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+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2009 Claire Maurice
+// Copyright (C) 2009 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_COMPLEX_SCHUR_H
+#define EIGEN_COMPLEX_SCHUR_H
+
+#include "./HessenbergDecomposition.h"
+
+namespace Eigen { 
+
+namespace internal {
+template<typename MatrixType, bool IsComplex> struct complex_schur_reduce_to_hessenberg;
+}
+
+/** \eigenvalues_module \ingroup Eigenvalues_Module
+  *
+  *
+  * \class ComplexSchur
+  *
+  * \brief Performs a complex Schur decomposition of a real or complex square matrix
+  *
+  * \tparam _MatrixType the type of the matrix of which we are
+  * computing the Schur decomposition; this is expected to be an
+  * instantiation of the Matrix class template.
+  *
+  * Given a real or complex square matrix A, this class computes the
+  * Schur decomposition: \f$ A = U T U^*\f$ where U is a unitary
+  * complex matrix, and T is a complex upper triangular matrix.  The
+  * diagonal of the matrix T corresponds to the eigenvalues of the
+  * matrix A.
+  *
+  * Call the function compute() to compute the Schur decomposition of
+  * a given matrix. Alternatively, you can use the 
+  * ComplexSchur(const MatrixType&, bool) constructor which computes
+  * the Schur decomposition at construction time. Once the
+  * decomposition is computed, you can use the matrixU() and matrixT()
+  * functions to retrieve the matrices U and V in the decomposition.
+  *
+  * \note This code is inspired from Jampack
+  *
+  * \sa class RealSchur, class EigenSolver, class ComplexEigenSolver
+  */
+template<typename _MatrixType> class ComplexSchur
+{
+  public:
+    typedef _MatrixType MatrixType;
+    enum {
+      RowsAtCompileTime = MatrixType::RowsAtCompileTime,
+      ColsAtCompileTime = MatrixType::ColsAtCompileTime,
+      Options = MatrixType::Options,
+      MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
+      MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
+    };
+
+    /** \brief Scalar type for matrices of type \p _MatrixType. */
+    typedef typename MatrixType::Scalar Scalar;
+    typedef typename NumTraits<Scalar>::Real RealScalar;
+    typedef typename MatrixType::Index Index;
+
+    /** \brief Complex scalar type for \p _MatrixType. 
+      *
+      * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
+      * \c float or \c double) and just \c Scalar if #Scalar is
+      * complex.
+      */
+    typedef std::complex<RealScalar> ComplexScalar;
+
+    /** \brief Type for the matrices in the Schur decomposition.
+      *
+      * This is a square matrix with entries of type #ComplexScalar. 
+      * The size is the same as the size of \p _MatrixType.
+      */
+    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrixType;
+
+    /** \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.
+      */
+    ComplexSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
+      : m_matT(size,size),
+        m_matU(size,size),
+        m_hess(size),
+        m_isInitialized(false),
+        m_matUisUptodate(false),
+        m_maxIters(-1)
+    {}
+
+    /** \brief Constructor; 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.
+      *
+      * This constructor calls compute() to compute the Schur decomposition.
+      *
+      * \sa matrixT() and matrixU() for examples.
+      */
+    ComplexSchur(const MatrixType& matrix, bool computeU = true)
+      : m_matT(matrix.rows(),matrix.cols()),
+        m_matU(matrix.rows(),matrix.cols()),
+        m_hess(matrix.rows()),
+        m_isInitialized(false),
+        m_matUisUptodate(false),
+        m_maxIters(-1)
+    {
+      compute(matrix, computeU);
+    }
+
+    /** \brief Returns the unitary matrix in the Schur decomposition. 
+      *
+      * \returns A const reference to the matrix U.
+      *
+      * It is assumed that either the constructor
+      * ComplexSchur(const MatrixType& matrix, bool computeU) or the
+      * member function compute(const MatrixType& matrix, bool computeU)
+      * has been called before to compute the Schur decomposition of a
+      * matrix, and that \p computeU was set to true (the default
+      * value).
+      *
+      * Example: \include ComplexSchur_matrixU.cpp
+      * Output: \verbinclude ComplexSchur_matrixU.out
+      */
+    const ComplexMatrixType& matrixU() const
+    {
+      eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
+      eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the ComplexSchur decomposition.");
+      return m_matU;
+    }
+
+    /** \brief Returns the triangular matrix in the Schur decomposition. 
+      *
+      * \returns A const reference to the matrix T.
+      *
+      * It is assumed that either the constructor
+      * ComplexSchur(const MatrixType& matrix, bool computeU) or the
+      * member function compute(const MatrixType& matrix, bool computeU)
+      * has been called before to compute the Schur decomposition of a
+      * matrix.
+      *
+      * Note that this function returns a plain square matrix. If you want to reference
+      * only the upper triangular part, use:
+      * \code schur.matrixT().triangularView<Upper>() \endcode 
+      *
+      * Example: \include ComplexSchur_matrixT.cpp
+      * Output: \verbinclude ComplexSchur_matrixT.out
+      */
+    const ComplexMatrixType& matrixT() const
+    {
+      eigen_assert(m_isInitialized && "ComplexSchur 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 QR iterations with a single
+      * shift. The cost of computing the Schur decomposition depends
+      * on the number of iterations; as a rough guide, it may be taken
+      * on the number of iterations; as a rough guide, it may be taken
+      * to be \f$25n^3\f$ complex flops, or \f$10n^3\f$ complex flops
+      * if \a computeU is false.
+      *
+      * Example: \include ComplexSchur_compute.cpp
+      * Output: \verbinclude ComplexSchur_compute.out
+      *
+      * \sa compute(const MatrixType&, bool, Index)
+      */
+    ComplexSchur& compute(const MatrixType& matrix, bool computeU = true);
+    
+    /** \brief Compute Schur decomposition from a given Hessenberg matrix
+     *  \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>
+    ComplexSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ,  bool computeU=true);
+
+    /** \brief Reports whether previous computation was successful.
+      *
+      * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
+      */
+    ComputationInfo info() const
+    {
+      eigen_assert(m_isInitialized && "ComplexSchur 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.
+      */
+    ComplexSchur& 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 30.
+      */
+    static const int m_maxIterationsPerRow = 30;
+
+  protected:
+    ComplexMatrixType m_matT, m_matU;
+    HessenbergDecomposition<MatrixType> m_hess;
+    ComputationInfo m_info;
+    bool m_isInitialized;
+    bool m_matUisUptodate;
+    Index m_maxIters;
+
+  private:  
+    bool subdiagonalEntryIsNeglegible(Index i);
+    ComplexScalar computeShift(Index iu, Index iter);
+    void reduceToTriangularForm(bool computeU);
+    friend struct internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>;
+};
+
+/** If m_matT(i+1,i) is neglegible in floating point arithmetic
+  * compared to m_matT(i,i) and m_matT(j,j), then set it to zero and
+  * return true, else return false. */
+template<typename MatrixType>
+inline bool ComplexSchur<MatrixType>::subdiagonalEntryIsNeglegible(Index i)
+{
+  RealScalar d = numext::norm1(m_matT.coeff(i,i)) + numext::norm1(m_matT.coeff(i+1,i+1));
+  RealScalar sd = numext::norm1(m_matT.coeff(i+1,i));
+  if (internal::isMuchSmallerThan(sd, d, NumTraits<RealScalar>::epsilon()))
+  {
+    m_matT.coeffRef(i+1,i) = ComplexScalar(0);
+    return true;
+  }
+  return false;
+}
+
+
+/** Compute the shift in the current QR iteration. */
+template<typename MatrixType>
+typename ComplexSchur<MatrixType>::ComplexScalar ComplexSchur<MatrixType>::computeShift(Index iu, Index iter)
+{
+  using std::abs;
+  if (iter == 10 || iter == 20) 
+  {
+    // exceptional shift, taken from http://www.netlib.org/eispack/comqr.f
+    return abs(numext::real(m_matT.coeff(iu,iu-1))) + abs(numext::real(m_matT.coeff(iu-1,iu-2)));
+  }
+
+  // compute the shift as one of the eigenvalues of t, the 2x2
+  // diagonal block on the bottom of the active submatrix
+  Matrix<ComplexScalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
+  RealScalar normt = t.cwiseAbs().sum();
+  t /= normt;     // the normalization by sf is to avoid under/overflow
+
+  ComplexScalar b = t.coeff(0,1) * t.coeff(1,0);
+  ComplexScalar c = t.coeff(0,0) - t.coeff(1,1);
+  ComplexScalar disc = sqrt(c*c + RealScalar(4)*b);
+  ComplexScalar det = t.coeff(0,0) * t.coeff(1,1) - b;
+  ComplexScalar trace = t.coeff(0,0) + t.coeff(1,1);
+  ComplexScalar eival1 = (trace + disc) / RealScalar(2);
+  ComplexScalar eival2 = (trace - disc) / RealScalar(2);
+
+  if(numext::norm1(eival1) > numext::norm1(eival2))
+    eival2 = det / eival1;
+  else
+    eival1 = det / eival2;
+
+  // choose the eigenvalue closest to the bottom entry of the diagonal
+  if(numext::norm1(eival1-t.coeff(1,1)) < numext::norm1(eival2-t.coeff(1,1)))
+    return normt * eival1;
+  else
+    return normt * eival2;
+}
+
+
+template<typename MatrixType>
+ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
+{
+  m_matUisUptodate = false;
+  eigen_assert(matrix.cols() == matrix.rows());
+
+  if(matrix.cols() == 1)
+  {
+    m_matT = matrix.template cast<ComplexScalar>();
+    if(computeU)  m_matU = ComplexMatrixType::Identity(1,1);
+    m_info = Success;
+    m_isInitialized = true;
+    m_matUisUptodate = computeU;
+    return *this;
+  }
+
+  internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>::run(*this, matrix, computeU);
+  computeFromHessenberg(m_matT, m_matU, computeU);
+  return *this;
+}
+
+template<typename MatrixType>
+template<typename HessMatrixType, typename OrthMatrixType>
+ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
+{
+  m_matT = matrixH;
+  if(computeU)
+    m_matU = matrixQ;
+  reduceToTriangularForm(computeU);
+  return *this;
+}
+namespace internal {
+
+/* Reduce given matrix to Hessenberg form */
+template<typename MatrixType, bool IsComplex>
+struct complex_schur_reduce_to_hessenberg
+{
+  // this is the implementation for the case IsComplex = true
+  static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
+  {
+    _this.m_hess.compute(matrix);
+    _this.m_matT = _this.m_hess.matrixH();
+    if(computeU)  _this.m_matU = _this.m_hess.matrixQ();
+  }
+};
+
+template<typename MatrixType>
+struct complex_schur_reduce_to_hessenberg<MatrixType, false>
+{
+  static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
+  {
+    typedef typename ComplexSchur<MatrixType>::ComplexScalar ComplexScalar;
+
+    // Note: m_hess is over RealScalar; m_matT and m_matU is over ComplexScalar
+    _this.m_hess.compute(matrix);
+    _this.m_matT = _this.m_hess.matrixH().template cast<ComplexScalar>();
+    if(computeU)  
+    {
+      // This may cause an allocation which seems to be avoidable
+      MatrixType Q = _this.m_hess.matrixQ(); 
+      _this.m_matU = Q.template cast<ComplexScalar>();
+    }
+  }
+};
+
+} // end namespace internal
+
+// Reduce the Hessenberg matrix m_matT to triangular form by QR iteration.
+template<typename MatrixType>
+void ComplexSchur<MatrixType>::reduceToTriangularForm(bool computeU)
+{  
+  Index maxIters = m_maxIters;
+  if (maxIters == -1)
+    maxIters = m_maxIterationsPerRow * m_matT.rows();
+
+  // 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 submatrix).
+  // Rows iu+1,...,end are already brought in triangular form.
+  Index iu = m_matT.cols() - 1;
+  Index il;
+  Index iter = 0; // number of iterations we are working on the (iu,iu) element
+  Index totalIter = 0; // number of iterations for whole matrix
+
+  while(true)
+  {
+    // find iu, the bottom row of the active submatrix
+    while(iu > 0)
+    {
+      if(!subdiagonalEntryIsNeglegible(iu-1)) break;
+      iter = 0;
+      --iu;
+    }
+
+    // if iu is zero then we are done; the whole matrix is triangularized
+    if(iu==0) break;
+
+    // if we spent too many iterations, we give up
+    iter++;
+    totalIter++;
+    if(totalIter > maxIters) break;
+
+    // find il, the top row of the active submatrix
+    il = iu-1;
+    while(il > 0 && !subdiagonalEntryIsNeglegible(il-1))
+    {
+      --il;
+    }
+
+    /* perform the QR step using Givens rotations. The first rotation
+       creates a bulge; the (il+2,il) element becomes nonzero. This
+       bulge is chased down to the bottom of the active submatrix. */
+
+    ComplexScalar shift = computeShift(iu, iter);
+    JacobiRotation<ComplexScalar> rot;
+    rot.makeGivens(m_matT.coeff(il,il) - shift, m_matT.coeff(il+1,il));
+    m_matT.rightCols(m_matT.cols()-il).applyOnTheLeft(il, il+1, rot.adjoint());
+    m_matT.topRows((std::min)(il+2,iu)+1).applyOnTheRight(il, il+1, rot);
+    if(computeU) m_matU.applyOnTheRight(il, il+1, rot);
+
+    for(Index i=il+1 ; i<iu ; i++)
+    {
+      rot.makeGivens(m_matT.coeffRef(i,i-1), m_matT.coeffRef(i+1,i-1), &m_matT.coeffRef(i,i-1));
+      m_matT.coeffRef(i+1,i-1) = ComplexScalar(0);
+      m_matT.rightCols(m_matT.cols()-i).applyOnTheLeft(i, i+1, rot.adjoint());
+      m_matT.topRows((std::min)(i+2,iu)+1).applyOnTheRight(i, i+1, rot);
+      if(computeU) m_matU.applyOnTheRight(i, i+1, rot);
+    }
+  }
+
+  if(totalIter <= maxIters)
+    m_info = Success;
+  else
+    m_info = NoConvergence;
+
+  m_isInitialized = true;
+  m_matUisUptodate = computeU;
+}
+
+} // end namespace Eigen
+
+#endif // EIGEN_COMPLEX_SCHUR_H