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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
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
namespace Eigen {
namespace internal {
template <typename MatrixType_, typename PermutationIndex_>
struct traits<ColPivHouseholderQR<MatrixType_, PermutationIndex_>> : traits<MatrixType_> {
typedef MatrixXpr XprKind;
typedef SolverStorage StorageKind;
typedef PermutationIndex_ PermutationIndex;
enum { Flags = 0 };
};
} // end namespace internal
/** \ingroup QR_Module
*
* \class ColPivHouseholderQR
*
* \brief Householder rank-revealing QR decomposition of a matrix with column-pivoting
*
* \tparam 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.
*
* This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
*
* \sa MatrixBase::colPivHouseholderQr()
*/
template <typename MatrixType_, typename PermutationIndex_>
class ColPivHouseholderQR : public SolverBase<ColPivHouseholderQR<MatrixType_, PermutationIndex_>> {
public:
typedef MatrixType_ MatrixType;
typedef SolverBase<ColPivHouseholderQR> Base;
friend class SolverBase<ColPivHouseholderQR>;
typedef PermutationIndex_ PermutationIndex;
EIGEN_GENERIC_PUBLIC_INTERFACE(ColPivHouseholderQR)
enum {
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex> PermutationType;
typedef typename internal::plain_row_type<MatrixType, PermutationIndex>::type IntRowVectorType;
typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
typedef HouseholderSequence<MatrixType, internal::remove_all_t<typename HCoeffsType::ConjugateReturnType>>
HouseholderSequenceType;
typedef typename MatrixType::PlainObject PlainObject;
private:
void init(Index rows, Index cols) {
Index diag = numext::mini(rows, cols);
m_hCoeffs.resize(diag);
m_colsPermutation.resize(cols);
m_colsTranspositions.resize(cols);
m_temp.resize(cols);
m_colNormsUpdated.resize(cols);
m_colNormsDirect.resize(cols);
m_isInitialized = false;
m_usePrescribedThreshold = false;
}
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_colNormsUpdated(),
m_colNormsDirect(),
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) { init(rows, cols); }
/** \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()
*/
template <typename InputType>
explicit ColPivHouseholderQR(const EigenBase<InputType>& matrix) : m_qr(matrix.rows(), matrix.cols()) {
init(matrix.rows(), matrix.cols());
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 ColPivHouseholderQR(const EigenBase&)
*/
template <typename InputType>
explicit ColPivHouseholderQR(EigenBase<InputType>& matrix) : m_qr(matrix.derived()) {
init(matrix.rows(), matrix.cols());
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 ColPivHouseholderQR_solve.cpp
* Output: \verbinclude ColPivHouseholderQR_solve.out
*/
template <typename Rhs>
inline const Solve<ColPivHouseholderQR, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif
HouseholderSequenceType householderQ() const;
HouseholderSequenceType matrixQ() 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;
}
template <typename InputType>
ColPivHouseholderQR& compute(const EigenBase<InputType>& 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 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 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.
*
* \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.
*
* \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.
*
* \sa determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
*/
typename MatrixType::Scalar signDeterminant() 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 Inverse<ColPivHouseholderQR> inverse() const {
eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
return Inverse<ColPivHouseholderQR>(*this);
}
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 successful.
*
* \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;
}
#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:
friend class CompleteOrthogonalDecomposition<MatrixType, PermutationIndex>;
EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)
void computeInPlace();
MatrixType m_qr;
HCoeffsType m_hCoeffs;
PermutationType m_colsPermutation;
IntRowVectorType m_colsTranspositions;
RowVectorType m_temp;
RealRowVectorType m_colNormsUpdated;
RealRowVectorType m_colNormsDirect;
bool m_isInitialized, m_usePrescribedThreshold;
RealScalar m_prescribedThreshold, m_maxpivot;
Index m_nonzero_pivots;
Index m_det_p;
};
template <typename MatrixType, typename PermutationIndex>
typename MatrixType::Scalar ColPivHouseholderQR<MatrixType, PermutationIndex>::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 isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().prod() : Scalar(0);
}
template <typename MatrixType, typename PermutationIndex>
typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType, PermutationIndex>::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 isInjective() ? abs(m_qr.diagonal().prod()) : RealScalar(0);
}
template <typename MatrixType, typename PermutationIndex>
typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType, PermutationIndex>::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 isInjective() ? m_qr.diagonal().cwiseAbs().array().log().sum() : -NumTraits<RealScalar>::infinity();
}
template <typename MatrixType, typename PermutationIndex>
typename MatrixType::Scalar ColPivHouseholderQR<MatrixType, PermutationIndex>::signDeterminant() 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!");
Scalar detQ;
internal::householder_determinant<HCoeffsType, Scalar, NumTraits<Scalar>::IsComplex>::run(m_hCoeffs, detQ);
return isInjective() ? (detQ * Scalar(m_det_p)) * m_qr.diagonal().array().sign().prod() : Scalar(0);
}
/** 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, typename PermutationIndex>
template <typename InputType>
ColPivHouseholderQR<MatrixType, PermutationIndex>& ColPivHouseholderQR<MatrixType, PermutationIndex>::compute(
const EigenBase<InputType>& matrix) {
m_qr = matrix.derived();
computeInPlace();
return *this;
}
template <typename MatrixType, typename PermutationIndex>
void ColPivHouseholderQR<MatrixType, PermutationIndex>::computeInPlace() {
eigen_assert(m_qr.cols() <= NumTraits<PermutationIndex>::highest());
using std::abs;
Index rows = m_qr.rows();
Index cols = m_qr.cols();
Index size = m_qr.diagonalSize();
m_hCoeffs.resize(size);
m_temp.resize(cols);
m_colsTranspositions.resize(m_qr.cols());
Index number_of_transpositions = 0;
m_colNormsUpdated.resize(cols);
m_colNormsDirect.resize(cols);
for (Index k = 0; k < cols; ++k) {
// colNormsDirect(k) caches the most recent directly computed norm of
// column k.
m_colNormsDirect.coeffRef(k) = m_qr.col(k).norm();
m_colNormsUpdated.coeffRef(k) = m_colNormsDirect.coeffRef(k);
}
RealScalar threshold_helper =
numext::abs2<RealScalar>(m_colNormsUpdated.maxCoeff() * NumTraits<RealScalar>::epsilon()) / RealScalar(rows);
RealScalar norm_downdate_threshold = numext::sqrt(NumTraits<RealScalar>::epsilon());
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_colNormsUpdated which column has the biggest norm
Index biggest_col_index;
RealScalar biggest_col_sq_norm = numext::abs2(m_colNormsUpdated.tail(cols - k).maxCoeff(&biggest_col_index));
biggest_col_index += k;
// 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) = static_cast<PermutationIndex>(biggest_col_index);
if (k != biggest_col_index) {
m_qr.col(k).swap(m_qr.col(biggest_col_index));
std::swap(m_colNormsUpdated.coeffRef(k), m_colNormsUpdated.coeffRef(biggest_col_index));
std::swap(m_colNormsDirect.coeffRef(k), m_colNormsDirect.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 norms of the columns
for (Index j = k + 1; j < cols; ++j) {
// The following implements the stable norm downgrade step discussed in
// http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
// and used in LAPACK routines xGEQPF and xGEQP3.
// See lines 278-297 in http://www.netlib.org/lapack/explore-html/dc/df4/sgeqpf_8f_source.html
if (!numext::is_exactly_zero(m_colNormsUpdated.coeffRef(j))) {
RealScalar temp = abs(m_qr.coeffRef(k, j)) / m_colNormsUpdated.coeffRef(j);
temp = (RealScalar(1) + temp) * (RealScalar(1) - temp);
temp = temp < RealScalar(0) ? RealScalar(0) : temp;
RealScalar temp2 =
temp * numext::abs2<RealScalar>(m_colNormsUpdated.coeffRef(j) / m_colNormsDirect.coeffRef(j));
if (temp2 <= norm_downdate_threshold) {
// The updated norm has become too inaccurate so re-compute the column
// norm directly.
m_colNormsDirect.coeffRef(j) = m_qr.col(j).tail(rows - k - 1).norm();
m_colNormsUpdated.coeffRef(j) = m_colNormsDirect.coeffRef(j);
} else {
m_colNormsUpdated.coeffRef(j) *= numext::sqrt(temp);
}
}
}
}
m_colsPermutation.setIdentity(cols);
for (Index k = 0; k < size /*m_nonzero_pivots*/; ++k)
m_colsPermutation.applyTranspositionOnTheRight(k, static_cast<Index>(m_colsTranspositions.coeff(k)));
m_det_p = (number_of_transpositions % 2) ? -1 : 1;
m_isInitialized = true;
}
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename MatrixType_, typename PermutationIndex_>
template <typename RhsType, typename DstType>
void ColPivHouseholderQR<MatrixType_, PermutationIndex_>::_solve_impl(const RhsType& rhs, DstType& dst) const {
const Index nonzero_pivots = nonzeroPivots();
if (nonzero_pivots == 0) {
dst.setZero();
return;
}
typename RhsType::PlainObject c(rhs);
c.applyOnTheLeft(householderQ().setLength(nonzero_pivots).adjoint());
m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
.template triangularView<Upper>()
.solveInPlace(c.topRows(nonzero_pivots));
for (Index i = 0; i < nonzero_pivots; ++i) dst.row(m_colsPermutation.indices().coeff(i)) = c.row(i);
for (Index i = nonzero_pivots; i < cols(); ++i) dst.row(m_colsPermutation.indices().coeff(i)).setZero();
}
template <typename MatrixType_, typename PermutationIndex_>
template <bool Conjugate, typename RhsType, typename DstType>
void ColPivHouseholderQR<MatrixType_, PermutationIndex_>::_solve_impl_transposed(const RhsType& rhs,
DstType& dst) const {
const Index nonzero_pivots = nonzeroPivots();
if (nonzero_pivots == 0) {
dst.setZero();
return;
}
typename RhsType::PlainObject c(m_colsPermutation.transpose() * rhs);
m_qr.topLeftCorner(nonzero_pivots, nonzero_pivots)
.template triangularView<Upper>()
.transpose()
.template conjugateIf<Conjugate>()
.solveInPlace(c.topRows(nonzero_pivots));
dst.topRows(nonzero_pivots) = c.topRows(nonzero_pivots);
dst.bottomRows(rows() - nonzero_pivots).setZero();
dst.applyOnTheLeft(householderQ().setLength(nonzero_pivots).template conjugateIf<!Conjugate>());
}
#endif
namespace internal {
template <typename DstXprType, typename MatrixType, typename PermutationIndex>
struct Assignment<DstXprType, Inverse<ColPivHouseholderQR<MatrixType, PermutationIndex>>,
internal::assign_op<typename DstXprType::Scalar,
typename ColPivHouseholderQR<MatrixType, PermutationIndex>::Scalar>,
Dense2Dense> {
typedef ColPivHouseholderQR<MatrixType, PermutationIndex> QrType;
typedef Inverse<QrType> SrcXprType;
static void run(DstXprType& dst, const SrcXprType& src,
const internal::assign_op<typename DstXprType::Scalar, typename QrType::Scalar>&) {
dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
}
};
} // 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 PermutationIndex>
typename ColPivHouseholderQR<MatrixType, PermutationIndex>::HouseholderSequenceType
ColPivHouseholderQR<MatrixType, PermutationIndex>::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>
template <typename PermutationIndexType>
const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject, PermutationIndexType>
MatrixBase<Derived>::colPivHouseholderQr() const {
return ColPivHouseholderQR<PlainObject, PermutationIndexType>(eval());
}
} // end namespace Eigen
#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H