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-// This file is part of Eigen, a lightweight C++ template library
-// for linear algebra.
-//
-// Copyright (C) 2008 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_LLT_H
-#define EIGEN_LLT_H
-
-namespace Eigen { 
-
-namespace internal{
-template<typename MatrixType, int UpLo> struct LLT_Traits;
-}
-
-/** \ingroup Cholesky_Module
-  *
-  * \class LLT
-  *
-  * \brief Standard Cholesky decomposition (LL^T) of a matrix and associated features
-  *
-  * \param MatrixType the type of the matrix of which we are computing the LL^T Cholesky decomposition
-  * \param UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
-  *             The other triangular part won't be read.
-  *
-  * This class performs a LL^T Cholesky decomposition of a symmetric, positive definite
-  * matrix A such that A = LL^* = U^*U, where L is lower triangular.
-  *
-  * While the Cholesky decomposition is particularly useful to solve selfadjoint problems like  D^*D x = b,
-  * for that purpose, we recommend the Cholesky decomposition without square root which is more stable
-  * and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other
-  * situations like generalised eigen problems with hermitian matrices.
-  *
-  * Remember that Cholesky decompositions are not rank-revealing. This LLT decomposition is only stable on positive definite matrices,
-  * use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations
-  * has a solution.
-  *
-  * Example: \include LLT_example.cpp
-  * Output: \verbinclude LLT_example.out
-  *    
-  * \sa MatrixBase::llt(), class LDLT
-  */
- /* HEY THIS DOX IS DISABLED BECAUSE THERE's A BUG EITHER HERE OR IN LDLT ABOUT THAT (OR BOTH)
-  * Note that during the decomposition, only the upper triangular part of A is considered. Therefore,
-  * the strict lower part does not have to store correct values.
-  */
-template<typename _MatrixType, int _UpLo> class LLT
-{
-  public:
-    typedef _MatrixType MatrixType;
-    enum {
-      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
-      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
-      Options = MatrixType::Options,
-      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
-    };
-    typedef typename MatrixType::Scalar Scalar;
-    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
-    typedef typename MatrixType::Index Index;
-
-    enum {
-      PacketSize = internal::packet_traits<Scalar>::size,
-      AlignmentMask = int(PacketSize)-1,
-      UpLo = _UpLo
-    };
-
-    typedef internal::LLT_Traits<MatrixType,UpLo> Traits;
-
-    /**
-      * \brief Default Constructor.
-      *
-      * The default constructor is useful in cases in which the user intends to
-      * perform decompositions via LLT::compute(const MatrixType&).
-      */
-    LLT() : m_matrix(), 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 LLT()
-      */
-    LLT(Index size) : m_matrix(size, size),
-                    m_isInitialized(false) {}
-
-    LLT(const MatrixType& matrix)
-      : m_matrix(matrix.rows(), matrix.cols()),
-        m_isInitialized(false)
-    {
-      compute(matrix);
-    }
-
-    /** \returns a view of the upper triangular matrix U */
-    inline typename Traits::MatrixU matrixU() const
-    {
-      eigen_assert(m_isInitialized && "LLT 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 && "LLT is not initialized.");
-      return Traits::getL(m_matrix);
-    }
-
-    /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
-      *
-      * Since this LLT class assumes anyway that the matrix A is invertible, the solution
-      * theoretically exists and is unique regardless of b.
-      *
-      * Example: \include LLT_solve.cpp
-      * Output: \verbinclude LLT_solve.out
-      *
-      * \sa solveInPlace(), MatrixBase::llt()
-      */
-    template<typename Rhs>
-    inline const internal::solve_retval<LLT, Rhs>
-    solve(const MatrixBase<Rhs>& b) const
-    {
-      eigen_assert(m_isInitialized && "LLT is not initialized.");
-      eigen_assert(m_matrix.rows()==b.rows()
-                && "LLT::solve(): invalid number of rows of the right hand side matrix b");
-      return internal::solve_retval<LLT, Rhs>(*this, b.derived());
-    }
-
-    #ifdef EIGEN2_SUPPORT
-    template<typename OtherDerived, typename ResultType>
-    bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const
-    {
-      *result = this->solve(b);
-      return true;
-    }
-    
-    bool isPositiveDefinite() const { return true; }
-    #endif
-
-    template<typename Derived>
-    void solveInPlace(MatrixBase<Derived> &bAndX) const;
-
-    LLT& compute(const MatrixType& matrix);
-
-    /** \returns the LLT decomposition matrix
-      *
-      * TODO: document the storage layout
-      */
-    inline const MatrixType& matrixLLT() const
-    {
-      eigen_assert(m_isInitialized && "LLT is not initialized.");
-      return m_matrix;
-    }
-
-    MatrixType reconstructedMatrix() const;
-
-
-    /** \brief Reports whether previous computation was successful.
-      *
-      * \returns \c Success if computation was succesful,
-      *          \c NumericalIssue if the matrix.appears to be negative.
-      */
-    ComputationInfo info() const
-    {
-      eigen_assert(m_isInitialized && "LLT is not initialized.");
-      return m_info;
-    }
-
-    inline Index rows() const { return m_matrix.rows(); }
-    inline Index cols() const { return m_matrix.cols(); }
-
-    template<typename VectorType>
-    LLT rankUpdate(const VectorType& vec, const RealScalar& sigma = 1);
-
-  protected:
-    /** \internal
-      * Used to compute and store L
-      * The strict upper part is not used and even not initialized.
-      */
-    MatrixType m_matrix;
-    bool m_isInitialized;
-    ComputationInfo m_info;
-};
-
-namespace internal {
-
-template<typename Scalar, int UpLo> struct llt_inplace;
-
-template<typename MatrixType, typename VectorType>
-static typename MatrixType::Index llt_rank_update_lower(MatrixType& mat, const VectorType& vec, const typename MatrixType::RealScalar& sigma)
-{
-  using std::sqrt;
-  typedef typename MatrixType::Scalar Scalar;
-  typedef typename MatrixType::RealScalar RealScalar;
-  typedef typename MatrixType::Index Index;
-  typedef typename MatrixType::ColXpr ColXpr;
-  typedef typename internal::remove_all<ColXpr>::type ColXprCleaned;
-  typedef typename ColXprCleaned::SegmentReturnType ColXprSegment;
-  typedef Matrix<Scalar,Dynamic,1> TempVectorType;
-  typedef typename TempVectorType::SegmentReturnType TempVecSegment;
-
-  Index n = mat.cols();
-  eigen_assert(mat.rows()==n && vec.size()==n);
-
-  TempVectorType temp;
-
-  if(sigma>0)
-  {
-    // This version is based on Givens rotations.
-    // It is faster than the other one below, but only works for updates,
-    // i.e., for sigma > 0
-    temp = sqrt(sigma) * vec;
-
-    for(Index i=0; i<n; ++i)
-    {
-      JacobiRotation<Scalar> g;
-      g.makeGivens(mat(i,i), -temp(i), &mat(i,i));
-
-      Index rs = n-i-1;
-      if(rs>0)
-      {
-        ColXprSegment x(mat.col(i).tail(rs));
-        TempVecSegment y(temp.tail(rs));
-        apply_rotation_in_the_plane(x, y, g);
-      }
-    }
-  }
-  else
-  {
-    temp = vec;
-    RealScalar beta = 1;
-    for(Index j=0; j<n; ++j)
-    {
-      RealScalar Ljj = numext::real(mat.coeff(j,j));
-      RealScalar dj = numext::abs2(Ljj);
-      Scalar wj = temp.coeff(j);
-      RealScalar swj2 = sigma*numext::abs2(wj);
-      RealScalar gamma = dj*beta + swj2;
-
-      RealScalar x = dj + swj2/beta;
-      if (x<=RealScalar(0))
-        return j;
-      RealScalar nLjj = sqrt(x);
-      mat.coeffRef(j,j) = nLjj;
-      beta += swj2/dj;
-
-      // Update the terms of L
-      Index rs = n-j-1;
-      if(rs)
-      {
-        temp.tail(rs) -= (wj/Ljj) * mat.col(j).tail(rs);
-        if(gamma != 0)
-          mat.col(j).tail(rs) = (nLjj/Ljj) * mat.col(j).tail(rs) + (nLjj * sigma*numext::conj(wj)/gamma)*temp.tail(rs);
-      }
-    }
-  }
-  return -1;
-}
-
-template<typename Scalar> struct llt_inplace<Scalar, Lower>
-{
-  typedef typename NumTraits<Scalar>::Real RealScalar;
-  template<typename MatrixType>
-  static typename MatrixType::Index unblocked(MatrixType& mat)
-  {
-    using std::sqrt;
-    typedef typename MatrixType::Index Index;
-    
-    eigen_assert(mat.rows()==mat.cols());
-    const Index size = mat.rows();
-    for(Index k = 0; k < size; ++k)
-    {
-      Index rs = size-k-1; // remaining size
-
-      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);
-
-      RealScalar x = numext::real(mat.coeff(k,k));
-      if (k>0) x -= A10.squaredNorm();
-      if (x<=RealScalar(0))
-        return k;
-      mat.coeffRef(k,k) = x = sqrt(x);
-      if (k>0 && rs>0) A21.noalias() -= A20 * A10.adjoint();
-      if (rs>0) A21 *= RealScalar(1)/x;
-    }
-    return -1;
-  }
-
-  template<typename MatrixType>
-  static typename MatrixType::Index blocked(MatrixType& m)
-  {
-    typedef typename MatrixType::Index Index;
-    eigen_assert(m.rows()==m.cols());
-    Index size = m.rows();
-    if(size<32)
-      return unblocked(m);
-
-    Index blockSize = size/8;
-    blockSize = (blockSize/16)*16;
-    blockSize = (std::min)((std::max)(blockSize,Index(8)), Index(128));
-
-    for (Index k=0; k<size; k+=blockSize)
-    {
-      // partition the matrix:
-      //       A00 |  -  |  -
-      // lu  = A10 | A11 |  -
-      //       A20 | A21 | A22
-      Index bs = (std::min)(blockSize, size-k);
-      Index rs = size - k - bs;
-      Block<MatrixType,Dynamic,Dynamic> A11(m,k,   k,   bs,bs);
-      Block<MatrixType,Dynamic,Dynamic> A21(m,k+bs,k,   rs,bs);
-      Block<MatrixType,Dynamic,Dynamic> A22(m,k+bs,k+bs,rs,rs);
-
-      Index ret;
-      if((ret=unblocked(A11))>=0) return k+ret;
-      if(rs>0) A11.adjoint().template triangularView<Upper>().template solveInPlace<OnTheRight>(A21);
-      if(rs>0) A22.template selfadjointView<Lower>().rankUpdate(A21,-1); // bottleneck
-    }
-    return -1;
-  }
-
-  template<typename MatrixType, typename VectorType>
-  static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
-  {
-    return Eigen::internal::llt_rank_update_lower(mat, vec, sigma);
-  }
-};
-  
-template<typename Scalar> struct llt_inplace<Scalar, Upper>
-{
-  typedef typename NumTraits<Scalar>::Real RealScalar;
-
-  template<typename MatrixType>
-  static EIGEN_STRONG_INLINE typename MatrixType::Index unblocked(MatrixType& mat)
-  {
-    Transpose<MatrixType> matt(mat);
-    return llt_inplace<Scalar, Lower>::unblocked(matt);
-  }
-  template<typename MatrixType>
-  static EIGEN_STRONG_INLINE typename MatrixType::Index blocked(MatrixType& mat)
-  {
-    Transpose<MatrixType> matt(mat);
-    return llt_inplace<Scalar, Lower>::blocked(matt);
-  }
-  template<typename MatrixType, typename VectorType>
-  static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
-  {
-    Transpose<MatrixType> matt(mat);
-    return llt_inplace<Scalar, Lower>::rankUpdate(matt, vec.conjugate(), sigma);
-  }
-};
-
-template<typename MatrixType> struct LLT_Traits<MatrixType,Lower>
-{
-  typedef const TriangularView<const MatrixType, Lower> MatrixL;
-  typedef const TriangularView<const typename MatrixType::AdjointReturnType, Upper> MatrixU;
-  static inline MatrixL getL(const MatrixType& m) { return m; }
-  static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
-  static bool inplace_decomposition(MatrixType& m)
-  { return llt_inplace<typename MatrixType::Scalar, Lower>::blocked(m)==-1; }
-};
-
-template<typename MatrixType> struct LLT_Traits<MatrixType,Upper>
-{
-  typedef const TriangularView<const typename MatrixType::AdjointReturnType, Lower> MatrixL;
-  typedef const TriangularView<const MatrixType, Upper> MatrixU;
-  static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
-  static inline MatrixU getU(const MatrixType& m) { return m; }
-  static bool inplace_decomposition(MatrixType& m)
-  { return llt_inplace<typename MatrixType::Scalar, Upper>::blocked(m)==-1; }
-};
-
-} // end namespace internal
-
-/** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix
-  *
-  * \returns a reference to *this
-  *
-  * Example: \include TutorialLinAlgComputeTwice.cpp
-  * Output: \verbinclude TutorialLinAlgComputeTwice.out
-  */
-template<typename MatrixType, int _UpLo>
-LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const MatrixType& a)
-{
-  eigen_assert(a.rows()==a.cols());
-  const Index size = a.rows();
-  m_matrix.resize(size, size);
-  m_matrix = a;
-
-  m_isInitialized = true;
-  bool ok = Traits::inplace_decomposition(m_matrix);
-  m_info = ok ? Success : NumericalIssue;
-
-  return *this;
-}
-
-/** Performs a rank one update (or dowdate) of the current decomposition.
-  * If A = LL^* before the rank one update,
-  * then after it we have LL^* = A + sigma * v v^* where \a v must be a vector
-  * of same dimension.
-  */
-template<typename _MatrixType, int _UpLo>
-template<typename VectorType>
-LLT<_MatrixType,_UpLo> LLT<_MatrixType,_UpLo>::rankUpdate(const VectorType& v, const RealScalar& sigma)
-{
-  EIGEN_STATIC_ASSERT_VECTOR_ONLY(VectorType);
-  eigen_assert(v.size()==m_matrix.cols());
-  eigen_assert(m_isInitialized);
-  if(internal::llt_inplace<typename MatrixType::Scalar, UpLo>::rankUpdate(m_matrix,v,sigma)>=0)
-    m_info = NumericalIssue;
-  else
-    m_info = Success;
-
-  return *this;
-}
-    
-namespace internal {
-template<typename _MatrixType, int UpLo, typename Rhs>
-struct solve_retval<LLT<_MatrixType, UpLo>, Rhs>
-  : solve_retval_base<LLT<_MatrixType, UpLo>, Rhs>
-{
-  typedef LLT<_MatrixType,UpLo> LLTType;
-  EIGEN_MAKE_SOLVE_HELPERS(LLTType,Rhs)
-
-  template<typename Dest> void evalTo(Dest& dst) const
-  {
-    dst = rhs();
-    dec().solveInPlace(dst);
-  }
-};
-}
-
-/** \internal use x = llt_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 LLT::solve(), MatrixBase::llt()
-  */
-template<typename MatrixType, int _UpLo>
-template<typename Derived>
-void LLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
-{
-  eigen_assert(m_isInitialized && "LLT is not initialized.");
-  eigen_assert(m_matrix.rows()==bAndX.rows());
-  matrixL().solveInPlace(bAndX);
-  matrixU().solveInPlace(bAndX);
-}
-
-/** \returns the matrix represented by the decomposition,
- * i.e., it returns the product: L L^*.
- * This function is provided for debug purpose. */
-template<typename MatrixType, int _UpLo>
-MatrixType LLT<MatrixType,_UpLo>::reconstructedMatrix() const
-{
-  eigen_assert(m_isInitialized && "LLT is not initialized.");
-  return matrixL() * matrixL().adjoint().toDenseMatrix();
-}
-
-/** \cholesky_module
-  * \returns the LLT decomposition of \c *this
-  */
-template<typename Derived>
-inline const LLT<typename MatrixBase<Derived>::PlainObject>
-MatrixBase<Derived>::llt() const
-{
-  return LLT<PlainObject>(derived());
-}
-
-/** \cholesky_module
-  * \returns the LLT decomposition of \c *this
-  */
-template<typename MatrixType, unsigned int UpLo>
-inline const LLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
-SelfAdjointView<MatrixType, UpLo>::llt() const
-{
-  return LLT<PlainObject,UpLo>(m_matrix);
-}
-
-} // end namespace Eigen
-
-#endif // EIGEN_LLT_H