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      1 // This file is part of Eigen, a lightweight C++ template library
      2 // for linear algebra.
      3 //
      4 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud (at) inria.fr>
      5 // Copyright (C) 2010,2012 Jitse Niesen <jitse (at) maths.leeds.ac.uk>
      6 //
      7 // This Source Code Form is subject to the terms of the Mozilla
      8 // Public License v. 2.0. If a copy of the MPL was not distributed
      9 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
     10 
     11 #ifndef EIGEN_REAL_SCHUR_H
     12 #define EIGEN_REAL_SCHUR_H
     13 
     14 #include "./HessenbergDecomposition.h"
     15 
     16 namespace Eigen {
     17 
     18 /** \eigenvalues_module \ingroup Eigenvalues_Module
     19   *
     20   *
     21   * \class RealSchur
     22   *
     23   * \brief Performs a real Schur decomposition of a square matrix
     24   *
     25   * \tparam _MatrixType the type of the matrix of which we are computing the
     26   * real Schur decomposition; this is expected to be an instantiation of the
     27   * Matrix class template.
     28   *
     29   * Given a real square matrix A, this class computes the real Schur
     30   * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
     31   * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
     32   * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
     33   * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
     34   * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
     35   * blocks on the diagonal of T are the same as the eigenvalues of the matrix
     36   * A, and thus the real Schur decomposition is used in EigenSolver to compute
     37   * the eigendecomposition of a matrix.
     38   *
     39   * Call the function compute() to compute the real Schur decomposition of a
     40   * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
     41   * constructor which computes the real Schur decomposition at construction
     42   * time. Once the decomposition is computed, you can use the matrixU() and
     43   * matrixT() functions to retrieve the matrices U and T in the decomposition.
     44   *
     45   * The documentation of RealSchur(const MatrixType&, bool) contains an example
     46   * of the typical use of this class.
     47   *
     48   * \note The implementation is adapted from
     49   * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
     50   * Their code is based on EISPACK.
     51   *
     52   * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
     53   */
     54 template<typename _MatrixType> class RealSchur
     55 {
     56   public:
     57     typedef _MatrixType MatrixType;
     58     enum {
     59       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
     60       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
     61       Options = MatrixType::Options,
     62       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
     63       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
     64     };
     65     typedef typename MatrixType::Scalar Scalar;
     66     typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
     67     typedef typename MatrixType::Index Index;
     68 
     69     typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
     70     typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
     71 
     72     /** \brief Default constructor.
     73       *
     74       * \param [in] size  Positive integer, size of the matrix whose Schur decomposition will be computed.
     75       *
     76       * The default constructor is useful in cases in which the user intends to
     77       * perform decompositions via compute().  The \p size parameter is only
     78       * used as a hint. It is not an error to give a wrong \p size, but it may
     79       * impair performance.
     80       *
     81       * \sa compute() for an example.
     82       */
     83     RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
     84             : m_matT(size, size),
     85               m_matU(size, size),
     86               m_workspaceVector(size),
     87               m_hess(size),
     88               m_isInitialized(false),
     89               m_matUisUptodate(false),
     90               m_maxIters(-1)
     91     { }
     92 
     93     /** \brief Constructor; computes real Schur decomposition of given matrix.
     94       *
     95       * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
     96       * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
     97       *
     98       * This constructor calls compute() to compute the Schur decomposition.
     99       *
    100       * Example: \include RealSchur_RealSchur_MatrixType.cpp
    101       * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
    102       */
    103     RealSchur(const MatrixType& matrix, bool computeU = true)
    104             : m_matT(matrix.rows(),matrix.cols()),
    105               m_matU(matrix.rows(),matrix.cols()),
    106               m_workspaceVector(matrix.rows()),
    107               m_hess(matrix.rows()),
    108               m_isInitialized(false),
    109               m_matUisUptodate(false),
    110               m_maxIters(-1)
    111     {
    112       compute(matrix, computeU);
    113     }
    114 
    115     /** \brief Returns the orthogonal matrix in the Schur decomposition.
    116       *
    117       * \returns A const reference to the matrix U.
    118       *
    119       * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
    120       * member function compute(const MatrixType&, bool) has been called before
    121       * to compute the Schur decomposition of a matrix, and \p computeU was set
    122       * to true (the default value).
    123       *
    124       * \sa RealSchur(const MatrixType&, bool) for an example
    125       */
    126     const MatrixType& matrixU() const
    127     {
    128       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
    129       eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
    130       return m_matU;
    131     }
    132 
    133     /** \brief Returns the quasi-triangular matrix in the Schur decomposition.
    134       *
    135       * \returns A const reference to the matrix T.
    136       *
    137       * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
    138       * member function compute(const MatrixType&, bool) has been called before
    139       * to compute the Schur decomposition of a matrix.
    140       *
    141       * \sa RealSchur(const MatrixType&, bool) for an example
    142       */
    143     const MatrixType& matrixT() const
    144     {
    145       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
    146       return m_matT;
    147     }
    148 
    149     /** \brief Computes Schur decomposition of given matrix.
    150       *
    151       * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
    152       * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
    153       * \returns    Reference to \c *this
    154       *
    155       * The Schur decomposition is computed by first reducing the matrix to
    156       * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
    157       * matrix is then reduced to triangular form by performing Francis QR
    158       * iterations with implicit double shift. The cost of computing the Schur
    159       * decomposition depends on the number of iterations; as a rough guide, it
    160       * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
    161       * \f$10n^3\f$ flops if \a computeU is false.
    162       *
    163       * Example: \include RealSchur_compute.cpp
    164       * Output: \verbinclude RealSchur_compute.out
    165       *
    166       * \sa compute(const MatrixType&, bool, Index)
    167       */
    168     RealSchur& compute(const MatrixType& matrix, bool computeU = true);
    169 
    170     /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
    171      *  \param[in] matrixH Matrix in Hessenberg form H
    172      *  \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
    173      *  \param computeU Computes the matriX U of the Schur vectors
    174      * \return Reference to \c *this
    175      *
    176      *  This routine assumes that the matrix is already reduced in Hessenberg form matrixH
    177      *  using either the class HessenbergDecomposition or another mean.
    178      *  It computes the upper quasi-triangular matrix T of the Schur decomposition of H
    179      *  When computeU is true, this routine computes the matrix U such that
    180      *  A = U T U^T =  (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
    181      *
    182      * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
    183      * is not available, the user should give an identity matrix (Q.setIdentity())
    184      *
    185      * \sa compute(const MatrixType&, bool)
    186      */
    187     template<typename HessMatrixType, typename OrthMatrixType>
    188     RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU);
    189     /** \brief Reports whether previous computation was successful.
    190       *
    191       * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
    192       */
    193     ComputationInfo info() const
    194     {
    195       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
    196       return m_info;
    197     }
    198 
    199     /** \brief Sets the maximum number of iterations allowed.
    200       *
    201       * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
    202       * of the matrix.
    203       */
    204     RealSchur& setMaxIterations(Index maxIters)
    205     {
    206       m_maxIters = maxIters;
    207       return *this;
    208     }
    209 
    210     /** \brief Returns the maximum number of iterations. */
    211     Index getMaxIterations()
    212     {
    213       return m_maxIters;
    214     }
    215 
    216     /** \brief Maximum number of iterations per row.
    217       *
    218       * If not otherwise specified, the maximum number of iterations is this number times the size of the
    219       * matrix. It is currently set to 40.
    220       */
    221     static const int m_maxIterationsPerRow = 40;
    222 
    223   private:
    224 
    225     MatrixType m_matT;
    226     MatrixType m_matU;
    227     ColumnVectorType m_workspaceVector;
    228     HessenbergDecomposition<MatrixType> m_hess;
    229     ComputationInfo m_info;
    230     bool m_isInitialized;
    231     bool m_matUisUptodate;
    232     Index m_maxIters;
    233 
    234     typedef Matrix<Scalar,3,1> Vector3s;
    235 
    236     Scalar computeNormOfT();
    237     Index findSmallSubdiagEntry(Index iu);
    238     void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
    239     void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
    240     void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
    241     void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
    242 };
    243 
    244 
    245 template<typename MatrixType>
    246 RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
    247 {
    248   eigen_assert(matrix.cols() == matrix.rows());
    249   Index maxIters = m_maxIters;
    250   if (maxIters == -1)
    251     maxIters = m_maxIterationsPerRow * matrix.rows();
    252 
    253   // Step 1. Reduce to Hessenberg form
    254   m_hess.compute(matrix);
    255 
    256   // Step 2. Reduce to real Schur form
    257   computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
    258 
    259   return *this;
    260 }
    261 template<typename MatrixType>
    262 template<typename HessMatrixType, typename OrthMatrixType>
    263 RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU)
    264 {
    265   m_matT = matrixH;
    266   if(computeU)
    267     m_matU = matrixQ;
    268 
    269   Index maxIters = m_maxIters;
    270   if (maxIters == -1)
    271     maxIters = m_maxIterationsPerRow * matrixH.rows();
    272   m_workspaceVector.resize(m_matT.cols());
    273   Scalar* workspace = &m_workspaceVector.coeffRef(0);
    274 
    275   // The matrix m_matT is divided in three parts.
    276   // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
    277   // Rows il,...,iu is the part we are working on (the active window).
    278   // Rows iu+1,...,end are already brought in triangular form.
    279   Index iu = m_matT.cols() - 1;
    280   Index iter = 0;      // iteration count for current eigenvalue
    281   Index totalIter = 0; // iteration count for whole matrix
    282   Scalar exshift(0);   // sum of exceptional shifts
    283   Scalar norm = computeNormOfT();
    284 
    285   if(norm!=0)
    286   {
    287     while (iu >= 0)
    288     {
    289       Index il = findSmallSubdiagEntry(iu);
    290 
    291       // Check for convergence
    292       if (il == iu) // One root found
    293       {
    294         m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
    295         if (iu > 0)
    296           m_matT.coeffRef(iu, iu-1) = Scalar(0);
    297         iu--;
    298         iter = 0;
    299       }
    300       else if (il == iu-1) // Two roots found
    301       {
    302         splitOffTwoRows(iu, computeU, exshift);
    303         iu -= 2;
    304         iter = 0;
    305       }
    306       else // No convergence yet
    307       {
    308         // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
    309         Vector3s firstHouseholderVector(0,0,0), shiftInfo;
    310         computeShift(iu, iter, exshift, shiftInfo);
    311         iter = iter + 1;
    312         totalIter = totalIter + 1;
    313         if (totalIter > maxIters) break;
    314         Index im;
    315         initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
    316         performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
    317       }
    318     }
    319   }
    320   if(totalIter <= maxIters)
    321     m_info = Success;
    322   else
    323     m_info = NoConvergence;
    324 
    325   m_isInitialized = true;
    326   m_matUisUptodate = computeU;
    327   return *this;
    328 }
    329 
    330 /** \internal Computes and returns vector L1 norm of T */
    331 template<typename MatrixType>
    332 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
    333 {
    334   const Index size = m_matT.cols();
    335   // FIXME to be efficient the following would requires a triangular reduxion code
    336   // Scalar norm = m_matT.upper().cwiseAbs().sum()
    337   //               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
    338   Scalar norm(0);
    339   for (Index j = 0; j < size; ++j)
    340     norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
    341   return norm;
    342 }
    343 
    344 /** \internal Look for single small sub-diagonal element and returns its index */
    345 template<typename MatrixType>
    346 inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu)
    347 {
    348   using std::abs;
    349   Index res = iu;
    350   while (res > 0)
    351   {
    352     Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
    353     if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * s)
    354       break;
    355     res--;
    356   }
    357   return res;
    358 }
    359 
    360 /** \internal Update T given that rows iu-1 and iu decouple from the rest. */
    361 template<typename MatrixType>
    362 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
    363 {
    364   using std::sqrt;
    365   using std::abs;
    366   const Index size = m_matT.cols();
    367 
    368   // The eigenvalues of the 2x2 matrix [a b; c d] are
    369   // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
    370   Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
    371   Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);   // q = tr^2 / 4 - det = discr/4
    372   m_matT.coeffRef(iu,iu) += exshift;
    373   m_matT.coeffRef(iu-1,iu-1) += exshift;
    374 
    375   if (q >= Scalar(0)) // Two real eigenvalues
    376   {
    377     Scalar z = sqrt(abs(q));
    378     JacobiRotation<Scalar> rot;
    379     if (p >= Scalar(0))
    380       rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
    381     else
    382       rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
    383 
    384     m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
    385     m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
    386     m_matT.coeffRef(iu, iu-1) = Scalar(0);
    387     if (computeU)
    388       m_matU.applyOnTheRight(iu-1, iu, rot);
    389   }
    390 
    391   if (iu > 1)
    392     m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
    393 }
    394 
    395 /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
    396 template<typename MatrixType>
    397 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
    398 {
    399   using std::sqrt;
    400   using std::abs;
    401   shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
    402   shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
    403   shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
    404 
    405   // Wilkinson's original ad hoc shift
    406   if (iter == 10)
    407   {
    408     exshift += shiftInfo.coeff(0);
    409     for (Index i = 0; i <= iu; ++i)
    410       m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
    411     Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
    412     shiftInfo.coeffRef(0) = Scalar(0.75) * s;
    413     shiftInfo.coeffRef(1) = Scalar(0.75) * s;
    414     shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
    415   }
    416 
    417   // MATLAB's new ad hoc shift
    418   if (iter == 30)
    419   {
    420     Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
    421     s = s * s + shiftInfo.coeff(2);
    422     if (s > Scalar(0))
    423     {
    424       s = sqrt(s);
    425       if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
    426         s = -s;
    427       s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
    428       s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
    429       exshift += s;
    430       for (Index i = 0; i <= iu; ++i)
    431         m_matT.coeffRef(i,i) -= s;
    432       shiftInfo.setConstant(Scalar(0.964));
    433     }
    434   }
    435 }
    436 
    437 /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
    438 template<typename MatrixType>
    439 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
    440 {
    441   using std::abs;
    442   Vector3s& v = firstHouseholderVector; // alias to save typing
    443 
    444   for (im = iu-2; im >= il; --im)
    445   {
    446     const Scalar Tmm = m_matT.coeff(im,im);
    447     const Scalar r = shiftInfo.coeff(0) - Tmm;
    448     const Scalar s = shiftInfo.coeff(1) - Tmm;
    449     v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
    450     v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
    451     v.coeffRef(2) = m_matT.coeff(im+2,im+1);
    452     if (im == il) {
    453       break;
    454     }
    455     const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
    456     const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
    457     if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
    458       break;
    459   }
    460 }
    461 
    462 /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
    463 template<typename MatrixType>
    464 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
    465 {
    466   eigen_assert(im >= il);
    467   eigen_assert(im <= iu-2);
    468 
    469   const Index size = m_matT.cols();
    470 
    471   for (Index k = im; k <= iu-2; ++k)
    472   {
    473     bool firstIteration = (k == im);
    474 
    475     Vector3s v;
    476     if (firstIteration)
    477       v = firstHouseholderVector;
    478     else
    479       v = m_matT.template block<3,1>(k,k-1);
    480 
    481     Scalar tau, beta;
    482     Matrix<Scalar, 2, 1> ess;
    483     v.makeHouseholder(ess, tau, beta);
    484 
    485     if (beta != Scalar(0)) // if v is not zero
    486     {
    487       if (firstIteration && k > il)
    488         m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
    489       else if (!firstIteration)
    490         m_matT.coeffRef(k,k-1) = beta;
    491 
    492       // These Householder transformations form the O(n^3) part of the algorithm
    493       m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
    494       m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
    495       if (computeU)
    496         m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
    497     }
    498   }
    499 
    500   Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
    501   Scalar tau, beta;
    502   Matrix<Scalar, 1, 1> ess;
    503   v.makeHouseholder(ess, tau, beta);
    504 
    505   if (beta != Scalar(0)) // if v is not zero
    506   {
    507     m_matT.coeffRef(iu-1, iu-2) = beta;
    508     m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
    509     m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
    510     if (computeU)
    511       m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
    512   }
    513 
    514   // clean up pollution due to round-off errors
    515   for (Index i = im+2; i <= iu; ++i)
    516   {
    517     m_matT.coeffRef(i,i-2) = Scalar(0);
    518     if (i > im+2)
    519       m_matT.coeffRef(i,i-3) = Scalar(0);
    520   }
    521 }
    522 
    523 } // end namespace Eigen
    524 
    525 #endif // EIGEN_REAL_SCHUR_H
    526