Tridiagonalization.h 21.9 KB
Newer Older
LM's avatar
LM committed
1 2 3 4 5 6
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
Don Gagne's avatar
Don Gagne committed
7 8 9
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
LM's avatar
LM committed
10 11 12 13

#ifndef EIGEN_TRIDIAGONALIZATION_H
#define EIGEN_TRIDIAGONALIZATION_H

Don Gagne's avatar
Don Gagne committed
14 15
namespace Eigen { 

LM's avatar
LM committed
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
namespace internal {
  
template<typename MatrixType> struct TridiagonalizationMatrixTReturnType;
template<typename MatrixType>
struct traits<TridiagonalizationMatrixTReturnType<MatrixType> >
{
  typedef typename MatrixType::PlainObject ReturnType;
};

template<typename MatrixType, typename CoeffVectorType>
void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs);
}

/** \eigenvalues_module \ingroup Eigenvalues_Module
  *
  *
  * \class Tridiagonalization
  *
  * \brief Tridiagonal decomposition of a selfadjoint matrix
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the
  * tridiagonal decomposition; this is expected to be an instantiation of the
  * Matrix class template.
  *
  * This class performs a tridiagonal decomposition of a selfadjoint matrix \f$ A \f$ such that:
  * \f$ A = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real symmetric tridiagonal matrix.
  *
  * A tridiagonal matrix is a matrix which has nonzero elements only on the
  * main diagonal and the first diagonal below and above it. The Hessenberg
  * decomposition of a selfadjoint matrix is in fact a tridiagonal
  * decomposition. This class is used in SelfAdjointEigenSolver to compute the
  * eigenvalues and eigenvectors of a selfadjoint matrix.
  *
  * Call the function compute() to compute the tridiagonal decomposition of a
  * given matrix. Alternatively, you can use the Tridiagonalization(const MatrixType&)
  * constructor which computes the tridiagonal Schur decomposition at
  * construction time. Once the decomposition is computed, you can use the
  * matrixQ() and matrixT() functions to retrieve the matrices Q and T in the
  * decomposition.
  *
  * The documentation of Tridiagonalization(const MatrixType&) contains an
  * example of the typical use of this class.
  *
  * \sa class HessenbergDecomposition, class SelfAdjointEigenSolver
  */
template<typename _MatrixType> class Tridiagonalization
{
  public:

    /** \brief Synonym for the template parameter \p _MatrixType. */
    typedef _MatrixType MatrixType;

    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef typename MatrixType::Index Index;

    enum {
      Size = MatrixType::RowsAtCompileTime,
      SizeMinusOne = Size == Dynamic ? Dynamic : (Size > 1 ? Size - 1 : 1),
      Options = MatrixType::Options,
      MaxSize = MatrixType::MaxRowsAtCompileTime,
      MaxSizeMinusOne = MaxSize == Dynamic ? Dynamic : (MaxSize > 1 ? MaxSize - 1 : 1)
    };

    typedef Matrix<Scalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> CoeffVectorType;
    typedef typename internal::plain_col_type<MatrixType, RealScalar>::type DiagonalType;
    typedef Matrix<RealScalar, SizeMinusOne, 1, Options & ~RowMajor, MaxSizeMinusOne, 1> SubDiagonalType;
    typedef typename internal::remove_all<typename MatrixType::RealReturnType>::type MatrixTypeRealView;
    typedef internal::TridiagonalizationMatrixTReturnType<MatrixTypeRealView> MatrixTReturnType;

    typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
Don Gagne's avatar
Don Gagne committed
87
              typename internal::add_const_on_value_type<typename Diagonal<const MatrixType>::RealReturnType>::type,
LM's avatar
LM committed
88 89 90 91
              const Diagonal<const MatrixType>
            >::type DiagonalReturnType;

    typedef typename internal::conditional<NumTraits<Scalar>::IsComplex,
Don Gagne's avatar
Don Gagne committed
92 93
              typename internal::add_const_on_value_type<typename Diagonal<
                Block<const MatrixType,SizeMinusOne,SizeMinusOne> >::RealReturnType>::type,
LM's avatar
LM committed
94 95 96 97 98
              const Diagonal<
                Block<const MatrixType,SizeMinusOne,SizeMinusOne> >
            >::type SubDiagonalReturnType;

    /** \brief Return type of matrixQ() */
Don Gagne's avatar
Don Gagne committed
99
    typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename CoeffVectorType::ConjugateReturnType>::type> HouseholderSequenceType;
LM's avatar
LM committed
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347

    /** \brief Default constructor.
      *
      * \param [in]  size  Positive integer, size of the matrix whose tridiagonal
      * decomposition will be computed.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via compute().  The \p size parameter is only
      * used as a hint. It is not an error to give a wrong \p size, but it may
      * impair performance.
      *
      * \sa compute() for an example.
      */
    Tridiagonalization(Index size = Size==Dynamic ? 2 : Size)
      : m_matrix(size,size),
        m_hCoeffs(size > 1 ? size-1 : 1),
        m_isInitialized(false)
    {}

    /** \brief Constructor; computes tridiagonal decomposition of given matrix.
      *
      * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
      * is to be computed.
      *
      * This constructor calls compute() to compute the tridiagonal decomposition.
      *
      * Example: \include Tridiagonalization_Tridiagonalization_MatrixType.cpp
      * Output: \verbinclude Tridiagonalization_Tridiagonalization_MatrixType.out
      */
    Tridiagonalization(const MatrixType& matrix)
      : m_matrix(matrix),
        m_hCoeffs(matrix.cols() > 1 ? matrix.cols()-1 : 1),
        m_isInitialized(false)
    {
      internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
      m_isInitialized = true;
    }

    /** \brief Computes tridiagonal decomposition of given matrix.
      *
      * \param[in]  matrix  Selfadjoint matrix whose tridiagonal decomposition
      * is to be computed.
      * \returns    Reference to \c *this
      *
      * The tridiagonal decomposition is computed by bringing the columns of
      * the matrix successively in the required form using Householder
      * reflections. The cost is \f$ 4n^3/3 \f$ flops, where \f$ n \f$ denotes
      * the size of the given matrix.
      *
      * This method reuses of the allocated data in the Tridiagonalization
      * object, if the size of the matrix does not change.
      *
      * Example: \include Tridiagonalization_compute.cpp
      * Output: \verbinclude Tridiagonalization_compute.out
      */
    Tridiagonalization& compute(const MatrixType& matrix)
    {
      m_matrix = matrix;
      m_hCoeffs.resize(matrix.rows()-1, 1);
      internal::tridiagonalization_inplace(m_matrix, m_hCoeffs);
      m_isInitialized = true;
      return *this;
    }

    /** \brief Returns the Householder coefficients.
      *
      * \returns a const reference to the vector of Householder coefficients
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * The Householder coefficients allow the reconstruction of the matrix
      * \f$ Q \f$ in the tridiagonal decomposition from the packed data.
      *
      * Example: \include Tridiagonalization_householderCoefficients.cpp
      * Output: \verbinclude Tridiagonalization_householderCoefficients.out
      *
      * \sa packedMatrix(), \ref Householder_Module "Householder module"
      */
    inline CoeffVectorType householderCoefficients() const
    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return m_hCoeffs;
    }

    /** \brief Returns the internal representation of the decomposition
      *
      *	\returns a const reference to a matrix with the internal representation
      *	         of the decomposition.
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * The returned matrix contains the following information:
      *  - the strict upper triangular part is equal to the input matrix A.
      *  - the diagonal and lower sub-diagonal represent the real tridiagonal
      *    symmetric matrix T.
      *  - the rest of the lower part contains the Householder vectors that,
      *    combined with Householder coefficients returned by
      *    householderCoefficients(), allows to reconstruct the matrix Q as
      *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
      *    Here, the matrices \f$ H_i \f$ are the Householder transformations
      *       \f$ H_i = (I - h_i v_i v_i^T) \f$
      *    where \f$ h_i \f$ is the \f$ i \f$th Householder coefficient and
      *    \f$ v_i \f$ is the Householder vector defined by
      *       \f$ v_i = [ 0, \ldots, 0, 1, M(i+2,i), \ldots, M(N-1,i) ]^T \f$
      *    with M the matrix returned by this function.
      *
      * See LAPACK for further details on this packed storage.
      *
      * Example: \include Tridiagonalization_packedMatrix.cpp
      * Output: \verbinclude Tridiagonalization_packedMatrix.out
      *
      * \sa householderCoefficients()
      */
    inline const MatrixType& packedMatrix() const
    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return m_matrix;
    }

    /** \brief Returns the unitary matrix Q in the decomposition
      *
      * \returns object representing the matrix Q
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * This function returns a light-weight object of template class
      * HouseholderSequence. You can either apply it directly to a matrix or
      * you can convert it to a matrix of type #MatrixType.
      *
      * \sa Tridiagonalization(const MatrixType&) for an example,
      *     matrixT(), class HouseholderSequence
      */
    HouseholderSequenceType matrixQ() const
    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return HouseholderSequenceType(m_matrix, m_hCoeffs.conjugate())
             .setLength(m_matrix.rows() - 1)
             .setShift(1);
    }

    /** \brief Returns an expression of the tridiagonal matrix T in the decomposition
      *
      * \returns expression object representing the matrix T
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * Currently, this function can be used to extract the matrix T from internal
      * data and copy it to a dense matrix object. In most cases, it may be
      * sufficient to directly use the packed matrix or the vector expressions
      * returned by diagonal() and subDiagonal() instead of creating a new
      * dense copy matrix with this function.
      *
      * \sa Tridiagonalization(const MatrixType&) for an example,
      * matrixQ(), packedMatrix(), diagonal(), subDiagonal()
      */
    MatrixTReturnType matrixT() const
    {
      eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
      return MatrixTReturnType(m_matrix.real());
    }

    /** \brief Returns the diagonal of the tridiagonal matrix T in the decomposition.
      *
      * \returns expression representing the diagonal of T
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * Example: \include Tridiagonalization_diagonal.cpp
      * Output: \verbinclude Tridiagonalization_diagonal.out
      *
      * \sa matrixT(), subDiagonal()
      */
    DiagonalReturnType diagonal() const;

    /** \brief Returns the subdiagonal of the tridiagonal matrix T in the decomposition.
      *
      * \returns expression representing the subdiagonal of T
      *
      * \pre Either the constructor Tridiagonalization(const MatrixType&) or
      * the member function compute(const MatrixType&) has been called before
      * to compute the tridiagonal decomposition of a matrix.
      *
      * \sa diagonal() for an example, matrixT()
      */
    SubDiagonalReturnType subDiagonal() const;

  protected:

    MatrixType m_matrix;
    CoeffVectorType m_hCoeffs;
    bool m_isInitialized;
};

template<typename MatrixType>
typename Tridiagonalization<MatrixType>::DiagonalReturnType
Tridiagonalization<MatrixType>::diagonal() const
{
  eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
  return m_matrix.diagonal();
}

template<typename MatrixType>
typename Tridiagonalization<MatrixType>::SubDiagonalReturnType
Tridiagonalization<MatrixType>::subDiagonal() const
{
  eigen_assert(m_isInitialized && "Tridiagonalization is not initialized.");
  Index n = m_matrix.rows();
  return Block<const MatrixType,SizeMinusOne,SizeMinusOne>(m_matrix, 1, 0, n-1,n-1).diagonal();
}

namespace internal {

/** \internal
  * Performs a tridiagonal decomposition of the selfadjoint matrix \a matA in-place.
  *
  * \param[in,out] matA On input the selfadjoint matrix. Only the \b lower triangular part is referenced.
  *                     On output, the strict upper part is left unchanged, and the lower triangular part
  *                     represents the T and Q matrices in packed format has detailed below.
  * \param[out]    hCoeffs returned Householder coefficients (see below)
  *
  * On output, the tridiagonal selfadjoint matrix T is stored in the diagonal
  * and lower sub-diagonal of the matrix \a matA.
  * The unitary matrix Q is represented in a compact way as a product of
  * Householder reflectors \f$ H_i \f$ such that:
  *       \f$ Q = H_{N-1} \ldots H_1 H_0 \f$.
  * The Householder reflectors are defined as
  *       \f$ H_i = (I - h_i v_i v_i^T) \f$
  * where \f$ h_i = hCoeffs[i]\f$ is the \f$ i \f$th Householder coefficient and
  * \f$ v_i \f$ is the Householder vector defined by
  *       \f$ v_i = [ 0, \ldots, 0, 1, matA(i+2,i), \ldots, matA(N-1,i) ]^T \f$.
  *
  * Implemented from Golub's "Matrix Computations", algorithm 8.3.1.
  *
  * \sa Tridiagonalization::packedMatrix()
  */
template<typename MatrixType, typename CoeffVectorType>
void tridiagonalization_inplace(MatrixType& matA, CoeffVectorType& hCoeffs)
{
Don Gagne's avatar
Don Gagne committed
348
  using numext::conj;
LM's avatar
LM committed
349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468
  typedef typename MatrixType::Index Index;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  Index n = matA.rows();
  eigen_assert(n==matA.cols());
  eigen_assert(n==hCoeffs.size()+1 || n==1);
  
  for (Index i = 0; i<n-1; ++i)
  {
    Index remainingSize = n-i-1;
    RealScalar beta;
    Scalar h;
    matA.col(i).tail(remainingSize).makeHouseholderInPlace(h, beta);

    // Apply similarity transformation to remaining columns,
    // i.e., A = H A H' where H = I - h v v' and v = matA.col(i).tail(n-i-1)
    matA.col(i).coeffRef(i+1) = 1;

    hCoeffs.tail(n-i-1).noalias() = (matA.bottomRightCorner(remainingSize,remainingSize).template selfadjointView<Lower>()
                                  * (conj(h) * matA.col(i).tail(remainingSize)));

    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);

    matA.bottomRightCorner(remainingSize, remainingSize).template selfadjointView<Lower>()
      .rankUpdate(matA.col(i).tail(remainingSize), hCoeffs.tail(remainingSize), -1);

    matA.col(i).coeffRef(i+1) = beta;
    hCoeffs.coeffRef(i) = h;
  }
}

// forward declaration, implementation at the end of this file
template<typename MatrixType,
         int Size=MatrixType::ColsAtCompileTime,
         bool IsComplex=NumTraits<typename MatrixType::Scalar>::IsComplex>
struct tridiagonalization_inplace_selector;

/** \brief Performs a full tridiagonalization in place
  *
  * \param[in,out]  mat  On input, the selfadjoint matrix whose tridiagonal
  *    decomposition is to be computed. Only the lower triangular part referenced.
  *    The rest is left unchanged. On output, the orthogonal matrix Q
  *    in the decomposition if \p extractQ is true.
  * \param[out]  diag  The diagonal of the tridiagonal matrix T in the
  *    decomposition.
  * \param[out]  subdiag  The subdiagonal of the tridiagonal matrix T in
  *    the decomposition.
  * \param[in]  extractQ  If true, the orthogonal matrix Q in the
  *    decomposition is computed and stored in \p mat.
  *
  * Computes the tridiagonal decomposition of the selfadjoint matrix \p mat in place
  * such that \f$ mat = Q T Q^* \f$ where \f$ Q \f$ is unitary and \f$ T \f$ a real
  * symmetric tridiagonal matrix.
  *
  * The tridiagonal matrix T is passed to the output parameters \p diag and \p subdiag. If
  * \p extractQ is true, then the orthogonal matrix Q is passed to \p mat. Otherwise the lower
  * part of the matrix \p mat is destroyed.
  *
  * The vectors \p diag and \p subdiag are not resized. The function
  * assumes that they are already of the correct size. The length of the
  * vector \p diag should equal the number of rows in \p mat, and the
  * length of the vector \p subdiag should be one left.
  *
  * This implementation contains an optimized path for 3-by-3 matrices
  * which is especially useful for plane fitting.
  *
  * \note Currently, it requires two temporary vectors to hold the intermediate
  * Householder coefficients, and to reconstruct the matrix Q from the Householder
  * reflectors.
  *
  * Example (this uses the same matrix as the example in
  *    Tridiagonalization::Tridiagonalization(const MatrixType&)):
  *    \include Tridiagonalization_decomposeInPlace.cpp
  * Output: \verbinclude Tridiagonalization_decomposeInPlace.out
  *
  * \sa class Tridiagonalization
  */
template<typename MatrixType, typename DiagonalType, typename SubDiagonalType>
void tridiagonalization_inplace(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
{
  eigen_assert(mat.cols()==mat.rows() && diag.size()==mat.rows() && subdiag.size()==mat.rows()-1);
  tridiagonalization_inplace_selector<MatrixType>::run(mat, diag, subdiag, extractQ);
}

/** \internal
  * General full tridiagonalization
  */
template<typename MatrixType, int Size, bool IsComplex>
struct tridiagonalization_inplace_selector
{
  typedef typename Tridiagonalization<MatrixType>::CoeffVectorType CoeffVectorType;
  typedef typename Tridiagonalization<MatrixType>::HouseholderSequenceType HouseholderSequenceType;
  typedef typename MatrixType::Index Index;
  template<typename DiagonalType, typename SubDiagonalType>
  static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
  {
    CoeffVectorType hCoeffs(mat.cols()-1);
    tridiagonalization_inplace(mat,hCoeffs);
    diag = mat.diagonal().real();
    subdiag = mat.template diagonal<-1>().real();
    if(extractQ)
      mat = HouseholderSequenceType(mat, hCoeffs.conjugate())
            .setLength(mat.rows() - 1)
            .setShift(1);
  }
};

/** \internal
  * Specialization for 3x3 real matrices.
  * Especially useful for plane fitting.
  */
template<typename MatrixType>
struct tridiagonalization_inplace_selector<MatrixType,3,false>
{
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;

  template<typename DiagonalType, typename SubDiagonalType>
  static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType& subdiag, bool extractQ)
  {
Don Gagne's avatar
Don Gagne committed
469
    using std::sqrt;
LM's avatar
LM committed
470
    diag[0] = mat(0,0);
Don Gagne's avatar
Don Gagne committed
471
    RealScalar v1norm2 = numext::abs2(mat(2,0));
LM's avatar
LM committed
472 473 474 475 476 477 478 479 480 481 482
    if(v1norm2 == RealScalar(0))
    {
      diag[1] = mat(1,1);
      diag[2] = mat(2,2);
      subdiag[0] = mat(1,0);
      subdiag[1] = mat(2,1);
      if (extractQ)
        mat.setIdentity();
    }
    else
    {
Don Gagne's avatar
Don Gagne committed
483
      RealScalar beta = sqrt(numext::abs2(mat(1,0)) + v1norm2);
LM's avatar
LM committed
484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512
      RealScalar invBeta = RealScalar(1)/beta;
      Scalar m01 = mat(1,0) * invBeta;
      Scalar m02 = mat(2,0) * invBeta;
      Scalar q = RealScalar(2)*m01*mat(2,1) + m02*(mat(2,2) - mat(1,1));
      diag[1] = mat(1,1) + m02*q;
      diag[2] = mat(2,2) - m02*q;
      subdiag[0] = beta;
      subdiag[1] = mat(2,1) - m01 * q;
      if (extractQ)
      {
        mat << 1,   0,    0,
               0, m01,  m02,
               0, m02, -m01;
      }
    }
  }
};

/** \internal
  * Trivial specialization for 1x1 matrices
  */
template<typename MatrixType, bool IsComplex>
struct tridiagonalization_inplace_selector<MatrixType,1,IsComplex>
{
  typedef typename MatrixType::Scalar Scalar;

  template<typename DiagonalType, typename SubDiagonalType>
  static void run(MatrixType& mat, DiagonalType& diag, SubDiagonalType&, bool extractQ)
  {
Don Gagne's avatar
Don Gagne committed
513
    diag(0,0) = numext::real(mat(0,0));
LM's avatar
LM committed
514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549
    if(extractQ)
      mat(0,0) = Scalar(1);
  }
};

/** \internal
  * \eigenvalues_module \ingroup Eigenvalues_Module
  *
  * \brief Expression type for return value of Tridiagonalization::matrixT()
  *
  * \tparam MatrixType type of underlying dense matrix
  */
template<typename MatrixType> struct TridiagonalizationMatrixTReturnType
: public ReturnByValue<TridiagonalizationMatrixTReturnType<MatrixType> >
{
    typedef typename MatrixType::Index Index;
  public:
    /** \brief Constructor.
      *
      * \param[in] mat The underlying dense matrix
      */
    TridiagonalizationMatrixTReturnType(const MatrixType& mat) : m_matrix(mat) { }

    template <typename ResultType>
    inline void evalTo(ResultType& result) const
    {
      result.setZero();
      result.template diagonal<1>() = m_matrix.template diagonal<-1>().conjugate();
      result.diagonal() = m_matrix.diagonal();
      result.template diagonal<-1>() = m_matrix.template diagonal<-1>();
    }

    Index rows() const { return m_matrix.rows(); }
    Index cols() const { return m_matrix.cols(); }

  protected:
Don Gagne's avatar
Don Gagne committed
550
    typename MatrixType::Nested m_matrix;
LM's avatar
LM committed
551 552 553 554
};

} // end namespace internal

Don Gagne's avatar
Don Gagne committed
555 556
} // end namespace Eigen

LM's avatar
LM committed
557
#endif // EIGEN_TRIDIAGONALIZATION_H