SelfAdjointEigenSolver.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2010 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// Eigen is free software; you can redistribute it and/or
// modify it under the terms of the GNU Lesser General Public
// License as published by the Free Software Foundation; either
// version 3 of the License, or (at your option) any later version.
//
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// modify it under the terms of the GNU General Public License as
// published by the Free Software Foundation; either version 2 of
// the License, or (at your option) any later version.
//
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#ifndef EIGEN_SELFADJOINTEIGENSOLVER_H
#define EIGEN_SELFADJOINTEIGENSOLVER_H

#include "./EigenvaluesCommon.h"
#include "./Tridiagonalization.h"

template<typename _MatrixType>
class GeneralizedSelfAdjointEigenSolver;

/** \eigenvalues_module \ingroup Eigenvalues_Module
  *
  *
  * \class SelfAdjointEigenSolver
  *
  * \brief Computes eigenvalues and eigenvectors of selfadjoint matrices
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the
  * eigendecomposition; this is expected to be an instantiation of the Matrix
  * class template.
  *
  * A matrix \f$ A \f$ is selfadjoint if it equals its adjoint. For real
  * matrices, this means that the matrix is symmetric: it equals its
  * transpose. This class computes the eigenvalues and eigenvectors of a
  * selfadjoint matrix. These are the scalars \f$ \lambda \f$ and vectors
  * \f$ v \f$ such that \f$ Av = \lambda v \f$.  The eigenvalues of a
  * selfadjoint matrix are always real. If \f$ D \f$ is a diagonal matrix with
  * the eigenvalues on the diagonal, and \f$ V \f$ is a matrix with the
  * eigenvectors as its columns, then \f$ A = V D V^{-1} \f$ (for selfadjoint
  * matrices, the matrix \f$ V \f$ is always invertible). This is called the
  * eigendecomposition.
  *
  * The algorithm exploits the fact that the matrix is selfadjoint, making it
  * faster and more accurate than the general purpose eigenvalue algorithms
  * implemented in EigenSolver and ComplexEigenSolver.
  *
  * Only the \b lower \b triangular \b part of the input matrix is referenced.
  *
  * Call the function compute() to compute the eigenvalues and eigenvectors of
  * a given matrix. Alternatively, you can use the
  * SelfAdjointEigenSolver(const MatrixType&, int) constructor which computes
  * the eigenvalues and eigenvectors at construction time. Once the eigenvalue
  * and eigenvectors are computed, they can be retrieved with the eigenvalues()
  * and eigenvectors() functions.
  *
  * The documentation for SelfAdjointEigenSolver(const MatrixType&, int)
  * contains an example of the typical use of this class.
  *
  * To solve the \em generalized eigenvalue problem \f$ Av = \lambda Bv \f$ and
  * the likes, see the class GeneralizedSelfAdjointEigenSolver.
  *
  * \sa MatrixBase::eigenvalues(), class EigenSolver, class ComplexEigenSolver
  */
template<typename _MatrixType> class SelfAdjointEigenSolver
{
  public:

    typedef _MatrixType MatrixType;
    enum {
      Size = MatrixType::RowsAtCompileTime,
      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
      Options = MatrixType::Options,
      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

    /** \brief Scalar type for matrices of type \p _MatrixType. */
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::Index Index;

    /** \brief Real scalar type for \p _MatrixType.
      *
      * This is just \c Scalar if #Scalar is real (e.g., \c float or
      * \c double), and the type of the real part of \c Scalar if #Scalar is
      * complex.
      */
    typedef typename NumTraits<Scalar>::Real RealScalar;

    /** \brief Type for vector of eigenvalues as returned by eigenvalues().
      *
      * This is a column vector with entries of type #RealScalar.
      * The length of the vector is the size of \p _MatrixType.
      */
    typedef typename internal::plain_col_type<MatrixType, RealScalar>::type RealVectorType;
    typedef Tridiagonalization<MatrixType> TridiagonalizationType;

    /** \brief Default constructor for fixed-size matrices.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via compute(). This constructor
      * can only be used if \p _MatrixType is a fixed-size matrix; use
      * SelfAdjointEigenSolver(Index) for dynamic-size matrices.
      *
      * Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver.out
      */
    SelfAdjointEigenSolver()
        : m_eivec(),
          m_eivalues(),
          m_subdiag(),
          m_isInitialized(false)
    { }

    /** \brief Constructor, pre-allocates memory for dynamic-size matrices.
      *
      * \param [in]  size  Positive integer, size of the matrix whose
      * eigenvalues and eigenvectors will be computed.
      *
      * This constructor is useful for dynamic-size matrices, when 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
      */
    SelfAdjointEigenSolver(Index size)
        : m_eivec(size, size),
          m_eivalues(size),
          m_subdiag(size > 1 ? size - 1 : 1),
          m_isInitialized(false)
    {}

    /** \brief Constructor; computes eigendecomposition of given matrix.
      *
      * \param[in]  matrix  Selfadjoint matrix whose eigendecomposition is to
      *    be computed. Only the lower triangular part of the matrix is referenced.
      * \param[in]  options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
      *
      * This constructor calls compute(const MatrixType&, int) to compute the
      * eigenvalues of the matrix \p matrix. The eigenvectors are computed if
      * \p options equals #ComputeEigenvectors.
      *
      * Example: \include SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_SelfAdjointEigenSolver_MatrixType.out
      *
      * \sa compute(const MatrixType&, int)
      */
    SelfAdjointEigenSolver(const MatrixType& matrix, int options = ComputeEigenvectors)
      : m_eivec(matrix.rows(), matrix.cols()),
        m_eivalues(matrix.cols()),
        m_subdiag(matrix.rows() > 1 ? matrix.rows() - 1 : 1),
        m_isInitialized(false)
    {
      compute(matrix, options);
    }

    /** \brief Computes eigendecomposition of given matrix.
      *
      * \param[in]  matrix  Selfadjoint matrix whose eigendecomposition is to
      *    be computed. Only the lower triangular part of the matrix is referenced.
      * \param[in]  options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
      * \returns    Reference to \c *this
      *
      * This function computes the eigenvalues of \p matrix.  The eigenvalues()
      * function can be used to retrieve them.  If \p options equals #ComputeEigenvectors,
      * then the eigenvectors are also computed and can be retrieved by
      * calling eigenvectors().
      *
      * This implementation uses a symmetric QR algorithm. The matrix is first
      * reduced to tridiagonal form using the Tridiagonalization class. The
      * tridiagonal matrix is then brought to diagonal form with implicit
      * symmetric QR steps with Wilkinson shift. Details can be found in
      * Section 8.3 of Golub \& Van Loan, <i>%Matrix Computations</i>.
      *
      * The cost of the computation is about \f$ 9n^3 \f$ if the eigenvectors
      * are required and \f$ 4n^3/3 \f$ if they are not required.
      *
      * This method reuses the memory in the SelfAdjointEigenSolver object that
      * was allocated when the object was constructed, if the size of the
      * matrix does not change.
      *
      * Example: \include SelfAdjointEigenSolver_compute_MatrixType.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_compute_MatrixType.out
      *
      * \sa SelfAdjointEigenSolver(const MatrixType&, int)
      */
    SelfAdjointEigenSolver& compute(const MatrixType& matrix, int options = ComputeEigenvectors);

    /** \brief Returns the eigenvectors of given matrix.
      *
      * \returns  A const reference to the matrix whose columns are the eigenvectors.
      *
      * \pre The eigenvectors have been computed before.
      *
      * Column \f$ k \f$ of the returned matrix is an eigenvector corresponding
      * to eigenvalue number \f$ k \f$ as returned by eigenvalues().  The
      * eigenvectors are normalized to have (Euclidean) norm equal to one. If
      * this object was used to solve the eigenproblem for the selfadjoint
      * matrix \f$ A \f$, then the matrix returned by this function is the
      * matrix \f$ V \f$ in the eigendecomposition \f$ A = V D V^{-1} \f$.
      *
      * Example: \include SelfAdjointEigenSolver_eigenvectors.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out
      *
      * \sa eigenvalues()
      */
    const MatrixType& eigenvectors() const
    {
      eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
      eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
      return m_eivec;
    }

    /** \brief Returns the eigenvalues of given matrix.
      *
      * \returns A const reference to the column vector containing the eigenvalues.
      *
      * \pre The eigenvalues have been computed before.
      *
      * The eigenvalues are repeated according to their algebraic multiplicity,
      * so there are as many eigenvalues as rows in the matrix. The eigenvalues
      * are sorted in increasing order.
      *
      * Example: \include SelfAdjointEigenSolver_eigenvalues.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_eigenvalues.out
      *
      * \sa eigenvectors(), MatrixBase::eigenvalues()
      */
    const RealVectorType& eigenvalues() const
    {
      eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
      return m_eivalues;
    }

    /** \brief Computes the positive-definite square root of the matrix.
      *
      * \returns the positive-definite square root of the matrix
      *
      * \pre The eigenvalues and eigenvectors of a positive-definite matrix
      * have been computed before.
      *
      * The square root of a positive-definite matrix \f$ A \f$ is the
      * positive-definite matrix whose square equals \f$ A \f$. This function
      * uses the eigendecomposition \f$ A = V D V^{-1} \f$ to compute the
      * square root as \f$ A^{1/2} = V D^{1/2} V^{-1} \f$.
      *
      * Example: \include SelfAdjointEigenSolver_operatorSqrt.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_operatorSqrt.out
      *
      * \sa operatorInverseSqrt(),
      *     \ref MatrixFunctions_Module "MatrixFunctions Module"
      */
    MatrixType operatorSqrt() const
    {
      eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
      eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
      return m_eivec * m_eivalues.cwiseSqrt().asDiagonal() * m_eivec.adjoint();
    }

    /** \brief Computes the inverse square root of the matrix.
      *
      * \returns the inverse positive-definite square root of the matrix
      *
      * \pre The eigenvalues and eigenvectors of a positive-definite matrix
      * have been computed before.
      *
      * This function uses the eigendecomposition \f$ A = V D V^{-1} \f$ to
      * compute the inverse square root as \f$ V D^{-1/2} V^{-1} \f$. This is
      * cheaper than first computing the square root with operatorSqrt() and
      * then its inverse with MatrixBase::inverse().
      *
      * Example: \include SelfAdjointEigenSolver_operatorInverseSqrt.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_operatorInverseSqrt.out
      *
      * \sa operatorSqrt(), MatrixBase::inverse(),
      *     \ref MatrixFunctions_Module "MatrixFunctions Module"
      */
    MatrixType operatorInverseSqrt() const
    {
      eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
      eigen_assert(m_eigenvectorsOk && "The eigenvectors have not been computed together with the eigenvalues.");
      return m_eivec * m_eivalues.cwiseInverse().cwiseSqrt().asDiagonal() * m_eivec.adjoint();
    }

    /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
      */
    ComputationInfo info() const
    {
      eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
      return m_info;
    }

    /** \brief Maximum number of iterations.
      *
      * Maximum number of iterations allowed for an eigenvalue to converge.
      */
    static const int m_maxIterations = 30;

    #ifdef EIGEN2_SUPPORT
    SelfAdjointEigenSolver(const MatrixType& matrix, bool computeEigenvectors)
      : m_eivec(matrix.rows(), matrix.cols()),
        m_eivalues(matrix.cols()),
        m_subdiag(matrix.rows() > 1 ? matrix.rows() - 1 : 1),
        m_isInitialized(false)
    {
      compute(matrix, computeEigenvectors);
    }
    
    SelfAdjointEigenSolver(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors = true)
        : m_eivec(matA.cols(), matA.cols()),
          m_eivalues(matA.cols()),
          m_subdiag(matA.cols() > 1 ? matA.cols() - 1 : 1),
          m_isInitialized(false)
    {
      static_cast<GeneralizedSelfAdjointEigenSolver<MatrixType>*>(this)->compute(matA, matB, computeEigenvectors ? ComputeEigenvectors : EigenvaluesOnly);
    }
    
    void compute(const MatrixType& matrix, bool computeEigenvectors)
    {
      compute(matrix, computeEigenvectors ? ComputeEigenvectors : EigenvaluesOnly);
    }

    void compute(const MatrixType& matA, const MatrixType& matB, bool computeEigenvectors = true)
    {
      compute(matA, matB, computeEigenvectors ? ComputeEigenvectors : EigenvaluesOnly);
    }
    #endif // EIGEN2_SUPPORT

  protected:
    MatrixType m_eivec;
    RealVectorType m_eivalues;
    typename TridiagonalizationType::SubDiagonalType m_subdiag;
    ComputationInfo m_info;
    bool m_isInitialized;
    bool m_eigenvectorsOk;
};

/** \internal
  *
  * \eigenvalues_module \ingroup Eigenvalues_Module
  *
  * Performs a QR step on a tridiagonal symmetric matrix represented as a
  * pair of two vectors \a diag and \a subdiag.
  *
  * \param matA the input selfadjoint matrix
  * \param hCoeffs returned Householder coefficients
  *
  * For compilation efficiency reasons, this procedure does not use eigen expression
  * for its arguments.
  *
  * Implemented from Golub's "Matrix Computations", algorithm 8.3.2:
  * "implicit symmetric QR step with Wilkinson shift"
  */
namespace internal {
template<int StorageOrder,typename RealScalar, typename Scalar, typename Index>
static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n);
}

template<typename MatrixType>
SelfAdjointEigenSolver<MatrixType>& SelfAdjointEigenSolver<MatrixType>
::compute(const MatrixType& matrix, int options)
{
  eigen_assert(matrix.cols() == matrix.rows());
  eigen_assert((options&~(EigVecMask|GenEigMask))==0
          && (options&EigVecMask)!=EigVecMask
          && "invalid option parameter");
  bool computeEigenvectors = (options&ComputeEigenvectors)==ComputeEigenvectors;
  Index n = matrix.cols();
  m_eivalues.resize(n,1);

  if(n==1)
  {
    m_eivalues.coeffRef(0,0) = internal::real(matrix.coeff(0,0));
    if(computeEigenvectors)
      m_eivec.setOnes();
    m_info = Success;
    m_isInitialized = true;
    m_eigenvectorsOk = computeEigenvectors;
    return *this;
  }

  // declare some aliases
  RealVectorType& diag = m_eivalues;
  MatrixType& mat = m_eivec;

  // map the matrix coefficients to [-1:1] to avoid over- and underflow.
  RealScalar scale = matrix.cwiseAbs().maxCoeff();
  if(scale==Scalar(0)) scale = 1;
  mat = matrix / scale;
  m_subdiag.resize(n-1);
  internal::tridiagonalization_inplace(mat, diag, m_subdiag, computeEigenvectors);
  
  Index end = n-1;
  Index start = 0;
  Index iter = 0; // number of iterations we are working on one element

  while (end>0)
  {
    for (Index i = start; i<end; ++i)
      if (internal::isMuchSmallerThan(internal::abs(m_subdiag[i]),(internal::abs(diag[i])+internal::abs(diag[i+1]))))
        m_subdiag[i] = 0;

    // find the largest unreduced block
    while (end>0 && m_subdiag[end-1]==0)
    {
      iter = 0;
      end--;
    }
    if (end<=0)
      break;

    // if we spent too many iterations on the current element, we give up
    iter++;
    if(iter > m_maxIterations) break;

    start = end - 1;
    while (start>0 && m_subdiag[start-1]!=0)
      start--;

    internal::tridiagonal_qr_step<MatrixType::Flags&RowMajorBit ? RowMajor : ColMajor>(diag.data(), m_subdiag.data(), start, end, computeEigenvectors ? m_eivec.data() : (Scalar*)0, n);
  }

  if (iter <= m_maxIterations)
    m_info = Success;
  else
    m_info = NoConvergence;

  // Sort eigenvalues and corresponding vectors.
  // TODO make the sort optional ?
  // TODO use a better sort algorithm !!
  if (m_info == Success)
  {
    for (Index i = 0; i < n-1; ++i)
    {
      Index k;
      m_eivalues.segment(i,n-i).minCoeff(&k);
      if (k > 0)
      {
        std::swap(m_eivalues[i], m_eivalues[k+i]);
        if(computeEigenvectors)
          m_eivec.col(i).swap(m_eivec.col(k+i));
      }
    }
  }
  
  // scale back the eigen values
  m_eivalues *= scale;

  m_isInitialized = true;
  m_eigenvectorsOk = computeEigenvectors;
  return *this;
}

namespace internal {
template<int StorageOrder,typename RealScalar, typename Scalar, typename Index>
static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n)
{
  // NOTE this version avoids over & underflow, however since the matrix is prescaled, overflow cannot occur,
  // and underflows should be meaningless anyway. So I don't any reason to enable this version, but I keep
  // it here for reference:
//   RealScalar td = (diag[end-1] - diag[end])*RealScalar(0.5);
//   RealScalar e = subdiag[end-1];
//   RealScalar mu = diag[end] - (e / (td + (td>0 ? 1 : -1))) * (e / hypot(td,e));
  RealScalar td = (diag[end-1] - diag[end])*RealScalar(0.5);
  RealScalar e2 = abs2(subdiag[end-1]);
  RealScalar mu = diag[end] - e2 / (td + (td>0 ? 1 : -1) * sqrt(td*td + e2));
  RealScalar x = diag[start] - mu;
  RealScalar z = subdiag[start];
  for (Index k = start; k < end; ++k)
  {
    JacobiRotation<RealScalar> rot;
    rot.makeGivens(x, z);

    // do T = G' T G
    RealScalar sdk = rot.s() * diag[k] + rot.c() * subdiag[k];
    RealScalar dkp1 = rot.s() * subdiag[k] + rot.c() * diag[k+1];

    diag[k] = rot.c() * (rot.c() * diag[k] - rot.s() * subdiag[k]) - rot.s() * (rot.c() * subdiag[k] - rot.s() * diag[k+1]);
    diag[k+1] = rot.s() * sdk + rot.c() * dkp1;
    subdiag[k] = rot.c() * sdk - rot.s() * dkp1;
    

    if (k > start)
      subdiag[k - 1] = rot.c() * subdiag[k-1] - rot.s() * z;

    x = subdiag[k];

    if (k < end - 1)
    {
      z = -rot.s() * subdiag[k+1];
      subdiag[k + 1] = rot.c() * subdiag[k+1];
    }
    
    // apply the givens rotation to the unit matrix Q = Q * G
    if (matrixQ)
    {
      // FIXME if StorageOrder == RowMajor this operation is not very efficient
      Map<Matrix<Scalar,Dynamic,Dynamic,StorageOrder> > q(matrixQ,n,n);
      q.applyOnTheRight(k,k+1,rot);
    }
  }
}
} // end namespace internal

#endif // EIGEN_SELFADJOINTEIGENSOLVER_H