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 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_TRIDIAGONALIZATION_H
12 #define EIGEN_TRIDIAGONALIZATION_H
13 
14 namespace Eigen {
15 
16 namespace internal {
17 
18 template<typename MatrixType> struct TridiagonalizationMatrixTReturnType;
19 template<typename MatrixType>
20 struct traits<TridiagonalizationMatrixTReturnType<MatrixType> >
21 {
22   typedef typename MatrixType::PlainObject ReturnType;
23 };
24 
25 template<typename MatrixType, typename CoeffVectorType>
26 void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
27 }
28 
29 /** \eigenvalues_module \ingroup Eigenvalues_Module
30   *
31   *
32   * \class Tridiagonalization
33   *
34   * \brief Tridiagonal decomposition of a selfadjoint matrix
35   *
36   * \tparam _MatrixType the type of the matrix of which we are computing the
37   * tridiagonal decomposition; this is expected to be an instantiation of the
38   * Matrix class template.
39   *
40   * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
41   * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
42   *
43   * A tridiagonal matrix is a matrix which has nonzero elements only on the
44   * main diagonal and the first diagonal below and above it. The Hessenberg
45   * decomposition of a selfadjoint matrix is in fact a tridiagonal
46   * decomposition. This class is used in SelfAdjointEigenSolver to compute the
47   * eigenvalues and eigenvectors of a selfadjoint matrix.
48   *
49   * Call the function compute() to compute the tridiagonal decomposition of a
50   * given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
51   * constructor which computes the tridiagonal Schur decomposition at
52   * construction time. Once the decomposition is computed, you can use the
53   * matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
54   * decomposition.
55   *
56   * The documentation of Tridiagonalization(const MatrixType&) contains an
57   * example of the typical use of this class.
58   *
59   * \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
60   */
61 template<typename _MatrixType> class Tridiagonalization
62 {
63   public:
64 
65     /** \brief Synonym for the template parameter \p _MatrixType. */
66     typedef _MatrixType MatrixType;
67 
68     typedef typename MatrixType::Scalar Scalar;
69     typedef typename NumTraits<Scalar>::Real RealScalar;
70     typedef typename MatrixType::Index Index;
71 
72     enum {
73       Size = MatrixType::RowsAtCompileTime,
74       SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
75       Options = MatrixType::Options,
76       MaxSize = MatrixType::MaxRowsAtCompileTime,
77       MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
78     };
79 
80     typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
81     typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
82     typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
83     typedef typename internal::remove_all<typename MatrixType::RealReturnType>::type MatrixTypeRealView;
84     typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;
85 
86     typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
87               typename internal::add_const_on_value_type<typename Diagonal<const MatrixType>::RealReturnType>::type,
88               const Diagonal<const MatrixType>
89             >::type DiagonalReturnType;
90 
91     typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
92               typename internal::add_const_on_value_type<typename Diagonal<
93                 Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType>::type,
94               const Diagonal<
95                 Block<const MatrixType,SizeMinusOne,SizeMinusOne> >
96             >::type SubDiagonalReturnType;
97 
98     /** \brief Return type of matrixQ() */
99     typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;
100 
101     /** \brief Default constructor.
102       *
103       * \param [in]  size  Positive integer, size of the matrix whose tridiagonal
104       * decomposition will be computed.
105       *
106       * The default constructor is useful in cases in which the user intends to
107       * perform decompositions via compute().  The \p size parameter is only
108       * used as a hint. It is not an error to give a wrong \p size, but it may
109       * impair performance.
110       *
111       * \sa compute() for an example.
112       */
113     Tridiagonalization(Index size = Size==Dynamic ? 2 : Size)
114       : m_matrix(size,size),
115         m_hCoeffs(size > 1 ? size-1 : 1),
116         m_isInitialized(false)
117     {}
118 
119     /** \brief Constructor; computes tridiagonal decomposition of given matrix.
120       *
121       * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
122       * is to be computed.
123       *
124       * This constructor calls compute() to compute the tridiagonal decomposition.
125       *
126       * Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
127       * Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
128       */
129     Tridiagonalization(const MatrixType& matrix)
130       : m_matrix(matrix),
131         m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
132         m_isInitialized(false)
133     {
134       internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
135       m_isInitialized = true;
136     }
137 
138     /** \brief Computes tridiagonal decomposition of given matrix.
139       *
140       * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
141       * is to be computed.
142       * \returns    Reference to \c *this
143       *
144       * The tridiagonal decomposition is computed by bringing the columns of
145       * the matrix successively in the required form using Householder
146       * reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
147       * the size of the given matrix.
148       *
149       * This method reuses of the allocated data in the Tridiagonalization
150       * object, if the size of the matrix does not change.
151       *
152       * Example: \include Tridiagonalization_compute.cpp
153       * Output: \verbinclude Tridiagonalization_compute.out
154       */
155     Tridiagonalization& compute(const MatrixType& matrix)
156     {
157       m_matrix = matrix;
158       m_hCoeffs.resize(matrix.rows()-1, 1);
159       internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
160       m_isInitialized = true;
161       return *this;
162     }
163 
164     /** \brief Returns the Householder coefficients.
165       *
166       * \returns a const reference to the vector of Householder coefficients
167       *
168       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
169       * the member function compute(const MatrixType&) has been called before
170       * to compute the tridiagonal decomposition of a matrix.
171       *
172       * The Householder coefficients allow the reconstruction of the matrix
173       * \f$ Q \f$ in the tridiagonal decomposition from the packed data.
174       *
175       * Example: \include Tridiagonalization_householderCoefficients.cpp
176       * Output: \verbinclude Tridiagonalization_householderCoefficients.out
177       *
178       * \sa packedMatrix(), \ref Householder_Module "Householder module"
179       */
180     inline CoeffVectorType householderCoefficients() const
181     {
182       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
183       return m_hCoeffs;
184     }
185 
186     /** \brief Returns the internal representation of the decomposition
187       *
188       *	\returns a const reference to a matrix with the internal representation
189       *	         of the decomposition.
190       *
191       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
192       * the member function compute(const MatrixType&) has been called before
193       * to compute the tridiagonal decomposition of a matrix.
194       *
195       * The returned matrix contains the following information:
196       *  - the strict upper triangular part is equal to the input matrix A.
197       *  - the diagonal and lower sub-diagonal represent the real tridiagonal
198       *    symmetric matrix T.
199       *  - the rest of the lower part contains the Householder vectors that,
200       *    combined with Householder coefficients returned by
201       *    householderCoefficients(), allows to reconstruct the matrix Q as
202       *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
203       *    Here, the matrices \f$ H_i \f$ are the Householder transformations
204       *       \f$ H_i = (I - h_i v_i v_i^T) \f$
205       *    where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
206       *    \f$ v_i \f$ is the Householder vector defined by
207       *       \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
208       *    with M the matrix returned by this function.
209       *
210       * See LAPACK for further details on this packed storage.
211       *
212       * Example: \include Tridiagonalization_packedMatrix.cpp
213       * Output: \verbinclude Tridiagonalization_packedMatrix.out
214       *
215       * \sa householderCoefficients()
216       */
217     inline const MatrixType& packedMatrix() const
218     {
219       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
220       return m_matrix;
221     }
222 
223     /** \brief Returns the unitary matrix Q in the decomposition
224       *
225       * \returns object representing the matrix Q
226       *
227       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
228       * the member function compute(const MatrixType&) has been called before
229       * to compute the tridiagonal decomposition of a matrix.
230       *
231       * This function returns a light-weight object of template class
232       * HouseholderSequence. You can either apply it directly to a matrix or
233       * you can convert it to a matrix of type #MatrixType.
234       *
235       * \sa Tridiagonalization(const MatrixType&) for an example,
236       *     matrixT(), class HouseholderSequence
237       */
238     HouseholderSequenceType matrixQ() const
239     {
240       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
241       return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
242              .setLength(m_matrix.rows() - 1)
243              .setShift(1);
244     }
245 
246     /** \brief Returns an expression of the tridiagonal matrix T in the decomposition
247       *
248       * \returns expression object representing the matrix T
249       *
250       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
251       * the member function compute(const MatrixType&) has been called before
252       * to compute the tridiagonal decomposition of a matrix.
253       *
254       * Currently, this function can be used to extract the matrix T from internal
255       * data and copy it to a dense matrix object. In most cases, it may be
256       * sufficient to directly use the packed matrix or the vector expressions
257       * returned by diagonal() and subDiagonal() instead of creating a new
258       * dense copy matrix with this function.
259       *
260       * \sa Tridiagonalization(const MatrixType&) for an example,
261       * matrixQ(), packedMatrix(), diagonal(), subDiagonal()
262       */
263     MatrixTReturnType matrixT() const
264     {
265       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
266       return MatrixTReturnType(m_matrix.real());
267     }
268 
269     /** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
270       *
271       * \returns expression representing the diagonal of T
272       *
273       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
274       * the member function compute(const MatrixType&) has been called before
275       * to compute the tridiagonal decomposition of a matrix.
276       *
277       * Example: \include Tridiagonalization_diagonal.cpp
278       * Output: \verbinclude Tridiagonalization_diagonal.out
279       *
280       * \sa matrixT(), subDiagonal()
281       */
282     DiagonalReturnType diagonal() const;
283 
284     /** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
285       *
286       * \returns expression representing the subdiagonal of T
287       *
288       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
289       * the member function compute(const MatrixType&) has been called before
290       * to compute the tridiagonal decomposition of a matrix.
291       *
292       * \sa diagonal() for an example, matrixT()
293       */
294     SubDiagonalReturnType subDiagonal() const;
295 
296   protected:
297 
298     MatrixType m_matrix;
299     CoeffVectorType m_hCoeffs;
300     bool m_isInitialized;
301 };
302 
303 template<typename MatrixType>
304 typename Tridiagonalization<MatrixType>::DiagonalReturnType
305 Tridiagonalization<MatrixType>::diagonal() const
306 {
307   eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
308   return m_matrix.diagonal();
309 }
310 
311 template<typename MatrixType>
312 typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
313 Tridiagonalization<MatrixType>::subDiagonal() const
314 {
315   eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
316   Index n = m_matrix.rows();
317   return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
318 }
319 
320 namespace internal {
321 
322 /** \internal
323   * Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
324   *
325   * \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
326   *                     On output, the strict upper part is left unchanged, and the lower triangular part
327   *                     represents the T and Q matrices in packed format has detailed below.
328   * \param[out]    hCoeffs returned Householder coefficients (see below)
329   *
330   * On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
331   * and lower sub-diagonal of the matrix \a matA.
332   * The unitary matrix Q is represented in a compact way as a product of
333   * Householder reflectors \f$ H_i \f$ such that:
334   *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
335   * The Householder reflectors are defined as
336   *       \f$ H_i = (I - h_i v_i v_i^T) \f$
337   * where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
338   * \f$ v_i \f$ is the Householder vector defined by
339   *       \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
340   *
341   * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
342   *
343   * \sa Tridiagonalization::packedMatrix()
344   */
345 template<typename MatrixType, typename CoeffVectorType>
346 void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
347 {
348   using numext::conj;
349   typedef typename MatrixType::Index Index;
350   typedef typename MatrixType::Scalar Scalar;
351   typedef typename MatrixType::RealScalar RealScalar;
352   Index n = matA.rows();
353   eigen_assert(n==matA.cols());
354   eigen_assert(n==hCoeffs.size()+1 || n==1);
355 
356   for (Index i = 0; i<n-1; ++i)
357   {
358     Index remainingSize = n-i-1;
359     RealScalar beta;
360     Scalar h;
361     matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
362 
363     // Apply similarity transformation to remaining columns,
364     // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
365     matA.col(i).coeffRef(i+1) = 1;
366 
367     hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
368                                   * (conj(h) * matA.col(i).tail(remainingSize)));
369 
370     hCoeffs.tail(n-i-1) += (conj(h)*Scalar(-0.5)*(hCoeffs.tail(remainingSize).dot(matA.col(i).tail(remainingSize)))) * matA.col(i).tail(n-i-1);
371 
372     matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
373       .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);
374 
375     matA.col(i).coeffRef(i+1) = beta;
376     hCoeffs.coeffRef(i) = h;
377   }
378 }
379 
380 // forward declaration, implementation at the end of this file
381 template<typename MatrixType,
382          int Size=MatrixType::ColsAtCompileTime,
383          bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
384 struct tridiagonalization_inplace_selector;
385 
386 /** \brief Performs a full tridiagonalization in place
387   *
388   * \param[in,out]  mat  On input, the selfadjoint matrix whose tridiagonal
389   *    decomposition is to be computed. Only the lower triangular part referenced.
390   *    The rest is left unchanged. On output, the orthogonal matrix Q
391   *    in the decomposition if \p extractQ is true.
392   * \param[out]  diag  The diagonal of the tridiagonal matrix T in the
393   *    decomposition.
394   * \param[out]  subdiag  The subdiagonal of the tridiagonal matrix T in
395   *    the decomposition.
396   * \param[in]  extractQ  If true, the orthogonal matrix Q in the
397   *    decomposition is computed and stored in \p mat.
398   *
399   * Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
400   * such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
401   * symmetric tridiagonal matrix.
402   *
403   * The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
404   * \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
405   * part of the matrix \p mat is destroyed.
406   *
407   * The vectors \p diag and \p subdiag are not resized. The function
408   * assumes that they are already of the correct size. The length of the
409   * vector \p diag should equal the number of rows in \p mat, and the
410   * length of the vector \p subdiag should be one left.
411   *
412   * This implementation contains an optimized path for 3-by-3 matrices
413   * which is especially useful for plane fitting.
414   *
415   * \note Currently, it requires two temporary vectors to hold the intermediate
416   * Householder coefficients, and to reconstruct the matrix Q from the Householder
417   * reflectors.
418   *
419   * Example (this uses the same matrix as the example in
420   *    Tridiagonalization::Tridiagonalization(const MatrixType&)):
421   *    \include Tridiagonalization_decomposeInPlace.cpp
422   * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
423   *
424   * \sa class Tridiagonalization
425   */
426 template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
427 void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
428 {
429   eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
430   tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
431 }
432 
433 /** \internal
434   * General full tridiagonalization
435   */
436 template<typename MatrixType, int Size, bool IsComplex>
437 struct tridiagonalization_inplace_selector
438 {
439   typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
440   typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
441   typedef typename MatrixType::Index Index;
442   template<typename DiagonalType, typename SubDiagonalType>
443   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
444   {
445     CoeffVectorType hCoeffs(mat.cols()-1);
446     tridiagonalization_inplace(mat,hCoeffs);
447     diag = mat.diagonal().real();
448     subdiag = mat.template diagonal<-1>().real();
449     if(extractQ)
450       mat = HouseholderSequenceType(mat, hCoeffs.conjugate())
451             .setLength(mat.rows() - 1)
452             .setShift(1);
453   }
454 };
455 
456 /** \internal
457   * Specialization for 3x3 real matrices.
458   * Especially useful for plane fitting.
459   */
460 template<typename MatrixType>
461 struct tridiagonalization_inplace_selector<MatrixType,3,false>
462 {
463   typedef typename MatrixType::Scalar Scalar;
464   typedef typename MatrixType::RealScalar RealScalar;
465 
466   template<typename DiagonalType, typename SubDiagonalType>
467   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
468   {
469     using std::sqrt;
470     diag[0] = mat(0,0);
471     RealScalar v1norm2 = numext::abs2(mat(2,0));
472     if(v1norm2 == RealScalar(0))
473     {
474       diag[1] = mat(1,1);
475       diag[2] = mat(2,2);
476       subdiag[0] = mat(1,0);
477       subdiag[1] = mat(2,1);
478       if (extractQ)
479         mat.setIdentity();
480     }
481     else
482     {
483       RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
484       RealScalar invBeta = RealScalar(1)/beta;
485       Scalar m01 = mat(1,0) * invBeta;
486       Scalar m02 = mat(2,0) * invBeta;
487       Scalar q = RealScalar(2)*m01*mat(2,1) + m02*(mat(2,2) - mat(1,1));
488       diag[1] = mat(1,1) + m02*q;
489       diag[2] = mat(2,2) - m02*q;
490       subdiag[0] = beta;
491       subdiag[1] = mat(2,1) - m01 * q;
492       if (extractQ)
493       {
494         mat << 1,   0,    0,
495                0, m01,  m02,
496                0, m02, -m01;
497       }
498     }
499   }
500 };
501 
502 /** \internal
503   * Trivial specialization for 1x1 matrices
504   */
505 template<typename MatrixType, bool IsComplex>
506 struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
507 {
508   typedef typename MatrixType::Scalar Scalar;
509 
510   template<typename DiagonalType, typename SubDiagonalType>
511   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
512   {
513     diag(0,0) = numext::real(mat(0,0));
514     if(extractQ)
515       mat(0,0) = Scalar(1);
516   }
517 };
518 
519 /** \internal
520   * \eigenvalues_module \ingroup Eigenvalues_Module
521   *
522   * \brief Expression type for return value of Tridiagonalization::matrixT()
523   *
524   * \tparam MatrixType type of underlying dense matrix
525   */
526 template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
527 : public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
528 {
529     typedef typename MatrixType::Index Index;
530   public:
531     /** \brief Constructor.
532       *
533       * \param[in] mat The underlying dense matrix
534       */
535     TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }
536 
537     template <typename ResultType>
538     inline void evalTo(ResultType& result) const
539     {
540       result.setZero();
541       result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
542       result.diagonal() = m_matrix.diagonal();
543       result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
544     }
545 
546     Index rows() const { return m_matrix.rows(); }
547     Index cols() const { return m_matrix.cols(); }
548 
549   protected:
550     typename MatrixType::Nested m_matrix;
551 };
552 
553 } // end namespace internal
554 
555 } // end namespace Eigen
556 
557 #endif // EIGEN_TRIDIAGONALIZATION_H
558