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-// This file is part of Eigen, a lightweight C++ template library
-// for linear algebra.
-//
-// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
-// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
-//
-// This Source Code Form is subject to the terms of the Mozilla
-// Public License v. 2.0. If a copy of the MPL was not distributed
-// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
-
-#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
-#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
-
-namespace Eigen {
-
-namespace internal {
-
-template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType;
-
-template<typename MatrixType>
-struct traits<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
-{
- typedef typename MatrixType::PlainObject ReturnType;
-};
-
-}
-
-/** \ingroup QR_Module
- *
- * \class FullPivHouseholderQR
- *
- * \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
- *
- * \param MatrixType the type of the matrix of which we are computing the QR decomposition
- *
- * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b P', \b Q and \b R
- * such that
- * \f[
- * \mathbf{P} \, \mathbf{A} \, \mathbf{P}' = \mathbf{Q} \, \mathbf{R}
- * \f]
- * by using Householder transformations. Here, \b P and \b P' are permutation matrices, \b Q a unitary matrix
- * and \b R an upper triangular matrix.
- *
- * This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal
- * numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
- *
- * \sa MatrixBase::fullPivHouseholderQr()
- */
-template<typename _MatrixType> class FullPivHouseholderQR
-{
- public:
-
- typedef _MatrixType MatrixType;
- enum {
- RowsAtCompileTime = MatrixType::RowsAtCompileTime,
- ColsAtCompileTime = MatrixType::ColsAtCompileTime,
- Options = MatrixType::Options,
- MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
- MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
- };
- typedef typename MatrixType::Scalar Scalar;
- typedef typename MatrixType::RealScalar RealScalar;
- typedef typename MatrixType::Index Index;
- typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> MatrixQReturnType;
- typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
- typedef Matrix<Index, 1,
- EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1,
- EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime,MaxRowsAtCompileTime)> IntDiagSizeVectorType;
- typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
- typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
- typedef typename internal::plain_col_type<MatrixType>::type ColVectorType;
-
- /** \brief Default Constructor.
- *
- * The default constructor is useful in cases in which the user intends to
- * perform decompositions via FullPivHouseholderQR::compute(const MatrixType&).
- */
- FullPivHouseholderQR()
- : m_qr(),
- m_hCoeffs(),
- m_rows_transpositions(),
- m_cols_transpositions(),
- m_cols_permutation(),
- m_temp(),
- m_isInitialized(false),
- m_usePrescribedThreshold(false) {}
-
- /** \brief Default Constructor with memory preallocation
- *
- * Like the default constructor but with preallocation of the internal data
- * according to the specified problem \a size.
- * \sa FullPivHouseholderQR()
- */
- FullPivHouseholderQR(Index rows, Index cols)
- : m_qr(rows, cols),
- m_hCoeffs((std::min)(rows,cols)),
- m_rows_transpositions((std::min)(rows,cols)),
- m_cols_transpositions((std::min)(rows,cols)),
- m_cols_permutation(cols),
- m_temp(cols),
- m_isInitialized(false),
- m_usePrescribedThreshold(false) {}
-
- /** \brief Constructs a QR factorization from a given matrix
- *
- * This constructor computes the QR factorization of the matrix \a matrix by calling
- * the method compute(). It is a short cut for:
- *
- * \code
- * FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
- * qr.compute(matrix);
- * \endcode
- *
- * \sa compute()
- */
- FullPivHouseholderQR(const MatrixType& matrix)
- : m_qr(matrix.rows(), matrix.cols()),
- m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
- m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
- m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
- m_cols_permutation(matrix.cols()),
- m_temp(matrix.cols()),
- m_isInitialized(false),
- m_usePrescribedThreshold(false)
- {
- compute(matrix);
- }
-
- /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
- * \c *this is the QR decomposition.
- *
- * \param b the right-hand-side of the equation to solve.
- *
- * \returns the exact or least-square solution if the rank is greater or equal to the number of columns of A,
- * and an arbitrary solution otherwise.
- *
- * \note The case where b is a matrix is not yet implemented. Also, this
- * code is space inefficient.
- *
- * \note_about_checking_solutions
- *
- * \note_about_arbitrary_choice_of_solution
- *
- * Example: \include FullPivHouseholderQR_solve.cpp
- * Output: \verbinclude FullPivHouseholderQR_solve.out
- */
- template<typename Rhs>
- inline const internal::solve_retval<FullPivHouseholderQR, Rhs>
- solve(const MatrixBase<Rhs>& b) const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return internal::solve_retval<FullPivHouseholderQR, Rhs>(*this, b.derived());
- }
-
- /** \returns Expression object representing the matrix Q
- */
- MatrixQReturnType matrixQ(void) const;
-
- /** \returns a reference to the matrix where the Householder QR decomposition is stored
- */
- const MatrixType& matrixQR() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return m_qr;
- }
-
- FullPivHouseholderQR& compute(const MatrixType& matrix);
-
- /** \returns a const reference to the column permutation matrix */
- const PermutationType& colsPermutation() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return m_cols_permutation;
- }
-
- /** \returns a const reference to the vector of indices representing the rows transpositions */
- const IntDiagSizeVectorType& rowsTranspositions() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return m_rows_transpositions;
- }
-
- /** \returns the absolute value of the determinant of the matrix of which
- * *this is the QR decomposition. It has only linear complexity
- * (that is, O(n) where n is the dimension of the square matrix)
- * as the QR decomposition has already been computed.
- *
- * \note This is only for square matrices.
- *
- * \warning a determinant can be very big or small, so for matrices
- * of large enough dimension, there is a risk of overflow/underflow.
- * One way to work around that is to use logAbsDeterminant() instead.
- *
- * \sa logAbsDeterminant(), MatrixBase::determinant()
- */
- typename MatrixType::RealScalar absDeterminant() const;
-
- /** \returns the natural log of the absolute value of the determinant of the matrix of which
- * *this is the QR decomposition. It has only linear complexity
- * (that is, O(n) where n is the dimension of the square matrix)
- * as the QR decomposition has already been computed.
- *
- * \note This is only for square matrices.
- *
- * \note This method is useful to work around the risk of overflow/underflow that's inherent
- * to determinant computation.
- *
- * \sa absDeterminant(), MatrixBase::determinant()
- */
- typename MatrixType::RealScalar logAbsDeterminant() const;
-
- /** \returns the rank of the matrix of which *this is the QR decomposition.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline Index rank() const
- {
- using std::abs;
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
- Index result = 0;
- for(Index i = 0; i < m_nonzero_pivots; ++i)
- result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
- return result;
- }
-
- /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline Index dimensionOfKernel() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return cols() - rank();
- }
-
- /** \returns true if the matrix of which *this is the QR decomposition represents an injective
- * linear map, i.e. has trivial kernel; false otherwise.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline bool isInjective() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return rank() == cols();
- }
-
- /** \returns true if the matrix of which *this is the QR decomposition represents a surjective
- * linear map; false otherwise.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline bool isSurjective() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return rank() == rows();
- }
-
- /** \returns true if the matrix of which *this is the QR decomposition is invertible.
- *
- * \note This method has to determine which pivots should be considered nonzero.
- * For that, it uses the threshold value that you can control by calling
- * setThreshold(const RealScalar&).
- */
- inline bool isInvertible() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return isInjective() && isSurjective();
- }
-
- /** \returns the inverse of the matrix of which *this is the QR decomposition.
- *
- * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
- * Use isInvertible() to first determine whether this matrix is invertible.
- */ inline const
- internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType>
- inverse() const
- {
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return internal::solve_retval<FullPivHouseholderQR,typename MatrixType::IdentityReturnType>
- (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
- }
-
- inline Index rows() const { return m_qr.rows(); }
- inline Index cols() const { return m_qr.cols(); }
-
- /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
- *
- * For advanced uses only.
- */
- const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
-
- /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
- * who need to determine when pivots are to be considered nonzero. This is not used for the
- * QR decomposition itself.
- *
- * When it needs to get the threshold value, Eigen calls threshold(). By default, this
- * uses a formula to automatically determine a reasonable threshold.
- * Once you have called the present method setThreshold(const RealScalar&),
- * your value is used instead.
- *
- * \param threshold The new value to use as the threshold.
- *
- * A pivot will be considered nonzero if its absolute value is strictly greater than
- * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
- * where maxpivot is the biggest pivot.
- *
- * If you want to come back to the default behavior, call setThreshold(Default_t)
- */
- FullPivHouseholderQR& setThreshold(const RealScalar& threshold)
- {
- m_usePrescribedThreshold = true;
- m_prescribedThreshold = threshold;
- return *this;
- }
-
- /** Allows to come back to the default behavior, letting Eigen use its default formula for
- * determining the threshold.
- *
- * You should pass the special object Eigen::Default as parameter here.
- * \code qr.setThreshold(Eigen::Default); \endcode
- *
- * See the documentation of setThreshold(const RealScalar&).
- */
- FullPivHouseholderQR& setThreshold(Default_t)
- {
- m_usePrescribedThreshold = false;
- return *this;
- }
-
- /** Returns the threshold that will be used by certain methods such as rank().
- *
- * See the documentation of setThreshold(const RealScalar&).
- */
- RealScalar threshold() const
- {
- eigen_assert(m_isInitialized || m_usePrescribedThreshold);
- return m_usePrescribedThreshold ? m_prescribedThreshold
- // this formula comes from experimenting (see "LU precision tuning" thread on the list)
- // and turns out to be identical to Higham's formula used already in LDLt.
- : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
- }
-
- /** \returns the number of nonzero pivots in the QR decomposition.
- * Here nonzero is meant in the exact sense, not in a fuzzy sense.
- * So that notion isn't really intrinsically interesting, but it is
- * still useful when implementing algorithms.
- *
- * \sa rank()
- */
- inline Index nonzeroPivots() const
- {
- eigen_assert(m_isInitialized && "LU is not initialized.");
- return m_nonzero_pivots;
- }
-
- /** \returns the absolute value of the biggest pivot, i.e. the biggest
- * diagonal coefficient of U.
- */
- RealScalar maxPivot() const { return m_maxpivot; }
-
- protected:
- MatrixType m_qr;
- HCoeffsType m_hCoeffs;
- IntDiagSizeVectorType m_rows_transpositions;
- IntDiagSizeVectorType m_cols_transpositions;
- PermutationType m_cols_permutation;
- RowVectorType m_temp;
- bool m_isInitialized, m_usePrescribedThreshold;
- RealScalar m_prescribedThreshold, m_maxpivot;
- Index m_nonzero_pivots;
- RealScalar m_precision;
- Index m_det_pq;
-};
-
-template<typename MatrixType>
-typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::absDeterminant() const
-{
- using std::abs;
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
- return abs(m_qr.diagonal().prod());
-}
-
-template<typename MatrixType>
-typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::logAbsDeterminant() const
-{
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
- return m_qr.diagonal().cwiseAbs().array().log().sum();
-}
-
-/** Performs the QR factorization of the given matrix \a matrix. The result of
- * the factorization is stored into \c *this, and a reference to \c *this
- * is returned.
- *
- * \sa class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)
- */
-template<typename MatrixType>
-FullPivHouseholderQR<MatrixType>& FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
-{
- using std::abs;
- Index rows = matrix.rows();
- Index cols = matrix.cols();
- Index size = (std::min)(rows,cols);
-
- m_qr = matrix;
- m_hCoeffs.resize(size);
-
- m_temp.resize(cols);
-
- m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);
-
- m_rows_transpositions.resize(size);
- m_cols_transpositions.resize(size);
- Index number_of_transpositions = 0;
-
- RealScalar biggest(0);
-
- m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
- m_maxpivot = RealScalar(0);
-
- for (Index k = 0; k < size; ++k)
- {
- Index row_of_biggest_in_corner, col_of_biggest_in_corner;
- RealScalar biggest_in_corner;
-
- biggest_in_corner = m_qr.bottomRightCorner(rows-k, cols-k)
- .cwiseAbs()
- .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
- row_of_biggest_in_corner += k;
- col_of_biggest_in_corner += k;
- if(k==0) biggest = biggest_in_corner;
-
- // if the corner is negligible, then we have less than full rank, and we can finish early
- if(internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
- {
- m_nonzero_pivots = k;
- for(Index i = k; i < size; i++)
- {
- m_rows_transpositions.coeffRef(i) = i;
- m_cols_transpositions.coeffRef(i) = i;
- m_hCoeffs.coeffRef(i) = Scalar(0);
- }
- break;
- }
-
- m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
- m_cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
- if(k != row_of_biggest_in_corner) {
- m_qr.row(k).tail(cols-k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols-k));
- ++number_of_transpositions;
- }
- if(k != col_of_biggest_in_corner) {
- m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
- ++number_of_transpositions;
- }
-
- RealScalar beta;
- m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
- m_qr.coeffRef(k,k) = beta;
-
- // remember the maximum absolute value of diagonal coefficients
- if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
-
- 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));
- }
-
- m_cols_permutation.setIdentity(cols);
- for(Index k = 0; k < size; ++k)
- m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.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<FullPivHouseholderQR<_MatrixType>, Rhs>
- : solve_retval_base<FullPivHouseholderQR<_MatrixType>, Rhs>
-{
- EIGEN_MAKE_SOLVE_HELPERS(FullPivHouseholderQR<_MatrixType>,Rhs)
-
- template<typename Dest> void evalTo(Dest& dst) const
- {
- const Index rows = dec().rows(), cols = dec().cols();
- eigen_assert(rhs().rows() == rows);
-
- // FIXME introduce nonzeroPivots() and use it here. and more generally,
- // make the same improvements in this dec as in FullPivLU.
- if(dec().rank()==0)
- {
- dst.setZero();
- return;
- }
-
- typename Rhs::PlainObject c(rhs());
-
- Matrix<Scalar,1,Rhs::ColsAtCompileTime> temp(rhs().cols());
- for (Index k = 0; k < dec().rank(); ++k)
- {
- Index remainingSize = rows-k;
- c.row(k).swap(c.row(dec().rowsTranspositions().coeff(k)));
- c.bottomRightCorner(remainingSize, rhs().cols())
- .applyHouseholderOnTheLeft(dec().matrixQR().col(k).tail(remainingSize-1),
- dec().hCoeffs().coeff(k), &temp.coeffRef(0));
- }
-
- dec().matrixQR()
- .topLeftCorner(dec().rank(), dec().rank())
- .template triangularView<Upper>()
- .solveInPlace(c.topRows(dec().rank()));
-
- for(Index i = 0; i < dec().rank(); ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
- for(Index i = dec().rank(); i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
- }
-};
-
-/** \ingroup QR_Module
- *
- * \brief Expression type for return value of FullPivHouseholderQR::matrixQ()
- *
- * \tparam MatrixType type of underlying dense matrix
- */
-template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType
- : public ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
-{
-public:
- typedef typename MatrixType::Index Index;
- typedef typename FullPivHouseholderQR<MatrixType>::IntDiagSizeVectorType IntDiagSizeVectorType;
- typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
- typedef Matrix<typename MatrixType::Scalar, 1, MatrixType::RowsAtCompileTime, RowMajor, 1,
- MatrixType::MaxRowsAtCompileTime> WorkVectorType;
-
- FullPivHouseholderQRMatrixQReturnType(const MatrixType& qr,
- const HCoeffsType& hCoeffs,
- const IntDiagSizeVectorType& rowsTranspositions)
- : m_qr(qr),
- m_hCoeffs(hCoeffs),
- m_rowsTranspositions(rowsTranspositions)
- {}
-
- template <typename ResultType>
- void evalTo(ResultType& result) const
- {
- const Index rows = m_qr.rows();
- WorkVectorType workspace(rows);
- evalTo(result, workspace);
- }
-
- template <typename ResultType>
- void evalTo(ResultType& result, WorkVectorType& workspace) const
- {
- using numext::conj;
- // compute the product H'_0 H'_1 ... H'_n-1,
- // where H_k is the k-th Householder transformation I - h_k v_k v_k'
- // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
- const Index rows = m_qr.rows();
- const Index cols = m_qr.cols();
- const Index size = (std::min)(rows, cols);
- workspace.resize(rows);
- result.setIdentity(rows, rows);
- for (Index k = size-1; k >= 0; k--)
- {
- result.block(k, k, rows-k, rows-k)
- .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), conj(m_hCoeffs.coeff(k)), &workspace.coeffRef(k));
- result.row(k).swap(result.row(m_rowsTranspositions.coeff(k)));
- }
- }
-
- Index rows() const { return m_qr.rows(); }
- Index cols() const { return m_qr.rows(); }
-
-protected:
- typename MatrixType::Nested m_qr;
- typename HCoeffsType::Nested m_hCoeffs;
- typename IntDiagSizeVectorType::Nested m_rowsTranspositions;
-};
-
-} // end namespace internal
-
-template<typename MatrixType>
-inline typename FullPivHouseholderQR<MatrixType>::MatrixQReturnType FullPivHouseholderQR<MatrixType>::matrixQ() const
-{
- eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
- return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
-}
-
-#ifndef __CUDACC__
-/** \return the full-pivoting Householder QR decomposition of \c *this.
- *
- * \sa class FullPivHouseholderQR
- */
-template<typename Derived>
-const FullPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
-MatrixBase<Derived>::fullPivHouseholderQr() const
-{
- return FullPivHouseholderQR<PlainObject>(eval());
-}
-#endif // __CUDACC__
-
-} // end namespace Eigen
-
-#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H