Eigen/src/Jacobi/Jacobi.h - eigen - Git at Google

 // 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

 // IWYU pragma: private
 #include "./InternalHeaderCheck.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. */
   EIGEN_DEVICE_FUNC JacobiRotation() {}

   /** Construct a planar rotation from a cosine-sine pair (\a c, \c s). */
   EIGEN_DEVICE_FUNC JacobiRotation(const Scalar& c, const Scalar& s) : m_c(c), m_s(s) {}

   EIGEN_DEVICE_FUNC Scalar& c() { return m_c; }
   EIGEN_DEVICE_FUNC Scalar c() const { return m_c; }
   EIGEN_DEVICE_FUNC Scalar& s() { return m_s; }
   EIGEN_DEVICE_FUNC Scalar s() const { return m_s; }

   /** Concatenates two planar rotation */
   EIGEN_DEVICE_FUNC 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 */
   EIGEN_DEVICE_FUNC JacobiRotation transpose() const {
     using numext::conj;
     return JacobiRotation(m_c, -conj(m_s));
   }

   /** Returns the adjoint transformation */
   EIGEN_DEVICE_FUNC JacobiRotation adjoint() const {
     using numext::conj;
     return JacobiRotation(conj(m_c), -m_s);
   }

   template <typename Derived>
   EIGEN_DEVICE_FUNC bool makeJacobi(const MatrixBase<Derived>&, Index p, Index q);
   EIGEN_DEVICE_FUNC bool makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z);

   EIGEN_DEVICE_FUNC void makeGivens(const Scalar& p, const Scalar& q, Scalar* r = 0);

  protected:
   EIGEN_DEVICE_FUNC void makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::true_type);
   EIGEN_DEVICE_FUNC void makeGivens(const Scalar& p, const Scalar& q, Scalar* r, 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>
 EIGEN_DEVICE_FUNC bool JacobiRotation<Scalar>::makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z) {
   using std::abs;
   using std::sqrt;

   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>
 EIGEN_DEVICE_FUNC 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 r is returned if \a r 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>
 EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r) {
   makeGivens(p, q, r, std::conditional_t<NumTraits<Scalar>::IsComplex, internal::true_type, internal::false_type>());
 }

 // specialization for complexes
 template <typename Scalar>
 EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r,
                                                           internal::true_type) {
   using numext::conj;
   using std::abs;
   using std::sqrt;

   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>
 EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r,
                                                           internal::false_type) {
   using std::abs;
   using std::sqrt;
   if (numext::is_exactly_zero(q)) {
     m_c = p < Scalar(0) ? Scalar(-1) : Scalar(1);
     m_s = Scalar(0);
     if (r) *r = abs(p);
   } else if (numext::is_exactly_zero(p)) {
     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 coordinates \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>
 EIGEN_DEVICE_FUNC void apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y,
                                                    const JacobiRotation<OtherScalar>& j);
 }  // namespace internal

 /** \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>
 EIGEN_DEVICE_FUNC 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);
 }

 /** \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>
 EIGEN_DEVICE_FUNC 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 Scalar, typename OtherScalar, int SizeAtCompileTime, int MinAlignment, bool Vectorizable>
 struct apply_rotation_in_the_plane_selector {
   static EIGEN_DEVICE_FUNC inline void run(Scalar* x, Index incrx, Scalar* y, Index incry, Index size, OtherScalar c,
                                            OtherScalar s) {
     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;
     }
   }
 };

 template <typename Scalar, typename OtherScalar, int SizeAtCompileTime, int MinAlignment>
 struct apply_rotation_in_the_plane_selector<Scalar, OtherScalar, SizeAtCompileTime, MinAlignment,
                                             true /* vectorizable */> {
   static inline void run(Scalar* x, Index incrx, Scalar* y, Index incry, Index size, OtherScalar c, OtherScalar s) {
     typedef typename packet_traits<Scalar>::type Packet;
     typedef typename packet_traits<OtherScalar>::type OtherPacket;

     constexpr int RequiredAlignment =
         (std::max)(unpacket_traits<Packet>::alignment, unpacket_traits<OtherPacket>::alignment);
     constexpr Index PacketSize = packet_traits<Scalar>::size;

     /*** dynamic-size vectorized paths ***/
     if (size >= 2 * PacketSize && SizeAtCompileTime == Dynamic && ((incrx == 1 && incry == 1) || PacketSize == 1)) {
       // both vectors are sequentially stored in memory => vectorization
       constexpr Index 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 (SizeAtCompileTime != Dynamic && MinAlignment >= RequiredAlignment) {
       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;
       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 {
       apply_rotation_in_the_plane_selector<Scalar, OtherScalar, SizeAtCompileTime, MinAlignment, false>::run(
           x, incrx, y, incry, size, c, s);
     }
   }
 };

 template <typename VectorX, typename VectorY, typename OtherScalar>
 EIGEN_DEVICE_FUNC void inline apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y,
                                                           const JacobiRotation<OtherScalar>& j) {
   typedef typename VectorX::Scalar Scalar;
   constexpr bool Vectorizable = (int(evaluator<VectorX>::Flags) & int(evaluator<VectorY>::Flags) & PacketAccessBit) &&
                                 (int(packet_traits<Scalar>::size) == int(packet_traits<OtherScalar>::size));

   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 (numext::is_exactly_one(c) && numext::is_exactly_zero(s)) return;

   constexpr int Alignment = (std::min)(int(evaluator<VectorX>::Alignment), int(evaluator<VectorY>::Alignment));
   apply_rotation_in_the_plane_selector<Scalar, OtherScalar, VectorX::SizeAtCompileTime, Alignment, Vectorizable>::run(
       x, incrx, y, incry, size, c, s);
 }

 }  // end namespace internal

 }  // end namespace Eigen

 #endif  // EIGEN_JACOBI_H
	// 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

	// IWYU pragma: private
	#include "./InternalHeaderCheck.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. */
	EIGEN_DEVICE_FUNC JacobiRotation() {}

	/** Construct a planar rotation from a cosine-sine pair (\a c, \c s). */
	EIGEN_DEVICE_FUNC JacobiRotation(const Scalar& c, const Scalar& s) : m_c(c), m_s(s) {}

	EIGEN_DEVICE_FUNC Scalar& c() { return m_c; }
	EIGEN_DEVICE_FUNC Scalar c() const { return m_c; }
	EIGEN_DEVICE_FUNC Scalar& s() { return m_s; }
	EIGEN_DEVICE_FUNC Scalar s() const { return m_s; }

	/** Concatenates two planar rotation */
	EIGEN_DEVICE_FUNC 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 */
	EIGEN_DEVICE_FUNC JacobiRotation transpose() const {
	using numext::conj;
	return JacobiRotation(m_c, -conj(m_s));
	}

	/** Returns the adjoint transformation */
	EIGEN_DEVICE_FUNC JacobiRotation adjoint() const {
	using numext::conj;
	return JacobiRotation(conj(m_c), -m_s);
	}

	template <typename Derived>
	EIGEN_DEVICE_FUNC bool makeJacobi(const MatrixBase<Derived>&, Index p, Index q);
	EIGEN_DEVICE_FUNC bool makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z);

	EIGEN_DEVICE_FUNC void makeGivens(const Scalar& p, const Scalar& q, Scalar* r = 0);

	protected:
	EIGEN_DEVICE_FUNC void makeGivens(const Scalar& p, const Scalar& q, Scalar* r, internal::true_type);
	EIGEN_DEVICE_FUNC void makeGivens(const Scalar& p, const Scalar& q, Scalar* r, 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>
	EIGEN_DEVICE_FUNC bool JacobiRotation<Scalar>::makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z) {
	using std::abs;
	using std::sqrt;

	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>
	EIGEN_DEVICE_FUNC 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 r is returned if \a r 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>
	EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r) {
	makeGivens(p, q, r, std::conditional_t<NumTraits<Scalar>::IsComplex, internal::true_type, internal::false_type>());
	}

	// specialization for complexes
	template <typename Scalar>
	EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r,
	internal::true_type) {
	using numext::conj;
	using std::abs;
	using std::sqrt;

	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>
	EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r,
	internal::false_type) {
	using std::abs;
	using std::sqrt;
	if (numext::is_exactly_zero(q)) {
	m_c = p < Scalar(0) ? Scalar(-1) : Scalar(1);
	m_s = Scalar(0);
	if (r) *r = abs(p);
	} else if (numext::is_exactly_zero(p)) {
	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 coordinates \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>
	EIGEN_DEVICE_FUNC void apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y,
	const JacobiRotation<OtherScalar>& j);
	} // namespace internal

	/** \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>
	EIGEN_DEVICE_FUNC 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);
	}

	/** \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>
	EIGEN_DEVICE_FUNC 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 Scalar, typename OtherScalar, int SizeAtCompileTime, int MinAlignment, bool Vectorizable>
	struct apply_rotation_in_the_plane_selector {
	static EIGEN_DEVICE_FUNC inline void run(Scalar* x, Index incrx, Scalar* y, Index incry, Index size, OtherScalar c,
	OtherScalar s) {
	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;
	}
	}
	};

	template <typename Scalar, typename OtherScalar, int SizeAtCompileTime, int MinAlignment>
	struct apply_rotation_in_the_plane_selector<Scalar, OtherScalar, SizeAtCompileTime, MinAlignment,
	true /* vectorizable */> {
	static inline void run(Scalar* x, Index incrx, Scalar* y, Index incry, Index size, OtherScalar c, OtherScalar s) {
	typedef typename packet_traits<Scalar>::type Packet;
	typedef typename packet_traits<OtherScalar>::type OtherPacket;

	constexpr int RequiredAlignment =
	(std::max)(unpacket_traits<Packet>::alignment, unpacket_traits<OtherPacket>::alignment);
	constexpr Index PacketSize = packet_traits<Scalar>::size;

	/* dynamic-size vectorized paths */
	if (size >= 2 * PacketSize && SizeAtCompileTime == Dynamic && ((incrx == 1 && incry == 1) \|\| PacketSize == 1)) {
	// both vectors are sequentially stored in memory => vectorization
	constexpr Index 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 (SizeAtCompileTime != Dynamic && MinAlignment >= RequiredAlignment) {
	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;
	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 {
	apply_rotation_in_the_plane_selector<Scalar, OtherScalar, SizeAtCompileTime, MinAlignment, false>::run(
	x, incrx, y, incry, size, c, s);
	}
	}
	};

	template <typename VectorX, typename VectorY, typename OtherScalar>
	EIGEN_DEVICE_FUNC void inline apply_rotation_in_the_plane(DenseBase<VectorX>& xpr_x, DenseBase<VectorY>& xpr_y,
	const JacobiRotation<OtherScalar>& j) {
	typedef typename VectorX::Scalar Scalar;
	constexpr bool Vectorizable = (int(evaluator<VectorX>::Flags) & int(evaluator<VectorY>::Flags) & PacketAccessBit) &&
	(int(packet_traits<Scalar>::size) == int(packet_traits<OtherScalar>::size));

	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 (numext::is_exactly_one(c) && numext::is_exactly_zero(s)) return;

	constexpr int Alignment = (std::min)(int(evaluator<VectorX>::Alignment), int(evaluator<VectorY>::Alignment));
	apply_rotation_in_the_plane_selector<Scalar, OtherScalar, VectorX::SizeAtCompileTime, Alignment, Vectorizable>::run(
	x, incrx, y, incry, size, c, s);
	}

	} // end namespace internal

	} // end namespace Eigen

	#endif // EIGEN_JACOBI_H