Detailed Description

template<typename Scalar>
class JacobiRotation< Scalar >

Rotation given by a cosine-sine pair.

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: $J = \left ( \begin{array}{cc} c & \overline s \\ -s & \overline c \end{array} \right )$

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

 v.applyOnTheLeft(J.adjoint());

See also:: MatrixBase::applyOnTheLeft(), MatrixBase::applyOnTheRight()

Definition at line 47 of file Jacobi.h.

#include <src/Jacobi/Jacobi.h>

List of all members.

Public Types
typedef NumTraits< Scalar >::Real	RealScalar
Public Member Functions
	JacobiRotation ()
	Default constructor without any initialization.
	JacobiRotation (const Scalar &c, const Scalar &s)
	Construct a planar rotation from a cosine-sine pair (c, `s`).
Scalar &	c ()
Scalar	c () const
Scalar &	s ()
Scalar	s () const
JacobiRotation	operator* (const JacobiRotation &other)
	Concatenates two planar rotation.
JacobiRotation	transpose () const
	Returns the transposed transformation.
JacobiRotation	adjoint () const
	Returns the adjoint transformation.
template<typename Derived >
bool	makeJacobi (const MatrixBase< Derived > &, typename Derived::Index p, typename Derived::Index q)
	Makes `this` as a Jacobi rotation `J` such that applying J on both the right and left sides of the 2x2 selfadjoint matrix $B = \left ( \begin{array}{cc} \text{this}_{pp} & \text{this}_{pq} \\ (\text{this}_{pq})^ & \text{this}_{qq} \end{array} \right )$ yields a diagonal matrix .
bool	makeJacobi (RealScalar x, Scalar y, 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 $B = \left ( \begin{array}{cc} x & y \\ \overline y & z \end{array} \right )$ yields a diagonal matrix .
void	makeGivens (const Scalar &p, const Scalar &q, Scalar *z=0)
	Makes `this` as a Givens rotation `G` such that applying to the left of the vector $V = \left ( \begin{array}{c} p \\ q \end{array} \right )$ yields: $G^ V = \left ( \begin{array}{c} r \\ 0 \end{array} \right )$ .
Protected Member Functions
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)
Protected Attributes
Scalar	m_c
Scalar	m_s

Member Typedef Documentation

template<typename Scalar>

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

Definition at line 50 of file Jacobi.h.

Constructor & Destructor Documentation

template<typename Scalar>

JacobiRotation< Scalar >::JacobiRotation ( ) [inline]

Default constructor without any initialization.

Definition at line 53 of file Jacobi.h.

Referenced by JacobiRotation< Scalar >::adjoint(), JacobiRotation< Scalar >::operator*(), and JacobiRotation< Scalar >::transpose().

template<typename Scalar>

JacobiRotation< Scalar >::JacobiRotation	(	const Scalar &	c,
		const Scalar &	s
	)		`[inline]`

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

Definition at line 56 of file Jacobi.h.

Member Function Documentation

template<typename Scalar>

JacobiRotation JacobiRotation< Scalar >::adjoint ( ) const [inline]

Returns the adjoint transformation.

Definition at line 74 of file Jacobi.h.

References JacobiRotation< Scalar >::JacobiRotation(), JacobiRotation< Scalar >::m_c, and JacobiRotation< Scalar >::m_s.

Referenced by ComplexSchur< MatrixType >::reduceToTriangularForm(), internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run(), and RealSchur< MatrixType >::splitOffTwoRows().

template<typename Scalar>

Scalar JacobiRotation< Scalar >::c ( ) const [inline]

Definition at line 59 of file Jacobi.h.

References JacobiRotation< Scalar >::m_c.

template<typename Scalar>

Scalar& JacobiRotation< Scalar >::c ( ) [inline]

Definition at line 58 of file Jacobi.h.

References JacobiRotation< Scalar >::m_c.

Referenced by internal::apply_rotation_in_the_plane(), internal::svd_precondition_2x2_block_to_be_real< MatrixType, QRPreconditioner, true >::run(), and internal::tridiagonal_qr_step().

template<typename Scalar >

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

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

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

Example:

Output:

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 158 of file Jacobi.h.

Referenced by ComplexSchur< MatrixType >::reduceToTriangularForm(), RealSchur< MatrixType >::splitOffTwoRows(), and internal::tridiagonal_qr_step().

template<typename Scalar >

void JacobiRotation< Scalar >::makeGivens	(	const Scalar &	p,
		const Scalar &	q,
		Scalar *	z,
		internal::true_type
	)		`[protected]`

Definition at line 166 of file Jacobi.h.

References abs(), abs2(), real(), and internal::sqrt().

template<typename Scalar >

void JacobiRotation< Scalar >::makeGivens	(	const Scalar &	p,
		const Scalar &	q,
		Scalar *	z,
		internal::false_type
	)		`[protected]`

Definition at line 221 of file Jacobi.h.

References abs(), abs2(), internal::sqrt(), and t().

template<typename Scalar >

bool JacobiRotation< Scalar >::makeJacobi	(	RealScalar	x,
		Scalar	y,
		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 $B = \left ( \begin{array}{cc} x & y \\ \overline y & z \end{array} \right )$ yields a diagonal matrix $ A = J^* B J $ .

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

Definition at line 95 of file Jacobi.h.

References abs(), abs2(), internal::sqrt(), and t().

template<typename Scalar >

template<typename Derived >

bool JacobiRotation< Scalar >::makeJacobi	(	const MatrixBase< Derived > &	m,
		typename Derived::Index	p,
		typename Derived::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 $B = \left ( \begin{array}{cc} \text{this}_{pp} & \text{this}_{pq} \\ (\text{this}_{pq})^* & \text{this}_{qq} \end{array} \right )$ yields a diagonal matrix $ A = J^* B J $ .

Example:

Output: