cart-elc

FullPivLU.h (32383B)

---

 // 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>
 //
 // 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_LU_H
 #define EIGEN_LU_H
 
 namespace Eigen {
 
 namespace internal {
 template<typename _MatrixType> struct traits<FullPivLU<_MatrixType> >
  : traits<_MatrixType>
 {
   typedef MatrixXpr XprKind;
   typedef SolverStorage StorageKind;
   typedef int StorageIndex;
   enum { Flags = 0 };
 };
 
 } // end namespace internal
 
 /** \ingroup LU_Module
   *
   * \class FullPivLU
   *
   * \brief LU decomposition of a matrix with complete pivoting, and related features
   *
   * \tparam _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 \f$ A = P^{-1} L U Q^{-1} \f$ 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 example, here is how the original matrix can be retrieved:
   * \include class_FullPivLU.cpp
   * Output: \verbinclude class_FullPivLU.out
   *
   * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
   *
   * \sa MatrixBase::fullPivLu(), MatrixBase::determinant(), MatrixBase::inverse()
   */
 template<typename _MatrixType> class FullPivLU
   : public SolverBase<FullPivLU<_MatrixType> >
 {
   public:
     typedef _MatrixType MatrixType;
     typedef SolverBase<FullPivLU> Base;
     friend class SolverBase<FullPivLU>;
 
     EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivLU)
     enum {
       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
     };
     typedef typename internal::plain_row_type<MatrixType, StorageIndex>::type IntRowVectorType;
     typedef typename internal::plain_col_type<MatrixType, StorageIndex>::type IntColVectorType;
     typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType;
     typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType;
     typedef typename MatrixType::PlainObject PlainObject;
 
     /**
       * \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.
       */
     template<typename InputType>
     explicit FullPivLU(const EigenBase<InputType>& matrix);
 
     /** \brief Constructs a LU factorization from a given matrix
       *
       * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
       *
       * \sa FullPivLU(const EigenBase&)
       */
     template<typename InputType>
     explicit FullPivLU(EigenBase<InputType>& 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
       */
     template<typename InputType>
     FullPivLU& compute(const EigenBase<InputType>& matrix) {
       m_lu = matrix.derived();
       computeInPlace();
       return *this;
     }
 
     /** \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()
       */
     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()
       */
     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; }
 
     /** \returns the permutation matrix P
       *
       * \sa permutationQ()
       */
     EIGEN_DEVICE_FUNC inline const PermutationPType& permutationP() const
     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return m_p;
     }
 
     /** \returns the permutation matrix Q
       *
       * \sa permutationP()
       */
     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()
       */
     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 image (column-space).
       *
       * \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>
       image(const MatrixType& originalMatrix) const
     {
       eigen_assert(m_isInitialized && "LU is not initialized.");
       return internal::image_retval<FullPivLU>(*this, originalMatrix);
     }
 
     #ifdef EIGEN_PARSED_BY_DOXYGEN
     /** \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.
       *
       * \note_about_checking_solutions
       *
       * \note_about_arbitrary_choice_of_solution
       * \note_about_using_kernel_to_study_multiple_solutions
       *
       * Example: \include FullPivLU_solve.cpp
       * Output: \verbinclude FullPivLU_solve.out
       *
       * \sa TriangularView::solve(), kernel(), inverse()
       */
     template<typename Rhs>
     inline const Solve<FullPivLU, Rhs>
     solve(const MatrixBase<Rhs>& b) const;
     #endif
 
     /** \returns an estimate of the reciprocal condition number of the matrix of which \c *this is
         the LU decomposition.
       */
     inline RealScalar rcond() const
     {
       eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
       return internal::rcond_estimate_helper(m_l1_norm, *this);
     }
 
     /** \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&),
       * 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)
       */
     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&).
       */
     FullPivLU& 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_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&).
       */
     inline Index rank() const
     {
       using std::abs;
       eigen_assert(m_isInitialized && "LU 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_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&).
       */
     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&).
       */
     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&).
       */
     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&).
       */
     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()
       */
     inline const Inverse<FullPivLU> 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 Inverse<FullPivLU>(*this);
     }
 
     MatrixType reconstructedMatrix() const;
 
     EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
     inline Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); }
     EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
     inline Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); }
 
     #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:
 
     static void check_template_parameters()
     {
       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
     }
 
     void computeInPlace();
 
     MatrixType m_lu;
     PermutationPType m_p;
     PermutationQType m_q;
     IntColVectorType m_rowsTranspositions;
     IntRowVectorType m_colsTranspositions;
     Index m_nonzero_pivots;
     RealScalar m_l1_norm;
     RealScalar m_maxpivot, m_prescribedThreshold;
     signed char m_det_pq;
     bool m_isInitialized, m_usePrescribedThreshold;
 };
 
 template<typename MatrixType>
 FullPivLU<MatrixType>::FullPivLU()
   : m_isInitialized(false), m_usePrescribedThreshold(false)
 {
 }
 
 template<typename MatrixType>
 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>
 template<typename InputType>
 FullPivLU<MatrixType>::FullPivLU(const EigenBase<InputType>& 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.derived());
 }
 
 template<typename MatrixType>
 template<typename InputType>
 FullPivLU<MatrixType>::FullPivLU(EigenBase<InputType>& matrix)
   : m_lu(matrix.derived()),
     m_p(matrix.rows()),
     m_q(matrix.cols()),
     m_rowsTranspositions(matrix.rows()),
     m_colsTranspositions(matrix.cols()),
     m_isInitialized(false),
     m_usePrescribedThreshold(false)
 {
   computeInPlace();
 }
 
 template<typename MatrixType>
 void FullPivLU<MatrixType>::computeInPlace()
 {
   check_template_parameters();
 
   // the permutations are stored as int indices, so just to be sure:
   eigen_assert(m_lu.rows()<=NumTraits<int>::highest() && m_lu.cols()<=NumTraits<int>::highest());
 
   m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
 
   const Index size = m_lu.diagonalSize();
   const Index rows = m_lu.rows();
   const Index cols = m_lu.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(m_lu.rows());
   m_colsTranspositions.resize(m_lu.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);
 
   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;
     typedef internal::scalar_score_coeff_op<Scalar> Scoring;
     typedef typename Scoring::result_type Score;
     Score biggest_in_corner;
     biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
                         .unaryExpr(Scoring())
                         .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.
 
     if(biggest_in_corner==Score(0))
     {
       // 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) = internal::convert_index<StorageIndex>(i);
         m_colsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
       }
       break;
     }
 
     RealScalar abs_pivot = internal::abs_knowing_score<Scalar>()(m_lu(row_of_biggest_in_corner, col_of_biggest_in_corner), biggest_in_corner);
     if(abs_pivot > m_maxpivot) m_maxpivot = abs_pivot;
 
     // 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) = internal::convert_index<StorageIndex>(row_of_biggest_in_corner);
     m_colsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(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;
 
   m_isInitialized = true;
 }
 
 template<typename MatrixType>
 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: \f$ P^{-1} L U Q^{-1} \f$.
  * This function is provided for debug purposes. */
 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>
 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
   {
     using std::abs;
     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)
     {
       if(i) m.row(i).head(i).setZero();
       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)));
 
     // see the negative sign in the next line, that's what we were talking about above.
     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>
 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
   {
     using std::abs;
     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() *****************************************************/
 
 } // end namespace internal
 
 #ifndef EIGEN_PARSED_BY_DOXYGEN
 template<typename _MatrixType>
 template<typename RhsType, typename DstType>
 void FullPivLU<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &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 = this->rows(),
               cols = this->cols(),
               nonzero_pivots = this->rank();
   const Index smalldim = (std::min)(rows, cols);
 
   if(nonzero_pivots == 0)
   {
     dst.setZero();
     return;
   }
 
   typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
 
   // Step 1
   c = permutationP() * rhs;
 
   // Step 2
   m_lu.topLeftCorner(smalldim,smalldim)
       .template triangularView<UnitLower>()
       .solveInPlace(c.topRows(smalldim));
   if(rows>cols)
     c.bottomRows(rows-cols) -= m_lu.bottomRows(rows-cols) * c.topRows(cols);
 
   // Step 3
   m_lu.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(permutationQ().indices().coeff(i)) = c.row(i);
   for(Index i = nonzero_pivots; i < m_lu.cols(); ++i)
     dst.row(permutationQ().indices().coeff(i)).setZero();
 }
 
 template<typename _MatrixType>
 template<bool Conjugate, typename RhsType, typename DstType>
 void FullPivLU<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
 {
   /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1},
    * and since permutations are real and unitary, we can write this
    * as   A^T = Q U^T L^T P,
    * So we proceed as follows:
    * Step 1: compute c = Q^T rhs.
    * Step 2: replace c by the solution x to U^T x = c. May or may not exist.
    * Step 3: replace c by the solution x to L^T x = c.
    * Step 4: result = P^T c.
    * If Conjugate is true, replace "^T" by "^*" above.
    */
 
   const Index rows = this->rows(), cols = this->cols(),
     nonzero_pivots = this->rank();
   const Index smalldim = (std::min)(rows, cols);
 
   if(nonzero_pivots == 0)
   {
     dst.setZero();
     return;
   }
 
   typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
 
   // Step 1
   c = permutationQ().inverse() * rhs;
 
   // Step 2
   m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
       .template triangularView<Upper>()
       .transpose()
       .template conjugateIf<Conjugate>()
       .solveInPlace(c.topRows(nonzero_pivots));
 
   // Step 3
   m_lu.topLeftCorner(smalldim, smalldim)
       .template triangularView<UnitLower>()
       .transpose()
       .template conjugateIf<Conjugate>()
       .solveInPlace(c.topRows(smalldim));
 
   // Step 4
   PermutationPType invp = permutationP().inverse().eval();
   for(Index i = 0; i < smalldim; ++i)
     dst.row(invp.indices().coeff(i)) = c.row(i);
   for(Index i = smalldim; i < rows; ++i)
     dst.row(invp.indices().coeff(i)).setZero();
 }
 
 #endif
 
 namespace internal {
 
 
 /***** Implementation of inverse() *****************************************************/
 template<typename DstXprType, typename MatrixType>
 struct Assignment<DstXprType, Inverse<FullPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename FullPivLU<MatrixType>::Scalar>, Dense2Dense>
 {
   typedef FullPivLU<MatrixType> LuType;
   typedef Inverse<LuType> SrcXprType;
   static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename MatrixType::Scalar> &)
   {
     dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
   }
 };
 } // 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>
 MatrixBase<Derived>::fullPivLu() const
 {
   return FullPivLU<PlainObject>(eval());
 }
 
 } // end namespace Eigen
 
 #endif // EIGEN_LU_H