LDLT.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2009 Keir Mierle <mierle@gmail.com>
// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2011 Timothy E. Holy <tim.holy@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_LDLT_H
#define EIGEN_LDLT_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.h"
 
namespace Eigen {
 
namespace internal {
template <typename MatrixType_, int UpLo_>
struct traits<LDLT<MatrixType_, UpLo_> > : traits<MatrixType_> {
  using XprKind = MatrixXpr;
  using StorageKind = SolverStorage;
  using StorageIndex = int;
  enum { Flags = 0 };
};
 
template <typename MatrixType, int UpLo>
struct LDLT_Traits;
 
// PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef
enum SignMatrix { PositiveSemiDef, NegativeSemiDef, ZeroSign, Indefinite };
}  // namespace internal

template <typename MatrixType_, int UpLo_>
class LDLT : public SolverBase<LDLT<MatrixType_, UpLo_> > {
 public:
  using MatrixType = MatrixType_;
  using Base = SolverBase<LDLT>;
  friend class SolverBase<LDLT>;
 
  EIGEN_GENERIC_PUBLIC_INTERFACE(LDLT)
  enum {
    MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
    MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
    UpLo = UpLo_
  };
  using TmpMatrixType = Matrix<Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1>;
 
  using TranspositionType = Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime>;
  using PermutationType = PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime>;
 
  using Traits = internal::LDLT_Traits<MatrixType, UpLo>;

  LDLT()
      : m_matrix(),
        m_l1_norm(0),
        m_transpositions(),
        m_sign(internal::ZeroSign),
        m_isInitialized(false),
        m_info(InvalidInput) {}

  explicit LDLT(Index size)
      : m_matrix(size, size),
        m_l1_norm(0),
        m_transpositions(size),
        m_temporary(size),
        m_sign(internal::ZeroSign),
        m_isInitialized(false),
        m_info(InvalidInput) {}

  template <typename InputType>
  explicit LDLT(const EigenBase<InputType>& matrix)
      : m_matrix(matrix.rows(), matrix.cols()),
        m_l1_norm(0),
        m_transpositions(matrix.rows()),
        m_temporary(matrix.rows()),
        m_sign(internal::ZeroSign),
        m_isInitialized(false),
        m_info(InvalidInput) {
    compute(matrix.derived());
  }

  template <typename InputType>
  explicit LDLT(EigenBase<InputType>& matrix)
      : m_matrix(matrix.derived()),
        m_l1_norm(0),
        m_transpositions(matrix.rows()),
        m_temporary(matrix.rows()),
        m_sign(internal::ZeroSign),
        m_isInitialized(false),
        m_info(InvalidInput) {
    compute(matrix.derived());
  }

  void setZero() { m_isInitialized = false; }

  inline typename Traits::MatrixU matrixU() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return Traits::getU(m_matrix);
  }

  inline typename Traits::MatrixL matrixL() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return Traits::getL(m_matrix);
  }

  inline const TranspositionType& transpositionsP() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return m_transpositions;
  }

  inline Diagonal<const MatrixType> vectorD() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return m_matrix.diagonal();
  }

  inline bool isPositive() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
  }

  inline bool isNegative(void) const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign;
  }
 
#ifdef EIGEN_PARSED_BY_DOXYGEN
  template <typename Rhs>
  inline Solve<LDLT, Rhs> solve(const MatrixBase<Rhs>& b) const;
#endif
 
  template <typename Derived>
  bool solveInPlace(MatrixBase<Derived>& bAndX) const;
 
  template <typename InputType>
  LDLT& compute(const EigenBase<InputType>& matrix);

  RealScalar rcond() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return internal::rcond_estimate_helper(m_l1_norm, *this);
  }
 
  template <typename Derived>
  LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha = 1);

  Scalar determinant() const;

  RealScalar absDeterminant() const;

  RealScalar logAbsDeterminant() const;

  Scalar signDeterminant() const;

  inline const MatrixType& matrixLDLT() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return m_matrix;
  }
 
  MatrixType reconstructedMatrix() const;

  const LDLT& adjoint() const { return *this; }
 
  EIGEN_DEVICE_FUNC constexpr Index rows() const noexcept { return m_matrix.rows(); }
  EIGEN_DEVICE_FUNC constexpr Index cols() const noexcept { return m_matrix.cols(); }

  ComputationInfo info() const {
    eigen_assert(m_isInitialized && "LDLT is not initialized.");
    return m_info;
  }
 
#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)
 
  
  MatrixType m_matrix;
  RealScalar m_l1_norm;
  TranspositionType m_transpositions;
  TmpMatrixType m_temporary;
  internal::SignMatrix m_sign;
  bool m_isInitialized;
  ComputationInfo m_info;
};
 
namespace internal {
 
template <int UpLo>
struct ldlt_inplace;
 
template <>
struct ldlt_inplace<Lower> {
  template <typename MatrixType, typename TranspositionType, typename Workspace>
  static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign) {
    using std::abs;
    using Scalar = typename MatrixType::Scalar;
    using RealScalar = typename MatrixType::RealScalar;
    using IndexType = typename TranspositionType::StorageIndex;
    eigen_assert(mat.rows() == mat.cols());
    const Index size = mat.rows();
    bool found_zero_pivot = false;
    bool ret = true;
 
    if (size <= 1) {
      transpositions.setIdentity();
      if (size == 0)
        sign = ZeroSign;
      else if (numext::real(mat.coeff(0, 0)) > static_cast<RealScalar>(0))
        sign = PositiveSemiDef;
      else if (numext::real(mat.coeff(0, 0)) < static_cast<RealScalar>(0))
        sign = NegativeSemiDef;
      else
        sign = ZeroSign;
      return true;
    }
 
    for (Index k = 0; k < size; ++k) {
      // Find largest diagonal element
      Index index_of_biggest_in_corner;
      mat.diagonal().tail(size - k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
      index_of_biggest_in_corner += k;
 
      transpositions.coeffRef(k) = IndexType(index_of_biggest_in_corner);
      if (k != index_of_biggest_in_corner) {
        // apply the transposition while taking care to consider only
        // the lower triangular part
        Index s = size - index_of_biggest_in_corner - 1;  // trailing size after the biggest element
        mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
        mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
        std::swap(mat.coeffRef(k, k), mat.coeffRef(index_of_biggest_in_corner, index_of_biggest_in_corner));
        for (Index i = k + 1; i < index_of_biggest_in_corner; ++i) {
          Scalar tmp = mat.coeffRef(i, k);
          mat.coeffRef(i, k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner, i));
          mat.coeffRef(index_of_biggest_in_corner, i) = numext::conj(tmp);
        }
        EIGEN_IF_CONSTEXPR (NumTraits<Scalar>::IsComplex)
          mat.coeffRef(index_of_biggest_in_corner, k) = numext::conj(mat.coeff(index_of_biggest_in_corner, k));
      }
 
      // partition the matrix:
      //       A00 |  -  |  -
      // lu  = A10 | A11 |  -
      //       A20 | A21 | A22
      Index rs = size - k - 1;
      Block<MatrixType, Dynamic, 1> A21(mat, k + 1, k, rs, 1);
      Block<MatrixType, 1, Dynamic> A10(mat, k, 0, 1, k);
      Block<MatrixType, Dynamic, Dynamic> A20(mat, k + 1, 0, rs, k);
 
      if (k > 0) {
        temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint();
        mat.coeffRef(k, k) -= (A10 * temp.head(k)).value();
        if (rs > 0) A21.noalias() -= A20 * temp.head(k);
      }
 
      // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot
      // was smaller than the cutoff value. However, since LDLT is not rank-revealing
      // we should only make sure that we do not introduce INF or NaN values.
      // Remark that LAPACK also uses 0 as the cutoff value.
      RealScalar realAkk = numext::real(mat.coeffRef(k, k));
      bool pivot_is_valid = (abs(realAkk) > RealScalar(0));
 
      if (k == 0 && !pivot_is_valid) {
        // The entire diagonal is zero, there is nothing more to do
        // except filling the transpositions, and checking whether the matrix is zero.
        sign = ZeroSign;
        for (Index j = 0; j < size; ++j) {
          transpositions.coeffRef(j) = IndexType(j);
          ret = ret && (mat.col(j).tail(size - j - 1).array() == Scalar(0)).all();
        }
        return ret;
      }
 
      if ((rs > 0) && pivot_is_valid)
        A21 /= realAkk;
      else if (rs > 0)
        ret = ret && (A21.array() == Scalar(0)).all();
 
      if (found_zero_pivot && pivot_is_valid)
        ret = false;  // factorization failed
      else if (!pivot_is_valid)
        found_zero_pivot = true;
 
      if (sign == PositiveSemiDef) {
        if (realAkk < static_cast<RealScalar>(0)) sign = Indefinite;
      } else if (sign == NegativeSemiDef) {
        if (realAkk > static_cast<RealScalar>(0)) sign = Indefinite;
      } else if (sign == ZeroSign) {
        if (realAkk > static_cast<RealScalar>(0))
          sign = PositiveSemiDef;
        else if (realAkk < static_cast<RealScalar>(0))
          sign = NegativeSemiDef;
      }
    }
 
    return ret;
  }
 
  // Reference for the algorithm: Davis and Hager, "Multiple Rank
  // Modifications of a Sparse Cholesky Factorization" (Algorithm 1)
  // Trivial rearrangements of their computations (Timothy E. Holy)
  // allow their algorithm to work for rank-1 updates even if the
  // original matrix is not of full rank.
  // Here only rank-1 updates are implemented, to reduce the
  // requirement for intermediate storage and improve accuracy
  template <typename MatrixType, typename WDerived>
  static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w,
                            const typename MatrixType::RealScalar& sigma = 1) {
    using numext::isfinite;
    using Scalar = typename MatrixType::Scalar;
    using RealScalar = typename MatrixType::RealScalar;
 
    const Index size = mat.rows();
    eigen_assert(mat.cols() == size && w.size() == size);
 
    RealScalar alpha = 1;
 
    // Apply the update
    for (Index j = 0; j < size; j++) {
      // Check for termination due to an original decomposition of low-rank
      if (!(isfinite)(alpha)) break;
 
      // Update the diagonal terms
      RealScalar dj = numext::real(mat.coeff(j, j));
      Scalar wj = w.coeff(j);
      RealScalar swj2 = sigma * numext::abs2(wj);
      RealScalar gamma = dj * alpha + swj2;
 
      // A zero contribution leaves both quantities unchanged, but spelling that out as an addition of swj2/alpha and
      // swj2/dj evaluates 0/0 when alpha or the pivot is zero. The resulting NaN reaches the termination test above
      // on the next iteration, which reads it as a low-rank signal and abandons the remainder of the update. Skip the
      // no-op instead. A zero pivot with a nonzero contribution still yields an infinite alpha, so genuine low-rank
      // termination is unaffected.
      if (!numext::is_exactly_zero(swj2)) {
        mat.coeffRef(j, j) += swj2 / alpha;
        alpha += swj2 / dj;
      }
 
      // Update the terms of L
      Index rs = size - j - 1;
      w.tail(rs) -= wj * mat.col(j).tail(rs);
      if (!numext::is_exactly_zero(gamma)) mat.col(j).tail(rs) += (sigma * numext::conj(wj) / gamma) * w.tail(rs);
    }
    return true;
  }
 
  template <typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
  static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w,
                     const typename MatrixType::RealScalar& sigma = 1) {
    // Apply the permutation to the input w
    tmp = transpositions * w;
 
    return ldlt_inplace<Lower>::updateInPlace(mat, tmp, sigma);
  }
};
 
template <>
struct ldlt_inplace<Upper> {
  template <typename MatrixType, typename TranspositionType, typename Workspace>
  static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp,
                                            SignMatrix& sign) {
    Transpose<MatrixType> matt(mat);
    return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
  }
 
  template <typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
  static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w,
                                         const typename MatrixType::RealScalar& sigma = 1) {
    Transpose<MatrixType> matt(mat);
    return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
  }
};
 
template <typename MatrixType>
struct LDLT_Traits<MatrixType, Lower> {
  using MatrixL = const TriangularView<const MatrixType, UnitLower>;
  using MatrixU = const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper>;
  static inline MatrixL getL(const MatrixType& m) { return MatrixL(m); }
  static inline MatrixU getU(const MatrixType& m) { return MatrixU(m.adjoint()); }
};
 
template <typename MatrixType>
struct LDLT_Traits<MatrixType, Upper> {
  using MatrixL = const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower>;
  using MatrixU = const TriangularView<const MatrixType, UnitUpper>;
  static inline MatrixL getL(const MatrixType& m) { return MatrixL(m.adjoint()); }
  static inline MatrixU getU(const MatrixType& m) { return MatrixU(m); }
};
 
}  // end namespace internal

template <typename MatrixType, int UpLo_>
template <typename InputType>
LDLT<MatrixType, UpLo_>& LDLT<MatrixType, UpLo_>::compute(const EigenBase<InputType>& a) {
  eigen_assert(a.rows() == a.cols());
  const Index size = a.rows();
 
  m_matrix = a.derived();
 
  // Compute matrix L1 norm = max abs column sum over the implicit self-adjoint matrix.
  m_l1_norm = m_matrix.template selfadjointView<UpLo_>().l1Norm();
 
  m_transpositions.resize(size);
  m_isInitialized = false;
  m_temporary.resize(size);
  m_sign = internal::ZeroSign;
 
  m_info = internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign) ? Success
                                                                                                    : NumericalIssue;
 
  m_isInitialized = true;
  return *this;
}

template <typename MatrixType, int UpLo_>
template <typename Derived>
LDLT<MatrixType, UpLo_>& LDLT<MatrixType, UpLo_>::rankUpdate(
    const MatrixBase<Derived>& w, const typename LDLT<MatrixType, UpLo_>::RealScalar& sigma) {
  using IndexType = typename TranspositionType::StorageIndex;
  const Index size = w.rows();
  if (m_isInitialized) {
    eigen_assert(m_matrix.rows() == size);
  } else {
    m_matrix.resize(size, size);
    m_matrix.setZero();
    m_transpositions.resize(size);
    for (Index i = 0; i < size; i++) m_transpositions.coeffRef(i) = IndexType(i);
    m_temporary.resize(size);
    m_sign = sigma >= 0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
    m_isInitialized = true;
    // Assigned here rather than after the branch: updating an existing factorization keeps the status it already has.
    m_info = Success;
  }
 
  internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
 
  return *this;
}
 
// A = P^T L D L^* P with L unit lower triangular and D real diagonal, so det(A) = prod(D_ii).
 
template <typename MatrixType_, int UpLo_>
typename LDLT<MatrixType_, UpLo_>::Scalar LDLT<MatrixType_, UpLo_>::determinant() const {
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  eigen_assert(m_info == Success && "LDLT failed because of a zero pivot.");
  return Scalar(vectorD().real().prod());
}
 
template <typename MatrixType_, int UpLo_>
typename LDLT<MatrixType_, UpLo_>::RealScalar LDLT<MatrixType_, UpLo_>::absDeterminant() const {
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  eigen_assert(m_info == Success && "LDLT failed because of a zero pivot.");
  return numext::abs(vectorD().real().prod());
}
 
template <typename MatrixType_, int UpLo_>
typename LDLT<MatrixType_, UpLo_>::RealScalar LDLT<MatrixType_, UpLo_>::logAbsDeterminant() const {
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  eigen_assert(m_info == Success && "LDLT failed because of a zero pivot.");
  return vectorD().real().cwiseAbs().array().log().sum();
}
 
template <typename MatrixType_, int UpLo_>
typename LDLT<MatrixType_, UpLo_>::Scalar LDLT<MatrixType_, UpLo_>::signDeterminant() const {
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  eigen_assert(m_info == Success && "LDLT failed because of a zero pivot.");
  return Scalar(vectorD().real().array().sign().prod());
}
 
#ifndef EIGEN_PARSED_BY_DOXYGEN
template <typename MatrixType_, int UpLo_>
template <typename RhsType, typename DstType>
void LDLT<MatrixType_, UpLo_>::_solve_impl(const RhsType& rhs, DstType& dst) const {
  _solve_impl_transposed<true>(rhs, dst);
}
 
template <typename MatrixType_, int UpLo_>
template <bool Conjugate, typename RhsType, typename DstType>
void LDLT<MatrixType_, UpLo_>::_solve_impl_transposed(const RhsType& rhs, DstType& dst) const {
  // dst = P b
  dst = m_transpositions * rhs;
 
  // dst = L^-1 (P b)
  // dst = L^-*T (P b)
  matrixL().template conjugateIf<!Conjugate>().solveInPlace(dst);
 
  // dst = D^-* (L^-1 P b)
  // dst = D^-1 (L^-*T P b)
  // more precisely, use pseudo-inverse of D (see bug 241)
  using std::abs;
  const typename Diagonal<const MatrixType>::RealReturnType vecD(vectorD());
  // In some previous versions, tolerance was set to the max of 1/highest (or rather numeric_limits::min())
  // and the maximal diagonal entry * epsilon as motivated by LAPACK's xGELSS:
  // RealScalar tolerance = numext::maxi(vecD.array().abs().maxCoeff() * NumTraits<RealScalar>::epsilon(),RealScalar(1)
  // / NumTraits<RealScalar>::highest()); However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the
  // highest diagonal element is not well justified and leads to numerical issues in some cases. Moreover, Lapack's
  // xSYTRS routines use 0 for the tolerance. Using numeric_limits::min() gives us more robustness to denormals.
  RealScalar tolerance = (std::numeric_limits<RealScalar>::min)();
  for (Index i = 0; i < vecD.size(); ++i) {
    if (abs(vecD(i)) > tolerance)
      dst.row(i) /= vecD(i);
    else
      dst.row(i).setZero();
  }
 
  // dst = L^-* (D^-* L^-1 P b)
  // dst = L^-T (D^-1 L^-*T P b)
  matrixL().transpose().template conjugateIf<Conjugate>().solveInPlace(dst);
 
  // dst = P^T (L^-* D^-* L^-1 P b) = A^-1 b
  // dst = P^-T (L^-T D^-1 L^-*T P b) = A^-1 b
  dst = m_transpositions.transpose() * dst;
}
#endif

template <typename MatrixType, int UpLo_>
template <typename Derived>
bool LDLT<MatrixType, UpLo_>::solveInPlace(MatrixBase<Derived>& bAndX) const {
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  eigen_assert(m_matrix.rows() == bAndX.rows());
 
  bAndX = this->solve(bAndX);
 
  return true;
}

template <typename MatrixType, int UpLo_>
MatrixType LDLT<MatrixType, UpLo_>::reconstructedMatrix() const {
  eigen_assert(m_isInitialized && "LDLT is not initialized.");
  const Index size = m_matrix.rows();
  MatrixType res(size, size);
 
  // P
  res.setIdentity();
  res = transpositionsP() * res;
  // L^* P
  res = matrixU() * res;
  // D(L^*P)
  res = vectorD().real().asDiagonal() * res;
  // L(DL^*P)
  res = matrixL() * res;
  // P^T (LDL^*P)
  res = transpositionsP().transpose() * res;
 
  return res;
}

template <typename MatrixType, unsigned int UpLo>
inline LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo> SelfAdjointView<MatrixType, UpLo>::ldlt()
    const {
  return LDLT<PlainObject, UpLo>(m_matrix);
}

template <typename Derived>
inline LDLT<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::ldlt() const {
  return LDLT<PlainObject>(derived());
}
 
}  // end namespace Eigen
 
#endif  // EIGEN_LDLT_H