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 DecompositionType>
struct kernel_retval;
template <typename DecompositionType>
struct image_retval;
 
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

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;

  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "FullPivLU is not initialized.");
    return Success;
  }

  FullPivLU();

  FullPivLU(Index rows, Index cols);

  template <typename InputType>
  explicit FullPivLU(const EigenBase<InputType>& matrix);

  template <typename InputType>
  explicit FullPivLU(EigenBase<InputType>& matrix);

  template <typename InputType>
  FullPivLU& compute(const EigenBase<InputType>& matrix) {
    m_lu = matrix.derived();
    computeInPlace();
    return *this;
  }

  inline const MatrixType& matrixLU() const {
    eigen_assert(m_isInitialized && "LU is not initialized.");
    return m_lu;
  }

  EIGEN_DEVICE_FUNC inline const PermutationPType& permutationP() const {
    eigen_assert(m_isInitialized && "LU is not initialized.");
    return m_p;
  }

  inline const PermutationQType& permutationQ() const {
    eigen_assert(m_isInitialized && "LU is not initialized.");
    return m_q;
  }

  inline const internal::kernel_retval<FullPivLU> kernel() const {
    eigen_assert(m_isInitialized && "LU is not initialized.");
    return internal::kernel_retval<FullPivLU>(*this);
  }

  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
  template <typename Rhs>
  inline Solve<FullPivLU, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif

  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);
  }

  typename internal::traits<MatrixType>::Scalar determinant() const;

  RealScalar absDeterminant() const;

  RealScalar logAbsDeterminant() const;

  Scalar signDeterminant() const;

  RealScalar pivotCoeff(Index i) const {
    using std::abs;
    return abs(m_lu.coeff(i, i));
  }

  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);
}

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.noalias() = 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>
auto fullpivlu_nonzero_pivots(const FullPivLU<MatrixType, PermutationIndex>& dec, Index rank) {
  constexpr int MaxSmallDimAtCompileTime =
      min_size_prefer_fixed(MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime);
  Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank);
  const typename MatrixType::RealScalar premultiplied_threshold = dec.maxPivot() * dec.threshold();
  Index p = 0;
  for (Index i = 0; i < dec.nonzeroPivots(); ++i)
    if (numext::abs(dec.matrixLU().coeff(i, i)) > premultiplied_threshold) pivots.coeffRef(p++) = i;
  eigen_internal_assert(p == rank);
  return pivots;
}
 
template <typename MatrixType_, typename PermutationIndex_>
struct traits<kernel_retval<FullPivLU<MatrixType_, PermutationIndex_>>> {
  using ReturnType = Matrix<typename MatrixType_::Scalar, MatrixType_::ColsAtCompileTime, Dynamic,
                            plain_object_options<MatrixType_>::value, MatrixType_::MaxColsAtCompileTime,
                            MatrixType_::MaxColsAtCompileTime>;
};
 
template <typename MatrixType_, typename PermutationIndex_>
struct kernel_retval<FullPivLU<MatrixType_, PermutationIndex_>>
    : ReturnByValue<kernel_retval<FullPivLU<MatrixType_, PermutationIndex_>>> {
  using DecompositionType = FullPivLU<MatrixType_, PermutationIndex_>;
  using MatrixType = MatrixType_;
  using Scalar = typename MatrixType::Scalar;
 
  static constexpr int MaxSmallDimAtCompileTime =
      min_size_prefer_fixed(MatrixType::MaxColsAtCompileTime, MatrixType::MaxRowsAtCompileTime);
 
  explicit kernel_retval(const DecompositionType& dec)
      : m_dec(dec), m_rank(dec.rank()), m_cols(m_rank == dec.cols() ? 1 : dec.cols() - m_rank) {}
 
  Index rows() const { return m_dec.cols(); }
  Index cols() const { return m_cols; }
 
  template <typename Dest>
  void evalTo(Dest& dst) const {
    const Index cols = m_dec.matrixLU().cols(), dimker = cols - m_rank;
    if (dimker == 0) {
      // Represent the zero space by a single zero column.
      dst.setZero();
      return;
    }
 
    // PAQ = LU implies ker(A) = Q ker(U).
    const auto pivots = fullpivlu_nonzero_pivots(m_dec, m_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, plain_object_options<MatrixType>::value,
           MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
        m(m_dec.matrixLU().block(0, 0, m_rank, cols));
    for (Index i = 0; i < m_rank; ++i) {
      if (i) m.row(i).head(i).setZero();
      m.row(i).tail(cols - i) = m_dec.matrixLU().row(pivots.coeff(i)).tail(cols - i);
    }
    m.block(0, 0, m_rank, m_rank).template triangularView<StrictlyLower>().setZero();
    for (Index i = 0; i < m_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(m_rank, m_rank).template triangularView<Upper>().solveInPlace(m.topRightCorner(m_rank, dimker));
 
    // now we must undo the column permutation that we had applied!
    for (Index i = m_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 < m_rank; ++i) dst.row(m_dec.permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
    for (Index i = m_rank; i < cols; ++i) dst.row(m_dec.permutationQ().indices().coeff(i)).setZero();
    for (Index k = 0; k < dimker; ++k) dst.coeffRef(m_dec.permutationQ().indices().coeff(m_rank + k), k) = Scalar(1);
  }
 
 private:
  const DecompositionType& m_dec;
  Index m_rank, m_cols;
};
 
/***** Implementation of image() *****************************************************/
 
template <typename MatrixType_, typename PermutationIndex_>
struct traits<image_retval<FullPivLU<MatrixType_, PermutationIndex_>>> {
  using ReturnType = Matrix<typename MatrixType_::Scalar, MatrixType_::RowsAtCompileTime, Dynamic,
                            plain_object_options<MatrixType_>::value, MatrixType_::MaxRowsAtCompileTime,
                            MatrixType_::MaxColsAtCompileTime>;
};
 
template <typename MatrixType_, typename PermutationIndex_>
struct image_retval<FullPivLU<MatrixType_, PermutationIndex_>>
    : ReturnByValue<image_retval<FullPivLU<MatrixType_, PermutationIndex_>>> {
  using DecompositionType = FullPivLU<MatrixType_, PermutationIndex_>;
  using MatrixType = MatrixType_;
 
  image_retval(const DecompositionType& dec, const MatrixType& originalMatrix)
      : m_dec(dec), m_rank(dec.rank()), m_originalMatrix(originalMatrix) {}
 
  Index rows() const { return m_dec.rows(); }
  Index cols() const { return m_rank == 0 ? 1 : m_rank; }
 
  template <typename Dest>
  void evalTo(Dest& dst) const {
    if (m_rank == 0) {
      // Represent the zero space by a single zero column.
      dst.setZero();
      return;
    }
 
    // Select pivots before writing: dst may overlap an inplace decomposition's storage.
    const auto pivots = fullpivlu_nonzero_pivots(m_dec, m_rank);
    for (Index i = 0; i < m_rank; ++i)
      dst.col(i) = m_originalMatrix.col(m_dec.permutationQ().indices().coeff(pivots.coeff(i)));
  }
 
 private:
  const DecompositionType& m_dec;
  Index m_rank;
  const MatrixType& m_originalMatrix;
};
 
/***** 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 *****************************************************************/

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