text: // 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 { 
  
  /** \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 \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 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()
    */
  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()
        */
      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()
        */
      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 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>
        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.
        *
        * \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 internal::solve_retval<FullPivLU, Rhs>
      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&),
        * 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() * 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 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:
      
      static void check_template_parameters()
      {
        EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
      }
      
      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>
  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>
  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>
  FullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const MatrixType& matrix)
  {
    check_template_parameters();
    
    // the permutations are stored as int indices, so just to be sure:
    eigen_assert(matrix.rows()<=NumTraits<int>::highest() && matrix.cols()<=NumTraits<int>::highest());
    
    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);
  
    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.
  
      if(biggest_in_corner==RealScalar(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) = 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>
  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() *****************************************************/
  
  template<typename _MatrixType, typename Rhs>
  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>
  MatrixBase<Derived>::fullPivLu() const
  {
    return FullPivLU<PlainObject>(eval());
  }
  
  } // end namespace Eigen
  
  #endif // EIGEN_LU_H
  