Eigen/src/Jacobi/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/.
// SPDX-License-Identifier: 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:
  using RealScalar = typename NumTraits<Scalar>::Real;

  /** 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 rotations */
  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, std::true_type);
  EIGEN_DEVICE_FUNC void makeGivens(const Scalar& p, const Scalar& q, Scalar* r, std::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;

  const RealScalar abs_y = abs(y);
  const RealScalar deno = RealScalar(2) * abs_y;
  if (deno < (std::numeric_limits<RealScalar>::min)()) {
    m_c = Scalar(1);
    m_s = Scalar(0);
    return false;
  } else {
    const RealScalar delta = x - z;
    const RealScalar abs_delta = abs(delta);
    // |t| = 1 / (|tau| + sqrt(1 + tau^2)), where tau = (x - z) / (2 * |y|).
    // Scale this numerator/denominator pair by min(1, 1/|tau|), taking scale = 1 at tau = 0.
    // Then numerator <= 1 and denominator <= 1 + sqrt(2); only a ratio <= 1 is squared.
    // Normalize the pair directly to save the division forming |t|.
    RealScalar numerator;
    RealScalar denominator;
    if (abs_delta > deno) {
      const RealScalar ratio = deno / abs_delta;
      numerator = ratio;
      denominator = sqrt(RealScalar(1) + numext::abs2(ratio)) + RealScalar(1);
    } else {
      const RealScalar ratio = abs_delta / deno;
      numerator = RealScalar(1);
      denominator = sqrt(RealScalar(1) + numext::abs2(ratio)) + ratio;
    }
    const RealScalar n = RealScalar(1) / sqrt(numext::abs2(numerator) + numext::abs2(denominator));
    const RealScalar sine = numerator * n;
    const RealScalar sign_t = delta > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
    EIGEN_IF_CONSTEXPR (NumTraits<Scalar>::IsComplex) {
      m_s = -sign_t * (numext::conj(y) / abs_y) * sine;
    } else {
      // For real y, conj(y) / abs(y) is just its sign.
      const RealScalar signed_sine = -sign_t * sine;
      m_s = numext::real(y) < RealScalar(0) ? -signed_sine : signed_sine;
    }
    m_c = denominator * 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, internal::bool_constant<NumTraits<Scalar>::IsComplex>());
}

// specialization for complexes
template <typename Scalar>
EIGEN_DEVICE_FUNC void JacobiRotation<Scalar>::makeGivens(const Scalar& p, const Scalar& q, Scalar* r, std::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,
                                                          std::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);
    return;
  }
  if (numext::is_exactly_zero(p)) {
    m_c = Scalar(0);
    m_s = q < Scalar(0) ? Scalar(1) : Scalar(-1);
    if (r) *r = abs(q);
    return;
  }

  // Safe-range thresholds following Anderson, "Algorithm 978: Safe Scaling
  // in the Level 1 BLAS", ACM TOMS 44(1), 2017.  When both |p| and |q| lie
  // in (rtmin, rtmax), the direct formula r = p * sqrt(1 + (q/p)^2) cannot
  // over- or underflow before the true result would.  Outside that range
  // we prescale by max(|p|, |q|) (clamped into [safmin, safmax]) so that
  // the squared sum stays in the representable range.  This preserves the
  // existing Eigen sign convention (r >= 0, sign carried in c).
  const Scalar safmin = (std::numeric_limits<Scalar>::min)();
  const Scalar safmax = Scalar(1) / safmin;
  const Scalar rtmin = sqrt(safmin);
  const Scalar rtmax = sqrt(safmax / Scalar(2));
  const Scalar abs_p = abs(p);
  const Scalar abs_q = abs(q);
  const Scalar mx = numext::maxi(abs_p, abs_q);
  const Scalar mn = numext::mini(abs_p, abs_q);

  if (EIGEN_PREDICT_TRUE(mx < rtmax && mn > rtmin)) {
    // Safe range: existing direct formulas are stable.
    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;
    }
  } else {
    // Out of safe range: prescale by max(|p|, |q|) clamped into [safmin, safmax].
    const Scalar scale = numext::mini(safmax, numext::maxi(safmin, numext::maxi(abs_p, abs_q)));
    const Scalar ps = p / scale;
    const Scalar qs = q / scale;
    if (abs_p > abs_q) {
      Scalar t = qs / ps;
      Scalar u = sqrt(Scalar(1) + numext::abs2(t));
      if (ps < Scalar(0)) u = -u;
      m_c = Scalar(1) / u;
      m_s = -t * m_c;
      if (r) *r = (ps * u) * scale;
    } else {
      Scalar t = ps / qs;
      Scalar u = sqrt(Scalar(1) + numext::abs2(t));
      if (qs < Scalar(0)) u = -u;
      m_s = -Scalar(1) / u;
      m_c = -t * m_s;
      if (r) *r = (qs * u) * scale;
    }
  }
}

/****************************************************************************************
 *   Implementation of MatrixBase methods
 ****************************************************************************************/

namespace internal {
/** \jacobi_module
 * Applies the clockwise 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) {
    using Packet = typename packet_traits<Scalar>::type;
    using OtherPacket = typename packet_traits<OtherScalar>::type;

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

    EIGEN_IF_CONSTEXPR (SizeAtCompileTime == Dynamic) {
      /*** dynamic-size vectorized paths ***/
      if (size >= 2 * PacketSize && ((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, pm.pmadd(pc, xi, pcj.pmul(ps, yi)));
            pstore(py, pcj.pmsub(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, pm.pmadd(pc, xi, pcj.pmul(ps, yi)));
            pstoreu(px + PacketSize, pm.pmadd(pc, xi1, pcj.pmul(ps, yi1)));
            pstore(py, pcj.pmsub(pc, yi, pm.pmul(ps, xi)));
            pstore(py + PacketSize, pcj.pmsub(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, pm.pmadd(pc, xi, pcj.pmul(ps, yi)));
            pstore(y + peelingEnd, pcj.pmsub(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;
        }
        return;
      }
    }

    EIGEN_IF_CONSTEXPR (SizeAtCompileTime != Dynamic && MinAlignment >= RequiredAlignment) {
      /*** fixed-size vectorized path ***/
      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, pm.pmadd(pc, xi, pcj.pmul(ps, yi)));
        pstore(py, pcj.pmsub(pc, yi, pm.pmul(ps, xi)));
        px += PacketSize;
        py += PacketSize;
      }
      return;
    }

    /*** non-vectorized path ***/
    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) {
  using Scalar = typename VectorX::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);
}

template <typename MatrixType, typename RealScalar, typename Index>
EIGEN_DONT_INLINE void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
                                           JacobiRotation<RealScalar>* j_left, JacobiRotation<RealScalar>* j_right) {
  // Extract 2x2 submatrix into scalars (avoids Matrix construction on stack).
  const RealScalar m00 = numext::real(matrix.coeff(p, p));
  const RealScalar m01 = numext::real(matrix.coeff(p, q));
  const RealScalar m10 = numext::real(matrix.coeff(q, p));
  const RealScalar m11 = numext::real(matrix.coeff(q, q));

  // Compute the symmetrizing rotation rot1 such that rot1 * [m] is symmetric.
  const RealScalar t = m00 + m11;
  const RealScalar d = m10 - m01;

  RealScalar c1, s1;
  if (numext::abs(d) < (std::numeric_limits<RealScalar>::min)()) {
    c1 = RealScalar(1);
    s1 = RealScalar(0);
  } else {
    // If d!=0, then t/d cannot overflow because the magnitude of the
    // entries forming d are not too small compared to the ones forming t.
    RealScalar u = t / d;
    s1 = RealScalar(1) / numext::sqrt(RealScalar(1) + numext::abs2(u));
    c1 = u * s1;
  }

  // Apply rot1 to the 2x2 submatrix inline (avoids rotation dispatch overhead).
  // Result is symmetric, so we only need 3 values: a00, a01 (== a10), a11.
  const RealScalar a00 = c1 * m00 + s1 * m10;
  const RealScalar a01 = c1 * m01 + s1 * m11;
  const RealScalar a11 = -s1 * m01 + c1 * m11;

  // Compute the diagonalizing rotation j_right from the symmetrized matrix.
  j_right->makeJacobi(a00, a01, a11);

  // Compose j_left = rot1 * j_right^T inline (avoids template machinery overhead).
  const RealScalar jr_c = j_right->c();
  const RealScalar jr_s = j_right->s();
  j_left->c() = c1 * jr_c + s1 * jr_s;
  j_left->s() = s1 * jr_c - c1 * jr_s;
}

}  // end namespace internal

}  // end namespace Eigen

#endif  // EIGEN_JACOBI_H