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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
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
namespace Eigen {
namespace internal {
template <typename MatrixType_>
struct traits<HouseholderQR<MatrixType_>> : traits<MatrixType_> {
typedef MatrixXpr XprKind;
typedef SolverStorage StorageKind;
typedef int StorageIndex;
enum { Flags = 0 };
};
} // end namespace internal
/** \ingroup QR_Module
*
*
* \class HouseholderQR
*
* \brief Householder QR decomposition of a matrix
*
* \tparam 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.
*
* This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
*
* \sa MatrixBase::householderQr()
*/
template <typename MatrixType_>
class HouseholderQR : public SolverBase<HouseholderQR<MatrixType_>> {
public:
typedef MatrixType_ MatrixType;
typedef SolverBase<HouseholderQR> Base;
friend class SolverBase<HouseholderQR>;
EIGEN_GENERIC_PUBLIC_INTERFACE(HouseholderQR)
enum {
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
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, internal::remove_all_t<typename HCoeffsType::ConjugateReturnType>>
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()
*/
template <typename InputType>
explicit HouseholderQR(const EigenBase<InputType>& matrix)
: m_qr(matrix.rows(), matrix.cols()),
m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
m_temp(matrix.cols()),
m_isInitialized(false) {
compute(matrix.derived());
}
/** \brief Constructs a QR factorization from a given matrix
*
* This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when
* \c MatrixType is a Eigen::Ref.
*
* \sa HouseholderQR(const EigenBase&)
*/
template <typename InputType>
explicit HouseholderQR(EigenBase<InputType>& matrix)
: m_qr(matrix.derived()),
m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
m_temp(matrix.cols()),
m_isInitialized(false) {
computeInPlace();
}
#ifdef EIGEN_PARSED_BY_DOXYGEN
/** 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_about_checking_solutions
*
* \note_about_arbitrary_choice_of_solution
*
* Example: \include HouseholderQR_solve.cpp
* Output: \verbinclude HouseholderQR_solve.out
*/
template <typename Rhs>
inline const Solve<HouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif
/** 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;
}
template <typename InputType>
HouseholderQR& compute(const EigenBase<InputType>& matrix) {
m_qr = matrix.derived();
computeInPlace();
return *this;
}
/** \returns 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.
* Also, do not rely on the determinant being exactly zero for testing
* singularity or rank-deficiency.
*
* \sa absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
*/
typename MatrixType::Scalar determinant() const;
/** \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.
* Also, do not rely on the determinant being exactly zero for testing
* singularity or rank-deficiency.
*
* \sa determinant(), 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.
*
* \warning Do not rely on the determinant being exactly zero for testing
* singularity or rank-deficiency.
*
* \sa determinant(), absDeterminant(), MatrixBase::determinant()
*/
typename MatrixType::RealScalar logAbsDeterminant() const;
/** \returns the sign 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.
*
* \warning Do not rely on the determinant being exactly zero for testing
* singularity or rank-deficiency.
*
* \sa determinant(), absDeterminant(), MatrixBase::determinant()
*/
typename MatrixType::Scalar signDeterminant() 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; }
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename RhsType, typename DstType>
void _solve_impl(const RhsType& rhs, DstType& dst) const;
template <bool Conjugate, typename RhsType, typename DstType>
void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const;
#endif
protected:
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
void computeInPlace();
MatrixType m_qr;
HCoeffsType m_hCoeffs;
RowVectorType m_temp;
bool m_isInitialized;
};
namespace internal {
/** \internal */
template <typename HCoeffs, typename Scalar, bool IsComplex>
struct householder_determinant {
static void run(const HCoeffs& hCoeffs, Scalar& out_det) {
out_det = Scalar(1);
Index size = hCoeffs.rows();
for (Index i = 0; i < size; i++) {
// For each valid reflection Q_n,
// det(Q_n) = - conj(h_n) / h_n
// where h_n is the Householder coefficient.
if (hCoeffs(i) != Scalar(0)) out_det *= -numext::conj(hCoeffs(i)) / hCoeffs(i);
}
}
};
/** \internal */
template <typename HCoeffs, typename Scalar>
struct householder_determinant<HCoeffs, Scalar, false> {
static void run(const HCoeffs& hCoeffs, Scalar& out_det) {
bool negated = false;
Index size = hCoeffs.rows();
for (Index i = 0; i < size; i++) {
// Each valid reflection negates the determinant.
if (hCoeffs(i) != Scalar(0)) negated ^= true;
}
out_det = negated ? Scalar(-1) : Scalar(1);
}
};
} // end namespace internal
template <typename MatrixType>
typename MatrixType::Scalar HouseholderQR<MatrixType>::determinant() 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!");
Scalar detQ;
internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
return m_qr.diagonal().prod() * detQ;
}
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();
}
template <typename MatrixType>
typename MatrixType::Scalar HouseholderQR<MatrixType>::signDeterminant() 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!");
Scalar detQ;
internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
return detQ * m_qr.diagonal().array().sign().prod();
}
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::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);
}
}
// TODO: add a corresponding public API for updating a QR factorization
/** \internal
* Basically a modified copy of @c Eigen::internal::householder_qr_inplace_unblocked that
* performs a rank-1 update of the QR matrix in compact storage. This function assumes, that
* the first @c k-1 columns of the matrix @c mat contain the QR decomposition of \f$A^N\f$ up to
* column k-1. Then the QR decomposition of the k-th column (given by @c newColumn) is computed by
* applying the k-1 Householder projectors on it and finally compute the projector \f$H_k\f$ of
* it. On exit the matrix @c mat and the vector @c hCoeffs contain the QR decomposition of the
* first k columns of \f$A^N\f$. The \a tempData argument must point to at least mat.cols() scalars. */
template <typename MatrixQR, typename HCoeffs, typename VectorQR>
void householder_qr_inplace_update(MatrixQR& mat, HCoeffs& hCoeffs, const VectorQR& newColumn,
typename MatrixQR::Index k, typename MatrixQR::Scalar* tempData) {
typedef typename MatrixQR::Index Index;
typedef typename MatrixQR::RealScalar RealScalar;
Index rows = mat.rows();
eigen_assert(k < mat.cols());
eigen_assert(k < rows);
eigen_assert(hCoeffs.size() == mat.cols());
eigen_assert(newColumn.size() == rows);
eigen_assert(tempData);
// Store new column in mat at column k
mat.col(k) = newColumn;
// Apply H = H_1...H_{k-1} on newColumn (skip if k=0)
for (Index i = 0; i < k; ++i) {
Index remainingRows = rows - i;
mat.col(k)
.tail(remainingRows)
.applyHouseholderOnTheLeft(mat.col(i).tail(remainingRows - 1), hCoeffs.coeffRef(i), tempData + i + 1);
}
// Construct Householder projector in-place in column k
RealScalar beta;
mat.col(k).tail(rows - k).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
mat.coeffRef(k, k) = beta;
}
/** \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 LAPACK-supported Scalar types in HouseholderQR_LAPACKE.h
static void run(MatrixQR& mat, HCoeffs& hCoeffs, Index maxBlockSize = 32, typename MatrixQR::Scalar* tempData = 0) {
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, false); // false == backward
}
}
}
};
} // end namespace internal
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename MatrixType_>
template <typename RhsType, typename DstType>
void HouseholderQR<MatrixType_>::_solve_impl(const RhsType& rhs, DstType& dst) const {
const Index rank = (std::min)(rows(), cols());
typename RhsType::PlainObject c(rhs);
c.applyOnTheLeft(householderQ().setLength(rank).adjoint());
m_qr.topLeftCorner(rank, rank).template triangularView<Upper>().solveInPlace(c.topRows(rank));
dst.topRows(rank) = c.topRows(rank);
dst.bottomRows(cols() - rank).setZero();
}
template <typename MatrixType_>
template <bool Conjugate, typename RhsType, typename DstType>
void HouseholderQR<MatrixType_>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const {
const Index rank = (std::min)(rows(), cols());
typename RhsType::PlainObject c(rhs);
m_qr.topLeftCorner(rank, rank)
.template triangularView<Upper>()
.transpose()
.template conjugateIf<Conjugate>()
.solveInPlace(c.topRows(rank));
dst.topRows(rank) = c.topRows(rank);
dst.bottomRows(rows() - rank).setZero();
dst.applyOnTheLeft(householderQ().setLength(rank).template conjugateIf<!Conjugate>());
}
#endif
/** 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>
void HouseholderQR<MatrixType>::computeInPlace() {
Index rows = m_qr.rows();
Index cols = m_qr.cols();
Index size = (std::min)(rows, cols);
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 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