RealSchur.h

00001 // This file is part of Eigen, a lightweight C++ template library
00002 // for linear algebra.
00003 //
00004 // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
00005 // Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
00006 //
00007 // This Source Code Form is subject to the terms of the Mozilla
00008 // Public License v. 2.0. If a copy of the MPL was not distributed
00009 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
00010 
00011 #ifndef EIGEN_REAL_SCHUR_H
00012 #define EIGEN_REAL_SCHUR_H
00013 
00014 #include "./HessenbergDecomposition.h"
00015 
00016 namespace Eigen { 
00017 
00018 /** \eigenvalues_module \ingroup Eigenvalues_Module
00019   *
00020   *
00021   * \class RealSchur
00022   *
00023   * \brief Performs a real Schur decomposition of a square matrix
00024   *
00025   * \tparam _MatrixType the type of the matrix of which we are computing the
00026   * real Schur decomposition; this is expected to be an instantiation of the
00027   * Matrix class template.
00028   *
00029   * Given a real square matrix A, this class computes the real Schur
00030   * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
00031   * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
00032   * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
00033   * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
00034   * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
00035   * blocks on the diagonal of T are the same as the eigenvalues of the matrix
00036   * A, and thus the real Schur decomposition is used in EigenSolver to compute
00037   * the eigendecomposition of a matrix.
00038   *
00039   * Call the function compute() to compute the real Schur decomposition of a
00040   * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
00041   * constructor which computes the real Schur decomposition at construction
00042   * time. Once the decomposition is computed, you can use the matrixU() and
00043   * matrixT() functions to retrieve the matrices U and T in the decomposition.
00044   *
00045   * The documentation of RealSchur(const MatrixType&, bool) contains an example
00046   * of the typical use of this class.
00047   *
00048   * \note The implementation is adapted from
00049   * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
00050   * Their code is based on EISPACK.
00051   *
00052   * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
00053   */
00054 template<typename _MatrixType> class RealSchur 
00055 {
00056   public:
00057     typedef _MatrixType MatrixType;
00058     enum {
00059       RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00060       ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00061       Options = MatrixType::Options,
00062       MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00063       MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
00064     };
00065     typedef typename MatrixType::Scalar Scalar;
00066     typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
00067     typedef typename MatrixType::Index Index;
00068 
00069     typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1>  EigenvalueType ;
00070     typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1>  ColumnVectorType ;
00071 
00072     /** \brief Default constructor.
00073       *
00074       * \param [in] size  Positive integer, size of the matrix whose Schur decomposition will be computed.
00075       *
00076       * The default constructor is useful in cases in which the user intends to
00077       * perform decompositions via compute().  The \p size parameter is only
00078       * used as a hint. It is not an error to give a wrong \p size, but it may
00079       * impair performance.
00080       *
00081       * \sa compute() for an example.
00082       */
00083     RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
00084             : m_matT(size, size),
00085               m_matU(size, size),
00086               m_workspaceVector(size),
00087               m_hess(size),
00088               m_isInitialized(false),
00089               m_matUisUptodate(false),
00090               m_maxIters(-1)
00091     { }
00092 
00093     /** \brief Constructor; computes real Schur decomposition of given matrix. 
00094       * 
00095       * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
00096       * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
00097       *
00098       * This constructor calls compute() to compute the Schur decomposition.
00099       *
00100       * Example: \include RealSchur_RealSchur_MatrixType.cpp
00101       * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
00102       */
00103     RealSchur(const MatrixType& matrix, bool computeU = true)
00104             : m_matT(matrix.rows(),matrix.cols()),
00105               m_matU(matrix.rows(),matrix.cols()),
00106               m_workspaceVector(matrix.rows()),
00107               m_hess(matrix.rows()),
00108               m_isInitialized(false),
00109               m_matUisUptodate(false),
00110               m_maxIters(-1)
00111     {
00112       compute(matrix, computeU);
00113     }
00114 
00115     /** \brief Returns the orthogonal matrix in the Schur decomposition. 
00116       *
00117       * \returns A const reference to the matrix U.
00118       *
00119       * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
00120       * member function compute(const MatrixType&, bool) has been called before
00121       * to compute the Schur decomposition of a matrix, and \p computeU was set
00122       * to true (the default value).
00123       *
00124       * \sa RealSchur(const MatrixType&, bool) for an example
00125       */
00126     const MatrixType& matrixU() const
00127     {
00128       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00129       eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
00130       return m_matU;
00131     }
00132 
00133     /** \brief Returns the quasi-triangular matrix in the Schur decomposition. 
00134       *
00135       * \returns A const reference to the matrix T.
00136       *
00137       * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
00138       * member function compute(const MatrixType&, bool) has been called before
00139       * to compute the Schur decomposition of a matrix.
00140       *
00141       * \sa RealSchur(const MatrixType&, bool) for an example
00142       */
00143     const MatrixType& matrixT() const
00144     {
00145       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00146       return m_matT;
00147     }
00148   
00149     /** \brief Computes Schur decomposition of given matrix. 
00150       * 
00151       * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
00152       * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
00153       * \returns    Reference to \c *this
00154       *
00155       * The Schur decomposition is computed by first reducing the matrix to
00156       * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
00157       * matrix is then reduced to triangular form by performing Francis QR
00158       * iterations with implicit double shift. The cost of computing the Schur
00159       * decomposition depends on the number of iterations; as a rough guide, it
00160       * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
00161       * \f$10n^3\f$ flops if \a computeU is false.
00162       *
00163       * Example: \include RealSchur_compute.cpp
00164       * Output: \verbinclude RealSchur_compute.out
00165       *
00166       * \sa compute(const MatrixType&, bool, Index)
00167       */
00168     RealSchur & compute(const MatrixType& matrix, bool computeU = true);
00169 
00170     /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
00171      *  \param[in] matrixH Matrix in Hessenberg form H
00172      *  \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
00173      *  \param computeU Computes the matriX U of the Schur vectors
00174      * \return Reference to \c *this
00175      * 
00176      *  This routine assumes that the matrix is already reduced in Hessenberg form matrixH
00177      *  using either the class HessenbergDecomposition or another mean. 
00178      *  It computes the upper quasi-triangular matrix T of the Schur decomposition of H
00179      *  When computeU is true, this routine computes the matrix U such that 
00180      *  A = U T U^T =  (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
00181      * 
00182      * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
00183      * is not available, the user should give an identity matrix (Q.setIdentity())
00184      * 
00185      * \sa compute(const MatrixType&, bool)
00186      */
00187     template<typename HessMatrixType, typename OrthMatrixType>
00188     RealSchur & computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU);
00189     /** \brief Reports whether previous computation was successful.
00190       *
00191       * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
00192       */
00193     ComputationInfo info() const
00194     {
00195       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
00196       return m_info;
00197     }
00198 
00199     /** \brief Sets the maximum number of iterations allowed. 
00200       *
00201       * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
00202       * of the matrix.
00203       */
00204     RealSchur & setMaxIterations(Index maxIters)
00205     {
00206       m_maxIters = maxIters;
00207       return *this;
00208     }
00209 
00210     /** \brief Returns the maximum number of iterations. */
00211     Index getMaxIterations()
00212     {
00213       return m_maxIters;
00214     }
00215 
00216     /** \brief Maximum number of iterations per row.
00217       *
00218       * If not otherwise specified, the maximum number of iterations is this number times the size of the
00219       * matrix. It is currently set to 40.
00220       */
00221     static const int m_maxIterationsPerRow = 40;
00222 
00223   private:
00224     
00225     MatrixType m_matT;
00226     MatrixType m_matU;
00227     ColumnVectorType  m_workspaceVector;
00228     HessenbergDecomposition<MatrixType>  m_hess;
00229     ComputationInfo m_info;
00230     bool m_isInitialized;
00231     bool m_matUisUptodate;
00232     Index m_maxIters;
00233 
00234     typedef Matrix<Scalar,3,1>  Vector3s ;
00235 
00236     Scalar computeNormOfT();
00237     Index findSmallSubdiagEntry(Index iu);
00238     void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
00239     void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s & shiftInfo);
00240     void initFrancisQRStep(Index il, Index iu, const Vector3s & shiftInfo, Index& im, Vector3s & firstHouseholderVector);
00241     void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s & firstHouseholderVector, Scalar* workspace);
00242 };
00243 
00244 
00245 template<typename MatrixType>
00246 RealSchur<MatrixType> & RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
00247 {
00248   eigen_assert(matrix.cols() == matrix.rows());
00249   Index maxIters = m_maxIters;
00250   if (maxIters == -1)
00251     maxIters = m_maxIterationsPerRow * matrix.rows();
00252 
00253   // Step 1. Reduce to Hessenberg form
00254   m_hess.compute(matrix);
00255 
00256   // Step 2. Reduce to real Schur form  
00257   computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
00258   
00259   return *this;
00260 }
00261 template<typename MatrixType>
00262 template<typename HessMatrixType, typename OrthMatrixType>
00263 RealSchur<MatrixType> & RealSchur<MatrixType>::computeFromHessenberg (const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU)
00264 {  
00265   m_matT = matrixH; 
00266   if(computeU)
00267     m_matU = matrixQ;
00268   
00269   Index maxIters = m_maxIters;
00270   if (maxIters == -1)
00271     maxIters = m_maxIterationsPerRow * matrixH.rows();
00272   m_workspaceVector.resize(m_matT.cols());
00273   Scalar* workspace = &m_workspaceVector.coeffRef(0);
00274 
00275   // The matrix m_matT is divided in three parts. 
00276   // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero. 
00277   // Rows il,...,iu is the part we are working on (the active window).
00278   // Rows iu+1,...,end are already brought in triangular form.
00279   Index iu = m_matT.cols() - 1;
00280   Index iter = 0;      // iteration count for current eigenvalue
00281   Index totalIter = 0; // iteration count for whole matrix
00282   Scalar exshift(0);   // sum of exceptional shifts
00283   Scalar norm = computeNormOfT();
00284 
00285   if(norm!=0)
00286   {
00287     while (iu >= 0)
00288     {
00289       Index il = findSmallSubdiagEntry(iu);
00290 
00291       // Check for convergence
00292       if (il == iu) // One root found
00293       {
00294         m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
00295         if (iu > 0)
00296           m_matT.coeffRef(iu, iu-1) = Scalar(0);
00297         iu--;
00298         iter = 0;
00299       }
00300       else if (il == iu-1) // Two roots found
00301       {
00302         splitOffTwoRows(iu, computeU, exshift);
00303         iu -= 2;
00304         iter = 0;
00305       }
00306       else // No convergence yet
00307       {
00308         // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
00309         Vector3s  firstHouseholderVector(0,0,0), shiftInfo;
00310         computeShift(iu, iter, exshift, shiftInfo);
00311         iter = iter + 1;
00312         totalIter = totalIter + 1;
00313         if (totalIter > maxIters) break;
00314         Index im;
00315         initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
00316         performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
00317       }
00318     }
00319   }
00320   if(totalIter <= maxIters)
00321     m_info = Success;
00322   else
00323     m_info = NoConvergence;
00324 
00325   m_isInitialized = true;
00326   m_matUisUptodate = computeU;
00327   return *this;
00328 }
00329 
00330 /** \internal Computes and returns vector L1 norm of T */
00331 template<typename MatrixType>
00332 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT ()
00333 {
00334   const Index size = m_matT.cols();
00335   // FIXME to be efficient the following would requires a triangular reduxion code
00336   // Scalar norm = m_matT.upper().cwiseAbs().sum() 
00337   //               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
00338   Scalar norm(0);
00339   for (Index j = 0; j < size; ++j)
00340     norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
00341   return norm;
00342 }
00343 
00344 /** \internal Look for single small sub-diagonal element and returns its index */
00345 template<typename MatrixType>
00346 inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu)
00347 {
00348   using std::abs;
00349   Index res = iu;
00350   while (res > 0)
00351   {
00352     Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
00353     if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * s)
00354       break;
00355     res--;
00356   }
00357   return res;
00358 }
00359 
00360 /** \internal Update T given that rows iu-1 and iu decouple from the rest. */
00361 template<typename MatrixType>
00362 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
00363 {
00364   using std::sqrt;
00365   using std::abs;
00366   const Index size = m_matT.cols();
00367 
00368   // The eigenvalues of the 2x2 matrix [a b; c d] are 
00369   // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
00370   Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
00371   Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);   // q = tr^2 / 4 - det = discr/4
00372   m_matT.coeffRef(iu,iu) += exshift;
00373   m_matT.coeffRef(iu-1,iu-1) += exshift;
00374 
00375   if (q >= Scalar(0)) // Two real eigenvalues
00376   {
00377     Scalar z = sqrt(abs(q));
00378     JacobiRotation<Scalar> rot;
00379     if (p >= Scalar(0))
00380       rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
00381     else
00382       rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
00383 
00384     m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
00385     m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
00386     m_matT.coeffRef(iu, iu-1) = Scalar(0); 
00387     if (computeU)
00388       m_matU.applyOnTheRight(iu-1, iu, rot);
00389   }
00390 
00391   if (iu > 1) 
00392     m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
00393 }
00394 
00395 /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
00396 template<typename MatrixType>
00397 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
00398 {
00399   using std::sqrt;
00400   using std::abs;
00401   shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
00402   shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
00403   shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
00404 
00405   // Wilkinson's original ad hoc shift
00406   if (iter == 10)
00407   {
00408     exshift += shiftInfo.coeff(0);
00409     for (Index i = 0; i <= iu; ++i)
00410       m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
00411     Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
00412     shiftInfo.coeffRef(0) = Scalar(0.75) * s;
00413     shiftInfo.coeffRef(1) = Scalar(0.75) * s;
00414     shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
00415   }
00416 
00417   // MATLAB's new ad hoc shift
00418   if (iter == 30)
00419   {
00420     Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
00421     s = s * s + shiftInfo.coeff(2);
00422     if (s > Scalar(0))
00423     {
00424       s = sqrt(s);
00425       if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
00426         s = -s;
00427       s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
00428       s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
00429       exshift += s;
00430       for (Index i = 0; i <= iu; ++i)
00431         m_matT.coeffRef(i,i) -= s;
00432       shiftInfo.setConstant(Scalar(0.964));
00433     }
00434   }
00435 }
00436 
00437 /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
00438 template<typename MatrixType>
00439 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
00440 {
00441   using std::abs;
00442   Vector3s& v = firstHouseholderVector; // alias to save typing
00443 
00444   for (im = iu-2; im >= il; --im)
00445   {
00446     const Scalar Tmm = m_matT.coeff(im,im);
00447     const Scalar r = shiftInfo.coeff(0) - Tmm;
00448     const Scalar s = shiftInfo.coeff(1) - Tmm;
00449     v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
00450     v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
00451     v.coeffRef(2) = m_matT.coeff(im+2,im+1);
00452     if (im == il) {
00453       break;
00454     }
00455     const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
00456     const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
00457     if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
00458       break;
00459   }
00460 }
00461 
00462 /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
00463 template<typename MatrixType>
00464 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
00465 {
00466   eigen_assert(im >= il);
00467   eigen_assert(im <= iu-2);
00468 
00469   const Index size = m_matT.cols();
00470 
00471   for (Index k = im; k <= iu-2; ++k)
00472   {
00473     bool firstIteration = (k == im);
00474 
00475     Vector3s v;
00476     if (firstIteration)
00477       v = firstHouseholderVector;
00478     else
00479       v = m_matT.template block<3,1>(k,k-1);
00480 
00481     Scalar tau, beta;
00482     Matrix<Scalar, 2, 1> ess;
00483     v.makeHouseholder(ess, tau, beta);
00484     
00485     if (beta != Scalar(0)) // if v is not zero
00486     {
00487       if (firstIteration && k > il)
00488         m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
00489       else if (!firstIteration)
00490         m_matT.coeffRef(k,k-1) = beta;
00491 
00492       // These Householder transformations form the O(n^3) part of the algorithm
00493       m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
00494       m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
00495       if (computeU)
00496         m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
00497     }
00498   }
00499 
00500   Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
00501   Scalar tau, beta;
00502   Matrix<Scalar, 1, 1> ess;
00503   v.makeHouseholder(ess, tau, beta);
00504 
00505   if (beta != Scalar(0)) // if v is not zero
00506   {
00507     m_matT.coeffRef(iu-1, iu-2) = beta;
00508     m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
00509     m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
00510     if (computeU)
00511       m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
00512   }
00513 
00514   // clean up pollution due to round-off errors
00515   for (Index i = im+2; i <= iu; ++i)
00516   {
00517     m_matT.coeffRef(i,i-2) = Scalar(0);
00518     if (i > im+2)
00519       m_matT.coeffRef(i,i-3) = Scalar(0);
00520   }
00521 }
00522 
00523 } // end namespace Eigen
00524 
00525 #endif // EIGEN_REAL_SCHUR_H