out
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*/
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template<typename Rhs>
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inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
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solve(const MatrixBase<Rhs>& b) const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
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}
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HouseholderSequenceType householderQ(void) const;
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HouseholderSequenceType matrixQ(void) const
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{
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return householderQ();
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}
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/** \returns a reference to the matrix where the Householder QR decomposition is stored
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*/
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const MatrixType& matrixQR() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return m_qr;
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}
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/** \returns a reference to the matrix where the result Householder QR is stored
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* \warning The strict lower part 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 matrixR().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& matrixR() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return m_qr;
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}
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ColPivHouseholderQR& compute(const MatrixType& matrix);
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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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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return m_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 QR decomposition. It has only linear complexity
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* (that is, O(n) where n is the dimension of the square matrix)
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* as the QR 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 matrix of which
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* *this is the QR decomposition. It has only linear complexity
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* (that is, O(n) where n is the dimension of the square matrix)
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* as the QR 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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* \note This method is useful to work around the risk of overflow/underflow that's inherent
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* 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 QR decomposition.
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*
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* \note This method has to determine which pivots should be considered nonzero.
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* For that, it uses the threshold value that you can control by calling
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* setThreshold(const RealScalar&).
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*/
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inline Index rank() const
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{
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using std::abs;
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
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Index result = 0;
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for(Index i = 0; i < m_nonzero_pivots; ++i)
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result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
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return result;
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}
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/** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
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*
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* \note This method has to determine which pivots should be considered nonzero.
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* For that, it uses the threshold value that you can control by calling
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* setThreshold(const RealScalar&).
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*/
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inline Index dimensionOfKernel() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return cols() - rank();
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}
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/** \returns true if the matrix of which *this is the QR decomposition represents an injective
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* 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 nonzero.
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* For that, it uses the threshold value that you can control by calling
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* setThreshold(const RealScalar&).
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*/
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inline bool isInjective() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return rank() == cols();
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}
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/** \returns true if the matrix of which *this is the QR decomposition represents a surjective
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* linear map; false otherwise.
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*
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* \note This method has to determine which pivots should be considered nonzero.
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* For that, it uses the threshold value that you can control by calling
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* setThreshold(const RealScalar&).
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*/
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inline bool isSurjective() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return rank() == rows();
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}
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/** \returns true if the matrix of which *this is the QR decomposition is invertible.
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*
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* \note This method has to determine which pivots should be considered nonzero.
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* For that, it uses the threshold value that you can control by calling
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* setThreshold(const RealScalar&).
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*/
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inline bool isInvertible() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return isInjective() && isSurjective();
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}
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/** \returns the inverse of the matrix of which *this is the QR decomposition.
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*
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* \note If this matrix is not invertible, the returned matrix has undefined coefficients.
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* Use isInvertible() to first determine whether this matrix is invertible.
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*/
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inline const
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internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
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inverse() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return internal::solve_retval<ColPivHouseholderQR,typename MatrixType::IdentityReturnType>
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(*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
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}
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inline Index rows() const { return m_qr.rows(); }
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inline Index cols() const { return m_qr.cols(); }
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/** \returns a const reference to the vector of Householder coefficients used 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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const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
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/** Allows to prescribe a threshold to be used by certain methods, such as rank(),
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* who need to determine when pivots are to be considered nonzero. This is not used for the
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* QR decomposition itself.
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*
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* When it needs to get the threshold value, Eigen calls threshold(). By default, this
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* uses a formula to automatically determine a reasonable threshold.
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* Once you have called the present method setThreshold(const RealScalar&),
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* 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 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 setThreshold(Default_t)
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*/
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ColPivHouseholderQR& setThreshold(const RealScalar& threshold)
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{
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m_usePrescribedThreshold = true;
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m_prescribedThreshold = 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 formula for
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* 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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ColPivHouseholderQR& setThreshold(Default_t)
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{
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m_usePrescribedThreshold = false;
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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
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{
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eigen_assert(m_isInitialized || m_usePrescribedThreshold);
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return m_usePrescribedThreshold ? m_prescribedThreshold
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// this formula comes from experimenting (see "LU precision tuning" thread on the list)
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// and turns out to be identical to Higham's formula used already in LDLt.
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: NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
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}
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/** \returns the number of nonzero pivots in the QR decomposition.
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* Here nonzero is meant in the exact sense, not in a fuzzy sense.
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* So that notion isn't really intrinsically interesting, but it is
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* 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
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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return m_nonzero_pivots;
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}
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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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RealScalar maxPivot() const { return m_maxpivot; }
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/** \brief Reports whether the QR factorization was succesful.
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*
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* \note This function always returns \c Success. It is provided for 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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{
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eigen_assert(m_isInitialized && "Decomposition is not initialized.");
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return Success;
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}
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protected:
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static void check_template_parameters()
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{
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EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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}
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MatrixType m_qr;
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HCoeffsType m_hCoeffs;
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PermutationType m_colsPermutation;
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IntRowVectorType m_colsTranspositions;
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RowVectorType m_temp;
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RealRowVectorType m_colSqNorms;
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bool m_isInitialized, m_usePrescribedThreshold;
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RealScalar m_prescribedThreshold, m_maxpivot;
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Index m_nonzero_pivots;
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Index m_det_pq;
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};
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template<typename MatrixType>
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typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
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{
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using std::abs;
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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return abs(m_qr.diagonal().prod());
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}
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template<typename MatrixType>
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typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
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{
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eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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return m_qr.diagonal().cwiseAbs().array().log().sum();
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}
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/** Performs the QR factorization of the given matrix \a matrix. The result of
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* the factorization is stored into \c *this, and a reference to \c *this
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* is returned.
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*
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* \sa class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
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*/
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template<typename MatrixType>
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ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
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{
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check_template_parameters();
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using std::abs;
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Index rows = matrix.rows();
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Index cols = matrix.cols();
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Index size = matrix.diagonalSize();
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// the column permutation is stored as int indices, so just to be sure:
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eigen_assert(cols<=NumTraits<int>::highest());
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m_qr = matrix;
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m_hCoeffs.resize(size);
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m_temp.resize(cols);
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m_colsTranspositions.resize(matrix.cols());
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Index number_of_transpositions = 0;
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m_colSqNorms.resize(cols);
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for(Index k = 0; k < cols; ++k)
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m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();
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RealScalar threshold_helper = m_colSqNorms.maxCoeff() * numext::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);
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m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
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m_maxpivot = RealScalar(0);
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for(Index k = 0; k < size; ++k)
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{
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// first, we look up in our table m_colSqNorms which column has the biggest squared norm
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Index biggest_col_index;
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RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
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biggest_col_index += k;
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// since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
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// the actual squared norm of the selected column.
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// Note that not doing so does result in solve() sometimes returning inf/nan values
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// when running the unit test with 1000 repetitions.
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biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();
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// we store that back into our table: it can't hurt to correct our table.
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m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;
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// Track the number of meaningful pivots but do not stop the decomposition to make
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// sure that the initial matrix is properly reproduced. See bug 941.
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if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
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m_nonzero_pivots = k;
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// apply the transposition to the columns
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m_colsTranspositions.coeffRef(k) = biggest_col_index;
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if(k != biggest_col_index) {
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m_qr.col(k).swap(m_qr.col(biggest_col_index));
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std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.coeffRef(biggest_col_index));
|
|
++number_of_transpositions;
|
|
}
|
|
|
|
// generate the householder vector, store it below the diagonal
|
|
RealScalar beta;
|
|
m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
|
|
|
|
// apply the householder transformation to the diagonal coefficient
|
|
m_qr.coeffRef(k,k) = beta;
|
|
|
|
// remember the maximum absolute value of diagonal coefficients
|
|
if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
|
|
|
|
// apply the householder transformation
|
|
m_qr.bottomRightCorner(rows-k, cols-k-1)
|
|
.applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
|
|
|
|
// update our table of squared norms of the columns
|
|
m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
|
|
}
|
|
|
|
m_colsPermutation.setIdentity(PermIndexType(cols));
|
|
for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
|
|
m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
|
|
|
|
m_det_pq = (number_of_transpositions%2) ? -1 : 1;
|
|
m_isInitialized = true;
|
|
|
|
return *this;
|
|
}
|
|
|
|
namespace internal {
|
|
|
|
template<typename _MatrixType, typename Rhs>
|
|
struct solve_retval<ColPivHouseholderQR<_MatrixType>, Rhs>
|
|
: solve_retval_base<ColPivHouseholderQR<_MatrixType>, Rhs>
|
|
{
|
|
EIGEN_MAKE_SOLVE_HELPERS(ColPivHouseholderQR<_MatrixType>,Rhs)
|
|
|
|
template<typename Dest> void evalTo(Dest& dst) const
|
|
{
|
|
eigen_assert(rhs().rows() == dec().rows());
|
|
|
|
const Index cols = dec().cols(),
|
|
nonzero_pivots = dec().nonzeroPivots();
|
|
|
|
if(nonzero_pivots == 0)
|
|
{
|
|
dst.setZero();
|
|
return;
|
|
}
|
|
|
|
typename Rhs::PlainObject c(rhs());
|
|
|
|
// Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
|
|
c.applyOnTheLeft(householderSequence(dec().matrixQR(), dec().hCoeffs())
|
|
.setLength(dec().nonzeroPivots())
|
|
.transpose()
|
|
);
|
|
|
|
dec().matrixR()
|
|
.topLeftCorner(nonzero_pivots, nonzero_pivots)
|
|
.template triangularView<Upper>()
|
|
.solveInPlace(c.topRows(nonzero_pivots));
|
|
|
|
for(Index i = 0; i < nonzero_pivots; ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
|
|
for(Index i = nonzero_pivots; i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
|
|
}
|
|
};
|
|
|
|
} // end namespace internal
|
|
|
|
/** \returns the matrix Q as a sequence of householder transformations.
|
|
* You can extract the meaningful part only by using:
|
|
* \code qr.householderQ().setLength(qr.nonzeroPivots()) \endcode*/
|
|
template<typename MatrixType>
|
|
typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
|
|
::householderQ() const
|
|
{
|
|
eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
|
|
return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
|
|
}
|
|
|
|
/** \return the column-pivoting Householder QR decomposition of \c *this.
|
|
*
|
|
* \sa class ColPivHouseholderQR
|
|
*/
|
|
template<typename Derived>
|
|
const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
|
|
MatrixBase<Derived>::colPivHouseholderQr() const
|
|
{
|
|
return ColPivHouseholderQR<PlainObject>(eval());
|
|
}
|
|
|
|
} // end namespace Eigen
|
|
|
|
#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
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