FullPivHouseholderQR.h
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Created: Sun, Nov 17, 22:24

FullPivHouseholderQR.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_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 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 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:

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

	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)
	{
	check_template_parameters();

	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);
	}

	/** \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());
	}

	} // end namespace Eigen

	#endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H

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