Eigen/src/LU/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>
//
// 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/.
// SPDX-License-Identifier: MPL-2.0

#ifndef EIGEN_LU_H
#define EIGEN_LU_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {
template <typename MatrixType_, typename PermutationIndex_>
struct traits<FullPivLU<MatrixType_, PermutationIndex_> > : traits<MatrixType_> {
  using XprKind = MatrixXpr;
  using StorageKind = SolverStorage;
  using StorageIndex = PermutationIndex_;
  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_, typename PermutationIndex_>
class FullPivLU : public SolverBase<FullPivLU<MatrixType_, PermutationIndex_> >,
                  public RankRevealingBase<FullPivLU<MatrixType_, PermutationIndex_> > {
 public:
  using MatrixType = MatrixType_;
  using Base = SolverBase<FullPivLU>;
  using RankRevealingBase_ = RankRevealingBase<FullPivLU>;
  friend class SolverBase<FullPivLU>;
  friend class RankRevealingBase<FullPivLU>;
  using RankRevealingBase_::dimensionOfKernel;
  using RankRevealingBase_::isInjective;
  using RankRevealingBase_::isInvertible;
  using RankRevealingBase_::isSurjective;
  using RankRevealingBase_::maxPivot;
  using RankRevealingBase_::nonzeroPivots;
  using RankRevealingBase_::rank;
  using RankRevealingBase_::setThreshold;
  using RankRevealingBase_::threshold;

  EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivLU)
  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };
  using PermutationIndex = PermutationIndex_;
  using IntRowVectorType = typename internal::plain_row_type<MatrixType, PermutationIndex>::type;
  using IntColVectorType = typename internal::plain_col_type<MatrixType, PermutationIndex>::type;
  using PermutationQType = PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime, PermutationIndex>;
  using PermutationPType = PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex>;
  using PlainObject = typename MatrixType::PlainObject;

  /** \brief Reports whether the LU factorization was successful.
   *
   * \note This function always returns \c Success. It is provided for compatibility
   * with other factorization routines.
   * \returns \c Success
   */
  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "FullPivLU is not initialized.");
    return Success;
  }

  /**
   * \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 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 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 && "FullPivLU is not initialized.");
    if (!isInvertible()) {
      return RealScalar(0);
    }
    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.
   * One way to work around that is to use logAbsDeterminant() and signDeterminant() instead.
   * Also, do not rely on the determinant being exactly zero for testing
   * singularity or rank-deficiency.
   *
   * \sa absDeterminant(), logAbsDeterminant(), signDeterminant(), MatrixBase::determinant()
   */
  typename internal::traits<MatrixType>::Scalar determinant() const;

  /** \returns the absolute value of 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.
   *
   * \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.
   *
   * \note Returns exactly zero when rank() finds the decomposition rank-deficient, as the
   * rank-revealing QR decompositions do. determinant() is not gated that way: it returns the
   * product of the pivots whatever the rank.
   *
   * \sa determinant(), logAbsDeterminant(), signDeterminant(), MatrixBase::determinant()
   */
  RealScalar absDeterminant() const;

  /** \returns the natural log of the absolute value of 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 This method is useful to work around the risk of overflow/underflow that's inherent
   * to determinant computation.
   *
   * \note Returns \c -infinity when rank() finds the decomposition rank-deficient.
   *
   * \sa determinant(), absDeterminant(), signDeterminant(), MatrixBase::determinant()
   */
  RealScalar logAbsDeterminant() const;

  /** \returns the sign of 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 This method is useful to work around the risk of overflow/underflow that's inherent
   * to determinant computation.
   *
   * \note Returns zero when rank() finds the decomposition rank-deficient, matching the documented
   * sign of a singular matrix.
   *
   * \sa determinant(), absDeterminant(), logAbsDeterminant(), MatrixBase::determinant()
   */
  Scalar signDeterminant() const;

  /** \returns the absolute value of the i-th pivot coefficient (for RankRevealingBase). */
  RealScalar pivotCoeff(Index i) const {
    using std::abs;
    return abs(m_lu.coeff(i, i));
  }

  /** \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 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 constexpr Index rows() const noexcept { return m_lu.rows(); }
  EIGEN_DEVICE_FUNC constexpr Index cols() const 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:
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

  void computeInPlace();

  MatrixType m_lu;
  PermutationPType m_p;
  PermutationQType m_q;
  IntColVectorType m_rowsTranspositions;
  IntRowVectorType m_colsTranspositions;
  RealScalar m_l1_norm;
  signed char m_det_pq;
  bool m_isInitialized;
};

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

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

template <typename MatrixType, typename PermutationIndex>
template <typename InputType>
FullPivLU<MatrixType, PermutationIndex>::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) {
  compute(matrix.derived());
}

template <typename MatrixType, typename PermutationIndex>
template <typename InputType>
FullPivLU<MatrixType, PermutationIndex>::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) {
  computeInPlace();
}

template <typename MatrixType, typename PermutationIndex>
void FullPivLU<MatrixType, PermutationIndex>::computeInPlace() {
  eigen_assert(m_lu.rows() <= NumTraits<PermutationIndex>::highest() &&
               m_lu.cols() <= NumTraits<PermutationIndex>::highest());

  if (m_lu.cols() > 0)
    m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
  else
    m_l1_norm = RealScalar(0);

  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

  this->m_nonzero_pivots = size;  // the generic case is that in which all pivots are nonzero (invertible case)
  this->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;
    using Scoring = internal::scalar_score_coeff_op<Scalar>;
    using Score = typename Scoring::result_type;
    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 (numext::is_exactly_zero(biggest_in_corner)) {
      // before exiting, make sure to initialize the still uninitialized transpositions
      // in a sane state without destroying what we already have.
      this->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 > this->m_maxpivot) this->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 PermutationIndex>
typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType, PermutationIndex>::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());
}

template <typename MatrixType, typename PermutationIndex>
typename FullPivLU<MatrixType, PermutationIndex>::RealScalar FullPivLU<MatrixType, PermutationIndex>::absDeterminant()
    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 isInjective() ? numext::abs(m_lu.diagonal().prod()) : RealScalar(0);
}

template <typename MatrixType, typename PermutationIndex>
typename FullPivLU<MatrixType, PermutationIndex>::RealScalar
FullPivLU<MatrixType, PermutationIndex>::logAbsDeterminant() 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 isInjective() ? m_lu.diagonal().cwiseAbs().array().log().sum() : -NumTraits<RealScalar>::infinity();
}

template <typename MatrixType, typename PermutationIndex>
typename FullPivLU<MatrixType, PermutationIndex>::Scalar FullPivLU<MatrixType, PermutationIndex>::signDeterminant()
    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 isInjective() ? Scalar(m_det_pq) * m_lu.diagonal().array().sign().prod() : Scalar(0);
}

/** \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, typename PermutationIndex>
MatrixType FullPivLU<MatrixType, PermutationIndex>::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() calls 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_, typename PermutationIndex_>
struct kernel_retval<FullPivLU<MatrixType_, PermutationIndex_> >
    : kernel_retval_base<FullPivLU<MatrixType_, PermutationIndex_> > {
  using DecompositionType = FullPivLU<MatrixType_, PermutationIndex_>;
  EIGEN_MAKE_KERNEL_HELPERS(DecompositionType)

  enum {
    MaxSmallDimAtCompileTime = min_size_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());

    // Construct a temporary 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.
    // This is needed to apply our triangular solvers.
    // FIXME: simplify once triangularView supports rectangular matrices.
    Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, traits<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()).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_, typename PermutationIndex_>
struct image_retval<FullPivLU<MatrixType_, PermutationIndex_> >
    : image_retval_base<FullPivLU<MatrixType_, PermutationIndex_> > {
  using DecompositionType = FullPivLU<MatrixType_, PermutationIndex_>;
  EIGEN_MAKE_IMAGE_HELPERS(DecompositionType)

  enum {
    MaxSmallDimAtCompileTime = min_size_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_, typename PermutationIndex_>
template <typename RhsType, typename DstType>
void FullPivLU<MatrixType_, PermutationIndex_>::_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).noalias() -= 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_, typename PermutationIndex_>
template <bool Conjugate, typename RhsType, typename DstType>
void FullPivLU<MatrixType_, PermutationIndex_>::_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, typename PermutationIndex>
struct Assignment<
    DstXprType, Inverse<FullPivLU<MatrixType, PermutationIndex> >,
    internal::assign_op<typename DstXprType::Scalar, typename FullPivLU<MatrixType, PermutationIndex>::Scalar>,
    Dense2Dense> {
  using LuType = FullPivLU<MatrixType, PermutationIndex>;
  using SrcXprType = Inverse<LuType>;
  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>
template <typename PermutationIndex>
inline FullPivLU<typename MatrixBase<Derived>::PlainObject, PermutationIndex> MatrixBase<Derived>::fullPivLu() const {
  return FullPivLU<PlainObject, PermutationIndex>(eval());
}

}  // end namespace Eigen

#endif  // EIGEN_LU_H