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// This file is part of Eigen, a lightweight C++ template library
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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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// Eigen is free software; you can redistribute it and/or
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// modify it under the terms of the GNU Lesser General Public
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// License as published by the Free Software Foundation; either
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// version 3 of the License, or (at your option) any later version.
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// Alternatively, you can redistribute it and/or
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// modify it under the terms of the GNU General Public License as
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// published by the Free Software Foundation; either version 2 of
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// the License, or (at your option) any later version.
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// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
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// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
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// GNU General Public License for more details.
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// You should have received a copy of the GNU Lesser General Public
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// License and a copy of the GNU General Public License along with
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// Eigen. If not, see <http://www.gnu.org/licenses/>.
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#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
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#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
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/** \ingroup QR_Module
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* \class FullPivHouseholderQR
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* \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
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* \param MatrixType the type of the matrix of which we are computing the QR decomposition
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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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* \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
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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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* This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal
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* numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
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* \sa MatrixBase::fullPivHouseholderQr()
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template<typename _MatrixType> class FullPivHouseholderQR
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typedef _MatrixType MatrixType;
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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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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 Matrix<Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime> IntRowVectorType;
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typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
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typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
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typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
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typedef typename internal::plain_col_type<MatrixType>::type ColVectorType;
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/** \brief Default Constructor.
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* The default constructor is useful in cases in which the user intends to
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* perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).
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FullPivHouseholderQR()
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m_rows_transpositions(),
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m_cols_transpositions(),
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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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* 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 FullPivHouseholderQR()
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FullPivHouseholderQR(Index rows, Index cols)
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m_hCoeffs(std::min(rows,cols)),
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m_rows_transpositions(rows),
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m_cols_transpositions(cols),
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m_cols_permutation(cols),
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m_temp(std::min(rows,cols)),
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m_isInitialized(false),
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m_usePrescribedThreshold(false) {}
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FullPivHouseholderQR(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_rows_transpositions(matrix.rows()),
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m_cols_transpositions(matrix.cols()),
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m_cols_permutation(matrix.cols()),
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m_temp(std::min(matrix.rows(), matrix.cols())),
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m_isInitialized(false),
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m_usePrescribedThreshold(false)
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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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* \param b the right-hand-side of the equation to solve.
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* \returns a solution.
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* \note The case where b is a matrix is not yet implemented. Also, this
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* code is space inefficient.
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* \note_about_checking_solutions
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* \note_about_arbitrary_choice_of_solution
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* Example: \include FullPivHouseholderQR_solve.cpp
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* Output: \verbinclude FullPivHouseholderQR_solve.out
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template<typename Rhs>
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inline const internal::solve_retval<FullPivHouseholderQR, Rhs>
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solve(const MatrixBase<Rhs>& b) const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return internal::solve_retval<FullPivHouseholderQR, Rhs>(*this, b.derived());
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MatrixQType matrixQ(void) const;
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/** \returns a reference to the matrix where the Householder QR decomposition is stored
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const MatrixType& matrixQR() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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FullPivHouseholderQR& compute(const MatrixType& matrix);
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const PermutationType& colsPermutation() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return m_cols_permutation;
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const IntColVectorType& rowsTranspositions() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return m_rows_transpositions;
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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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* \note This is only for square matrices.
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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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* \sa logAbsDeterminant(), MatrixBase::determinant()
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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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* \note This is only for square matrices.
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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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* \sa absDeterminant(), MatrixBase::determinant()
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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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* \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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inline Index rank() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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RealScalar premultiplied_threshold = internal::abs(m_maxpivot) * threshold();
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for(Index i = 0; i < m_nonzero_pivots; ++i)
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result += (internal::abs(m_qr.coeff(i,i)) > premultiplied_threshold);
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/** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
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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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inline Index dimensionOfKernel() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return cols() - rank();
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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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* \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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inline bool isInjective() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return rank() == cols();
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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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* \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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inline bool isSurjective() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return rank() == rows();
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/** \returns true if the matrix of which *this is the QR decomposition is invertible.
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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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inline bool isInvertible() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return isInjective() && isSurjective();
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/** \returns the inverse of the matrix of which *this is the QR decomposition.
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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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internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType>
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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return internal::solve_retval<FullPivHouseholderQR,typename MatrixType::IdentityReturnType>
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(*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
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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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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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* 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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* \param threshold The new value to use as the threshold.
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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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* If you want to come back to the default behavior, call setThreshold(Default_t)
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FullPivHouseholderQR& setThreshold(const RealScalar& threshold)
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m_usePrescribedThreshold = true;
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m_prescribedThreshold = threshold;
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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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* 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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* See the documentation of setThreshold(const RealScalar&).
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FullPivHouseholderQR& setThreshold(Default_t)
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m_usePrescribedThreshold = false;
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/** Returns the threshold that will be used by certain methods such as rank().
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* See the documentation of setThreshold(const RealScalar&).
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RealScalar threshold() const
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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() * m_qr.diagonalSize();
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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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inline Index nonzeroPivots() const
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eigen_assert(m_isInitialized && "LU is not initialized.");
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return m_nonzero_pivots;
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/** \returns the absolute value of the biggest pivot, i.e. the biggest
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* diagonal coefficient of U.
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RealScalar maxPivot() const { return m_maxpivot; }
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HCoeffsType m_hCoeffs;
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IntColVectorType m_rows_transpositions;
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IntRowVectorType m_cols_transpositions;
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PermutationType m_cols_permutation;
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RowVectorType m_temp;
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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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RealScalar m_precision;
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template<typename MatrixType>
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typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::absDeterminant() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR 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 internal::abs(m_qr.diagonal().prod());
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template<typename MatrixType>
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typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::logAbsDeterminant() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR 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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template<typename MatrixType>
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FullPivHouseholderQR<MatrixType>& FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
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Index rows = matrix.rows();
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Index cols = matrix.cols();
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Index size = std::min(rows,cols);
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m_hCoeffs.resize(size);
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m_precision = NumTraits<Scalar>::epsilon() * size;
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m_rows_transpositions.resize(matrix.rows());
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m_cols_transpositions.resize(matrix.cols());
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Index number_of_transpositions = 0;
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RealScalar biggest(0);
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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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Index row_of_biggest_in_corner, col_of_biggest_in_corner;
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RealScalar biggest_in_corner;
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biggest_in_corner = m_qr.bottomRightCorner(rows-k, cols-k)
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.maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
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row_of_biggest_in_corner += k;
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col_of_biggest_in_corner += k;
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if(k==0) biggest = biggest_in_corner;
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// if the corner is negligible, then we have less than full rank, and we can finish early
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if(internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
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m_nonzero_pivots = k;
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for(Index i = k; i < size; i++)
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m_rows_transpositions.coeffRef(i) = i;
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m_cols_transpositions.coeffRef(i) = i;
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m_hCoeffs.coeffRef(i) = Scalar(0);
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m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
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m_cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
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if(k != row_of_biggest_in_corner) {
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m_qr.row(k).tail(cols-k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols-k));
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++number_of_transpositions;
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if(k != col_of_biggest_in_corner) {
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m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
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++number_of_transpositions;
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m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
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m_qr.coeffRef(k,k) = beta;
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// remember the maximum absolute value of diagonal coefficients
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if(internal::abs(beta) > m_maxpivot) m_maxpivot = internal::abs(beta);
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m_qr.bottomRightCorner(rows-k, cols-k-1)
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.applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
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m_cols_permutation.setIdentity(cols);
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for(Index k = 0; k < size; ++k)
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m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));
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m_det_pq = (number_of_transpositions%2) ? -1 : 1;
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m_isInitialized = true;
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template<typename _MatrixType, typename Rhs>
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struct solve_retval<FullPivHouseholderQR<_MatrixType>, Rhs>
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: solve_retval_base<FullPivHouseholderQR<_MatrixType>, Rhs>
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EIGEN_MAKE_SOLVE_HELPERS(FullPivHouseholderQR<_MatrixType>,Rhs)
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template<typename Dest> void evalTo(Dest& dst) const
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const Index rows = dec().rows(), cols = dec().cols();
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eigen_assert(rhs().rows() == rows);
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// FIXME introduce nonzeroPivots() and use it here. and more generally,
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// make the same improvements in this dec as in FullPivLU.
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typename Rhs::PlainObject c(rhs());
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Matrix<Scalar,1,Rhs::ColsAtCompileTime> temp(rhs().cols());
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for (Index k = 0; k < dec().rank(); ++k)
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Index remainingSize = rows-k;
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c.row(k).swap(c.row(dec().rowsTranspositions().coeff(k)));
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c.bottomRightCorner(remainingSize, rhs().cols())
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.applyHouseholderOnTheLeft(dec().matrixQR().col(k).tail(remainingSize-1),
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dec().hCoeffs().coeff(k), &temp.coeffRef(0));
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if(!dec().isSurjective())
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// is c is in the image of R ?
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RealScalar biggest_in_upper_part_of_c = c.topRows( dec().rank() ).cwiseAbs().maxCoeff();
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RealScalar biggest_in_lower_part_of_c = c.bottomRows(rows-dec().rank()).cwiseAbs().maxCoeff();
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const RealScalar m_precision = NumTraits<Scalar>::epsilon() * std::min(rows,cols);
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// this internal:: prefix is needed by at least gcc 3.4 and ICC
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if(!internal::isMuchSmallerThan(biggest_in_lower_part_of_c, biggest_in_upper_part_of_c, m_precision))
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.topLeftCorner(dec().rank(), dec().rank())
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.template triangularView<Upper>()
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.solveInPlace(c.topRows(dec().rank()));
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for(Index i = 0; i < dec().rank(); ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
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for(Index i = dec().rank(); i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
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} // end namespace internal
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/** \returns the matrix Q */
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template<typename MatrixType>
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typename FullPivHouseholderQR<MatrixType>::MatrixQType FullPivHouseholderQR<MatrixType>::matrixQ() const
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eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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// compute the product H'_0 H'_1 ... H'_n-1,
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// where H_k is the k-th Householder transformation I - h_k v_k v_k'
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// and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
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Index rows = m_qr.rows();
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Index cols = m_qr.cols();
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Index size = std::min(rows,cols);
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MatrixQType res = MatrixQType::Identity(rows, rows);
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Matrix<Scalar,1,MatrixType::RowsAtCompileTime> temp(rows);
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for (Index k = size-1; k >= 0; k--)
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res.block(k, k, rows-k, rows-k)
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.applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), internal::conj(m_hCoeffs.coeff(k)), &temp.coeffRef(k));
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res.row(k).swap(res.row(m_rows_transpositions.coeff(k)));
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/** \return the full-pivoting Householder QR decomposition of \c *this.
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* \sa class FullPivHouseholderQR
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template<typename Derived>
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const FullPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
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MatrixBase<Derived>::fullPivHouseholderQr() const
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return FullPivHouseholderQR<PlainObject>(eval());
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#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
added
removed
thirdparty/eigen3/Eigen/src/QR/FullPivHouseholderQR.h
