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 // 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/.
 
 #ifndef EIGEN_LDLT_H
 #define EIGEN_LDLT_H
 
 namespace Eigen {
 
 namespace internal {
   template<typename _MatrixType, int _UpLo> struct traits<LDLT<_MatrixType, _UpLo> >
    : traits<_MatrixType>
   {
     typedef MatrixXpr XprKind;
     typedef SolverStorage StorageKind;
     typedef int StorageIndex;
     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 };
 }
 
 /** \ingroup Cholesky_Module
   *
   * \class LDLT
   *
   * \brief Robust Cholesky decomposition of a matrix with pivoting
   *
   * \tparam _MatrixType the type of the matrix of which to compute the LDL^T Cholesky decomposition
   * \tparam _UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
   *             The other triangular part won't be read.
   *
   * Perform a robust Cholesky decomposition of a positive semidefinite or negative semidefinite
   * matrix \f$ A \f$ such that \f$ A =  P^TLDL^*P \f$, where P is a permutation matrix, L
   * is lower triangular with a unit diagonal and D is a diagonal matrix.
   *
   * The decomposition uses pivoting to ensure stability, so that D will have
   * zeros in the bottom right rank(A) - n submatrix. Avoiding the square root
   * on D also stabilizes the computation.
   *
   * Remember that Cholesky decompositions are not rank-revealing. Also, do not use a Cholesky
   * decomposition to determine whether a system of equations has a solution.
   *
   * This class supports the \link InplaceDecomposition inplace decomposition \endlink mechanism.
   *
   * \sa MatrixBase::ldlt(), SelfAdjointView::ldlt(), class LLT
   */
 template<typename _MatrixType, int _UpLo> class LDLT
         : public SolverBase<LDLT<_MatrixType, _UpLo> >
 {
   public:
     typedef _MatrixType MatrixType;
     typedef SolverBase<LDLT> Base;
     friend class SolverBase<LDLT>;
 
     EIGEN_GENERIC_PUBLIC_INTERFACE(LDLT)
     enum {
       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
       UpLo = _UpLo
     };
     typedef Matrix<Scalar, RowsAtCompileTime, 1, 0, MaxRowsAtCompileTime, 1> TmpMatrixType;
 
     typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
     typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
 
     typedef internal::LDLT_Traits<MatrixType,UpLo> Traits;
 
     /** \brief Default Constructor.
       *
       * The default constructor is useful in cases in which the user intends to
       * perform decompositions via LDLT::compute(const MatrixType&).
       */
     LDLT()
       : m_matrix(),
         m_transpositions(),
         m_sign(internal::ZeroSign),
         m_isInitialized(false)
     {}
 
     /** \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 LDLT()
       */
     explicit LDLT(Index size)
       : m_matrix(size, size),
         m_transpositions(size),
         m_temporary(size),
         m_sign(internal::ZeroSign),
         m_isInitialized(false)
     {}
 
     /** \brief Constructor with decomposition
       *
       * This calculates the decomposition for the input \a matrix.
       *
       * \sa LDLT(Index size)
       */
     template<typename InputType>
     explicit LDLT(const EigenBase<InputType>& matrix)
       : m_matrix(matrix.rows(), matrix.cols()),
         m_transpositions(matrix.rows()),
         m_temporary(matrix.rows()),
         m_sign(internal::ZeroSign),
         m_isInitialized(false)
     {
       compute(matrix.derived());
     }
 
     /** \brief Constructs a LDLT factorization from a given matrix
       *
       * This overloaded constructor is provided for \link InplaceDecomposition inplace decomposition \endlink when \c MatrixType is a Eigen::Ref.
       *
       * \sa LDLT(const EigenBase&)
       */
     template<typename InputType>
     explicit LDLT(EigenBase<InputType>& matrix)
       : m_matrix(matrix.derived()),
         m_transpositions(matrix.rows()),
         m_temporary(matrix.rows()),
         m_sign(internal::ZeroSign),
         m_isInitialized(false)
     {
       compute(matrix.derived());
     }
 
     /** Clear any existing decomposition
      * \sa rankUpdate(w,sigma)
      */
     void setZero()
     {
       m_isInitialized = false;
     }
 
     /** \returns a view of the upper triangular matrix U */
     inline typename Traits::MatrixU matrixU() const
     {
       eigen_assert(m_isInitialized && "LDLT is not initialized.");
       return Traits::getU(m_matrix);
     }
 
     /** \returns a view of the lower triangular matrix L */
     inline typename Traits::MatrixL matrixL() const
     {
       eigen_assert(m_isInitialized && "LDLT is not initialized.");
       return Traits::getL(m_matrix);
     }
 
     /** \returns the permutation matrix P as a transposition sequence.
       */
     inline const TranspositionType& transpositionsP() const
     {
       eigen_assert(m_isInitialized && "LDLT is not initialized.");
       return m_transpositions;
     }
 
     /** \returns the coefficients of the diagonal matrix D */
     inline Diagonal<const MatrixType> vectorD() const
     {
       eigen_assert(m_isInitialized && "LDLT is not initialized.");
       return m_matrix.diagonal();
     }
 
     /** \returns true if the matrix is positive (semidefinite) */
     inline bool isPositive() const
     {
       eigen_assert(m_isInitialized && "LDLT is not initialized.");
       return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
     }
 
     /** \returns true if the matrix is negative (semidefinite) */
     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
     /** \returns a solution x of \f$ A x = b \f$ using the current decomposition of A.
       *
       * This function also supports in-place solves using the syntax <tt>x = decompositionObject.solve(x)</tt> .
       *
       * \note_about_checking_solutions
       *
       * More precisely, this method solves \f$ A x = b \f$ using the decomposition \f$ A = P^T L D L^* P \f$
       * by solving the systems \f$ P^T y_1 = b \f$, \f$ L y_2 = y_1 \f$, \f$ D y_3 = y_2 \f$,
       * \f$ L^* y_4 = y_3 \f$ and \f$ P x = y_4 \f$ in succession. If the matrix \f$ A \f$ is singular, then
       * \f$ D \f$ will also be singular (all the other matrices are invertible). In that case, the
       * least-square solution of \f$ D y_3 = y_2 \f$ is computed. This does not mean that this function
       * computes the least-square solution of \f$ A x = b \f$ if \f$ A \f$ is singular.
       *
       * \sa MatrixBase::ldlt(), SelfAdjointView::ldlt()
       */
     template<typename Rhs>
     inline const 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);
 
     /** \returns an estimate of the reciprocal condition number of the matrix of
      *  which \c *this is the LDLT decomposition.
      */
     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);
 
     /** \returns the internal LDLT decomposition matrix
       *
       * TODO: document the storage layout
       */
     inline const MatrixType& matrixLDLT() const
     {
       eigen_assert(m_isInitialized && "LDLT is not initialized.");
       return m_matrix;
     }
 
     MatrixType reconstructedMatrix() const;
 
     /** \returns the adjoint of \c *this, that is, a const reference to the decomposition itself as the underlying matrix is self-adjoint.
       *
       * This method is provided for compatibility with other matrix decompositions, thus enabling generic code such as:
       * \code x = decomposition.adjoint().solve(b) \endcode
       */
     const LDLT& adjoint() const { return *this; };
 
     EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return m_matrix.rows(); }
     EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }
 
     /** \brief Reports whether previous computation was successful.
       *
       * \returns \c Success if computation was successful,
       *          \c NumericalIssue if the factorization failed because of a zero pivot.
       */
     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:
 
     static void check_template_parameters()
     {
       EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
     }
 
     /** \internal
       * Used to compute and store the Cholesky decomposition A = L D L^* = U^* D U.
       * The strict upper part is used during the decomposition, the strict lower
       * part correspond to the coefficients of L (its diagonal is equal to 1 and
       * is not stored), and the diagonal entries correspond to D.
       */
     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;
     typedef typename MatrixType::Scalar Scalar;
     typedef typename MatrixType::RealScalar RealScalar;
     typedef typename TranspositionType::StorageIndex IndexType;
     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);
         }
         if(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;
     typedef typename MatrixType::Scalar Scalar;
     typedef typename MatrixType::RealScalar 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;
 
       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(gamma != 0)
         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>
 {
   typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
   typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
   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>
 {
   typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
   typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
   static inline MatrixL getL(const MatrixType& m) { return MatrixL(m.adjoint()); }
   static inline MatrixU getU(const MatrixType& m) { return MatrixU(m); }
 };
 
 } // end namespace internal
 
 /** Compute / recompute the LDLT decomposition A = L D L^* = U^* D U of \a matrix
   */
 template<typename MatrixType, int _UpLo>
 template<typename InputType>
 LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const EigenBase<InputType>& a)
 {
   check_template_parameters();
 
   eigen_assert(a.rows()==a.cols());
   const Index size = a.rows();
 
   m_matrix = a.derived();
 
   // Compute matrix L1 norm = max abs column sum.
   m_l1_norm = RealScalar(0);
   // TODO move this code to SelfAdjointView
   for (Index col = 0; col < size; ++col) {
     RealScalar abs_col_sum;
     if (_UpLo == Lower)
       abs_col_sum = m_matrix.col(col).tail(size - col).template lpNorm<1>() + m_matrix.row(col).head(col).template lpNorm<1>();
     else
       abs_col_sum = m_matrix.col(col).head(col).template lpNorm<1>() + m_matrix.row(col).tail(size - col).template lpNorm<1>();
     if (abs_col_sum > m_l1_norm)
       m_l1_norm = abs_col_sum;
   }
 
   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;
 }
 
 /** Update the LDLT decomposition:  given A = L D L^T, efficiently compute the decomposition of A + sigma w w^T.
  * \param w a vector to be incorporated into the decomposition.
  * \param sigma a scalar, +1 for updates and -1 for "downdates," which correspond to removing previously-added column vectors. Optional; default value is +1.
  * \sa setZero()
   */
 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)
 {
   typedef typename TranspositionType::StorageIndex IndexType;
   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;
   }
 
   internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
 
   return *this;
 }
 
 #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
 
 /** \internal use x = ldlt_object.solve(x);
   *
   * This is the \em in-place version of solve().
   *
   * \param bAndX represents both the right-hand side matrix b and result x.
   *
   * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
   *
   * This version avoids a copy when the right hand side matrix b is not
   * needed anymore.
   *
   * \sa LDLT::solve(), MatrixBase::ldlt()
   */
 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;
 }
 
 /** \returns the matrix represented by the decomposition,
  * i.e., it returns the product: P^T L D L^* P.
  * This function is provided for debug purpose. */
 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;
 }
 
 /** \cholesky_module
   * \returns the Cholesky decomposition with full pivoting without square root of \c *this
   * \sa MatrixBase::ldlt()
   */
 template<typename MatrixType, unsigned int UpLo>
 inline const LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
 SelfAdjointView<MatrixType, UpLo>::ldlt() const
 {
   return LDLT<PlainObject,UpLo>(m_matrix);
 }
 
 /** \cholesky_module
   * \returns the Cholesky decomposition with full pivoting without square root of \c *this
   * \sa SelfAdjointView::ldlt()
   */
 template<typename Derived>
 inline const LDLT<typename MatrixBase<Derived>::PlainObject>
 MatrixBase<Derived>::ldlt() const
 {
   return LDLT<PlainObject>(derived());
 }
 
 } // end namespace Eigen
 
 #endif // EIGEN_LDLT_H

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