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diff --git a/extern/Eigen3/Eigen/src/QR/FullPivHouseholderQR.h b/extern/Eigen3/Eigen/src/QR/FullPivHouseholderQR.h new file mode 100644 index 00000000000..dde3013be9d --- /dev/null +++ b/extern/Eigen3/Eigen/src/QR/FullPivHouseholderQR.h @@ -0,0 +1,546 @@ +// 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> +// +// Eigen is free software; you can redistribute it and/or +// modify it under the terms of the GNU Lesser General Public +// License as published by the Free Software Foundation; either +// version 3 of the License, or (at your option) any later version. +// +// Alternatively, you can redistribute it and/or +// modify it under the terms of the GNU General Public License as +// published by the Free Software Foundation; either version 2 of +// the License, or (at your option) any later version. +// +// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY +// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS +// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the +// GNU General Public License for more details. +// +// You should have received a copy of the GNU Lesser General Public +// License and a copy of the GNU General Public License along with +// Eigen. If not, see <http://www.gnu.org/licenses/>. + +#ifndef EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H +#define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H + +/** \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 Q and \b R + * such that + * \f[ + * \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R} + * \f] + * by using Householder transformations. Here, \b P is a permutation matrix, \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 Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType; + typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType; + typedef Matrix<Index, 1, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime> IntRowVectorType; + typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType; + typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType; + 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(rows), + m_cols_transpositions(cols), + m_cols_permutation(cols), + m_temp((std::min)(rows,cols)), + m_isInitialized(false), + m_usePrescribedThreshold(false) {} + + FullPivHouseholderQR(const MatrixType& matrix) + : m_qr(matrix.rows(), matrix.cols()), + m_hCoeffs((std::min)(matrix.rows(), matrix.cols())), + m_rows_transpositions(matrix.rows()), + m_cols_transpositions(matrix.cols()), + m_cols_permutation(matrix.cols()), + m_temp((std::min)(matrix.rows(), 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 + * *this is the QR decomposition, if any exists. + * + * \param b the right-hand-side of the equation to solve. + * + * \returns a solution. + * + * \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()); + } + + MatrixQType 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); + + const PermutationType& colsPermutation() const + { + eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized."); + return m_cols_permutation; + } + + const IntColVectorType& 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 + { + eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized."); + RealScalar premultiplied_threshold = internal::abs(m_maxpivot) * threshold(); + Index result = 0; + for(Index i = 0; i < m_nonzero_pivots; ++i) + result += (internal::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(); } + 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() * 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; + IntColVectorType m_rows_transpositions; + IntRowVectorType 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 +{ + 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 internal::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(); +} + +template<typename MatrixType> +FullPivHouseholderQR<MatrixType>& FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix) +{ + 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() * size; + + m_rows_transpositions.resize(matrix.rows()); + m_cols_transpositions.resize(matrix.cols()); + 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(internal::abs(beta) > m_maxpivot) m_maxpivot = internal::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)); + } + + if(!dec().isSurjective()) + { + // is c is in the image of R ? + RealScalar biggest_in_upper_part_of_c = c.topRows( dec().rank() ).cwiseAbs().maxCoeff(); + RealScalar biggest_in_lower_part_of_c = c.bottomRows(rows-dec().rank()).cwiseAbs().maxCoeff(); + // FIXME brain dead + const RealScalar m_precision = NumTraits<Scalar>::epsilon() * (std::min)(rows,cols); + // this internal:: prefix is needed by at least gcc 3.4 and ICC + if(!internal::isMuchSmallerThan(biggest_in_lower_part_of_c, biggest_in_upper_part_of_c, m_precision)) + return; + } + 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(); + } +}; + +} // end namespace internal + +/** \returns the matrix Q */ +template<typename MatrixType> +typename FullPivHouseholderQR<MatrixType>::MatrixQType FullPivHouseholderQR<MatrixType>::matrixQ() const +{ + eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized."); + // 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), ...] + Index rows = m_qr.rows(); + Index cols = m_qr.cols(); + Index size = (std::min)(rows,cols); + MatrixQType res = MatrixQType::Identity(rows, rows); + Matrix<Scalar,1,MatrixType::RowsAtCompileTime> temp(rows); + for (Index k = size-1; k >= 0; k--) + { + res.block(k, k, rows-k, rows-k) + .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), internal::conj(m_hCoeffs.coeff(k)), &temp.coeffRef(k)); + res.row(k).swap(res.row(m_rows_transpositions.coeff(k))); + } + return res; +} + +/** \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 // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H |