Tridiagonalization.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 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_TRIDIAGONALIZATION_H
00012 #define EIGEN_TRIDIAGONALIZATION_H
00013 
00014 namespace Eigen { 
00015 
00016 namespace internal {
00017   
00018 template<typename MatrixType> struct TridiagonalizationMatrixTReturnType;
00019 template<typename MatrixType>
00020 struct traits<TridiagonalizationMatrixTReturnType<MatrixType> >
00021 {
00022   typedef typename MatrixType::PlainObject ReturnType;
00023 };
00024 
00025 template<typename MatrixType, typename CoeffVectorType>
00026 void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
00027 }
00028 
00029 /** \eigenvalues_module \ingroup Eigenvalues_Module
00030   *
00031   *
00032   * \class Tridiagonalization
00033   *
00034   * \brief Tridiagonal decomposition of a selfadjoint matrix
00035   *
00036   * \tparam _MatrixType the type of the matrix of which we are computing the
00037   * tridiagonal decomposition; this is expected to be an instantiation of the
00038   * Matrix class template.
00039   *
00040   * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
00041   * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
00042   *
00043   * A tridiagonal matrix is a matrix which has nonzero elements only on the
00044   * main diagonal and the first diagonal below and above it. The Hessenberg
00045   * decomposition of a selfadjoint matrix is in fact a tridiagonal
00046   * decomposition. This class is used in SelfAdjointEigenSolver to compute the
00047   * eigenvalues and eigenvectors of a selfadjoint matrix.
00048   *
00049   * Call the function compute() to compute the tridiagonal decomposition of a
00050   * given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
00051   * constructor which computes the tridiagonal Schur decomposition at
00052   * construction time. Once the decomposition is computed, you can use the
00053   * matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
00054   * decomposition.
00055   *
00056   * The documentation of Tridiagonalization(const MatrixType&) contains an
00057   * example of the typical use of this class.
00058   *
00059   * \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
00060   */
00061 template<typename _MatrixType> class Tridiagonalization 
00062 {
00063   public:
00064 
00065     /** \brief Synonym for the template parameter \p _MatrixType. */
00066     typedef _MatrixType MatrixType;
00067 
00068     typedef typename MatrixType::Scalar Scalar;
00069     typedef typename NumTraits<Scalar>::Real RealScalar;
00070     typedef typename MatrixType::Index Index;
00071 
00072     enum {
00073       Size = MatrixType::RowsAtCompileTime,
00074       SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
00075       Options = MatrixType::Options,
00076       MaxSize = MatrixType::MaxRowsAtCompileTime,
00077       MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
00078     };
00079 
00080     typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1>  CoeffVectorType;
00081     typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
00082     typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
00083     typedef typename internal::remove_all<typename MatrixType::RealReturnType>::type MatrixTypeRealView;
00084     typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;
00085 
00086     typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
00087               typename internal::add_const_on_value_type<typename Diagonal<const MatrixType>::RealReturnType>::type,
00088               const Diagonal<const MatrixType>
00089             >::type DiagonalReturnType;
00090 
00091     typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
00092               typename internal::add_const_on_value_type<typename Diagonal<
00093                 Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType>::type,
00094               const Diagonal<
00095                 Block<const MatrixType,SizeMinusOne,SizeMinusOne> >
00096             >::type SubDiagonalReturnType;
00097 
00098     /** \brief Return type of matrixQ() */
00099     typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type > HouseholderSequenceType;
00100 
00101     /** \brief Default constructor.
00102       *
00103       * \param [in]  size  Positive integer, size of the matrix whose tridiagonal
00104       * decomposition will be computed.
00105       *
00106       * The default constructor is useful in cases in which the user intends to
00107       * perform decompositions via compute().  The \p size parameter is only
00108       * used as a hint. It is not an error to give a wrong \p size, but it may
00109       * impair performance.
00110       *
00111       * \sa compute() for an example.
00112       */
00113     Tridiagonalization(Index size = Size==Dynamic ? 2 : Size)
00114       : m_matrix(size,size),
00115         m_hCoeffs(size > 1 ? size-1 : 1),
00116         m_isInitialized(false)
00117     {}
00118 
00119     /** \brief Constructor; computes tridiagonal decomposition of given matrix.
00120       *
00121       * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
00122       * is to be computed.
00123       *
00124       * This constructor calls compute() to compute the tridiagonal decomposition.
00125       *
00126       * Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
00127       * Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
00128       */
00129     Tridiagonalization(const MatrixType& matrix)
00130       : m_matrix(matrix),
00131         m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
00132         m_isInitialized(false)
00133     {
00134       internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
00135       m_isInitialized = true;
00136     }
00137 
00138     /** \brief Computes tridiagonal decomposition of given matrix.
00139       *
00140       * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
00141       * is to be computed.
00142       * \returns    Reference to \c *this
00143       *
00144       * The tridiagonal decomposition is computed by bringing the columns of
00145       * the matrix successively in the required form using Householder
00146       * reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
00147       * the size of the given matrix.
00148       *
00149       * This method reuses of the allocated data in the Tridiagonalization
00150       * object, if the size of the matrix does not change.
00151       *
00152       * Example: \include Tridiagonalization_compute.cpp
00153       * Output: \verbinclude Tridiagonalization_compute.out
00154       */
00155     Tridiagonalization & compute(const MatrixType& matrix)
00156     {
00157       m_matrix = matrix;
00158       m_hCoeffs.resize(matrix.rows()-1, 1);
00159       internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
00160       m_isInitialized = true;
00161       return *this;
00162     }
00163 
00164     /** \brief Returns the Householder coefficients.
00165       *
00166       * \returns a const reference to the vector of Householder coefficients
00167       *
00168       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
00169       * the member function compute(const MatrixType&) has been called before
00170       * to compute the tridiagonal decomposition of a matrix.
00171       *
00172       * The Householder coefficients allow the reconstruction of the matrix
00173       * \f$ Q \f$ in the tridiagonal decomposition from the packed data.
00174       *
00175       * Example: \include Tridiagonalization_householderCoefficients.cpp
00176       * Output: \verbinclude Tridiagonalization_householderCoefficients.out
00177       *
00178       * \sa packedMatrix(), \ref Householder_Module "Householder module"
00179       */
00180     inline CoeffVectorType  householderCoefficients() const
00181     {
00182       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
00183       return m_hCoeffs;
00184     }
00185 
00186     /** \brief Returns the internal representation of the decomposition
00187       *
00188       * \returns a const reference to a matrix with the internal representation
00189       *          of the decomposition.
00190       *
00191       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
00192       * the member function compute(const MatrixType&) has been called before
00193       * to compute the tridiagonal decomposition of a matrix.
00194       *
00195       * The returned matrix contains the following information:
00196       *  - the strict upper triangular part is equal to the input matrix A.
00197       *  - the diagonal and lower sub-diagonal represent the real tridiagonal
00198       *    symmetric matrix T.
00199       *  - the rest of the lower part contains the Householder vectors that,
00200       *    combined with Householder coefficients returned by
00201       *    householderCoefficients(), allows to reconstruct the matrix Q as
00202       *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
00203       *    Here, the matrices \f$ H_i \f$ are the Householder transformations
00204       *       \f$ H_i = (I - h_i v_i v_i^T) \f$
00205       *    where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
00206       *    \f$ v_i \f$ is the Householder vector defined by
00207       *       \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
00208       *    with M the matrix returned by this function.
00209       *
00210       * See LAPACK for further details on this packed storage.
00211       *
00212       * Example: \include Tridiagonalization_packedMatrix.cpp
00213       * Output: \verbinclude Tridiagonalization_packedMatrix.out
00214       *
00215       * \sa householderCoefficients()
00216       */
00217     inline const MatrixType& packedMatrix() const
00218     {
00219       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
00220       return m_matrix;
00221     }
00222 
00223     /** \brief Returns the unitary matrix Q in the decomposition
00224       *
00225       * \returns object representing the matrix Q
00226       *
00227       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
00228       * the member function compute(const MatrixType&) has been called before
00229       * to compute the tridiagonal decomposition of a matrix.
00230       *
00231       * This function returns a light-weight object of template class
00232       * HouseholderSequence. You can either apply it directly to a matrix or
00233       * you can convert it to a matrix of type #MatrixType.
00234       *
00235       * \sa Tridiagonalization(const MatrixType&) for an example,
00236       *     matrixT(), class HouseholderSequence
00237       */
00238     HouseholderSequenceType  matrixQ() const
00239     {
00240       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
00241       return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
00242              .setLength(m_matrix.rows() - 1)
00243              .setShift(1);
00244     }
00245 
00246     /** \brief Returns an expression of the tridiagonal matrix T in the decomposition
00247       *
00248       * \returns expression object representing the matrix T
00249       *
00250       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
00251       * the member function compute(const MatrixType&) has been called before
00252       * to compute the tridiagonal decomposition of a matrix.
00253       *
00254       * Currently, this function can be used to extract the matrix T from internal
00255       * data and copy it to a dense matrix object. In most cases, it may be
00256       * sufficient to directly use the packed matrix or the vector expressions
00257       * returned by diagonal() and subDiagonal() instead of creating a new
00258       * dense copy matrix with this function.
00259       *
00260       * \sa Tridiagonalization(const MatrixType&) for an example,
00261       * matrixQ(), packedMatrix(), diagonal(), subDiagonal()
00262       */
00263     MatrixTReturnType matrixT() const
00264     {
00265       eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
00266       return MatrixTReturnType(m_matrix.real());
00267     }
00268 
00269     /** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
00270       *
00271       * \returns expression representing the diagonal of T
00272       *
00273       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
00274       * the member function compute(const MatrixType&) has been called before
00275       * to compute the tridiagonal decomposition of a matrix.
00276       *
00277       * Example: \include Tridiagonalization_diagonal.cpp
00278       * Output: \verbinclude Tridiagonalization_diagonal.out
00279       *
00280       * \sa matrixT(), subDiagonal()
00281       */
00282     DiagonalReturnType diagonal() const;
00283 
00284     /** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
00285       *
00286       * \returns expression representing the subdiagonal of T
00287       *
00288       * \pre Either the constructor Tridiagonalization(const MatrixType&) or
00289       * the member function compute(const MatrixType&) has been called before
00290       * to compute the tridiagonal decomposition of a matrix.
00291       *
00292       * \sa diagonal() for an example, matrixT()
00293       */
00294     SubDiagonalReturnType subDiagonal() const;
00295 
00296   protected:
00297 
00298     MatrixType m_matrix;
00299     CoeffVectorType m_hCoeffs;
00300     bool m_isInitialized;
00301 };
00302 
00303 template<typename MatrixType>
00304 typename Tridiagonalization<MatrixType>::DiagonalReturnType
00305 Tridiagonalization<MatrixType>::diagonal() const
00306 {
00307   eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
00308   return m_matrix.diagonal();
00309 }
00310 
00311 template<typename MatrixType>
00312 typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
00313 Tridiagonalization<MatrixType>::subDiagonal() const
00314 {
00315   eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
00316   Index n = m_matrix.rows();
00317   return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
00318 }
00319 
00320 namespace internal {
00321 
00322 /** \internal
00323   * Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
00324   *
00325   * \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
00326   *                     On output, the strict upper part is left unchanged, and the lower triangular part
00327   *                     represents the T and Q matrices in packed format has detailed below.
00328   * \param[out]    hCoeffs returned Householder coefficients (see below)
00329   *
00330   * On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
00331   * and lower sub-diagonal of the matrix \a matA.
00332   * The unitary matrix Q is represented in a compact way as a product of
00333   * Householder reflectors \f$ H_i \f$ such that:
00334   *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
00335   * The Householder reflectors are defined as
00336   *       \f$ H_i = (I - h_i v_i v_i^T) \f$
00337   * where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
00338   * \f$ v_i \f$ is the Householder vector defined by
00339   *       \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
00340   *
00341   * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
00342   *
00343   * \sa Tridiagonalization::packedMatrix()
00344   */
00345 template<typename MatrixType, typename CoeffVectorType>
00346 void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
00347 {
00348   using numext::conj;
00349   typedef typename MatrixType::Index Index;
00350   typedef typename MatrixType::Scalar Scalar;
00351   typedef typename MatrixType::RealScalar RealScalar;
00352   Index n = matA.rows();
00353   eigen_assert(n==matA.cols());
00354   eigen_assert(n==hCoeffs.size()+1 || n==1);
00355   
00356   for (Index i = 0; i<n-1; ++i)
00357   {
00358     Index remainingSize = n-i-1;
00359     RealScalar beta;
00360     Scalar h;
00361     matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);
00362 
00363     // Apply similarity transformation to remaining columns,
00364     // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
00365     matA.col(i).coeffRef(i+1) = 1;
00366 
00367     hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
00368                                   * (conj(h) * matA.col(i).tail(remainingSize)));
00369 
00370     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);
00371 
00372     matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
00373       .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);
00374 
00375     matA.col(i).coeffRef(i+1) = beta;
00376     hCoeffs.coeffRef(i) = h;
00377   }
00378 }
00379 
00380 // forward declaration, implementation at the end of this file
00381 template<typename MatrixType,
00382          int Size=MatrixType::ColsAtCompileTime,
00383          bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
00384 struct tridiagonalization_inplace_selector;
00385 
00386 /** \brief Performs a full tridiagonalization in place
00387   *
00388   * \param[in,out]  mat  On input, the selfadjoint matrix whose tridiagonal
00389   *    decomposition is to be computed. Only the lower triangular part referenced.
00390   *    The rest is left unchanged. On output, the orthogonal matrix Q
00391   *    in the decomposition if \p extractQ is true.
00392   * \param[out]  diag  The diagonal of the tridiagonal matrix T in the
00393   *    decomposition.
00394   * \param[out]  subdiag  The subdiagonal of the tridiagonal matrix T in
00395   *    the decomposition.
00396   * \param[in]  extractQ  If true, the orthogonal matrix Q in the
00397   *    decomposition is computed and stored in \p mat.
00398   *
00399   * Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
00400   * such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
00401   * symmetric tridiagonal matrix.
00402   *
00403   * The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
00404   * \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
00405   * part of the matrix \p mat is destroyed.
00406   *
00407   * The vectors \p diag and \p subdiag are not resized. The function
00408   * assumes that they are already of the correct size. The length of the
00409   * vector \p diag should equal the number of rows in \p mat, and the
00410   * length of the vector \p subdiag should be one left.
00411   *
00412   * This implementation contains an optimized path for 3-by-3 matrices
00413   * which is especially useful for plane fitting.
00414   *
00415   * \note Currently, it requires two temporary vectors to hold the intermediate
00416   * Householder coefficients, and to reconstruct the matrix Q from the Householder
00417   * reflectors.
00418   *
00419   * Example (this uses the same matrix as the example in
00420   *    Tridiagonalization::Tridiagonalization(const MatrixType&)):
00421   *    \include Tridiagonalization_decomposeInPlace.cpp
00422   * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
00423   *
00424   * \sa class Tridiagonalization
00425   */
00426 template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
00427 void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
00428 {
00429   eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
00430   tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
00431 }
00432 
00433 /** \internal
00434   * General full tridiagonalization
00435   */
00436 template<typename MatrixType, int Size, bool IsComplex>
00437 struct tridiagonalization_inplace_selector
00438 {
00439   typedef typename Tridiagonalization<MatrixType>::CoeffVectorType  CoeffVectorType;
00440   typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType  HouseholderSequenceType;
00441   typedef typename MatrixType::Index Index;
00442   template<typename DiagonalType, typename SubDiagonalType>
00443   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
00444   {
00445     CoeffVectorType hCoeffs(mat.cols()-1);
00446     tridiagonalization_inplace(mat,hCoeffs);
00447     diag = mat.diagonal().real();
00448     subdiag = mat.template diagonal<-1>().real();
00449     if(extractQ)
00450       mat = HouseholderSequenceType(mat, hCoeffs.conjugate())
00451             .setLength(mat.rows() - 1)
00452             .setShift(1);
00453   }
00454 };
00455 
00456 /** \internal
00457   * Specialization for 3x3 real matrices.
00458   * Especially useful for plane fitting.
00459   */
00460 template<typename MatrixType>
00461 struct tridiagonalization_inplace_selector<MatrixType,3,false>
00462 {
00463   typedef typename MatrixType::Scalar Scalar;
00464   typedef typename MatrixType::RealScalar RealScalar;
00465 
00466   template<typename DiagonalType, typename SubDiagonalType>
00467   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
00468   {
00469     using std::sqrt;
00470     diag[0] = mat(0,0);
00471     RealScalar v1norm2 = numext::abs2(mat(2,0));
00472     if(v1norm2 == RealScalar(0))
00473     {
00474       diag[1] = mat(1,1);
00475       diag[2] = mat(2,2);
00476       subdiag[0] = mat(1,0);
00477       subdiag[1] = mat(2,1);
00478       if (extractQ)
00479         mat.setIdentity();
00480     }
00481     else
00482     {
00483       RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
00484       RealScalar invBeta = RealScalar(1)/beta;
00485       Scalar m01 = mat(1,0) * invBeta;
00486       Scalar m02 = mat(2,0) * invBeta;
00487       Scalar q = RealScalar(2)*m01*mat(2,1) + m02*(mat(2,2) - mat(1,1));
00488       diag[1] = mat(1,1) + m02*q;
00489       diag[2] = mat(2,2) - m02*q;
00490       subdiag[0] = beta;
00491       subdiag[1] = mat(2,1) - m01 * q;
00492       if (extractQ)
00493       {
00494         mat << 1,   0,    0,
00495                0, m01,  m02,
00496                0, m02, -m01;
00497       }
00498     }
00499   }
00500 };
00501 
00502 /** \internal
00503   * Trivial specialization for 1x1 matrices
00504   */
00505 template<typename MatrixType, bool IsComplex>
00506 struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
00507 {
00508   typedef typename MatrixType::Scalar Scalar;
00509 
00510   template<typename DiagonalType, typename SubDiagonalType>
00511   static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
00512   {
00513     diag(0,0) = numext::real(mat(0,0));
00514     if(extractQ)
00515       mat(0,0) = Scalar(1);
00516   }
00517 };
00518 
00519 /** \internal
00520   * \eigenvalues_module \ingroup Eigenvalues_Module
00521   *
00522   * \brief Expression type for return value of Tridiagonalization::matrixT()
00523   *
00524   * \tparam MatrixType type of underlying dense matrix
00525   */
00526 template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
00527 : public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
00528 {
00529     typedef typename MatrixType::Index Index;
00530   public:
00531     /** \brief Constructor.
00532       *
00533       * \param[in] mat The underlying dense matrix
00534       */
00535     TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }
00536 
00537     template <typename ResultType>
00538     inline void evalTo(ResultType& result) const
00539     {
00540       result.setZero();
00541       result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
00542       result.diagonal() = m_matrix.diagonal();
00543       result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
00544     }
00545 
00546     Index rows() const { return m_matrix.rows(); }
00547     Index cols() const { return m_matrix.cols(); }
00548 
00549   protected:
00550     typename MatrixType::Nested m_matrix;
00551 };
00552 
00553 } // end namespace internal
00554 
00555 } // end namespace Eigen
00556 
00557 #endif // EIGEN_TRIDIAGONALIZATION_H