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lammps/lib/smd/Eigen/src/QR/HouseholderQR.h
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HouseholderQR.h
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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2010 Vincent Lejeune
//
// 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_QR_H
#define EIGEN_QR_H
namespace Eigen {
/** \ingroup QR_Module
*
*
* \class HouseholderQR
*
* \brief Householder QR decomposition of a matrix
*
* \param MatrixType the type of the matrix of which we are computing the QR decomposition
*
* This class performs a QR decomposition of a matrix \b A into matrices \b Q and \b R
* such that
* \f[
* \mathbf{A} = \mathbf{Q} \, \mathbf{R}
* \f]
* by using Householder transformations. Here, \b Q a unitary matrix and \b R an upper triangular matrix.
* The result is stored in a compact way compatible with LAPACK.
*
* Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
* If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.
*
* This Householder QR decomposition is faster, but less numerically stable and less feature-full than
* FullPivHouseholderQR or ColPivHouseholderQR.
*
* \sa MatrixBase::householderQr()
*/
template<typename _MatrixType> class HouseholderQR
{
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, (MatrixType::Flags&RowMajorBit) ? RowMajor : ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via HouseholderQR::compute(const MatrixType&).
*/
HouseholderQR() : m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(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 HouseholderQR()
*/
HouseholderQR(Index rows, Index cols)
: m_qr(rows, cols),
m_hCoeffs((std::min)(rows,cols)),
m_temp(cols),
m_isInitialized(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
* HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
* qr.compute(matrix);
* \endcode
*
* \sa compute()
*/
HouseholderQR(const MatrixType& matrix)
: m_qr(matrix.rows(), matrix.cols()),
m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
m_temp(matrix.cols()),
m_isInitialized(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 HouseholderQR_solve.cpp
* Output: \verbinclude HouseholderQR_solve.out
*/
template<typename Rhs>
inline const internal::solve_retval<HouseholderQR, Rhs>
solve(const MatrixBase<Rhs>& b) const
{
eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
return internal::solve_retval<HouseholderQR, Rhs>(*this, b.derived());
}
/** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
*
* The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object.
* Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:
*
* Example: \include HouseholderQR_householderQ.cpp
* Output: \verbinclude HouseholderQR_householderQ.out
*/
HouseholderSequenceType householderQ() const
{
eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
}
/** \returns a reference to the matrix where the Householder QR decomposition is stored
* in a LAPACK-compatible way.
*/
const MatrixType& matrixQR() const
{
eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
return m_qr;
}
HouseholderQR& compute(const MatrixType& matrix);
/** \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;
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; }
protected:
static void check_template_parameters()
{
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
}
MatrixType m_qr;
HCoeffsType m_hCoeffs;
RowVectorType m_temp;
bool m_isInitialized;
};
template<typename MatrixType>
typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant() const
{
using std::abs;
eigen_assert(m_isInitialized && "HouseholderQR 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 HouseholderQR<MatrixType>::logAbsDeterminant() const
{
eigen_assert(m_isInitialized && "HouseholderQR 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();
}
namespace internal {
/** \internal */
template<typename MatrixQR, typename HCoeffs>
void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
{
typedef typename MatrixQR::Index Index;
typedef typename MatrixQR::Scalar Scalar;
typedef typename MatrixQR::RealScalar RealScalar;
Index rows = mat.rows();
Index cols = mat.cols();
Index size = (std::min)(rows,cols);
eigen_assert(hCoeffs.size() == size);
typedef Matrix<Scalar,MatrixQR::ColsAtCompileTime,1> TempType;
TempType tempVector;
if(tempData==0)
{
tempVector.resize(cols);
tempData = tempVector.data();
}
for(Index k = 0; k < size; ++k)
{
Index remainingRows = rows - k;
Index remainingCols = cols - k - 1;
RealScalar beta;
mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
mat.coeffRef(k,k) = beta;
// apply H to remaining part of m_qr from the left
mat.bottomRightCorner(remainingRows, remainingCols)
.applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), hCoeffs.coeffRef(k), tempData+k+1);
}
}
/** \internal */
template<typename MatrixQR, typename HCoeffs,
typename MatrixQRScalar = typename MatrixQR::Scalar,
bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
struct householder_qr_inplace_blocked
{
// This is specialized for MKL-supported Scalar types in HouseholderQR_MKL.h
static void run(MatrixQR& mat, HCoeffs& hCoeffs,
typename MatrixQR::Index maxBlockSize=32,
typename MatrixQR::Scalar* tempData = 0)
{
typedef typename MatrixQR::Index Index;
typedef typename MatrixQR::Scalar Scalar;
typedef Block<MatrixQR,Dynamic,Dynamic> BlockType;
Index rows = mat.rows();
Index cols = mat.cols();
Index size = (std::min)(rows, cols);
typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixQR::MaxColsAtCompileTime,1> TempType;
TempType tempVector;
if(tempData==0)
{
tempVector.resize(cols);
tempData = tempVector.data();
}
Index blockSize = (std::min)(maxBlockSize,size);
Index k = 0;
for (k = 0; k < size; k += blockSize)
{
Index bs = (std::min)(size-k,blockSize); // actual size of the block
Index tcols = cols - k - bs; // trailing columns
Index brows = rows-k; // rows of the block
// partition the matrix:
// A00 | A01 | A02
// mat = A10 | A11 | A12
// A20 | A21 | A22
// and performs the qr dec of [A11^T A12^T]^T
// and update [A21^T A22^T]^T using level 3 operations.
// Finally, the algorithm continue on A22
BlockType A11_21 = mat.block(k,k,brows,bs);
Block<HCoeffs,Dynamic,1> hCoeffsSegment = hCoeffs.segment(k,bs);
householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);
if(tcols)
{
BlockType A21_22 = mat.block(k,k+bs,brows,tcols);
apply_block_householder_on_the_left(A21_22,A11_21,hCoeffsSegment.adjoint());
}
}
}
};
template<typename _MatrixType, typename Rhs>
struct solve_retval<HouseholderQR<_MatrixType>, Rhs>
: solve_retval_base<HouseholderQR<_MatrixType>, Rhs>
{
EIGEN_MAKE_SOLVE_HELPERS(HouseholderQR<_MatrixType>,Rhs)
template<typename Dest> void evalTo(Dest& dst) const
{
const Index rows = dec().rows(), cols = dec().cols();
const Index rank = (std::min)(rows, cols);
eigen_assert(rhs().rows() == rows);
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().leftCols(rank),
dec().hCoeffs().head(rank)).transpose()
);
dec().matrixQR()
.topLeftCorner(rank, rank)
.template triangularView<Upper>()
.solveInPlace(c.topRows(rank));
dst.topRows(rank) = c.topRows(rank);
dst.bottomRows(cols-rank).setZero();
}
};
} // end namespace internal
/** 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 HouseholderQR, HouseholderQR(const MatrixType&)
*/
template<typename MatrixType>
HouseholderQR<MatrixType>& HouseholderQR<MatrixType>::compute(const MatrixType& matrix)
{
check_template_parameters();
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);
internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());
m_isInitialized = true;
return *this;
}
/** \return the Householder QR decomposition of \c *this.
*
* \sa class HouseholderQR
*/
template<typename Derived>
const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
MatrixBase<Derived>::householderQr() const
{
return HouseholderQR<PlainObject>(eval());
}
} // end namespace Eigen
#endif // EIGEN_QR_H
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