eigen/Eigen/src/LU/PartialPivLU.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>
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// 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_PARTIALLU_H
#define EIGEN_PARTIALLU_H

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

namespace Eigen {

namespace internal {
template <typename MatrixType_, typename PermutationIndex_>
struct traits<PartialPivLU<MatrixType_, PermutationIndex_> > : traits<MatrixType_> {
  typedef MatrixXpr XprKind;
  typedef SolverStorage StorageKind;
  typedef PermutationIndex_ StorageIndex;
  typedef traits<MatrixType_> BaseTraits;
  enum { Flags = BaseTraits::Flags & RowMajorBit, CoeffReadCost = Dynamic };
};

template <typename T, typename Derived>
struct enable_if_ref;
// {
//   typedef Derived type;
// };

template <typename T, typename Derived>
struct enable_if_ref<Ref<T>, Derived> {
  typedef Derived type;
};

}  // end namespace internal

/** \ingroup LU_Module
 *
 * \class PartialPivLU
 *
 * \brief LU decomposition of a matrix with partial 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 a \b square \b invertible matrix, with partial pivoting: the matrix A
 * is decomposed as A = PLU where L is unit-lower-triangular, U is upper-triangular, and P
 * is a permutation matrix.
 *
 * Typically, partial pivoting LU decomposition is only considered numerically stable for square invertible
 * matrices. Thus LAPACK's dgesv and dgesvx require the matrix to be square and invertible. The present class
 * does the same. It will assert that the matrix is square, but it won't (actually it can't) check that the
 * matrix is invertible: it is your task to check that you only use this decomposition on invertible matrices.
 *
 * The guaranteed safe alternative, working for all matrices, is the full pivoting LU decomposition, provided
 * by class FullPivLU.
 *
 * This is \b not a rank-revealing LU decomposition. Many features are intentionally absent from this class,
 * such as rank computation. If you need these features, use class FullPivLU.
 *
 * This LU decomposition is suitable to invert invertible matrices. It is what MatrixBase::inverse() uses
 * in the general case.
 * On the other hand, it is \b not suitable to determine whether a given matrix is invertible.
 *
 * The data of the LU decomposition can be directly accessed through the methods matrixLU(), permutationP().
 *
 * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
 *
 * \sa MatrixBase::partialPivLu(), MatrixBase::determinant(), MatrixBase::inverse(), MatrixBase::computeInverse(), class
 * FullPivLU
 */
template <typename MatrixType_, typename PermutationIndex_>
class PartialPivLU : public SolverBase<PartialPivLU<MatrixType_, PermutationIndex_> > {
 public:
  typedef MatrixType_ MatrixType;
  typedef SolverBase<PartialPivLU> Base;
  friend class SolverBase<PartialPivLU>;

  EIGEN_GENERIC_PUBLIC_INTERFACE(PartialPivLU)
  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
  };
  using PermutationIndex = PermutationIndex_;
  typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex> PermutationType;
  typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime, PermutationIndex> TranspositionType;
  typedef typename MatrixType::PlainObject PlainObject;

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

  /** \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 PartialPivLU()
   */
  explicit PartialPivLU(Index size);

  /** Constructor.
   *
   * \param matrix the matrix of which to compute the LU decomposition.
   *
   * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
   * If you need to deal with non-full rank, use class FullPivLU instead.
   */
  template <typename InputType>
  explicit PartialPivLU(const EigenBase<InputType>& matrix);

  /** Constructor for \link InplaceDecomposition inplace decomposition \endlink
   *
   * \param matrix the matrix of which to compute the LU decomposition.
   *
   * \warning The matrix should have full rank (e.g. if it's square, it should be invertible).
   * If you need to deal with non-full rank, use class FullPivLU instead.
   */
  template <typename InputType>
  explicit PartialPivLU(EigenBase<InputType>& matrix);

  template <typename InputType>
  PartialPivLU& compute(const EigenBase<InputType>& matrix) {
    m_lu = matrix.derived();
    compute();
    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 && "PartialPivLU is not initialized.");
    return m_lu;
  }

  /** \returns the permutation matrix P.
   */
  inline const PermutationType& permutationP() const {
    eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
    return m_p;
  }

#ifdef EIGEN_PARSED_BY_DOXYGEN
  /** This method returns the 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 the solution.
   *
   * Example: \include PartialPivLU_solve.cpp
   * Output: \verbinclude PartialPivLU_solve.out
   *
   * Since this PartialPivLU class assumes anyway that the matrix A is invertible, the solution
   * theoretically exists and is unique regardless of b.
   *
   * \sa TriangularView::solve(), inverse(), computeInverse()
   */
  template <typename Rhs>
  inline const Solve<PartialPivLU, 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 inverse of the matrix of which *this is the LU decomposition.
   *
   * \warning The matrix being decomposed here is assumed to be invertible. If you need to check for
   *          invertibility, use class FullPivLU instead.
   *
   * \sa MatrixBase::inverse(), LU::inverse()
   */
  inline const Inverse<PartialPivLU> inverse() const {
    eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
    return Inverse<PartialPivLU>(*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 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()
   */
  Scalar determinant() const;

  MatrixType reconstructedMatrix() const;

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

#ifndef EIGEN_PARSED_BY_DOXYGEN
  template <typename RhsType, typename DstType>
  EIGEN_DEVICE_FUNC void _solve_impl(const RhsType& rhs, DstType& dst) const {
    /* The decomposition PA = LU can be rewritten as A = P^{-1} L U.
     * So we proceed as follows:
     * Step 1: compute c = Pb.
     * Step 2: replace c by the solution x to Lx = c.
     * Step 3: replace c by the solution x to Ux = c.
     */

    // Step 1
    dst = permutationP() * rhs;

    // Step 2
    m_lu.template triangularView<UnitLower>().solveInPlace(dst);

    // Step 3
    m_lu.template triangularView<Upper>().solveInPlace(dst);
  }

  template <bool Conjugate, typename RhsType, typename DstType>
  EIGEN_DEVICE_FUNC void _solve_impl_transposed(const RhsType& rhs, DstType& dst) const {
    /* The decomposition PA = LU can be rewritten as A^T = U^T L^T P.
     * So we proceed as follows:
     * Step 1: compute c as the solution to L^T c = b
     * Step 2: replace c by the solution x to U^T x = c.
     * Step 3: update  c = P^-1 c.
     */

    eigen_assert(rhs.rows() == m_lu.cols());

    // Step 1
    dst = m_lu.template triangularView<Upper>().transpose().template conjugateIf<Conjugate>().solve(rhs);
    // Step 2
    m_lu.template triangularView<UnitLower>().transpose().template conjugateIf<Conjugate>().solveInPlace(dst);
    // Step 3
    dst = permutationP().transpose() * dst;
  }
#endif

 protected:
  EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar)

  void compute();

  MatrixType m_lu;
  PermutationType m_p;
  TranspositionType m_rowsTranspositions;
  RealScalar m_l1_norm;
  signed char m_det_p;
  bool m_isInitialized;
};

template <typename MatrixType, typename PermutationIndex>
PartialPivLU<MatrixType, PermutationIndex>::PartialPivLU()
    : m_lu(), m_p(), m_rowsTranspositions(), m_l1_norm(0), m_det_p(0), m_isInitialized(false) {}

template <typename MatrixType, typename PermutationIndex>
PartialPivLU<MatrixType, PermutationIndex>::PartialPivLU(Index size)
    : m_lu(size, size), m_p(size), m_rowsTranspositions(size), m_l1_norm(0), m_det_p(0), m_isInitialized(false) {}

template <typename MatrixType, typename PermutationIndex>
template <typename InputType>
PartialPivLU<MatrixType, PermutationIndex>::PartialPivLU(const EigenBase<InputType>& matrix)
    : m_lu(matrix.rows(), matrix.cols()),
      m_p(matrix.rows()),
      m_rowsTranspositions(matrix.rows()),
      m_l1_norm(0),
      m_det_p(0),
      m_isInitialized(false) {
  compute(matrix.derived());
}

template <typename MatrixType, typename PermutationIndex>
template <typename InputType>
PartialPivLU<MatrixType, PermutationIndex>::PartialPivLU(EigenBase<InputType>& matrix)
    : m_lu(matrix.derived()),
      m_p(matrix.rows()),
      m_rowsTranspositions(matrix.rows()),
      m_l1_norm(0),
      m_det_p(0),
      m_isInitialized(false) {
  compute();
}

namespace internal {

/** \internal This is the blocked version of fullpivlu_unblocked() */
template <typename Scalar, int StorageOrder, typename PivIndex, int SizeAtCompileTime = Dynamic>
struct partial_lu_impl {
  static constexpr int UnBlockedBound = 16;
  static constexpr bool UnBlockedAtCompileTime = SizeAtCompileTime != Dynamic && SizeAtCompileTime <= UnBlockedBound;
  static constexpr int ActualSizeAtCompileTime = UnBlockedAtCompileTime ? SizeAtCompileTime : Dynamic;
  // Remaining rows and columns at compile-time:
  static constexpr int RRows = SizeAtCompileTime == 2 ? 1 : Dynamic;
  static constexpr int RCols = SizeAtCompileTime == 2 ? 1 : Dynamic;
  typedef Matrix<Scalar, ActualSizeAtCompileTime, ActualSizeAtCompileTime, StorageOrder> MatrixType;
  typedef Ref<MatrixType> MatrixTypeRef;
  typedef Ref<Matrix<Scalar, Dynamic, Dynamic, StorageOrder> > BlockType;
  typedef typename MatrixType::RealScalar RealScalar;

  /** \internal performs the LU decomposition in-place of the matrix \a lu
   * using an unblocked algorithm.
   *
   * In addition, this function returns the row transpositions in the
   * vector \a row_transpositions which must have a size equal to the number
   * of columns of the matrix \a lu, and an integer \a nb_transpositions
   * which returns the actual number of transpositions.
   *
   * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
   */
  static Index unblocked_lu(MatrixTypeRef& lu, PivIndex* row_transpositions, PivIndex& nb_transpositions) {
    typedef scalar_score_coeff_op<Scalar> Scoring;
    typedef typename Scoring::result_type Score;
    const Index rows = lu.rows();
    const Index cols = lu.cols();
    const Index size = (std::min)(rows, cols);
    // For small compile-time matrices it is worth processing the last row separately:
    //  speedup: +100% for 2x2, +10% for others.
    const Index endk = UnBlockedAtCompileTime ? size - 1 : size;
    nb_transpositions = 0;
    Index first_zero_pivot = -1;
    for (Index k = 0; k < endk; ++k) {
      int rrows = internal::convert_index<int>(rows - k - 1);
      int rcols = internal::convert_index<int>(cols - k - 1);

      Index row_of_biggest_in_col;
      Score biggest_in_corner = lu.col(k).tail(rows - k).unaryExpr(Scoring()).maxCoeff(&row_of_biggest_in_col);
      row_of_biggest_in_col += k;

      row_transpositions[k] = PivIndex(row_of_biggest_in_col);

      if (!numext::is_exactly_zero(biggest_in_corner)) {
        if (k != row_of_biggest_in_col) {
          lu.row(k).swap(lu.row(row_of_biggest_in_col));
          ++nb_transpositions;
        }

        lu.col(k).tail(fix<RRows>(rrows)) /= lu.coeff(k, k);
      } else if (first_zero_pivot == -1) {
        // the pivot is exactly zero, we record the index of the first pivot which is exactly 0,
        // and continue the factorization such we still have A = PLU
        first_zero_pivot = k;
      }

      if (k < rows - 1)
        lu.bottomRightCorner(fix<RRows>(rrows), fix<RCols>(rcols)).noalias() -=
            lu.col(k).tail(fix<RRows>(rrows)) * lu.row(k).tail(fix<RCols>(rcols));
    }

    // special handling of the last entry
    if (UnBlockedAtCompileTime) {
      Index k = endk;
      row_transpositions[k] = PivIndex(k);
      if (numext::is_exactly_zero(Scoring()(lu(k, k))) && first_zero_pivot == -1) first_zero_pivot = k;
    }

    return first_zero_pivot;
  }

  /** \internal performs the LU decomposition in-place of the matrix represented
   * by the variables \a rows, \a cols, \a lu_data, and \a lu_stride using a
   * recursive, blocked algorithm.
   *
   * In addition, this function returns the row transpositions in the
   * vector \a row_transpositions which must have a size equal to the number
   * of columns of the matrix \a lu, and an integer \a nb_transpositions
   * which returns the actual number of transpositions.
   *
   * \returns The index of the first pivot which is exactly zero if any, or a negative number otherwise.
   *
   * \note This very low level interface using pointers, etc. is to:
   *   1 - reduce the number of instantiations to the strict minimum
   *   2 - avoid infinite recursion of the instantiations with Block<Block<Block<...> > >
   */
  static Index blocked_lu(Index rows, Index cols, Scalar* lu_data, Index luStride, PivIndex* row_transpositions,
                          PivIndex& nb_transpositions, Index maxBlockSize = 256) {
    MatrixTypeRef lu = MatrixType::Map(lu_data, rows, cols, OuterStride<>(luStride));

    const Index size = (std::min)(rows, cols);

    // if the matrix is too small, no blocking:
    if (UnBlockedAtCompileTime || size <= UnBlockedBound) {
      return unblocked_lu(lu, row_transpositions, nb_transpositions);
    }

    // automatically adjust the number of subdivisions to the size
    // of the matrix so that there is enough sub blocks:
    Index blockSize;
    {
      blockSize = size / 8;
      blockSize = (blockSize / 16) * 16;
      blockSize = (std::min)((std::max)(blockSize, Index(8)), maxBlockSize);
    }

    nb_transpositions = 0;
    Index first_zero_pivot = -1;
    for (Index k = 0; k < size; k += blockSize) {
      Index bs = (std::min)(size - k, blockSize);  // actual size of the block
      Index trows = rows - k - bs;                 // trailing rows
      Index tsize = size - k - bs;                 // trailing size

      // partition the matrix:
      //                          A00 | A01 | A02
      // lu  = A_0 | A_1 | A_2 =  A10 | A11 | A12
      //                          A20 | A21 | A22
      BlockType A_0 = lu.block(0, 0, rows, k);
      BlockType A_2 = lu.block(0, k + bs, rows, tsize);
      BlockType A11 = lu.block(k, k, bs, bs);
      BlockType A12 = lu.block(k, k + bs, bs, tsize);
      BlockType A21 = lu.block(k + bs, k, trows, bs);
      BlockType A22 = lu.block(k + bs, k + bs, trows, tsize);

      PivIndex nb_transpositions_in_panel;
      // recursively call the blocked LU algorithm on [A11^T A21^T]^T
      // with a very small blocking size:
      Index ret = blocked_lu(trows + bs, bs, &lu.coeffRef(k, k), luStride, row_transpositions + k,
                             nb_transpositions_in_panel, 16);
      if (ret >= 0 && first_zero_pivot == -1) first_zero_pivot = k + ret;

      nb_transpositions += nb_transpositions_in_panel;
      // update permutations and apply them to A_0
      for (Index i = k; i < k + bs; ++i) {
        Index piv = (row_transpositions[i] += internal::convert_index<PivIndex>(k));
        A_0.row(i).swap(A_0.row(piv));
      }

      if (trows) {
        // apply permutations to A_2
        for (Index i = k; i < k + bs; ++i) A_2.row(i).swap(A_2.row(row_transpositions[i]));

        // A12 = A11^-1 A12
        A11.template triangularView<UnitLower>().solveInPlace(A12);

        A22.noalias() -= A21 * A12;
      }
    }
    return first_zero_pivot;
  }
};

/** \internal performs the LU decomposition with partial pivoting in-place.
 */
template <typename MatrixType, typename TranspositionType>
void partial_lu_inplace(MatrixType& lu, TranspositionType& row_transpositions,
                        typename TranspositionType::StorageIndex& nb_transpositions) {
  // Special-case of zero matrix.
  if (lu.rows() == 0 || lu.cols() == 0) {
    nb_transpositions = 0;
    return;
  }
  eigen_assert(lu.cols() == row_transpositions.size());
  eigen_assert(row_transpositions.size() < 2 ||
               (&row_transpositions.coeffRef(1) - &row_transpositions.coeffRef(0)) == 1);

  partial_lu_impl<typename MatrixType::Scalar, MatrixType::Flags & RowMajorBit ? RowMajor : ColMajor,
                  typename TranspositionType::StorageIndex,
                  internal::min_size_prefer_fixed(MatrixType::RowsAtCompileTime, MatrixType::ColsAtCompileTime)>::
      blocked_lu(lu.rows(), lu.cols(), &lu.coeffRef(0, 0), lu.outerStride(), &row_transpositions.coeffRef(0),
                 nb_transpositions);
}

}  // end namespace internal

template <typename MatrixType, typename PermutationIndex>
void PartialPivLU<MatrixType, PermutationIndex>::compute() {
  eigen_assert(m_lu.rows() < NumTraits<PermutationIndex>::highest());

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

  eigen_assert(m_lu.rows() == m_lu.cols() && "PartialPivLU is only for square (and moreover invertible) matrices");
  const Index size = m_lu.rows();

  m_rowsTranspositions.resize(size);

  typename TranspositionType::StorageIndex nb_transpositions;
  internal::partial_lu_inplace(m_lu, m_rowsTranspositions, nb_transpositions);
  m_det_p = (nb_transpositions % 2) ? -1 : 1;

  m_p = m_rowsTranspositions;

  m_isInitialized = true;
}

template <typename MatrixType, typename PermutationIndex>
typename PartialPivLU<MatrixType, PermutationIndex>::Scalar PartialPivLU<MatrixType, PermutationIndex>::determinant()
    const {
  eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
  return Scalar(m_det_p) * m_lu.diagonal().prod();
}

/** \returns the matrix represented by the decomposition,
 * i.e., it returns the product: P^{-1} L U.
 * This function is provided for debug purpose. */
template <typename MatrixType, typename PermutationIndex>
MatrixType PartialPivLU<MatrixType, PermutationIndex>::reconstructedMatrix() const {
  eigen_assert(m_isInitialized && "LU is not initialized.");
  // LU
  MatrixType res = m_lu.template triangularView<UnitLower>().toDenseMatrix() * m_lu.template triangularView<Upper>();

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

  return res;
}

/***** Implementation details *****************************************************/

namespace internal {

/***** Implementation of inverse() *****************************************************/
template <typename DstXprType, typename MatrixType, typename PermutationIndex>
struct Assignment<
    DstXprType, Inverse<PartialPivLU<MatrixType, PermutationIndex> >,
    internal::assign_op<typename DstXprType::Scalar, typename PartialPivLU<MatrixType, PermutationIndex>::Scalar>,
    Dense2Dense> {
  typedef PartialPivLU<MatrixType, PermutationIndex> LuType;
  typedef Inverse<LuType> SrcXprType;
  static void run(DstXprType& dst, const SrcXprType& src,
                  const internal::assign_op<typename DstXprType::Scalar, typename LuType::Scalar>&) {
    dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
  }
};
}  // end namespace internal

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

/** \lu_module
 *
 * \return the partial-pivoting LU decomposition of \c *this.
 *
 * \sa class PartialPivLU
 */
template <typename Derived>
template <typename PermutationIndex>
inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject, PermutationIndex>
MatrixBase<Derived>::partialPivLu() const {
  return PartialPivLU<PlainObject, PermutationIndex>(eval());
}

/** \lu_module
 *
 * Synonym of partialPivLu().
 *
 * \return the partial-pivoting LU decomposition of \c *this.
 *
 * \sa class PartialPivLU
 */
template <typename Derived>
template <typename PermutationIndex>
inline const PartialPivLU<typename MatrixBase<Derived>::PlainObject, PermutationIndex> MatrixBase<Derived>::lu() const {
  return PartialPivLU<PlainObject, PermutationIndex>(eval());
}

}  // end namespace Eigen

#endif  // EIGEN_PARTIALLU_H