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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, ColsAtCompileTime, RowMajor, 1, MaxColsAtCompileTime> IntRowVectorType;
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
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(rows),
m_cols_transpositions(cols),
m_cols_permutation(cols),
m_temp((std::min)(rows,cols)),
m_isInitialized(false),
m_usePrescribedThreshold(false) {}
FullPivHouseholderQR(const MatrixType& matrix)
: m_qr(matrix.rows(), matrix.cols()),
m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
m_rows_transpositions(matrix.rows()),
m_cols_transpositions(matrix.cols()),
m_cols_permutation(matrix.cols()),
m_temp((std::min)(matrix.rows(), 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 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);
const PermutationType& colsPermutation() const
{
eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
return m_cols_permutation;
}
const IntColVectorType& 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
{
eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
RealScalar premultiplied_threshold = internal::abs(m_maxpivot) * threshold();
Index result = 0;
for(Index i = 0; i < m_nonzero_pivots; ++i)
result += (internal::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(); }
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() * 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:
MatrixType m_qr;
HCoeffsType m_hCoeffs;
IntColVectorType m_rows_transpositions;
IntRowVectorType 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
{
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 internal::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();
}
template<typename MatrixType>
FullPivHouseholderQR<MatrixType>& FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
{
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() * size;
m_rows_transpositions.resize(matrix.rows());
m_cols_transpositions.resize(matrix.cols());
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(internal::abs(beta) > m_maxpivot) m_maxpivot = internal::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));
}
if(!dec().isSurjective())
{
// is c is in the image of R ?
RealScalar biggest_in_upper_part_of_c = c.topRows( dec().rank() ).cwiseAbs().maxCoeff();
RealScalar biggest_in_lower_part_of_c = c.bottomRows(rows-dec().rank()).cwiseAbs().maxCoeff();
// FIXME brain dead
const RealScalar m_precision = NumTraits<Scalar>::epsilon() * (std::min)(rows,cols);
// this internal:: prefix is needed by at least gcc 3.4 and ICC
if(!internal::isMuchSmallerThan(biggest_in_lower_part_of_c, biggest_in_upper_part_of_c, m_precision))
return;
}
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 internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
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 IntColVectorType& 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
{
// 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), internal::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 IntColVectorType::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