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 {

template <typename Scalar>
class JacobiRotation {
 public:
  typedef typename NumTraits<Scalar>::Real RealScalar;

  EIGEN_DEVICE_FUNC JacobiRotation() {}

  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; }

  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)));
  }
  EIGEN_DEVICE_FUNC JacobiRotation operator*(const JacobiRotation& other) {…}

  EIGEN_DEVICE_FUNC JacobiRotation transpose() const {
    using numext::conj;
    return JacobiRotation(m_c, -conj(m_s));
  }
  EIGEN_DEVICE_FUNC JacobiRotation transpose() const  {…}

  EIGEN_DEVICE_FUNC JacobiRotation adjoint() const {
    using numext::conj;
    return JacobiRotation(conj(m_c), -m_s);
  }
  EIGEN_DEVICE_FUNC JacobiRotation adjoint() const  {…}
 
  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;
};
class JacobiRotation {…};

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;
  }
}
EIGEN_DEVICE_FUNC bool JacobiRotation<Scalar>::makeJacobi(const RealScalar& x, const Scalar& y, const RealScalar& z) {…}

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)));
}
EIGEN_DEVICE_FUNC inline bool JacobiRotation<Scalar>::makeJacobi(const MatrixBase<Derived>& m, Index p, Index q) {…}

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>());
}
EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r) {…}
 
// 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 {
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

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);
}
EIGEN_DEVICE_FUNC inline void MatrixBase<Derived>::applyOnTheLeft(Index p, Index q, {…}

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());
}
EIGEN_DEVICE_FUNC inline void MatrixBase<Derived>::applyOnTheRight(Index p, Index q, {…}
 
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