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 Sourcecode: eigen3 version 3.0.0-23.0.1-13.0.2-23.0.3-13.0.4-13.0.5-13.1.0~alpha1-1

ColPivHouseholderQR.h
// 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>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// version 3 of the License, or (at your option) any later version.
//
// Alternatively, you can redistribute it and/or
// modify it under the terms of the GNU General Public License as
//
// Eigen is distributed in the hope that it will be useful, but WITHOUT ANY
// WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
// FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License or the
// GNU General Public License for more details.
//
// You should have received a copy of the GNU Lesser General Public
// License and a copy of the GNU General Public License along with
// Eigen. If not, see <http://www.gnu.org/licenses/>.

#ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
#define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H

/** \ingroup QR_Module
*
* \class ColPivHouseholderQR
*
* \brief Householder rank-revealing QR decomposition of a matrix with column-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 column pivoting in order to be rank-revealing and improve
* numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
*
* \sa MatrixBase::colPivHouseholderQr()
*/
00050 template<typename _MatrixType> class ColPivHouseholderQR
{
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, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
typedef typename HouseholderSequence<MatrixType,HCoeffsType>::ConjugateReturnType HouseholderSequenceType;

/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
*/
00079     ColPivHouseholderQR()
: m_qr(),
m_hCoeffs(),
m_colsPermutation(),
m_colsTranspositions(),
m_temp(),
m_colSqNorms(),
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 ColPivHouseholderQR()
*/
00094     ColPivHouseholderQR(Index rows, Index cols)
: m_qr(rows, cols),
m_hCoeffs((std::min)(rows,cols)),
m_colsPermutation(cols),
m_colsTranspositions(cols),
m_temp(cols),
m_colSqNorms(cols),
m_isInitialized(false),
m_usePrescribedThreshold(false) {}

ColPivHouseholderQR(const MatrixType& matrix)
: m_qr(matrix.rows(), matrix.cols()),
m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
m_colsPermutation(matrix.cols()),
m_colsTranspositions(matrix.cols()),
m_temp(matrix.cols()),
m_colSqNorms(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.
*
*
*
* Example: \include ColPivHouseholderQR_solve.cpp
* Output: \verbinclude ColPivHouseholderQR_solve.out
*/
template<typename Rhs>
inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
00136     solve(const MatrixBase<Rhs>& b) const
{
eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
}

HouseholderSequenceType householderQ(void) const;

/** \returns a reference to the matrix where the Householder QR decomposition is stored
*/
00146     const MatrixType& matrixQR() const
{
eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
return m_qr;
}

ColPivHouseholderQR& compute(const MatrixType& matrix);

const PermutationType& colsPermutation() const
{
eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
return m_colsPermutation;
}

/** \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&).
*/
00195     inline Index rank() const
{
eigen_assert(m_isInitialized && "ColPivHouseholderQR 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&).
*/
00211     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&).
*/
00224     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&).
*/
00237     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&).
*/
00249     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
internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
00262     inverse() const
{
eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
return internal::solve_retval<ColPivHouseholderQR,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&),
*
* \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)
*/
00290     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&).
*/
00305     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&).
*/
00315     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()
*/
00331     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.
*/
00340     RealScalar maxPivot() const { return m_maxpivot; }

protected:
MatrixType m_qr;
HCoeffsType m_hCoeffs;
PermutationType m_colsPermutation;
IntRowVectorType m_colsTranspositions;
RowVectorType m_temp;
RealRowVectorType m_colSqNorms;
bool m_isInitialized, m_usePrescribedThreshold;
RealScalar m_prescribedThreshold, m_maxpivot;
Index m_nonzero_pivots;
Index m_det_pq;
};

template<typename MatrixType>
00356 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() 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 internal::abs(m_qr.diagonal().prod());
}

template<typename MatrixType>
00364 typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::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 m_qr.diagonal().cwiseAbs().array().log().sum();
}

template<typename MatrixType>
ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
{
Index rows = matrix.rows();
Index cols = matrix.cols();
Index size = matrix.diagonalSize();

m_qr = matrix;
m_hCoeffs.resize(size);

m_temp.resize(cols);

m_colsTranspositions.resize(matrix.cols());
Index number_of_transpositions = 0;

m_colSqNorms.resize(cols);
for(Index k = 0; k < cols; ++k)
m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();

RealScalar threshold_helper = m_colSqNorms.maxCoeff() * internal::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);

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_colSqNorms which column has the biggest squared norm
Index biggest_col_index;
RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
biggest_col_index += k;

// since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
// the actual squared norm of the selected column.
// Note that not doing so does result in solve() sometimes returning inf/nan values
// when running the unit test with 1000 repetitions.
biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();

// we store that back into our table: it can't hurt to correct our table.
m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;

// if the current biggest column is smaller than epsilon times the initial biggest column,
// terminate to avoid generating nan/inf values.
// Note that here, if we test instead for "biggest == 0", we get a failure every 1000 (or so)
// repetitions of the unit test, with the result of solve() filled with large values of the order
// of 1/(size*epsilon).
if(biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
{
m_nonzero_pivots = k;
m_hCoeffs.tail(size-k).setZero();
m_qr.bottomRightCorner(rows-k,cols-k)
.template triangularView<StrictlyLower>()
.setZero();
break;
}

// apply the transposition to the columns
m_colsTranspositions.coeffRef(k) = biggest_col_index;
if(k != biggest_col_index) {
m_qr.col(k).swap(m_qr.col(biggest_col_index));
std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.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(internal::abs(beta) > m_maxpivot) m_maxpivot = internal::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 squared norms of the columns
m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
}

m_colsPermutation.setIdentity(cols);
for(Index k = 0; k < m_nonzero_pivots; ++k)
m_colsPermutation.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));

m_det_pq = (number_of_transpositions%2) ? -1 : 1;
m_isInitialized = true;

return *this;
}

namespace internal {

template<typename _MatrixType, typename Rhs>
00465 struct solve_retval<ColPivHouseholderQR<_MatrixType>, Rhs>
: solve_retval_base<ColPivHouseholderQR<_MatrixType>, Rhs>
{
EIGEN_MAKE_SOLVE_HELPERS(ColPivHouseholderQR<_MatrixType>,Rhs)

template<typename Dest> void evalTo(Dest& dst) const
{
eigen_assert(rhs().rows() == dec().rows());

const int cols = dec().cols(),
nonzero_pivots = dec().nonzeroPivots();

if(nonzero_pivots == 0)
{
dst.setZero();
return;
}

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(), dec().hCoeffs())
.setLength(dec().nonzeroPivots())
.transpose()
);

dec().matrixQR()
.topLeftCorner(nonzero_pivots, nonzero_pivots)
.template triangularView<Upper>()
.solveInPlace(c.topRows(nonzero_pivots));

typename Rhs::PlainObject d(c);
d.topRows(nonzero_pivots)
= dec().matrixQR()
.topLeftCorner(nonzero_pivots, nonzero_pivots)
.template triangularView<Upper>()
* c.topRows(nonzero_pivots);

for(Index i = 0; i < nonzero_pivots; ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
for(Index i = nonzero_pivots; i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
}
};

} // end namespace internal

/** \returns the matrix Q as a sequence of householder transformations */
template<typename MatrixType>
typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
00514   ::householderQ() const
{
eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate()).setLength(m_nonzero_pivots);
}

/** \return the column-pivoting Householder QR decomposition of \c *this.
*
* \sa class ColPivHouseholderQR
*/
template<typename Derived>
const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
00526 MatrixBase<Derived>::colPivHouseholderQr() const
{
return ColPivHouseholderQR<PlainObject>(eval());
}

#endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H


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