// NOTE: This is the same as SelfAdjustEigenSolver without
// the automatic sorting of eigenvalues by magnitude.
//

// 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>
//
// 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/.

#ifndef EIGEN_SELFADJOINTEIGENSOLVERNOSORT_H
#define EIGEN_SELFADJOINTEIGENSOLVERNOSORT_H

#include "Tridiagonalization34.h"

#ifndef EIGEN_USING_STD
#define EIGEN_USING_STD(fn) using std::fn
#endif

namespace Eigen {

template<typename _MatrixType>
class GeneralizedSelfAdjointEigenSolver;

namespace internal {
template<typename SolverType,int Size,bool IsComplex> struct direct_selfadjoint_eigenvalues;

template<typename MatrixType, typename DiagType, typename SubDiagType>
EIGEN_DEVICE_FUNC
ComputationInfo computeFromTridiagonalNosort_impl(DiagType& diag, SubDiagType& subdiag, const Index maxIterations, bool computeEigenvectors, MatrixType& eivec);
}

/** \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$. This is called the
  * eigendecomposition.
  *
  * For a selfadjoint matrix, \f$ V \f$ is unitary, meaning its inverse is equal
  * to its adjoint, \f$ V^{-1} = V^{\dagger} \f$. If \f$ A \f$ is real, then
  * \f$ V \f$ is also real and therefore orthogonal, meaning its inverse is
  * equal to its transpose, \f$ V^{-1} = V^T \f$.
  *
  * The algorithm exploits the fact that the matrix is selfadjoint, making it
  * faster and more accurate than the general purpose eigenvalue algorithms
  * implemented in EigenSolver and ComplexEigenSolver.
  *
  * Only the \b lower \b triangular \b part of the input matrix is referenced.
  *
  * Call the function compute() to compute the eigenvalues and eigenvectors of
  * a given matrix. Alternatively, you can use the
  * SelfAdjointEigenSolver(const MatrixType&, int) constructor which computes
  * the eigenvalues and eigenvectors at construction time. Once the eigenvalue
  * and eigenvectors are computed, they can be retrieved with the eigenvalues()
  * and eigenvectors() functions.
  *
  * The documentation for SelfAdjointEigenSolver(const MatrixType&, int)
  * contains an example of the typical use of this class.
  *
  * To solve the \em generalized eigenvalue problem \f$ Av = \lambda Bv \f$ and
  * the likes, see the class GeneralizedSelfAdjointEigenSolver.
  *
  * \sa MatrixBase::eigenvalues(), class EigenSolver, class ComplexEigenSolver
  */
template<typename _MatrixType> class SelfAdjointEigenSolverNosort
{
  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 Eigen::Index Index; ///< \deprecated since Eigen 3.3

    typedef Matrix<Scalar,Size,Size,ColMajor,MaxColsAtCompileTime,MaxColsAtCompileTime> EigenvectorsType;

    /** \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;

    friend struct internal::direct_selfadjoint_eigenvalues<SelfAdjointEigenSolverNosort,Size,NumTraits<Scalar>::IsComplex>;

    /** \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;
    typedef typename TridiagonalizationType::SubDiagonalType SubDiagonalType;

    /** \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
      */
    EIGEN_DEVICE_FUNC
    SelfAdjointEigenSolverNosort()
        : m_eivec(),
          m_eivalues(),
          m_subdiag(),
          m_hcoeffs(),
          m_info(InvalidInput),
          m_isInitialized(false),
          m_eigenvectorsOk(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
      */
    EIGEN_DEVICE_FUNC
    explicit SelfAdjointEigenSolverNosort(Index size)
        : m_eivec(size, size),
          m_eivalues(size),
          m_subdiag(size > 1 ? size - 1 : 1),
          m_hcoeffs(size > 1 ? size - 1 : 1),
          m_isInitialized(false),
          m_eigenvectorsOk(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)
      */
    template<typename InputType>
    EIGEN_DEVICE_FUNC
    explicit SelfAdjointEigenSolverNosort(const EigenBase<InputType>& matrix, int options = ComputeEigenvectors)
      : m_eivec(matrix.rows(), matrix.cols()),
        m_eivalues(matrix.cols()),
        m_subdiag(matrix.rows() > 1 ? matrix.rows() - 1 : 1),
        m_hcoeffs(matrix.cols() > 1 ? matrix.cols() - 1 : 1),
        m_isInitialized(false),
        m_eigenvectorsOk(false)
    {
      compute(matrix.derived(), 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)
      */
    template<typename InputType>
    EIGEN_DEVICE_FUNC
    SelfAdjointEigenSolverNosort& compute(const EigenBase<InputType>& matrix, int options = ComputeEigenvectors);

    /** \brief Computes eigendecomposition of given matrix using a closed-form algorithm
      *
      * This is a variant of compute(const MatrixType&, int options) which
      * directly solves the underlying polynomial equation.
      *
      * Currently only 2x2 and 3x3 matrices for which the sizes are known at compile time are supported (e.g., Matrix3d).
      *
      * This method is usually significantly faster than the QR iterative algorithm
      * but it might also be less accurate. It is also worth noting that
      * for 3x3 matrices it involves trigonometric operations which are
      * not necessarily available for all scalar types.
      *
      * For the 3x3 case, we observed the following worst case relative error regarding the eigenvalues:
      *   - double: 1e-8
      *   - float:  1e-3
      *
      * \sa compute(const MatrixType&, int options)
      */
    EIGEN_DEVICE_FUNC
    SelfAdjointEigenSolverNosort& computeDirect(const MatrixType& matrix, int options = ComputeEigenvectors);

    /**
      *\brief Computes the eigen decomposition from a tridiagonal symmetric matrix
      *
      * \param[in] diag The vector containing the diagonal of the matrix.
      * \param[in] subdiag The subdiagonal of the matrix.
      * \param[in] options Can be #ComputeEigenvectors (default) or #EigenvaluesOnly.
      * \returns Reference to \c *this
      *
      * This function assumes that the matrix has been reduced to tridiagonal form.
      *
      * \sa compute(const MatrixType&, int) for more information
      */
    SelfAdjointEigenSolverNosort& computeFromTridiagonal(const RealVectorType& diag, const SubDiagonalType& subdiag , 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$.
      *
      * For a selfadjoint matrix, \f$ V \f$ is unitary, meaning its inverse is equal
      * to its adjoint, \f$ V^{-1} = V^{\dagger} \f$. If \f$ A \f$ is real, then
      * \f$ V \f$ is also real and therefore orthogonal, meaning its inverse is
      * equal to its transpose, \f$ V^{-1} = V^T \f$.
      *
      * Example: \include SelfAdjointEigenSolver_eigenvectors.cpp
      * Output: \verbinclude SelfAdjointEigenSolver_eigenvectors.out
      *
      * \sa eigenvalues()
      */
    EIGEN_DEVICE_FUNC
    const EigenvectorsType& 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()
      */
    EIGEN_DEVICE_FUNC
    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(), <a href="unsupported/group__MatrixFunctions__Module.html">MatrixFunctions Module</a>
      */
    EIGEN_DEVICE_FUNC
    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(), <a href="unsupported/group__MatrixFunctions__Module.html">MatrixFunctions Module</a>
      */
    EIGEN_DEVICE_FUNC
    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 successful, \c NoConvergence otherwise.
      */
    EIGEN_DEVICE_FUNC
    ComputationInfo info() const
    {
      eigen_assert(m_isInitialized && "SelfAdjointEigenSolver is not initialized.");
      return m_info;
    }

    /** \brief Maximum number of iterations.
      *
      * The algorithm terminates if it does not converge within m_maxIterations * n iterations, where n
      * denotes the size of the matrix. This value is currently set to 30 (copied from LAPACK).
      */
    static const int m_maxIterations = 30;

  protected:
    static EIGEN_DEVICE_FUNC
    void check_template_parameters()
    {
      EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
    }

    EigenvectorsType m_eivec;
    RealVectorType m_eivalues;
    typename TridiagonalizationType::SubDiagonalType m_subdiag;
    typename TridiagonalizationType::CoeffVectorType m_hcoeffs;
    ComputationInfo m_info;
    bool m_isInitialized;
    bool m_eigenvectorsOk;
};

namespace internal {
/** \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 diag the diagonal part of the input selfadjoint tridiagonal matrix
  * \param subdiag the sub-diagonal part of the input selfadjoint tridiagonal matrix
  * \param start starting index of the submatrix to work on
  * \param end last+1 index of the submatrix to work on
  * \param matrixQ pointer to the column-major matrix holding the eigenvectors, can be 0
  * \param n size of the input matrix
  *
  * 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"
  */
template<int StorageOrder,typename RealScalar, typename Scalar, typename Index>
EIGEN_DEVICE_FUNC
static void tridiagonal_qr_step(RealScalar* diag, RealScalar* subdiag, Index start, Index end, Scalar* matrixQ, Index n);
}

template<typename MatrixType>
template<typename InputType>
EIGEN_DEVICE_FUNC
SelfAdjointEigenSolverNosort<MatrixType>& SelfAdjointEigenSolverNosort<MatrixType>
::compute(const EigenBase<InputType>& a_matrix, int options)
{
  check_template_parameters();

  const InputType &matrix(a_matrix.derived());

  EIGEN_USING_STD(abs);
  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_eivec = matrix;
    m_eivalues.coeffRef(0,0) = numext::real(m_eivec.coeff(0,0));
    if(computeEigenvectors)
      m_eivec.setOnes(n,n);
    m_info = Success;
    m_isInitialized = true;
    m_eigenvectorsOk = computeEigenvectors;
    return *this;
  }

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

  // map the matrix coefficients to [-1:1] to avoid over- and underflow.
  mat = matrix.template triangularView<Lower>();
  RealScalar scale = mat.cwiseAbs().maxCoeff();
  if(scale==RealScalar(0)) scale = RealScalar(1);
  mat.template triangularView<Lower>() /= scale;
  m_subdiag.resize(n-1);
  m_hcoeffs.resize(n-1);
  // API changed from Eigen 3.3.x to 3.4
  internal::tridiagonalization_inplace_3_4(mat, diag, m_subdiag, m_hcoeffs, computeEigenvectors);

  m_info = internal::computeFromTridiagonalNosort_impl(diag, m_subdiag, m_maxIterations, computeEigenvectors, m_eivec);

  // scale back the eigen values
  m_eivalues *= scale;

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

template<typename MatrixType>
SelfAdjointEigenSolverNosort<MatrixType>& SelfAdjointEigenSolverNosort<MatrixType>
::computeFromTridiagonal(const RealVectorType& diag, const SubDiagonalType& subdiag , int options)
{
  //TODO : Add an option to scale the values beforehand
  bool computeEigenvectors = (options&ComputeEigenvectors)==ComputeEigenvectors;

  m_eivalues = diag;
  m_subdiag = subdiag;
  if (computeEigenvectors)
  {
    m_eivec.setIdentity(diag.size(), diag.size());
  }
  m_info = internal::computeFromTridiagonalNosort_impl(m_eivalues, m_subdiag, m_maxIterations, computeEigenvectors, m_eivec);

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

namespace internal {
/**
  * \internal
  * \brief Compute the eigendecomposition from a tridiagonal matrix
  *
  * \param[in,out] diag : On input, the diagonal of the matrix, on output the eigenvalues
  * \param[in,out] subdiag : The subdiagonal part of the matrix (entries are modified during the decomposition)
  * \param[in] maxIterations : the maximum number of iterations
  * \param[in] computeEigenvectors : whether the eigenvectors have to be computed or not
  * \param[out] eivec : The matrix to store the eigenvectors if computeEigenvectors==true. Must be allocated on input.
  * \returns \c Success or \c NoConvergence
  */
template<typename MatrixType, typename DiagType, typename SubDiagType>
EIGEN_DEVICE_FUNC
ComputationInfo computeFromTridiagonalNosort_impl(DiagType& diag, SubDiagType& subdiag, const Index maxIterations, bool computeEigenvectors, MatrixType& eivec)
{
  ComputationInfo info;
  typedef typename MatrixType::Scalar Scalar;

  Index n = diag.size();
  Index end = n-1;
  Index start = 0;
  Index iter = 0; // total number of iterations

  typedef typename DiagType::RealScalar RealScalar;
  const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
  const RealScalar precision_inv = RealScalar(1)/NumTraits<RealScalar>::epsilon();
  while (end>0)
  {
    for (Index i = start; i<end; ++i) {
      if (numext::abs(subdiag[i]) < considerAsZero) {
        subdiag[i] = RealScalar(0);
      } else {
        // abs(subdiag[i]) <= epsilon * sqrt(abs(diag[i]) + abs(diag[i+1]))
        // Scaled to prevent underflows.
        const RealScalar scaled_subdiag = precision_inv * subdiag[i];
        if (scaled_subdiag * scaled_subdiag <= (numext::abs(diag[i])+numext::abs(diag[i+1]))) {
          subdiag[i] = RealScalar(0);
        }
      }
    }

    // find the largest unreduced block at the end of the matrix.
    while (end>0 && subdiag[end-1]==RealScalar(0))
    {
      end--;
    }
    if (end<=0)
      break;

    // if we spent too many iterations, we give up
    iter++;
    if(iter > maxIterations * n) break;

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

    internal::tridiagonal_qr_step<MatrixType::Flags&RowMajorBit ? RowMajor : ColMajor>(diag.data(), subdiag.data(), start, end, computeEigenvectors ? eivec.data() : (Scalar*)0, n);
  }
  if (iter <= maxIterations * n)
    info = Success;
  else
    info = NoConvergence;

  return info;
}

}

} // end namespace Eigen

#endif // EIGEN_SELFADJOINTEIGENSOLVER_H
