JacobiRotation< Scalar > Class Template Reference

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

#include <Jacobi.h>

Public Member Functions
JacobiRotation	adjoint () const

	JacobiRotation ()

	JacobiRotation (const Scalar &c, const Scalar &s)

void	makeGivens (const Scalar &p, const Scalar &q, Scalar *z=0)

template<typename Derived>
bool	makeJacobi (const MatrixBase< Derived > &, typename Derived::Index p, typename Derived::Index q)

bool	makeJacobi (RealScalar x, Scalar y, RealScalar z)

JacobiRotation	operator* (const JacobiRotation &other)

JacobiRotation	transpose () const

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>

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()

Constructor & Destructor Documentation

◆ JacobiRotation() [1/2]

template<typename Scalar>

JacobiRotation ( )

inline

Default constructor without any initialization.

Referenced by adjoint(), operator*(), and transpose().

◆ JacobiRotation() [2/2]

template<typename Scalar>

JacobiRotation	(	const Scalar &	c,
		const Scalar &	s )

inline

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

Member Function Documentation

◆ adjoint()

template<typename Scalar>

JacobiRotation adjoint ( ) const

inline

Returns the adjoint transformation

References JacobiRotation().

◆ makeGivens()

template<typename Scalar>

void 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:

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;

Output:

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

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()

References makeGivens().

Referenced by makeGivens().

◆ makeJacobi() [1/2]

template<typename Scalar>

template<typename Derived>

bool 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:

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;

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

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

References makeJacobi().

Referenced by makeJacobi().

◆ makeJacobi() [2/2]

template<typename Scalar>

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 $ A = J^* B J $