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Jacobi.h
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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>&, typename Derived::Index p, typename Derived::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;
if(y == Scalar(0))
{
m_c = Scalar(1);
m_s = Scalar(0);
return false;
}
else
{
RealScalar tau = (x-z)/(RealScalar(2)*abs(y));
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, typename Derived::Index p, typename Derived::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
****************************************************************************************/
/** \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()
*/
namespace internal {
template<typename VectorX, typename VectorY, typename OtherScalar>
void apply_rotation_in_the_plane(VectorX& _x, VectorY& _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(VectorX& _x, VectorY& _y, const JacobiRotation<OtherScalar>& j)
{
typedef typename VectorX::Index Index;
typedef typename VectorX::Scalar Scalar;
enum { PacketSize = packet_traits<Scalar>::size };
typedef typename packet_traits<Scalar>::type Packet;
eigen_assert(_x.size() == _y.size());
Index size = _x.size();
Index incrx = _x.innerStride();
Index incry = _y.innerStride();
Scalar* EIGEN_RESTRICT x = &_x.coeffRef(0);
Scalar* EIGEN_RESTRICT y = &_y.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) &&
((incrx==1 && incry==1) || PacketSize == 1))
{
// both vectors are sequentially stored in memory => vectorization
enum { Peeling = 2 };
Index alignedStart = internal::first_aligned(y, size);
Index alignedEnd = alignedStart + ((size-alignedStart)/PacketSize)*PacketSize;
const Packet pc = pset1<Packet>(c);
const Packet ps = pset1<Packet>(s);
conj_helper<Packet,Packet,NumTraits<Scalar>::IsComplex,false> pcj;
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_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(pmul(pc,xi),pcj.pmul(ps,yi)));
pstore(py, psub(pcj.pmul(pc,yi),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(pmul(pc,xi),pcj.pmul(ps,yi)));
pstoreu(px+PacketSize, padd(pmul(pc,xi1),pcj.pmul(ps,yi1)));
pstore (py, psub(pcj.pmul(pc,yi),pmul(ps,xi)));
pstore (py+PacketSize, psub(pcj.pmul(pc,yi1),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(pmul(pc,xi),pcj.pmul(ps,yi)));
pstore (y+peelingEnd, psub(pcj.pmul(pc,yi),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) &&
(VectorX::Flags & VectorY::Flags & AlignedBit))
{
const Packet pc = pset1<Packet>(c);
const Packet ps = pset1<Packet>(s);
conj_helper<Packet,Packet,NumTraits<Scalar>::IsComplex,false> pcj;
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(pmul(pc,xi),pcj.pmul(ps,yi)));
pstore(py, psub(pcj.pmul(pc,yi),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

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