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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

# FullPivLU.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2006-2009 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_LU_H
#define EIGEN_LU_H

/** \ingroup LU_Module
*
* \class FullPivLU
*
* \brief LU decomposition of a matrix with complete pivoting, and related features
*
* \param MatrixType the type of the matrix of which we are computing the LU decomposition
*
* This class represents a LU decomposition of any matrix, with complete pivoting: the matrix A
* is decomposed as A = PLUQ where L is unit-lower-triangular, U is upper-triangular, and P and Q
* are permutation matrices. This is a rank-revealing LU decomposition. The eigenvalues (diagonal
* coefficients) of U are sorted in such a way that any zeros are at the end.
*
* This decomposition provides the generic approach to solving systems of linear equations, computing
* the rank, invertibility, inverse, kernel, and determinant.
*
* This LU decomposition is very stable and well tested with large matrices. However there are use cases where the SVD
* decomposition is inherently more stable and/or flexible. For example, when computing the kernel of a matrix,
* working with the SVD allows to select the smallest singular values of the matrix, something that
* the LU decomposition doesn't see.
*
* The data of the LU decomposition can be directly accessed through the methods matrixLU(),
* permutationP(), permutationQ().
*
* As an exemple, here is how the original matrix can be retrieved:
* \include class_FullPivLU.cpp
* Output: \verbinclude class_FullPivLU.out
*
* \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()
*/
00058 template<typename _MatrixType> class FullPivLU
{
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 NumTraits<typename MatrixType::Scalar>::Real RealScalar;
typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
typedef typename MatrixType::Index Index;
typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType;
typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType;

/**
* \brief Default Constructor.
*
* The default constructor is useful in cases in which the user intends to
* perform decompositions via LU::compute(const MatrixType&).
*/
FullPivLU();

/** \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 FullPivLU()
*/
FullPivLU(Index rows, Index cols);

/** Constructor.
*
* \param matrix the matrix of which to compute the LU decomposition.
*               It is required to be nonzero.
*/
FullPivLU(const MatrixType& matrix);

/** Computes the LU decomposition of the given matrix.
*
* \param matrix the matrix of which to compute the LU decomposition.
*               It is required to be nonzero.
*
* \returns a reference to *this
*/
FullPivLU& compute(const MatrixType& matrix);

/** \returns the LU decomposition matrix: the upper-triangular part is U, the
* unit-lower-triangular part is L (at least for square matrices; in the non-square
* case, special care is needed, see the documentation of class FullPivLU).
*
* \sa matrixL(), matrixU()
*/
00116     inline const MatrixType& matrixLU() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return m_lu;
}

/** \returns the number of nonzero pivots in the LU 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()
*/
00129     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.
*/
00138     RealScalar maxPivot() const { return m_maxpivot; }

/** \returns the permutation matrix P
*
* \sa permutationQ()
*/
00144     inline const PermutationPType& permutationP() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return m_p;
}

/** \returns the permutation matrix Q
*
* \sa permutationP()
*/
00154     inline const PermutationQType& permutationQ() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return m_q;
}

/** \returns the kernel of the matrix, also called its null-space. The columns of the returned matrix
* will form a basis of the kernel.
*
* \note If the kernel has dimension zero, then the returned matrix is a column-vector filled with zeros.
*
* \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&).
*
* Example: \include FullPivLU_kernel.cpp
* Output: \verbinclude FullPivLU_kernel.out
*
* \sa image()
*/
00174     inline const internal::kernel_retval<FullPivLU> kernel() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return internal::kernel_retval<FullPivLU>(*this);
}

/** \returns the image of the matrix, also called its column-space. The columns of the returned matrix
* will form a basis of the kernel.
*
* \param originalMatrix the original matrix, of which *this is the LU decomposition.
*                       The reason why it is needed to pass it here, is that this allows
*                       a large optimization, as otherwise this method would need to reconstruct it
*                       from the LU decomposition.
*
* \note If the image has dimension zero, then the returned matrix is a column-vector filled with zeros.
*
* \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&).
*
* Example: \include FullPivLU_image.cpp
* Output: \verbinclude FullPivLU_image.out
*
* \sa kernel()
*/
inline const internal::image_retval<FullPivLU>
00200       image(const MatrixType& originalMatrix) const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return internal::image_retval<FullPivLU>(*this, originalMatrix);
}

/** \return a solution x to the equation Ax=b, where A is the matrix of which
* *this is the LU decomposition.
*
* \param b the right-hand-side of the equation to solve. Can be a vector or a matrix,
*          the only requirement in order for the equation to make sense is that
*          b.rows()==A.rows(), where A is the matrix of which *this is the LU decomposition.
*
* \returns a solution.
*
*
*
* Example: \include FullPivLU_solve.cpp
* Output: \verbinclude FullPivLU_solve.out
*
* \sa TriangularView::solve(), kernel(), inverse()
*/
template<typename Rhs>
inline const internal::solve_retval<FullPivLU, Rhs>
00227     solve(const MatrixBase<Rhs>& b) const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return internal::solve_retval<FullPivLU, Rhs>(*this, b.derived());
}

/** \returns the determinant of the matrix of which
* *this is the LU decomposition. It has only linear complexity
* (that is, O(n) where n is the dimension of the square matrix)
* as the LU decomposition has already been computed.
*
* \note This is only for square matrices.
*
* \note For fixed-size matrices of size up to 4, MatrixBase::determinant() offers
*       optimized paths.
*
* \warning a determinant can be very big or small, so for matrices
* of large enough dimension, there is a risk of overflow/underflow.
*
* \sa MatrixBase::determinant()
*/
typename internal::traits<MatrixType>::Scalar determinant() const;

/** 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
* LU 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)
*/
00267     FullPivLU& 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 lu.setThreshold(Eigen::Default); \endcode
*
* See the documentation of setThreshold(const RealScalar&).
*/
00282     FullPivLU& setThreshold(Default_t)
{
m_usePrescribedThreshold = false;
}

/** Returns the threshold that will be used by certain methods such as rank().
*
* See the documentation of setThreshold(const RealScalar&).
*/
00291     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_lu.diagonalSize();
}

/** \returns the rank of the matrix of which *this is the LU 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&).
*/
00306     inline Index rank() const
{
eigen_assert(m_isInitialized && "LU 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_lu.coeff(i,i)) > premultiplied_threshold);
return result;
}

/** \returns the dimension of the kernel of the matrix of which *this is the LU 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&).
*/
00322     inline Index dimensionOfKernel() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return cols() - rank();
}

/** \returns true if the matrix of which *this is the LU 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&).
*/
00335     inline bool isInjective() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return rank() == cols();
}

/** \returns true if the matrix of which *this is the LU 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&).
*/
00348     inline bool isSurjective() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return rank() == rows();
}

/** \returns true if the matrix of which *this is the LU 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&).
*/
00360     inline bool isInvertible() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
return isInjective() && (m_lu.rows() == m_lu.cols());
}

/** \returns the inverse of the matrix of which *this is the LU decomposition.
*
* \note If this matrix is not invertible, the returned matrix has undefined coefficients.
*       Use isInvertible() to first determine whether this matrix is invertible.
*
* \sa MatrixBase::inverse()
*/
00373     inline const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> inverse() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
return internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType>
(*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
}

MatrixType reconstructedMatrix() const;

inline Index rows() const { return m_lu.rows(); }
inline Index cols() const { return m_lu.cols(); }

protected:
MatrixType m_lu;
PermutationPType m_p;
PermutationQType m_q;
IntColVectorType m_rowsTranspositions;
IntRowVectorType m_colsTranspositions;
Index m_det_pq, m_nonzero_pivots;
RealScalar m_maxpivot, m_prescribedThreshold;
bool m_isInitialized, m_usePrescribedThreshold;
};

template<typename MatrixType>
00398 FullPivLU<MatrixType>::FullPivLU()
: m_isInitialized(false), m_usePrescribedThreshold(false)
{
}

template<typename MatrixType>
00404 FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols)
: m_lu(rows, cols),
m_p(rows),
m_q(cols),
m_rowsTranspositions(rows),
m_colsTranspositions(cols),
m_isInitialized(false),
m_usePrescribedThreshold(false)
{
}

template<typename MatrixType>
00416 FullPivLU<MatrixType>::FullPivLU(const MatrixType& matrix)
: m_lu(matrix.rows(), matrix.cols()),
m_p(matrix.rows()),
m_q(matrix.cols()),
m_rowsTranspositions(matrix.rows()),
m_colsTranspositions(matrix.cols()),
m_isInitialized(false),
m_usePrescribedThreshold(false)
{
compute(matrix);
}

template<typename MatrixType>
00429 FullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const MatrixType& matrix)
{
m_isInitialized = true;
m_lu = matrix;

const Index size = matrix.diagonalSize();
const Index rows = matrix.rows();
const Index cols = matrix.cols();

// will store the transpositions, before we accumulate them at the end.
// can't accumulate on-the-fly because that will be done in reverse order for the rows.
m_rowsTranspositions.resize(matrix.rows());
m_colsTranspositions.resize(matrix.cols());
Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i

m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
m_maxpivot = RealScalar(0);
RealScalar cutoff(0);

for(Index k = 0; k < size; ++k)
{
// First, we need to find the pivot.

// biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
Index row_of_biggest_in_corner, col_of_biggest_in_corner;
RealScalar biggest_in_corner;
biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
.cwiseAbs()
.maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner,
col_of_biggest_in_corner += k; // need to add k to them.

// when k==0, biggest_in_corner is the biggest coeff absolute value in the original matrix
if(k == 0) cutoff = biggest_in_corner * NumTraits<Scalar>::epsilon();

// if the pivot (hence the corner) is "zero", terminate to avoid generating nan/inf values.
// Notice that using an exact comparison (biggest_in_corner==0) here, as Golub-van Loan do in
// their pseudo-code, results in numerical instability! The cutoff here has been validated
// by running the unit test 'lu' with many repetitions.
if(biggest_in_corner < cutoff)
{
// before exiting, make sure to initialize the still uninitialized transpositions
// in a sane state without destroying what we already have.
m_nonzero_pivots = k;
for(Index i = k; i < size; ++i)
{
m_rowsTranspositions.coeffRef(i) = i;
m_colsTranspositions.coeffRef(i) = i;
}
break;
}

if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner;

// Now that we've found the pivot, we need to apply the row/col swaps to
// bring it to the location (k,k).

m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner;
m_colsTranspositions.coeffRef(k) = col_of_biggest_in_corner;
if(k != row_of_biggest_in_corner) {
m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
++number_of_transpositions;
}
if(k != col_of_biggest_in_corner) {
m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
++number_of_transpositions;
}

// Now that the pivot is at the right location, we update the remaining
// bottom-right corner by Gaussian elimination.

if(k<rows-1)
m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
if(k<size-1)
m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1);
}

// the main loop is over, we still have to accumulate the transpositions to find the
// permutations P and Q

m_p.setIdentity(rows);
for(Index k = size-1; k >= 0; --k)
m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));

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

m_det_pq = (number_of_transpositions%2) ? -1 : 1;
return *this;
}

template<typename MatrixType>
00522 typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
}

/** \returns the matrix represented by the decomposition,
* i.e., it returns the product: P^{-1} L U Q^{-1}.
* This function is provided for debug purpose. */
template<typename MatrixType>
MatrixType FullPivLU<MatrixType>::reconstructedMatrix() const
{
eigen_assert(m_isInitialized && "LU is not initialized.");
const Index smalldim = std::min(m_lu.rows(), m_lu.cols());
// LU
MatrixType res(m_lu.rows(),m_lu.cols());
// FIXME the .toDenseMatrix() should not be needed...
res = m_lu.leftCols(smalldim)
.template triangularView<UnitLower>().toDenseMatrix()
* m_lu.topRows(smalldim)
.template triangularView<Upper>().toDenseMatrix();

// P^{-1}(LU)
res = m_p.inverse() * res;

// (P^{-1}LU)Q^{-1}
res = res * m_q.inverse();

return res;
}

/********* Implementation of kernel() **************************************************/

namespace internal {
template<typename _MatrixType>
00558 struct kernel_retval<FullPivLU<_MatrixType> >
: kernel_retval_base<FullPivLU<_MatrixType> >
{
EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>)

enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
MatrixType::MaxColsAtCompileTime,
MatrixType::MaxRowsAtCompileTime)
};

template<typename Dest> void evalTo(Dest& dst) const
{
const Index cols = dec().matrixLU().cols(), dimker = cols - rank();
if(dimker == 0)
{
// The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's
// avoid crashing/asserting as that depends on floating point calculations. Let's
// just return a single column vector filled with zeros.
dst.setZero();
return;
}

/* Let us use the following lemma:
*
* Lemma: If the matrix A has the LU decomposition PAQ = LU,
* then Ker A = Q(Ker U).
*
* Proof: trivial: just keep in mind that P, Q, L are invertible.
*/

/* Thus, all we need to do is to compute Ker U, and then apply Q.
*
* U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
* Thus, the diagonal of U ends with exactly
* dimKer zero's. Let us use that to construct dimKer linearly
* independent vectors in Ker U.
*/

Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
Index p = 0;
for(Index i = 0; i < dec().nonzeroPivots(); ++i)
if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
pivots.coeffRef(p++) = i;
eigen_internal_assert(p == rank());

// we construct a temporaty trapezoid matrix m, by taking the U matrix and
// permuting the rows and cols to bring the nonnegligible pivots to the top of
// the main diagonal. We need that to be able to apply our triangular solvers.
// FIXME when we get triangularView-for-rectangular-matrices, this can be simplified
Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
m(dec().matrixLU().block(0, 0, rank(), cols));
for(Index i = 0; i < rank(); ++i)
{
m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i);
}
m.block(0, 0, rank(), rank());
m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
for(Index i = 0; i < rank(); ++i)
m.col(i).swap(m.col(pivots.coeff(i)));

// ok, we have our trapezoid matrix, we can apply the triangular solver.
// notice that the math behind this suggests that we should apply this to the
// negative of the RHS, but for performance we just put the negative sign elsewhere, see below.
m.topLeftCorner(rank(), rank())
.template triangularView<Upper>().solveInPlace(
m.topRightCorner(rank(), dimker)
);

// now we must undo the column permutation that we had applied!
for(Index i = rank()-1; i >= 0; --i)
m.col(i).swap(m.col(pivots.coeff(i)));

for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero();
for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1);
}
};

/***** Implementation of image() *****************************************************/

template<typename _MatrixType>
00643 struct image_retval<FullPivLU<_MatrixType> >
: image_retval_base<FullPivLU<_MatrixType> >
{
EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)

enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
MatrixType::MaxColsAtCompileTime,
MatrixType::MaxRowsAtCompileTime)
};

template<typename Dest> void evalTo(Dest& dst) const
{
if(rank() == 0)
{
// The Image is just {0}, so it doesn't have a basis properly speaking, but let's
// avoid crashing/asserting as that depends on floating point calculations. Let's
// just return a single column vector filled with zeros.
dst.setZero();
return;
}

Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
Index p = 0;
for(Index i = 0; i < dec().nonzeroPivots(); ++i)
if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
pivots.coeffRef(p++) = i;
eigen_internal_assert(p == rank());

for(Index i = 0; i < rank(); ++i)
dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
}
};

/***** Implementation of solve() *****************************************************/

template<typename _MatrixType, typename Rhs>
00680 struct solve_retval<FullPivLU<_MatrixType>, Rhs>
: solve_retval_base<FullPivLU<_MatrixType>, Rhs>
{
EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs)

template<typename Dest> void evalTo(Dest& dst) const
{
/* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
* So we proceed as follows:
* Step 1: compute c = P * rhs.
* Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
* Step 3: replace c by the solution x to Ux = c. May or may not exist.
* Step 4: result = Q * c;
*/

const Index rows = dec().rows(), cols = dec().cols(),
nonzero_pivots = dec().nonzeroPivots();
eigen_assert(rhs().rows() == rows);
const Index smalldim = std::min(rows, cols);

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

typename Rhs::PlainObject c(rhs().rows(), rhs().cols());

// Step 1
c = dec().permutationP() * rhs();

// Step 2
dec().matrixLU()
.topLeftCorner(smalldim,smalldim)
.template triangularView<UnitLower>()
.solveInPlace(c.topRows(smalldim));
if(rows>cols)
{
c.bottomRows(rows-cols)
-= dec().matrixLU().bottomRows(rows-cols)
* c.topRows(cols);
}

// Step 3
dec().matrixLU()
.topLeftCorner(nonzero_pivots, nonzero_pivots)
.template triangularView<Upper>()
.solveInPlace(c.topRows(nonzero_pivots));

// Step 4
for(Index i = 0; i < nonzero_pivots; ++i)
dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i);
for(Index i = nonzero_pivots; i < dec().matrixLU().cols(); ++i)
dst.row(dec().permutationQ().indices().coeff(i)).setZero();
}
};

} // end namespace internal

/******* MatrixBase methods *****************************************************************/

/** \lu_module
*
* \return the full-pivoting LU decomposition of \c *this.
*
* \sa class FullPivLU
*/
template<typename Derived>
inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
00749 MatrixBase<Derived>::fullPivLu() const
{
return FullPivLU<PlainObject>(eval());
}

#endif // EIGEN_LU_H

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