Eigen::JacobiRotation< Scalar > Class Template Reference

Rotation given by a cosine-sine pair. More...

Public Types
typedef NumTraits< Scalar >::Real	RealScalar

Public Member Functions
JacobiRotation	adjoint () const
Scalar &	c ()
Scalar	c () const
	JacobiRotation ()
	JacobiRotation (const Scalar &c, const Scalar &s)
void	makeGivens (const Scalar &p, const Scalar &q, Scalar *r=0)
template<typename Derived >
bool	makeJacobi (const MatrixBase< Derived > &, Index p, Index q)
bool	makeJacobi (const RealScalar &x, const Scalar &y, const RealScalar &z)
JacobiRotation	operator* (const JacobiRotation &other)
Scalar &	s ()
Scalar	s () const
JacobiRotation	transpose () const

Protected Member Functions
void	makeGivens (const Scalar &p, const Scalar &q, Scalar *r, internal::false_type)
void	makeGivens (const Scalar &p, const Scalar &q, Scalar *r, internal::true_type)

Protected Attributes
Scalar	m_c
Scalar	m_s

Detailed Description

template<typename Scalar>
class Eigen::JacobiRotation< Scalar >

Rotation given by a cosine-sine pair.

This is defined in the Jacobi module.

#include <Eigen/Jacobi> 

Code

This class represents a Jacobi or Givens rotation. This is a 2D rotation in the plane J of angle $\theta$ defined by its cosine c and sine s as follow: $J=\left(\begin{array}{cc}c& \bar{s}\\ -s& \bar{c}\end{array}\right)$

You can apply the respective counter-clockwise rotation to a column vector v by applying its adjoint on the left: $v=J^{\ast }v$ that translates to the following Eigen code:

v.applyOnTheLeft(J.adjoint());

Code

See also

MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

Definition at line 36 of file Jacobi.h.

Member Typedef Documentation

RealScalar

template<typename Scalar >

typedef NumTraits<Scalar>::Real Eigen::JacobiRotation< Scalar >::RealScalar

Definition at line 39 of file Jacobi.h.

Constructor & Destructor Documentation

JacobiRotation() [1/2]

template<typename Scalar >

Eigen::JacobiRotation< Scalar >::JacobiRotation ( )

inline

Default constructor without any initialization.

Definition at line 43 of file Jacobi.h.

{}

Code

JacobiRotation() [2/2]

template<typename Scalar >

Eigen::JacobiRotation< Scalar >::JacobiRotation ( const Scalar &  c, const Scalar &  s  )

inline

Construct a planar rotation from a cosine-sine pair (c, s).

Definition at line 47 of file Jacobi.h.

: m_c(c), m_s(s) {}

Code

Member Function Documentation

adjoint()

template<typename Scalar >

JacobiRotation Eigen::JacobiRotation< Scalar >::adjoint ( ) const

inline

Returns the adjoint transformation

Definition at line 69 of file Jacobi.h.

{ using numext::conj; return JacobiRotation(conj(m_c), -m_s); }

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c() [1/2]

template<typename Scalar >

Scalar& Eigen::JacobiRotation< Scalar >::c ( )

inline

Definition at line 49 of file Jacobi.h.

{ return m_c; }

Code

c() [2/2]

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::c ( ) const

inline

Definition at line 50 of file Jacobi.h.

{ return m_c; }

Code

makeGivens() [1/3]

template<typename Scalar >

void Eigen::JacobiRotation< Scalar >::makeGivens ( const Scalar &  p, const Scalar &  q, Scalar *  r, internal::false_type    )

protected

Definition at line 233 of file Jacobi.h.

{
  using std::sqrt;
  using std::abs;
  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;
  }
 
}

Code

makeGivens() [2/3]

template<typename Scalar >

void Eigen::JacobiRotation< Scalar >::makeGivens ( const Scalar &  p, const Scalar &  q, Scalar *  r, internal::true_type    )

protected

Definition at line 173 of file Jacobi.h.

{
  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;
    }
  }
}

Code

makeGivens() [3/3]

template<typename Scalar >

void Eigen::JacobiRotation< Scalar >::makeGivens	(	const Scalar &	p,
		const Scalar &	q,
		Scalar *	r = 0
	)

Makes *this as a Givens rotation G such that applying $G^{\ast }$ to the left of the vector $V=\left(\begin{array}{c}p\\ q\end{array}\right)$ yields: $G^{\ast }V=\left(\begin{array}{c}r\\ 0\end{array}\right)$.

The value of r is returned if r is not null (the default is null). Also note that G is built such that the cosine is always real.

Example:

Vector2f v = Vector2f::Random();
JacobiRotation<float> G;
G.makeGivens(v.x(), v.y());
cout << "Here is the vector v:" << endl << v << endl;
v.applyOnTheLeft(0, 1, G.adjoint());
cout << "Here is the vector J' * v:" << endl << v << endl;

Code

Output:

Here is the vector v:
  0.68
-0.211
Here is the vector J' * v:
0.712
    0

Code

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.

See also

MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

Definition at line 164 of file Jacobi.h.

{
  makeGivens(p, q, r, std::conditional_t<NumTraits<Scalar>::IsComplex, internal::true_type, internal::false_type>());
}

Code

makeJacobi() [1/2]

template<typename Scalar >

template<typename Derived >

bool Eigen::JacobiRotation< Scalar >::makeJacobi ( const MatrixBase< Derived > &  m, Index  p, Index  q  )

inline

Makes *this as a Jacobi rotation J such that applying J on both the right and left sides of the 2x2 selfadjoint matrix $B=\left(\begin{array}{cc}{\text{this}}_{pp}& {\text{this}}_{pq}\\ ({\text{this}}_{pq}{)}^{\ast }& {\text{this}}_{qq}\end{array}\right)$ yields a diagonal matrix $A=J^{\ast }BJ$

Example:

Matrix2f m = Matrix2f::Random();
m = (m + m.adjoint()).eval();
JacobiRotation<float> J;
J.makeJacobi(m, 0, 1);
cout << "Here is the matrix m:" << endl << m << endl;
m.applyOnTheLeft(0, 1, J.adjoint());
m.applyOnTheRight(0, 1, J);
cout << "Here is the matrix J' * m * J:" << endl << m << endl;

Code

Output:

Here is the matrix m:
 1.36 0.355
0.355  1.19
Here is the matrix J' * m * J:
 1.64     0
    0 0.913

Code

See also

JacobiRotation::makeJacobi(RealScalar, Scalar, RealScalar), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

Definition at line 141 of file Jacobi.h.

{
  return makeJacobi(numext::real(m.coeff(p,p)), m.coeff(p,q), numext::real(m.coeff(q,q)));
}

Code

makeJacobi() [2/2]

template<typename Scalar >

bool Eigen::JacobiRotation< Scalar >::makeJacobi	(	const RealScalar &	x,
		const Scalar &	y,
		const RealScalar &	z
	)

Makes *this as a Jacobi rotation J such that applying J on both the right and left sides of the selfadjoint 2x2 matrix $B=\left(\begin{array}{cc}x& y\\ \bar{y}& z\end{array}\right)$ yields a diagonal matrix $A=J^{\ast }BJ$

See also

MatrixBase::makeJacobi(const MatrixBase<Derived>&, Index, Index), MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

Definition at line 96 of file Jacobi.h.

{
  using std::sqrt;
  using std::abs;
 
  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;
  }
}

Code

operator*()

template<typename Scalar >

JacobiRotation Eigen::JacobiRotation< Scalar >::operator* ( const JacobiRotation< Scalar > &  other )

inline

Concatenates two planar rotation

Definition at line 56 of file Jacobi.h.

    {
      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)));
    }

Code

s() [1/2]

template<typename Scalar >

Scalar& Eigen::JacobiRotation< Scalar >::s ( )

inline

Definition at line 51 of file Jacobi.h.

{ return m_s; }

Code

s() [2/2]

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::s ( ) const

inline

Definition at line 52 of file Jacobi.h.

{ return m_s; }

Code

transpose()

template<typename Scalar >

JacobiRotation Eigen::JacobiRotation< Scalar >::transpose ( ) const

inline

Returns the transposed transformation

Definition at line 65 of file Jacobi.h.

{ using numext::conj; return JacobiRotation(m_c, -conj(m_s)); }

Code

Member Data Documentation

m_c

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::m_c

protected

Definition at line 86 of file Jacobi.h.

m_s

template<typename Scalar >

Scalar Eigen::JacobiRotation< Scalar >::m_s

protected

Definition at line 86 of file Jacobi.h.

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The documentation for this class was generated from the following files:

• ForwardDeclarations.h
• Jacobi.h