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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
// 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