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+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
+// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
+//
+// 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_JACOBI_H
+#define EIGEN_JACOBI_H
+
+namespace Eigen { 
+
+/** \ingroup Jacobi_Module
+  * \jacobi_module
+  * \class JacobiRotation
+  * \brief Rotation given by a cosine-sine pair.
+  *
+  * This class represents a Jacobi or Givens rotation.
+  * This is a 2D rotation in the plane \c J of angle \f$ \theta \f$ defined by
+  * its cosine \c c and sine \c s as follow:
+  * \f$ J = \left ( \begin{array}{cc} c & \overline s \\ -s  & \overline c \end{array} \right ) \f$
+  *
+  * You can apply the respective counter-clockwise rotation to a column vector \c v by
+  * applying its adjoint on the left: \f$ v = J^* v \f$ that translates to the following Eigen code:
+  * \code
+  * v.applyOnTheLeft(J.adjoint());
+  * \endcode
+  *
+  * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
+  */
+template<typename Scalar> class JacobiRotation
+{
+  public:
+    typedef typename NumTraits<Scalar>::Real RealScalar;
+
+    /** Default constructor without any initialization. */
+    JacobiRotation() {}
+
+    /** Construct a planar rotation from a cosine-sine pair (\a c, \c s). */
+    JacobiRotation(const Scalar& c, const Scalar& s) : m_c(c), m_s(s) {}
+
+    Scalar& c() { return m_c; }
+    Scalar c() const { return m_c; }
+    Scalar& s() { return m_s; }
+    Scalar s() const { return m_s; }
+
+    /** Concatenates two planar rotation */
+    JacobiRotation operator*(const JacobiRotation& other)
+    {
+      using numext::conj;
+      return JacobiRotation(m_c * other.m_c - conj(m_s) * other.m_s,
+                            conj(m_c * conj(other.m_s) + conj(m_s) * conj(other.m_c)));
+    }
+
+    /** Returns the transposed transformation */
+    JacobiRotation transpose() const { using numext::conj; return JacobiRotation(m_c, -conj(m_s)); }
+
+    /** Returns the adjoint transformation */
+    JacobiRotation adjoint() const { using numext::conj; return JacobiRotation(conj(m_c), -m_s); }
+
+    template<typename Derived>
+    bool makeJacobi(const MatrixBase<Derived>&, Index p, Index q);
+    bool makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z);
+
+    void makeGivens(const Scalar& p, const Scalar& q, Scalar* z=0);
+
+  protected:
+    void makeGivens(const Scalar& p, const Scalar& q, Scalar* z, internal::true_type);
+    void makeGivens(const Scalar& p, const Scalar& q, Scalar* z, internal::false_type);
+
+    Scalar m_c, m_s;
+};
+
+/** Makes \c *this as a Jacobi rotation \a J such that applying \a J on both the right and left sides of the selfadjoint 2x2 matrix
+  * \f$ B = \left ( \begin{array}{cc} x & y \\ \overline y & z \end{array} \right )\f$ yields a diagonal matrix \f$ A = J^* B J \f$
+  *
+  * \sa MatrixBase::makeJacobi(const MatrixBase<Derived>&, Index, Index), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
+  */
+template<typename Scalar>
+bool JacobiRotation<Scalar>::makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z)
+{
+  using std::sqrt;
+  using std::abs;
+  typedef typename NumTraits<Scalar>::Real RealScalar;
+  RealScalar deno = RealScalar(2)*abs(y);
+  if(deno < (std::numeric_limits<RealScalar>::min)())
+  {
+    m_c = Scalar(1);
+    m_s = Scalar(0);
+    return false;
+  }
+  else
+  {
+    RealScalar tau = (x-z)/deno;
+    RealScalar w = sqrt(numext::abs2(tau) + RealScalar(1));
+    RealScalar t;
+    if(tau>RealScalar(0))
+    {
+      t = RealScalar(1) / (tau + w);
+    }
+    else
+    {
+      t = RealScalar(1) / (tau - w);
+    }
+    RealScalar sign_t = t > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
+    RealScalar n = RealScalar(1) / sqrt(numext::abs2(t)+RealScalar(1));
+    m_s = - sign_t * (numext::conj(y) / abs(y)) * abs(t) * n;
+    m_c = n;
+    return true;
+  }
+}
+
+/** Makes \c *this as a Jacobi rotation \c J such that applying \a J on both the right and left sides of the 2x2 selfadjoint matrix
+  * \f$ B = \left ( \begin{array}{cc} \text{this}_{pp} & \text{this}_{pq} \\ (\text{this}_{pq})^* & \text{this}_{qq} \end{array} \right )\f$ yields
+  * a diagonal matrix \f$ A = J^* B J \f$
+  *
+  * Example: \include Jacobi_makeJacobi.cpp
+  * Output: \verbinclude Jacobi_makeJacobi.out
+  *
+  * \sa JacobiRotation::makeJacobi(RealScalar, Scalar, RealScalar), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
+  */
+template<typename Scalar>
+template<typename Derived>
+inline bool JacobiRotation<Scalar>::makeJacobi(const MatrixBase<Derived>& m, Index p, Index q)
+{
+  return makeJacobi(numext::real(m.coeff(p,p)), m.coeff(p,q), numext::real(m.coeff(q,q)));
+}
+
+/** Makes \c *this as a Givens rotation \c G such that applying \f$ G^* \f$ to the left of the vector
+  * \f$ V = \left ( \begin{array}{c} p \\ q \end{array} \right )\f$ yields:
+  * \f$ G^* V = \left ( \begin{array}{c} r \\ 0 \end{array} \right )\f$.
+  *
+  * The value of \a z is returned if \a z is not null (the default is null).
+  * Also note that G is built such that the cosine is always real.
+  *
+  * Example: \include Jacobi_makeGivens.cpp
+  * Output: \verbinclude Jacobi_makeGivens.out
+  *
+  * This function implements the continuous Givens rotation generation algorithm
+  * found in Anderson (2000), Discontinuous Plane Rotations and the Symmetric Eigenvalue Problem.
+  * LAPACK Working Note 150, University of Tennessee, UT-CS-00-454, December 4, 2000.
+  *
+  * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
+  */
+template<typename Scalar>
+void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* z)
+{
+  makeGivens(p, q, z, typename internal::conditional<NumTraits<Scalar>::IsComplex, internal::true_type, internal::false_type>::type());
+}
+
+
+// specialization for complexes
+template<typename Scalar>
+void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::true_type)
+{
+  using std::sqrt;
+  using std::abs;
+  using numext::conj;
+  
+  if(q==Scalar(0))
+  {
+    m_c = numext::real(p)<0 ? Scalar(-1) : Scalar(1);
+    m_s = 0;
+    if(r) *r = m_c * p;
+  }
+  else if(p==Scalar(0))
+  {
+    m_c = 0;
+    m_s = -q/abs(q);
+    if(r) *r = abs(q);
+  }
+  else
+  {
+    RealScalar p1 = numext::norm1(p);
+    RealScalar q1 = numext::norm1(q);
+    if(p1>=q1)
+    {
+      Scalar ps = p / p1;
+      RealScalar p2 = numext::abs2(ps);
+      Scalar qs = q / p1;
+      RealScalar q2 = numext::abs2(qs);
+
+      RealScalar u = sqrt(RealScalar(1) + q2/p2);
+      if(numext::real(p)<RealScalar(0))
+        u = -u;
+
+      m_c = Scalar(1)/u;
+      m_s = -qs*conj(ps)*(m_c/p2);
+      if(r) *r = p * u;
+    }
+    else
+    {
+      Scalar ps = p / q1;
+      RealScalar p2 = numext::abs2(ps);
+      Scalar qs = q / q1;
+      RealScalar q2 = numext::abs2(qs);
+
+      RealScalar u = q1 * sqrt(p2 + q2);
+      if(numext::real(p)<RealScalar(0))
+        u = -u;
+
+      p1 = abs(p);
+      ps = p/p1;
+      m_c = p1/u;
+      m_s = -conj(ps) * (q/u);
+      if(r) *r = ps * u;
+    }
+  }
+}
+
+// specialization for reals
+template<typename Scalar>
+void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::false_type)
+{
+  using std::sqrt;
+  using std::abs;
+  if(q==Scalar(0))
+  {
+    m_c = p<Scalar(0) ? Scalar(-1) : Scalar(1);
+    m_s = Scalar(0);
+    if(r) *r = abs(p);
+  }
+  else if(p==Scalar(0))
+  {
+    m_c = Scalar(0);
+    m_s = q<Scalar(0) ? Scalar(1) : Scalar(-1);
+    if(r) *r = abs(q);
+  }
+  else if(abs(p) > abs(q))
+  {
+    Scalar t = q/p;
+    Scalar u = sqrt(Scalar(1) + numext::abs2(t));
+    if(p<Scalar(0))
+      u = -u;
+    m_c = Scalar(1)/u;
+    m_s = -t * m_c;
+    if(r) *r = p * u;
+  }
+  else
+  {
+    Scalar t = p/q;
+    Scalar u = sqrt(Scalar(1) + numext::abs2(t));
+    if(q<Scalar(0))
+      u = -u;
+    m_s = -Scalar(1)/u;
+    m_c = -t * m_s;
+    if(r) *r = q * u;
+  }
+
+}
+
+/****************************************************************************************
+*   Implementation of MatrixBase methods
+****************************************************************************************/
+
+namespace internal {
+/** \jacobi_module
+  * Applies the clock wise 2D rotation \a j to the set of 2D vectors of cordinates \a x and \a y:
+  * \f$ \left ( \begin{array}{cc} x \\ y \end{array} \right )  =  J \left ( \begin{array}{cc} x \\ y \end{array} \right ) \f$
+  *
+  * \sa MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()
+  */
+template<typename VectorX, typename VectorY, typename OtherScalar>
+void apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y, const JacobiRotation<OtherScalar>& j);
+}
+
+/** \jacobi_module
+  * Applies the rotation in the plane \a j to the rows \a p and \a q of \c *this, i.e., it computes B = J * B,
+  * with \f$ B = \left ( \begin{array}{cc} \text{*this.row}(p) \\ \text{*this.row}(q) \end{array} \right ) \f$.
+  *
+  * \sa class JacobiRotation, MatrixBase::applyOnTheRight(), internal::apply_rotation_in_the_plane()
+  */
+template<typename Derived>
+template<typename OtherScalar>
+inline void MatrixBase<Derived>::applyOnTheLeft(Index p, Index q, const JacobiRotation<OtherScalar>& j)
+{
+  RowXpr x(this->row(p));
+  RowXpr y(this->row(q));
+  internal::apply_rotation_in_the_plane(x, y, j);
+}
+
+/** \ingroup Jacobi_Module
+  * Applies the rotation in the plane \a j to the columns \a p and \a q of \c *this, i.e., it computes B = B * J
+  * with \f$ B = \left ( \begin{array}{cc} \text{*this.col}(p) & \text{*this.col}(q) \end{array} \right ) \f$.
+  *
+  * \sa class JacobiRotation, MatrixBase::applyOnTheLeft(), internal::apply_rotation_in_the_plane()
+  */
+template<typename Derived>
+template<typename OtherScalar>
+inline void MatrixBase<Derived>::applyOnTheRight(Index p, Index q, const JacobiRotation<OtherScalar>& j)
+{
+  ColXpr x(this->col(p));
+  ColXpr y(this->col(q));
+  internal::apply_rotation_in_the_plane(x, y, j.transpose());
+}
+
+namespace internal {
+template<typename VectorX, typename VectorY, typename OtherScalar>
+void /*EIGEN_DONT_INLINE*/ apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y, const JacobiRotation<OtherScalar>& j)
+{
+  typedef typename VectorX::Scalar Scalar;
+  enum {
+    PacketSize = packet_traits<Scalar>::size,
+    OtherPacketSize = packet_traits<OtherScalar>::size
+  };
+  typedef typename packet_traits<Scalar>::type Packet;
+  typedef typename packet_traits<OtherScalar>::type OtherPacket;
+  eigen_assert(xpr_x.size() == xpr_y.size());
+  Index size = xpr_x.size();
+  Index incrx = xpr_x.derived().innerStride();
+  Index incry = xpr_y.derived().innerStride();
+
+  Scalar* EIGEN_RESTRICT x = &xpr_x.derived().coeffRef(0);
+  Scalar* EIGEN_RESTRICT y = &xpr_y.derived().coeffRef(0);
+  
+  OtherScalar c = j.c();
+  OtherScalar s = j.s();
+  if (c==OtherScalar(1) && s==OtherScalar(0))
+    return;
+
+  /*** dynamic-size vectorized paths ***/
+
+  if(VectorX::SizeAtCompileTime == Dynamic &&
+    (VectorX::Flags & VectorY::Flags & PacketAccessBit) &&
+    (PacketSize == OtherPacketSize) &&
+    ((incrx==1 && incry==1) || PacketSize == 1))
+  {
+    // both vectors are sequentially stored in memory => vectorization
+    enum { Peeling = 2 };
+
+    Index alignedStart = internal::first_default_aligned(y, size);
+    Index alignedEnd = alignedStart + ((size-alignedStart)/PacketSize)*PacketSize;
+
+    const OtherPacket pc = pset1<OtherPacket>(c);
+    const OtherPacket ps = pset1<OtherPacket>(s);
+    conj_helper<OtherPacket,Packet,NumTraits<OtherScalar>::IsComplex,false> pcj;
+    conj_helper<OtherPacket,Packet,false,false> pm;
+
+    for(Index i=0; i<alignedStart; ++i)
+    {
+      Scalar xi = x[i];
+      Scalar yi = y[i];
+      x[i] =  c * xi + numext::conj(s) * yi;
+      y[i] = -s * xi + numext::conj(c) * yi;
+    }
+
+    Scalar* EIGEN_RESTRICT px = x + alignedStart;
+    Scalar* EIGEN_RESTRICT py = y + alignedStart;
+
+    if(internal::first_default_aligned(x, size)==alignedStart)
+    {
+      for(Index i=alignedStart; i<alignedEnd; i+=PacketSize)
+      {
+        Packet xi = pload<Packet>(px);
+        Packet yi = pload<Packet>(py);
+        pstore(px, padd(pm.pmul(pc,xi),pcj.pmul(ps,yi)));
+        pstore(py, psub(pcj.pmul(pc,yi),pm.pmul(ps,xi)));
+        px += PacketSize;
+        py += PacketSize;
+      }
+    }
+    else
+    {
+      Index peelingEnd = alignedStart + ((size-alignedStart)/(Peeling*PacketSize))*(Peeling*PacketSize);
+      for(Index i=alignedStart; i<peelingEnd; i+=Peeling*PacketSize)
+      {
+        Packet xi   = ploadu<Packet>(px);
+        Packet xi1  = ploadu<Packet>(px+PacketSize);
+        Packet yi   = pload <Packet>(py);
+        Packet yi1  = pload <Packet>(py+PacketSize);
+        pstoreu(px, padd(pm.pmul(pc,xi),pcj.pmul(ps,yi)));
+        pstoreu(px+PacketSize, padd(pm.pmul(pc,xi1),pcj.pmul(ps,yi1)));
+        pstore (py, psub(pcj.pmul(pc,yi),pm.pmul(ps,xi)));
+        pstore (py+PacketSize, psub(pcj.pmul(pc,yi1),pm.pmul(ps,xi1)));
+        px += Peeling*PacketSize;
+        py += Peeling*PacketSize;
+      }
+      if(alignedEnd!=peelingEnd)
+      {
+        Packet xi = ploadu<Packet>(x+peelingEnd);
+        Packet yi = pload <Packet>(y+peelingEnd);
+        pstoreu(x+peelingEnd, padd(pm.pmul(pc,xi),pcj.pmul(ps,yi)));
+        pstore (y+peelingEnd, psub(pcj.pmul(pc,yi),pm.pmul(ps,xi)));
+      }
+    }
+
+    for(Index i=alignedEnd; i<size; ++i)
+    {
+      Scalar xi = x[i];
+      Scalar yi = y[i];
+      x[i] =  c * xi + numext::conj(s) * yi;
+      y[i] = -s * xi + numext::conj(c) * yi;
+    }
+  }
+
+  /*** fixed-size vectorized path ***/
+  else if(VectorX::SizeAtCompileTime != Dynamic &&
+          (VectorX::Flags & VectorY::Flags & PacketAccessBit) &&
+          (PacketSize == OtherPacketSize) &&
+          (EIGEN_PLAIN_ENUM_MIN(evaluator<VectorX>::Alignment, evaluator<VectorY>::Alignment)>0)) // FIXME should be compared to the required alignment
+  {
+    const OtherPacket pc = pset1<OtherPacket>(c);
+    const OtherPacket ps = pset1<OtherPacket>(s);
+    conj_helper<OtherPacket,Packet,NumTraits<OtherPacket>::IsComplex,false> pcj;
+    conj_helper<OtherPacket,Packet,false,false> pm;
+    Scalar* EIGEN_RESTRICT px = x;
+    Scalar* EIGEN_RESTRICT py = y;
+    for(Index i=0; i<size; i+=PacketSize)
+    {
+      Packet xi = pload<Packet>(px);
+      Packet yi = pload<Packet>(py);
+      pstore(px, padd(pm.pmul(pc,xi),pcj.pmul(ps,yi)));
+      pstore(py, psub(pcj.pmul(pc,yi),pm.pmul(ps,xi)));
+      px += PacketSize;
+      py += PacketSize;
+    }
+  }
+
+  /*** non-vectorized path ***/
+  else
+  {
+    for(Index i=0; i<size; ++i)
+    {
+      Scalar xi = *x;
+      Scalar yi = *y;
+      *x =  c * xi + numext::conj(s) * yi;
+      *y = -s * xi + numext::conj(c) * yi;
+      x += incrx;
+      y += incry;
+    }
+  }
+}
+
+} // end namespace internal
+
+} // end namespace Eigen
+
+#endif // EIGEN_JACOBI_H