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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@inria.fr>
5 // Copyright (C) 2010,2012 Jitse Niesen <jitse@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 Eigen::Index Index; ///< \deprecated since Eigen 3.3
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     explicit RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
m_matT(size,size)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     template<typename InputType>
104     explicit RealSchur(const EigenBase<InputType>& matrix, bool computeU = true)
105             : m_matT(matrix.rows(),matrix.cols()),
106               m_matU(matrix.rows(),matrix.cols()),
107               m_workspaceVector(matrix.rows()),
108               m_hess(matrix.rows()),
109               m_isInitialized(false),
110               m_matUisUptodate(false),
111               m_maxIters(-1)
112     {
113       compute(matrix.derived(), computeU);
114     }
115 
116     /** \brief Returns the orthogonal matrix in the Schur decomposition.
117       *
118       * \returns A const reference to the matrix U.
119       *
120       * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
121       * member function compute(const MatrixType&, bool) has been called before
122       * to compute the Schur decomposition of a matrix, and \p computeU was set
123       * to true (the default value).
124       *
125       * \sa RealSchur(const MatrixType&, bool) for an example
126       */
matrixU()127     const MatrixType& matrixU() const
128     {
129       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
130       eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
131       return m_matU;
132     }
133 
134     /** \brief Returns the quasi-triangular matrix in the Schur decomposition.
135       *
136       * \returns A const reference to the matrix T.
137       *
138       * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
139       * member function compute(const MatrixType&, bool) has been called before
140       * to compute the Schur decomposition of a matrix.
141       *
142       * \sa RealSchur(const MatrixType&, bool) for an example
143       */
matrixT()144     const MatrixType& matrixT() const
145     {
146       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
147       return m_matT;
148     }
149 
150     /** \brief Computes Schur decomposition of given matrix.
151       *
152       * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
153       * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
154       * \returns    Reference to \c *this
155       *
156       * The Schur decomposition is computed by first reducing the matrix to
157       * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
158       * matrix is then reduced to triangular form by performing Francis QR
159       * iterations with implicit double shift. The cost of computing the Schur
160       * decomposition depends on the number of iterations; as a rough guide, it
161       * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
162       * \f$10n^3\f$ flops if \a computeU is false.
163       *
164       * Example: \include RealSchur_compute.cpp
165       * Output: \verbinclude RealSchur_compute.out
166       *
167       * \sa compute(const MatrixType&, bool, Index)
168       */
169     template<typename InputType>
170     RealSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true);
171 
172     /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
173      *  \param[in] matrixH Matrix in Hessenberg form H
174      *  \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
175      *  \param computeU Computes the matriX U of the Schur vectors
176      * \return Reference to \c *this
177      *
178      *  This routine assumes that the matrix is already reduced in Hessenberg form matrixH
179      *  using either the class HessenbergDecomposition or another mean.
180      *  It computes the upper quasi-triangular matrix T of the Schur decomposition of H
181      *  When computeU is true, this routine computes the matrix U such that
182      *  A = U T U^T =  (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
183      *
184      * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
185      * is not available, the user should give an identity matrix (Q.setIdentity())
186      *
187      * \sa compute(const MatrixType&, bool)
188      */
189     template<typename HessMatrixType, typename OrthMatrixType>
190     RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU);
191     /** \brief Reports whether previous computation was successful.
192       *
193       * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
194       */
info()195     ComputationInfo info() const
196     {
197       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
198       return m_info;
199     }
200 
201     /** \brief Sets the maximum number of iterations allowed.
202       *
203       * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
204       * of the matrix.
205       */
setMaxIterations(Index maxIters)206     RealSchur& setMaxIterations(Index maxIters)
207     {
208       m_maxIters = maxIters;
209       return *this;
210     }
211 
212     /** \brief Returns the maximum number of iterations. */
getMaxIterations()213     Index getMaxIterations()
214     {
215       return m_maxIters;
216     }
217 
218     /** \brief Maximum number of iterations per row.
219       *
220       * If not otherwise specified, the maximum number of iterations is this number times the size of the
221       * matrix. It is currently set to 40.
222       */
223     static const int m_maxIterationsPerRow = 40;
224 
225   private:
226 
227     MatrixType m_matT;
228     MatrixType m_matU;
229     ColumnVectorType m_workspaceVector;
230     HessenbergDecomposition<MatrixType> m_hess;
231     ComputationInfo m_info;
232     bool m_isInitialized;
233     bool m_matUisUptodate;
234     Index m_maxIters;
235 
236     typedef Matrix<Scalar,3,1> Vector3s;
237 
238     Scalar computeNormOfT();
239     Index findSmallSubdiagEntry(Index iu);
240     void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
241     void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
242     void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
243     void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
244 };
245 
246 
247 template<typename MatrixType>
248 template<typename InputType>
compute(const EigenBase<InputType> & matrix,bool computeU)249 RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU)
250 {
251   const Scalar considerAsZero = (std::numeric_limits<Scalar>::min)();
252 
253   eigen_assert(matrix.cols() == matrix.rows());
254   Index maxIters = m_maxIters;
255   if (maxIters == -1)
256     maxIters = m_maxIterationsPerRow * matrix.rows();
257 
258   Scalar scale = matrix.derived().cwiseAbs().maxCoeff();
259   if(scale<considerAsZero)
260   {
261     m_matT.setZero(matrix.rows(),matrix.cols());
262     if(computeU)
263       m_matU.setIdentity(matrix.rows(),matrix.cols());
264     m_info = Success;
265     m_isInitialized = true;
266     m_matUisUptodate = computeU;
267     return *this;
268   }
269 
270   // Step 1. Reduce to Hessenberg form
271   m_hess.compute(matrix.derived()/scale);
272 
273   // Step 2. Reduce to real Schur form
274   computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
275 
276   m_matT *= scale;
277 
278   return *this;
279 }
280 template<typename MatrixType>
281 template<typename HessMatrixType, typename OrthMatrixType>
computeFromHessenberg(const HessMatrixType & matrixH,const OrthMatrixType & matrixQ,bool computeU)282 RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU)
283 {
284   using std::abs;
285 
286   m_matT = matrixH;
287   if(computeU)
288     m_matU = matrixQ;
289 
290   Index maxIters = m_maxIters;
291   if (maxIters == -1)
292     maxIters = m_maxIterationsPerRow * matrixH.rows();
293   m_workspaceVector.resize(m_matT.cols());
294   Scalar* workspace = &m_workspaceVector.coeffRef(0);
295 
296   // The matrix m_matT is divided in three parts.
297   // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
298   // Rows il,...,iu is the part we are working on (the active window).
299   // Rows iu+1,...,end are already brought in triangular form.
300   Index iu = m_matT.cols() - 1;
301   Index iter = 0;      // iteration count for current eigenvalue
302   Index totalIter = 0; // iteration count for whole matrix
303   Scalar exshift(0);   // sum of exceptional shifts
304   Scalar norm = computeNormOfT();
305 
306   if(norm!=0)
307   {
308     while (iu >= 0)
309     {
310       Index il = findSmallSubdiagEntry(iu);
311 
312       // Check for convergence
313       if (il == iu) // One root found
314       {
315         m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
316         if (iu > 0)
317           m_matT.coeffRef(iu, iu-1) = Scalar(0);
318         iu--;
319         iter = 0;
320       }
321       else if (il == iu-1) // Two roots found
322       {
323         splitOffTwoRows(iu, computeU, exshift);
324         iu -= 2;
325         iter = 0;
326       }
327       else // No convergence yet
328       {
329         // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
330         Vector3s firstHouseholderVector(0,0,0), shiftInfo;
331         computeShift(iu, iter, exshift, shiftInfo);
332         iter = iter + 1;
333         totalIter = totalIter + 1;
334         if (totalIter > maxIters) break;
335         Index im;
336         initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
337         performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
338       }
339     }
340   }
341   if(totalIter <= maxIters)
342     m_info = Success;
343   else
344     m_info = NoConvergence;
345 
346   m_isInitialized = true;
347   m_matUisUptodate = computeU;
348   return *this;
349 }
350 
351 /** \internal Computes and returns vector L1 norm of T */
352 template<typename MatrixType>
computeNormOfT()353 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
354 {
355   const Index size = m_matT.cols();
356   // FIXME to be efficient the following would requires a triangular reduxion code
357   // Scalar norm = m_matT.upper().cwiseAbs().sum()
358   //               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
359   Scalar norm(0);
360   for (Index j = 0; j < size; ++j)
361     norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
362   return norm;
363 }
364 
365 /** \internal Look for single small sub-diagonal element and returns its index */
366 template<typename MatrixType>
findSmallSubdiagEntry(Index iu)367 inline Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu)
368 {
369   using std::abs;
370   Index res = iu;
371   while (res > 0)
372   {
373     Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
374     if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * s)
375       break;
376     res--;
377   }
378   return res;
379 }
380 
381 /** \internal Update T given that rows iu-1 and iu decouple from the rest. */
382 template<typename MatrixType>
splitOffTwoRows(Index iu,bool computeU,const Scalar & exshift)383 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
384 {
385   using std::sqrt;
386   using std::abs;
387   const Index size = m_matT.cols();
388 
389   // The eigenvalues of the 2x2 matrix [a b; c d] are
390   // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
391   Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
392   Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);   // q = tr^2 / 4 - det = discr/4
393   m_matT.coeffRef(iu,iu) += exshift;
394   m_matT.coeffRef(iu-1,iu-1) += exshift;
395 
396   if (q >= Scalar(0)) // Two real eigenvalues
397   {
398     Scalar z = sqrt(abs(q));
399     JacobiRotation<Scalar> rot;
400     if (p >= Scalar(0))
401       rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
402     else
403       rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
404 
405     m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
406     m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
407     m_matT.coeffRef(iu, iu-1) = Scalar(0);
408     if (computeU)
409       m_matU.applyOnTheRight(iu-1, iu, rot);
410   }
411 
412   if (iu > 1)
413     m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
414 }
415 
416 /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
417 template<typename MatrixType>
computeShift(Index iu,Index iter,Scalar & exshift,Vector3s & shiftInfo)418 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
419 {
420   using std::sqrt;
421   using std::abs;
422   shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
423   shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
424   shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
425 
426   // Wilkinson's original ad hoc shift
427   if (iter == 10)
428   {
429     exshift += shiftInfo.coeff(0);
430     for (Index i = 0; i <= iu; ++i)
431       m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
432     Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
433     shiftInfo.coeffRef(0) = Scalar(0.75) * s;
434     shiftInfo.coeffRef(1) = Scalar(0.75) * s;
435     shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
436   }
437 
438   // MATLAB's new ad hoc shift
439   if (iter == 30)
440   {
441     Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
442     s = s * s + shiftInfo.coeff(2);
443     if (s > Scalar(0))
444     {
445       s = sqrt(s);
446       if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
447         s = -s;
448       s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
449       s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
450       exshift += s;
451       for (Index i = 0; i <= iu; ++i)
452         m_matT.coeffRef(i,i) -= s;
453       shiftInfo.setConstant(Scalar(0.964));
454     }
455   }
456 }
457 
458 /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
459 template<typename MatrixType>
initFrancisQRStep(Index il,Index iu,const Vector3s & shiftInfo,Index & im,Vector3s & firstHouseholderVector)460 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
461 {
462   using std::abs;
463   Vector3s& v = firstHouseholderVector; // alias to save typing
464 
465   for (im = iu-2; im >= il; --im)
466   {
467     const Scalar Tmm = m_matT.coeff(im,im);
468     const Scalar r = shiftInfo.coeff(0) - Tmm;
469     const Scalar s = shiftInfo.coeff(1) - Tmm;
470     v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
471     v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
472     v.coeffRef(2) = m_matT.coeff(im+2,im+1);
473     if (im == il) {
474       break;
475     }
476     const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
477     const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
478     if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
479       break;
480   }
481 }
482 
483 /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
484 template<typename MatrixType>
performFrancisQRStep(Index il,Index im,Index iu,bool computeU,const Vector3s & firstHouseholderVector,Scalar * workspace)485 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
486 {
487   eigen_assert(im >= il);
488   eigen_assert(im <= iu-2);
489 
490   const Index size = m_matT.cols();
491 
492   for (Index k = im; k <= iu-2; ++k)
493   {
494     bool firstIteration = (k == im);
495 
496     Vector3s v;
497     if (firstIteration)
498       v = firstHouseholderVector;
499     else
500       v = m_matT.template block<3,1>(k,k-1);
501 
502     Scalar tau, beta;
503     Matrix<Scalar, 2, 1> ess;
504     v.makeHouseholder(ess, tau, beta);
505 
506     if (beta != Scalar(0)) // if v is not zero
507     {
508       if (firstIteration && k > il)
509         m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
510       else if (!firstIteration)
511         m_matT.coeffRef(k,k-1) = beta;
512 
513       // These Householder transformations form the O(n^3) part of the algorithm
514       m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
515       m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
516       if (computeU)
517         m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
518     }
519   }
520 
521   Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
522   Scalar tau, beta;
523   Matrix<Scalar, 1, 1> ess;
524   v.makeHouseholder(ess, tau, beta);
525 
526   if (beta != Scalar(0)) // if v is not zero
527   {
528     m_matT.coeffRef(iu-1, iu-2) = beta;
529     m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
530     m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
531     if (computeU)
532       m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
533   }
534 
535   // clean up pollution due to round-off errors
536   for (Index i = im+2; i <= iu; ++i)
537   {
538     m_matT.coeffRef(i,i-2) = Scalar(0);
539     if (i > im+2)
540       m_matT.coeffRef(i,i-3) = Scalar(0);
541   }
542 }
543 
544 } // end namespace Eigen
545 
546 #endif // EIGEN_REAL_SCHUR_H
547