ColPivHouseholderQR.h
943cc0ac196clammm-devel

ColPivHouseholderQR.h
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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_COLPIVOTINGHOUSEHOLDERQR_H
			#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

			namespace Eigen {

			/** \ingroup QR_Module
			*
			* \class ColPivHouseholderQR
			*
			* \brief Householder rank-revealing QR decomposition of a matrix with column-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 column pivoting in order to be rank-revealing and improve
			* numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
			*
			* \sa MatrixBase::colPivHouseholderQr()
			*/
			template<typename _MatrixType> class ColPivHouseholderQR
			{
			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 PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
			typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
			typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
			typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
			typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;

			private:

			typedef typename PermutationType::Index PermIndexType;

			public:

			/**
			* \brief Default Constructor.
			*
			* The default constructor is useful in cases in which the user intends to
			* perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
			*/
			ColPivHouseholderQR()
			: m_qr(),
			m_hCoeffs(),
			m_colsPermutation(),
			m_colsTranspositions(),
			m_temp(),
			m_colSqNorms(),
			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 ColPivHouseholderQR()
			*/
			ColPivHouseholderQR(Index rows, Index cols)
			: m_qr(rows, cols),
			m_hCoeffs((std::min)(rows,cols)),
			m_colsPermutation(PermIndexType(cols)),
			m_colsTranspositions(cols),
			m_temp(cols),
			m_colSqNorms(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
			* ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
			* qr.compute(matrix);
			* \endcode
			*
			* \sa compute()
			*/
			ColPivHouseholderQR(const MatrixType& matrix)
			: m_qr(matrix.rows(), matrix.cols()),
			m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
			m_colsPermutation(PermIndexType(matrix.cols())),
			m_colsTranspositions(matrix.cols()),
			m_temp(matrix.cols()),
			m_colSqNorms(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 ColPivHouseholderQR_solve.cpp
			* Output: \verbinclude ColPivHouseholderQR_solve.out
			*/
			template<typename Rhs>
			inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
			solve(const MatrixBase<Rhs>& b) const
			{
			eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
			return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
			}

			HouseholderSequenceType householderQ(void) const;
			HouseholderSequenceType matrixQ(void) const
			{
			return householderQ();
			}

			/** \returns a reference to the matrix where the Householder QR decomposition is stored
			*/
			const MatrixType& matrixQR() const
			{
			eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
			return m_qr;
			}

			/** \returns a reference to the matrix where the result Householder QR is stored
			* \warning The strict lower part of this matrix contains internal values.
			* Only the upper triangular part should be referenced. To get it, use
			* \code matrixR().template triangularView<Upper>() \endcode
			* For rank-deficient matrices, use
			* \code
			* matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
			* \endcode
			*/
			const MatrixType& matrixR() const
			{
			eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
			return m_qr;
			}

			ColPivHouseholderQR& compute(const MatrixType& matrix);

			/** \returns a const reference to the column permutation matrix */
			const PermutationType& colsPermutation() const
			{
			eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
			return m_colsPermutation;
			}

			/** \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 && "ColPivHouseholderQR 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 && "ColPivHouseholderQR 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 && "ColPivHouseholderQR 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 && "ColPivHouseholderQR 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 && "ColPivHouseholderQR 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<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
			inverse() const
			{
			eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
			return internal::solve_retval<ColPivHouseholderQR,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)
			*/
			ColPivHouseholderQR& 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&).
			*/
			ColPivHouseholderQR& 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 && "ColPivHouseholderQR is not initialized.");
			return m_nonzero_pivots;
			}

			/** \returns the absolute value of the biggest pivot, i.e. the biggest
			* diagonal coefficient of R.
			*/
			RealScalar maxPivot() const { return m_maxpivot; }

			/** \brief Reports whether the QR factorization was succesful.
			*
			* \note This function always returns \c Success. It is provided for compatibility
			* with other factorization routines.
			* \returns \c Success
			*/
			ComputationInfo info() const
			{
			eigen_assert(m_isInitialized && "Decomposition is not initialized.");
			return Success;
			}

			protected:

			static void check_template_parameters()
			{
			EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
			}

			MatrixType m_qr;
			HCoeffsType m_hCoeffs;
			PermutationType m_colsPermutation;
			IntRowVectorType m_colsTranspositions;
			RowVectorType m_temp;
			RealRowVectorType m_colSqNorms;
			bool m_isInitialized, m_usePrescribedThreshold;
			RealScalar m_prescribedThreshold, m_maxpivot;
			Index m_nonzero_pivots;
			Index m_det_pq;
			};

			template<typename MatrixType>
			typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
			{
			using std::abs;
			eigen_assert(m_isInitialized && "ColPivHouseholderQR 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 ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
			{
			eigen_assert(m_isInitialized && "ColPivHouseholderQR 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 ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
			*/
			template<typename MatrixType>
			ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
			{
			check_template_parameters();

			using std::abs;
			Index rows = matrix.rows();
			Index cols = matrix.cols();
			Index size = matrix.diagonalSize();

			// the column permutation is stored as int indices, so just to be sure:
			eigen_assert(cols<=NumTraits<int>::highest());

			m_qr = matrix;
			m_hCoeffs.resize(size);

			m_temp.resize(cols);

			m_colsTranspositions.resize(matrix.cols());
			Index number_of_transpositions = 0;

			m_colSqNorms.resize(cols);
			for(Index k = 0; k < cols; ++k)
			m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();

			RealScalar threshold_helper = m_colSqNorms.maxCoeff() * numext::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);

			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)
			{
			// first, we look up in our table m_colSqNorms which column has the biggest squared norm
			Index biggest_col_index;
			RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
			biggest_col_index += k;

			// since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
			// the actual squared norm of the selected column.
			// Note that not doing so does result in solve() sometimes returning inf/nan values
			// when running the unit test with 1000 repetitions.
			biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();

			// we store that back into our table: it can't hurt to correct our table.
			m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;

			// Track the number of meaningful pivots but do not stop the decomposition to make
			// sure that the initial matrix is properly reproduced. See bug 941.
			if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
			m_nonzero_pivots = k;

			// apply the transposition to the columns
			m_colsTranspositions.coeffRef(k) = biggest_col_index;
			if(k != biggest_col_index) {
			m_qr.col(k).swap(m_qr.col(biggest_col_index));
			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

lammps/lib/smd/Eigen/src/QR/ColPivHouseholderQR.h943cc0ac196clammm-devel

ColPivHouseholderQR.hView Options

lammps/lib/smd/Eigen/src/QR/ColPivHouseholderQR.h
943cc0ac196clammm-devel

ColPivHouseholderQR.h
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