Computes eigenvalues and eigenvectors of selfadjoint matrices. More...

#include <SelfAdjointEigenSolver.h>

Inheritance diagram for SelfAdjointEigenSolver< _MatrixType >:

Public Types
enum	{ Size = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
typedef MatrixType::Index	Index
typedef _MatrixType	MatrixType
typedef NumTraits< Scalar >::Real	RealScalar
	Real scalar type for `_MatrixType`.
typedef internal::plain_col_type < MatrixType, RealScalar > ::type	RealVectorType
	Type for vector of eigenvalues as returned by eigenvalues().
typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type `_MatrixType`.
typedef Tridiagonalization < MatrixType >	TridiagonalizationType
Public Member Functions
SelfAdjointEigenSolver &	compute (const MatrixType &matrix, int options=ComputeEigenvectors)
	Computes eigendecomposition of given matrix.
const RealVectorType &	eigenvalues () const
	Returns the eigenvalues of given matrix.
const MatrixType &	eigenvectors () const
	Returns the eigenvectors of given matrix.
ComputationInfo	info () const
	Reports whether previous computation was successful.
MatrixType	operatorInverseSqrt () const
	Computes the inverse square root of the matrix.
MatrixType	operatorSqrt () const
	Computes the positive-definite square root of the matrix.
	SelfAdjointEigenSolver ()
	Default constructor for fixed-size matrices.
	SelfAdjointEigenSolver (Index size)
	Constructor, pre-allocates memory for dynamic-size matrices.
	SelfAdjointEigenSolver (const MatrixType &matrix, int options=ComputeEigenvectors)
	Constructor; computes eigendecomposition of given matrix.
Static Public Attributes
static const int	m_maxIterations = 30
	Maximum number of iterations.
Protected Attributes
bool	m_eigenvectorsOk
RealVectorType	m_eivalues
MatrixType	m_eivec
ComputationInfo	m_info
bool	m_isInitialized
TridiagonalizationType::SubDiagonalType	m_subdiag

Detailed Description

template<typename _MatrixType>
class SelfAdjointEigenSolver< _MatrixType >

Computes eigenvalues and eigenvectors of selfadjoint matrices.

Template Parameters:

_MatrixType the type of the matrix of which we are computing the eigendecomposition; this is expected to be an instantiation of the Matrix class template.

A matrix $ A $ is selfadjoint if it equals its adjoint. For real matrices, this means that the matrix is symmetric: it equals its transpose. This class computes the eigenvalues and eigenvectors of a selfadjoint matrix. These are the scalars $\lambda$ and vectors $ v $ such that $Av = \lambda v$ . The eigenvalues of a selfadjoint matrix are always real. If $ D $ is a diagonal matrix with the eigenvalues on the diagonal, and $ V $ is a matrix with the eigenvectors as its columns, then $A = V D V^{-1}$ (for selfadjoint matrices, the matrix $ V $ is always invertible). This is called the eigendecomposition.

The algorithm exploits the fact that the matrix is selfadjoint, making it faster and more accurate than the general purpose eigenvalue algorithms implemented in EigenSolver and ComplexEigenSolver.

Only the lower triangular part of the input matrix is referenced.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the SelfAdjointEigenSolver(const MatrixType&, int) constructor which computes the eigenvalues and eigenvectors at construction time. Once the eigenvalue and eigenvectors are computed, they can be retrieved with the eigenvalues() and eigenvectors() functions.

The documentation for SelfAdjointEigenSolver(const MatrixType&, int) contains an example of the typical use of this class.

To solve the generalized eigenvalue problem $Av = \lambda Bv$ and the likes, see the class GeneralizedSelfAdjointEigenSolver.

See also:: MatrixBase::eigenvalues(), class EigenSolver, class ComplexEigenSolver

Definition at line 78 of file SelfAdjointEigenSolver.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef MatrixType::Index SelfAdjointEigenSolver< _MatrixType >::Index

Reimplemented in GeneralizedSelfAdjointEigenSolver< _MatrixType >.

Definition at line 92 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

typedef _MatrixType SelfAdjointEigenSolver< _MatrixType >::MatrixType

Reimplemented in GeneralizedSelfAdjointEigenSolver< _MatrixType >.

Definition at line 82 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

typedef NumTraits<Scalar>::Real SelfAdjointEigenSolver< _MatrixType >::RealScalar

Real scalar type for _MatrixType.

This is just Scalar if Scalar is real (e.g., float or double), and the type of the real part of Scalar if Scalar is complex.

Definition at line 100 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

typedef internal::plain_col_type<MatrixType, RealScalar>::type SelfAdjointEigenSolver< _MatrixType >::RealVectorType

Type for vector of eigenvalues as returned by eigenvalues().

This is a column vector with entries of type RealScalar. The length of the vector is the size of _MatrixType.

Definition at line 107 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

typedef MatrixType::Scalar SelfAdjointEigenSolver< _MatrixType >::Scalar

Scalar type for matrices of type _MatrixType.

Definition at line 91 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

typedef Tridiagonalization<MatrixType> SelfAdjointEigenSolver< _MatrixType >::TridiagonalizationType

Definition at line 108 of file SelfAdjointEigenSolver.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator:

Size
ColsAtCompileTime
Options
MaxColsAtCompileTime

Definition at line 83 of file SelfAdjointEigenSolver.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

SelfAdjointEigenSolver< _MatrixType >::SelfAdjointEigenSolver ( ) [inline]

Default constructor for fixed-size matrices.

The default constructor is useful in cases in which the user intends to perform decompositions via compute(). This constructor can only be used if _MatrixType is a fixed-size matrix; use SelfAdjointEigenSolver(Index) for dynamic-size matrices.

Example:

SelfAdjointEigenSolver<Matrix4f> es;
Matrix4f X = Matrix4f::Random(4,4);
Matrix4f A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + Matrix4f::Identity(4,4)); // re-use es to compute eigenvalues of A+I
cout << "The eigenvalues of A+I are: " << es.eigenvalues().transpose() << endl;

Output:

Definition at line 120 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

SelfAdjointEigenSolver< _MatrixType >::SelfAdjointEigenSolver ( Index size ) [inline]

Constructor, pre-allocates memory for dynamic-size matrices.

Parameters:

[in] size Positive integer, size of the matrix whose eigenvalues and eigenvectors will be computed.

This constructor is useful for dynamic-size matrices, when the user intends to perform decompositions via compute(). The size parameter is only used as a hint. It is not an error to give a wrong size, but it may impair performance.

See also:: compute() for an example

Definition at line 139 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

SelfAdjointEigenSolver< _MatrixType >::SelfAdjointEigenSolver	(	const MatrixType &	matrix,
		int	options = `ComputeEigenvectors`
	)		`[inline]`

Constructor; computes eigendecomposition of given matrix.

Parameters:

[in]	matrix	Selfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

This constructor calls compute(const MatrixType&, int) to compute the eigenvalues of the matrix matrix. The eigenvectors are computed if options equals ComputeEigenvectors.

Example:

MatrixXd X = MatrixXd::Random(5,5);
MatrixXd A = X + X.transpose();
cout << "Here is a random symmetric 5x5 matrix, A:" << endl << A << endl << endl;

SelfAdjointEigenSolver<MatrixXd> es(A);
cout << "The eigenvalues of A are:" << endl << es.eigenvalues() << endl;
cout << "The matrix of eigenvectors, V, is:" << endl << es.eigenvectors() << endl << endl;

double lambda = es.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXd v = es.eigenvectors().col(0);
cout << "If v is the corresponding eigenvector, then lambda * v = " << endl << lambda * v << endl;
cout << "... and A * v = " << endl << A * v << endl << endl;

MatrixXd D = es.eigenvalues().asDiagonal();
MatrixXd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

See also:: compute(const MatrixType&, int)

Definition at line 161 of file SelfAdjointEigenSolver.h.

Member Function Documentation

template<typename MatrixType >

SelfAdjointEigenSolver< MatrixType > & SelfAdjointEigenSolver< MatrixType >::compute	(	const MatrixType &	matrix,
		int	options = `ComputeEigenvectors`
	)

Computes eigendecomposition of given matrix.

Parameters:

[in]	matrix	Selfadjoint matrix whose eigendecomposition is to be computed. Only the lower triangular part of the matrix is referenced.
[in]	options	Can be ComputeEigenvectors (default) or EigenvaluesOnly.

Returns:: Reference to *this

This function computes the eigenvalues of matrix. The eigenvalues() function can be used to retrieve them. If options equals ComputeEigenvectors, then the eigenvectors are also computed and can be retrieved by calling eigenvectors().

This implementation uses a symmetric QR algorithm. The matrix is first reduced to tridiagonal form using the Tridiagonalization class. The tridiagonal matrix is then brought to diagonal form with implicit symmetric QR steps with Wilkinson shift. Details can be found in Section 8.3 of Golub & Van Loan, Matrix Computations.

The cost of the computation is about $ 9n^3 $ if the eigenvectors are required and $ 4n^3/3 $ if they are not required.

This method reuses the memory in the SelfAdjointEigenSolver object that was allocated when the object was constructed, if the size of the matrix does not change.

Example:

SelfAdjointEigenSolver<MatrixXf> es(4);
MatrixXf X = MatrixXf::Random(4,4);
MatrixXf A = X + X.transpose();
es.compute(A);
cout << "The eigenvalues of A are: " << es.eigenvalues().transpose() << endl;
es.compute(A + MatrixXf::Identity(4,4)); // re-use es to compute eigenvalues of A+I
cout << "The eigenvalues of A+I are: " << es.eigenvalues().transpose() << endl;

Output:

See also:: SelfAdjointEigenSolver(const MatrixType&, int)

Definition at line 376 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

const RealVectorType& SelfAdjointEigenSolver< _MatrixType >::eigenvalues ( ) const [inline]

Returns the eigenvalues of given matrix.

Returns:: A const reference to the column vector containing the eigenvalues.

Precondition:: The eigenvalues have been computed before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are sorted in increasing order.

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
SelfAdjointEigenSolver<MatrixXd> es(ones);
cout << "The eigenvalues of the 3x3 matrix of ones are:" 
     << endl << es.eigenvalues() << endl;

Output:

See also:: eigenvectors(), MatrixBase::eigenvalues()

Definition at line 242 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

const MatrixType& SelfAdjointEigenSolver< _MatrixType >::eigenvectors ( ) const [inline]

Returns the eigenvectors of given matrix.

Returns:: A const reference to the matrix whose columns are the eigenvectors.

Precondition:: The eigenvectors have been computed before.

Column $ k $ of the returned matrix is an eigenvector corresponding to eigenvalue number $ k $ as returned by eigenvalues(). The eigenvectors are normalized to have (Euclidean) norm equal to one. If this object was used to solve the eigenproblem for the selfadjoint matrix $ A $ , then the matrix returned by this function is the matrix $ V $ in the eigendecomposition $A = V D V^{-1}$ .

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
SelfAdjointEigenSolver<MatrixXd> es(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:" 
     << endl << es.eigenvectors().col(1) << endl;

Output:

See also:: eigenvalues()

Definition at line 220 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

ComputationInfo SelfAdjointEigenSolver< _MatrixType >::info ( ) const [inline]

Reports whether previous computation was successful.

Returns:: Success if computation was succesful, NoConvergence otherwise.

Definition at line 302 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

MatrixType SelfAdjointEigenSolver< _MatrixType >::operatorInverseSqrt ( ) const [inline]

Computes the inverse square root of the matrix.

Returns:: the inverse positive-definite square root of the matrix

Precondition:: The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

This function uses the eigendecomposition $A = V D V^{-1}$ to compute the inverse square root as $V D^{-1/2} V^{-1}$ . This is cheaper than first computing the square root with operatorSqrt() and then its inverse with MatrixBase::inverse().

Example:

MatrixXd X = MatrixXd::Random(4,4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;

SelfAdjointEigenSolver<MatrixXd> es(A);
cout << "The inverse square root of A is: " << endl;
cout << es.operatorInverseSqrt() << endl;
cout << "We can also compute it with operatorSqrt() and inverse(). That yields: " << endl;
cout << es.operatorSqrt().inverse() << endl;

Output:

See also:: operatorSqrt(), MatrixBase::inverse(), MatrixFunctions Module

Definition at line 291 of file SelfAdjointEigenSolver.h.

template<typename _MatrixType>

MatrixType SelfAdjointEigenSolver< _MatrixType >::operatorSqrt ( ) const [inline]

Computes the positive-definite square root of the matrix.

Returns:: the positive-definite square root of the matrix

Precondition:: The eigenvalues and eigenvectors of a positive-definite matrix have been computed before.

The square root of a positive-definite matrix $ A $ is the positive-definite matrix whose square equals $ A $ . This function uses the eigendecomposition $A = V D V^{-1}$ to compute the square root as $A^{1/2} = V D^{1/2} V^{-1}$ .

Example:

MatrixXd X = MatrixXd::Random(4,4);
MatrixXd A = X * X.transpose();
cout << "Here is a random positive-definite matrix, A:" << endl << A << endl << endl;

SelfAdjointEigenSolver<MatrixXd> es(A);
MatrixXd sqrtA = es.operatorSqrt();
cout << "The square root of A is: " << endl << sqrtA << endl;
cout << "If we square this, we get: " << endl << sqrtA*sqrtA << endl;

Output: