don't index Eigen

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diff --git a/eigen/Eigen/src/Eigenvalues/RealSchur.h b/eigen/Eigen/src/Eigenvalues/RealSchur.h
deleted file mode 100644
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--- a/eigen/Eigen/src/Eigenvalues/RealSchur.h
+++ /dev/null
@@ -1,546 +0,0 @@
-// 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) 2010,2012 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_REAL_SCHUR_H
-#define EIGEN_REAL_SCHUR_H
-
-#include "./HessenbergDecomposition.h"
-
-namespace Eigen { 
-
-/** \eigenvalues_module \ingroup Eigenvalues_Module
-  *
-  *
-  * \class RealSchur
-  *
-  * \brief Performs a real Schur decomposition of a square matrix
-  *
-  * \tparam _MatrixType the type of the matrix of which we are computing the
-  * real Schur decomposition; this is expected to be an instantiation of the
-  * Matrix class template.
-  *
-  * Given a real square matrix A, this class computes the real Schur
-  * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
-  * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
-  * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
-  * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
-  * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
-  * blocks on the diagonal of T are the same as the eigenvalues of the matrix
-  * A, and thus the real Schur decomposition is used in EigenSolver to compute
-  * the eigendecomposition of a matrix.
-  *
-  * Call the function compute() to compute the real Schur decomposition of a
-  * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
-  * constructor which computes the real Schur decomposition at construction
-  * time. Once the decomposition is computed, you can use the matrixU() and
-  * matrixT() functions to retrieve the matrices U and T in the decomposition.
-  *
-  * The documentation of RealSchur(const MatrixType&, bool) contains an example
-  * of the typical use of this class.
-  *
-  * \note The implementation is adapted from
-  * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
-  * Their code is based on EISPACK.
-  *
-  * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
-  */
-template<typename _MatrixType> class RealSchur
-{
-  public:
-    typedef _MatrixType MatrixType;
-    enum {
-      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
-      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
-      Options = MatrixType::Options,
-      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
-      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
-    };
-    typedef typename MatrixType::Scalar Scalar;
-    typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
-    typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
-
-    typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
-    typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
-
-    /** \brief Default constructor.
-      *
-      * \param [in] size  Positive integer, size of the matrix whose Schur 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.
-      */
-    explicit RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
-            : m_matT(size, size),
-              m_matU(size, size),
-              m_workspaceVector(size),
-              m_hess(size),
-              m_isInitialized(false),
-              m_matUisUptodate(false),
-              m_maxIters(-1)
-    { }
-
-    /** \brief Constructor; computes real Schur decomposition of given matrix. 
-      * 
-      * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
-      * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
-      *
-      * This constructor calls compute() to compute the Schur decomposition.
-      *
-      * Example: \include RealSchur_RealSchur_MatrixType.cpp
-      * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
-      */
-    template<typename InputType>
-    explicit RealSchur(const EigenBase<InputType>& matrix, bool computeU = true)
-            : m_matT(matrix.rows(),matrix.cols()),
-              m_matU(matrix.rows(),matrix.cols()),
-              m_workspaceVector(matrix.rows()),
-              m_hess(matrix.rows()),
-              m_isInitialized(false),
-              m_matUisUptodate(false),
-              m_maxIters(-1)
-    {
-      compute(matrix.derived(), computeU);
-    }
-
-    /** \brief Returns the orthogonal matrix in the Schur decomposition. 
-      *
-      * \returns A const reference to the matrix U.
-      *
-      * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
-      * member function compute(const MatrixType&, bool) has been called before
-      * to compute the Schur decomposition of a matrix, and \p computeU was set
-      * to true (the default value).
-      *
-      * \sa RealSchur(const MatrixType&, bool) for an example
-      */
-    const MatrixType& matrixU() const
-    {
-      eigen_assert(m_isInitialized && "RealSchur is not initialized.");
-      eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
-      return m_matU;
-    }
-
-    /** \brief Returns the quasi-triangular matrix in the Schur decomposition. 
-      *
-      * \returns A const reference to the matrix T.
-      *
-      * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
-      * member function compute(const MatrixType&, bool) has been called before
-      * to compute the Schur decomposition of a matrix.
-      *
-      * \sa RealSchur(const MatrixType&, bool) for an example
-      */
-    const MatrixType& matrixT() const
-    {
-      eigen_assert(m_isInitialized && "RealSchur is not initialized.");
-      return m_matT;
-    }
-  
-    /** \brief Computes Schur decomposition of given matrix. 
-      * 
-      * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
-      * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
-      * \returns    Reference to \c *this
-      *
-      * The Schur decomposition is computed by first reducing the matrix to
-      * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
-      * matrix is then reduced to triangular form by performing Francis QR
-      * iterations with implicit double shift. The cost of computing the Schur
-      * decomposition depends on the number of iterations; as a rough guide, it
-      * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
-      * \f$10n^3\f$ flops if \a computeU is false.
-      *
-      * Example: \include RealSchur_compute.cpp
-      * Output: \verbinclude RealSchur_compute.out
-      *
-      * \sa compute(const MatrixType&, bool, Index)
-      */
-    template<typename InputType>
-    RealSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true);
-
-    /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
-     *  \param[in] matrixH Matrix in Hessenberg form H
-     *  \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
-     *  \param computeU Computes the matriX U of the Schur vectors
-     * \return Reference to \c *this
-     * 
-     *  This routine assumes that the matrix is already reduced in Hessenberg form matrixH
-     *  using either the class HessenbergDecomposition or another mean. 
-     *  It computes the upper quasi-triangular matrix T of the Schur decomposition of H
-     *  When computeU is true, this routine computes the matrix U such that 
-     *  A = U T U^T =  (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
-     * 
-     * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
-     * is not available, the user should give an identity matrix (Q.setIdentity())
-     * 
-     * \sa compute(const MatrixType&, bool)
-     */
-    template<typename HessMatrixType, typename OrthMatrixType>
-    RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU);
-    /** \brief Reports whether previous computation was successful.
-      *
-      * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
-      */
-    ComputationInfo info() const
-    {
-      eigen_assert(m_isInitialized && "RealSchur is not initialized.");
-      return m_info;
-    }
-
-    /** \brief Sets the maximum number of iterations allowed. 
-      *
-      * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
-      * of the matrix.
-      */
-    RealSchur& setMaxIterations(Index maxIters)
-    {
-      m_maxIters = maxIters;
-      return *this;
-    }
-
-    /** \brief Returns the maximum number of iterations. */
-    Index getMaxIterations()
-    {
-      return m_maxIters;
-    }
-
-    /** \brief Maximum number of iterations per row.
-      *
-      * If not otherwise specified, the maximum number of iterations is this number times the size of the
-      * matrix. It is currently set to 40.
-      */
-    static const int m_maxIterationsPerRow = 40;
-
-  private:
-    
-    MatrixType m_matT;
-    MatrixType m_matU;
-    ColumnVectorType m_workspaceVector;
-    HessenbergDecomposition<MatrixType> m_hess;
-    ComputationInfo m_info;
-    bool m_isInitialized;
-    bool m_matUisUptodate;
-    Index m_maxIters;
-
-    typedef Matrix<Scalar,3,1> Vector3s;
-
-    Scalar computeNormOfT();
-    Index findSmallSubdiagEntry(Index iu);
-    void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
-    void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
-    void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
-    void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
-};
-
-
-template<typename MatrixType>
-template<typename InputType>
-RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU)
-{
-  const Scalar considerAsZero = (std::numeric_limits<Scalar>::min)();
-
-  eigen_assert(matrix.cols() == matrix.rows());
-  Index maxIters = m_maxIters;
-  if (maxIters == -1)
-    maxIters = m_maxIterationsPerRow * matrix.rows();
-
-  Scalar scale = matrix.derived().cwiseAbs().maxCoeff();
-  if(scale<considerAsZero)
-  {
-    m_matT.setZero(matrix.rows(),matrix.cols());
-    if(computeU)
-      m_matU.setIdentity(matrix.rows(),matrix.cols());
-    m_info = Success;
-    m_isInitialized = true;
-    m_matUisUptodate = computeU;
-    return *this;
-  }
-
-  // Step 1. Reduce to Hessenberg form
-  m_hess.compute(matrix.derived()/scale);
-
-  // Step 2. Reduce to real Schur form  
-  computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
-
-  m_matT *= scale;
-  
-  return *this;
-}
-template<typename MatrixType>
-template<typename HessMatrixType, typename OrthMatrixType>
-RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU)
-{
-  using std::abs;
-
-  m_matT = matrixH;
-  if(computeU)
-    m_matU = matrixQ;
-  
-  Index maxIters = m_maxIters;
-  if (maxIters == -1)
-    maxIters = m_maxIterationsPerRow * matrixH.rows();
-  m_workspaceVector.resize(m_matT.cols());
-  Scalar* workspace = &m_workspaceVector.coeffRef(0);
-
-  // The matrix m_matT is divided in three parts. 
-  // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero. 
-  // Rows il,...,iu is the part we are working on (the active window).
-  // Rows iu+1,...,end are already brought in triangular form.
-  Index iu = m_matT.cols() - 1;
-  Index iter = 0;      // iteration count for current eigenvalue
-  Index totalIter = 0; // iteration count for whole matrix
-  Scalar exshift(0);   // sum of exceptional shifts
-  Scalar norm = computeNormOfT();
-
-  if(norm!=Scalar(0))
-  {
-    while (iu >= 0)
-    {
-      Index il = findSmallSubdiagEntry(iu);
-
-      // Check for convergence
-      if (il == iu) // One root found
-      {
-        m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
-        if (iu > 0)
-          m_matT.coeffRef(iu, iu-1) = Scalar(0);
-        iu--;
-        iter = 0;
-      }
-      else if (il == iu-1) // Two roots found
-      {
-        splitOffTwoRows(iu, computeU, exshift);
-        iu -= 2;
-        iter = 0;
-      }
-      else // No convergence yet
-      {
-        // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
-        Vector3s firstHouseholderVector = Vector3s::Zero(), shiftInfo;
-        computeShift(iu, iter, exshift, shiftInfo);
-        iter = iter + 1;
-        totalIter = totalIter + 1;
-        if (totalIter > maxIters) break;
-        Index im;
-        initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
-        performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
-      }
-    }
-  }
-  if(totalIter <= maxIters)
-    m_info = Success;
-  else
-    m_info = NoConvergence;
-
-  m_isInitialized = true;
-  m_matUisUptodate = computeU;
-  return *this;
-}
-
-/** \internal Computes and returns vector L1 norm of T */
-template<typename MatrixType>
-inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
-{
-  const Index size = m_matT.cols();
-  // FIXME to be efficient the following would requires a triangular reduxion code
-  // Scalar norm = m_matT.upper().cwiseAbs().sum() 
-  //               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
-  Scalar norm(0);
-  for (Index j = 0; j < size; ++j)
-    norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
-  return norm;
-}
-
-/** \internal Look for single small sub-diagonal element and returns its index */
-template<typename MatrixType>
-inline Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu)
-{
-  using std::abs;
-  Index res = iu;
-  while (res > 0)
-  {
-    Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
-    if (abs(m_matT.coeff(res,res-1)) <= NumTraits<Scalar>::epsilon() * s)
-      break;
-    res--;
-  }
-  return res;
-}
-
-/** \internal Update T given that rows iu-1 and iu decouple from the rest. */
-template<typename MatrixType>
-inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
-{
-  using std::sqrt;
-  using std::abs;
-  const Index size = m_matT.cols();
-
-  // The eigenvalues of the 2x2 matrix [a b; c d] are 
-  // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
-  Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
-  Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);   // q = tr^2 / 4 - det = discr/4
-  m_matT.coeffRef(iu,iu) += exshift;
-  m_matT.coeffRef(iu-1,iu-1) += exshift;
-
-  if (q >= Scalar(0)) // Two real eigenvalues
-  {
-    Scalar z = sqrt(abs(q));
-    JacobiRotation<Scalar> rot;
-    if (p >= Scalar(0))
-      rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
-    else
-      rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
-
-    m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
-    m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
-    m_matT.coeffRef(iu, iu-1) = Scalar(0); 
-    if (computeU)
-      m_matU.applyOnTheRight(iu-1, iu, rot);
-  }
-
-  if (iu > 1) 
-    m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
-}
-
-/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
-template<typename MatrixType>
-inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
-{
-  using std::sqrt;
-  using std::abs;
-  shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
-  shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
-  shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
-
-  // Wilkinson's original ad hoc shift
-  if (iter == 10)
-  {
-    exshift += shiftInfo.coeff(0);
-    for (Index i = 0; i <= iu; ++i)
-      m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
-    Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
-    shiftInfo.coeffRef(0) = Scalar(0.75) * s;
-    shiftInfo.coeffRef(1) = Scalar(0.75) * s;
-    shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
-  }
-
-  // MATLAB's new ad hoc shift
-  if (iter == 30)
-  {
-    Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
-    s = s * s + shiftInfo.coeff(2);
-    if (s > Scalar(0))
-    {
-      s = sqrt(s);
-      if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
-        s = -s;
-      s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
-      s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
-      exshift += s;
-      for (Index i = 0; i <= iu; ++i)
-        m_matT.coeffRef(i,i) -= s;
-      shiftInfo.setConstant(Scalar(0.964));
-    }
-  }
-}
-
-/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
-template<typename MatrixType>
-inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
-{
-  using std::abs;
-  Vector3s& v = firstHouseholderVector; // alias to save typing
-
-  for (im = iu-2; im >= il; --im)
-  {
-    const Scalar Tmm = m_matT.coeff(im,im);
-    const Scalar r = shiftInfo.coeff(0) - Tmm;
-    const Scalar s = shiftInfo.coeff(1) - Tmm;
-    v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
-    v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
-    v.coeffRef(2) = m_matT.coeff(im+2,im+1);
-    if (im == il) {
-      break;
-    }
-    const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
-    const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
-    if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
-      break;
-  }
-}
-
-/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
-template<typename MatrixType>
-inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
-{
-  eigen_assert(im >= il);
-  eigen_assert(im <= iu-2);
-
-  const Index size = m_matT.cols();
-
-  for (Index k = im; k <= iu-2; ++k)
-  {
-    bool firstIteration = (k == im);
-
-    Vector3s v;
-    if (firstIteration)
-      v = firstHouseholderVector;
-    else
-      v = m_matT.template block<3,1>(k,k-1);
-
-    Scalar tau, beta;
-    Matrix<Scalar, 2, 1> ess;
-    v.makeHouseholder(ess, tau, beta);
-    
-    if (beta != Scalar(0)) // if v is not zero
-    {
-      if (firstIteration && k > il)
-        m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
-      else if (!firstIteration)
-        m_matT.coeffRef(k,k-1) = beta;
-
-      // These Householder transformations form the O(n^3) part of the algorithm
-      m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
-      m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
-      if (computeU)
-        m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
-    }
-  }
-
-  Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
-  Scalar tau, beta;
-  Matrix<Scalar, 1, 1> ess;
-  v.makeHouseholder(ess, tau, beta);
-
-  if (beta != Scalar(0)) // if v is not zero
-  {
-    m_matT.coeffRef(iu-1, iu-2) = beta;
-    m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
-    m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
-    if (computeU)
-      m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
-  }
-
-  // clean up pollution due to round-off errors
-  for (Index i = im+2; i <= iu; ++i)
-  {
-    m_matT.coeffRef(i,i-2) = Scalar(0);
-    if (i > im+2)
-      m_matT.coeffRef(i,i-3) = Scalar(0);
-  }
-}
-
-} // end namespace Eigen
-
-#endif // EIGEN_REAL_SCHUR_H