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

#include <EigenSolver.h>

Public Types
enum	{ RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }

typedef std::complex< RealScalar >	ComplexScalar
	Complex scalar type for MatrixType. More...

typedef Matrix< ComplexScalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	EigenvalueType
	Type for vector of eigenvalues as returned by eigenvalues(). More...

typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >	EigenvectorsType
	Type for matrix of eigenvectors as returned by eigenvectors(). More...

typedef Eigen::Index	Index

typedef _MatrixType	MatrixType
	Synonym for the template parameter `_MatrixType`. More...

typedef NumTraits< Scalar >::Real	RealScalar

typedef MatrixType::Scalar	Scalar
	Scalar type for matrices of type MatrixType. More...

Public Member Functions
template<typename InputType >
EigenSolver &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Computes eigendecomposition of given matrix. More...

template<typename InputType >
EigenSolver< MatrixType > &	compute (const EigenBase< InputType > &matrix, bool computeEigenvectors)

	EigenSolver ()
	Default constructor. More...

	EigenSolver (Index size)
	Default constructor with memory preallocation. More...

template<typename InputType >
	EigenSolver (const EigenBase< InputType > &matrix, bool computeEigenvectors=true)
	Constructor; computes eigendecomposition of given matrix. More...

const EigenvalueType &	eigenvalues () const
	Returns the eigenvalues of given matrix. More...

EigenvectorsType	eigenvectors () const
	Returns the eigenvectors of given matrix. More...

Index	getMaxIterations ()
	Returns the maximum number of iterations. More...

ComputationInfo	info () const

MatrixType	pseudoEigenvalueMatrix () const
	Returns the block-diagonal matrix in the pseudo-eigendecomposition. More...

const MatrixType &	pseudoEigenvectors () const
	Returns the pseudo-eigenvectors of given matrix. More...

EigenSolver &	setMaxIterations (Index maxIters)
	Sets the maximum number of iterations allowed. More...

Protected Types
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 >	ColumnVectorType

Static Protected Member Functions
static void	check_template_parameters ()

Protected Attributes
bool	m_eigenvectorsOk

EigenvalueType	m_eivalues

MatrixType	m_eivec

ComputationInfo	m_info

bool	m_isInitialized

MatrixType	m_matT

RealSchur< MatrixType >	m_realSchur

ColumnVectorType	m_tmp

Private Member Functions
void	doComputeEigenvectors ()

Detailed Description

template<typename _MatrixType>
class Eigen::EigenSolver< _MatrixType >

Computes eigenvalues and eigenvectors of general 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. Currently, only real matrices are supported.

The eigenvalues and eigenvectors of a matrix $ A $ are scalars $\lambda$ and vectors $ v $ such that $Av = \lambda v$ . 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 = V D $ . The matrix $ V $ is almost always invertible, in which case we have $A = V D V^{-1}$ . This is called the eigendecomposition.

The eigenvalues and eigenvectors of a matrix may be complex, even when the matrix is real. However, we can choose real matrices $ V $ and $ D $ satisfying $ A V = V D $ , just like the eigendecomposition, if the matrix $ D $ is not required to be diagonal, but if it is allowed to have blocks of the form

$\begin{bmatrix} u & v \\ -v & u \end{bmatrix}$

(where $ u $ and $ v $ are real numbers) on the diagonal. These blocks correspond to complex eigenvalue pairs $u \pm iv$ . We call this variant of the eigendecomposition the pseudo-eigendecomposition.

Call the function compute() to compute the eigenvalues and eigenvectors of a given matrix. Alternatively, you can use the EigenSolver(const MatrixType&, bool) 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 pseudoEigenvalueMatrix() and pseudoEigenvectors() methods allow the construction of the pseudo-eigendecomposition.

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

Note: The implementation is adapted from JAMA (public domain). Their code is based on EISPACK.

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

Definition at line 64 of file EigenSolver.h.

Member Typedef Documentation

template<typename _MatrixType>

typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< _MatrixType >::ColumnVectorType

protected

Definition at line 319 of file EigenSolver.h.

template<typename _MatrixType>

typedef std::complex<RealScalar> Eigen::EigenSolver< _MatrixType >::ComplexScalar

Complex scalar type for MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

Definition at line 90 of file EigenSolver.h.

template<typename _MatrixType>

typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> Eigen::EigenSolver< _MatrixType >::EigenvalueType

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

This is a column vector with entries of type ComplexScalar. The length of the vector is the size of MatrixType.

Definition at line 97 of file EigenSolver.h.

template<typename _MatrixType>

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::EigenSolver< _MatrixType >::EigenvectorsType

Type for matrix of eigenvectors as returned by eigenvectors().

This is a square matrix with entries of type ComplexScalar. The size is the same as the size of MatrixType.

Definition at line 104 of file EigenSolver.h.

template<typename _MatrixType>

typedef Eigen::Index Eigen::EigenSolver< _MatrixType >::Index

Deprecated:: since Eigen 3.3

Definition at line 82 of file EigenSolver.h.

template<typename _MatrixType>

typedef _MatrixType Eigen::EigenSolver< _MatrixType >::MatrixType

Synonym for the template parameter _MatrixType.

Definition at line 69 of file EigenSolver.h.

template<typename _MatrixType>

typedef NumTraits<Scalar>::Real Eigen::EigenSolver< _MatrixType >::RealScalar

Definition at line 81 of file EigenSolver.h.

template<typename _MatrixType>

typedef MatrixType::Scalar Eigen::EigenSolver< _MatrixType >::Scalar

Scalar type for matrices of type MatrixType.

Definition at line 80 of file EigenSolver.h.

Member Enumeration Documentation

template<typename _MatrixType>

anonymous enum

Enumerator
RowsAtCompileTime
ColsAtCompileTime
Options
MaxRowsAtCompileTime
MaxColsAtCompileTime

Definition at line 71 of file EigenSolver.h.

Constructor & Destructor Documentation

template<typename _MatrixType>

Eigen::EigenSolver< _MatrixType >::EigenSolver ( )

inline

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via EigenSolver::compute(const MatrixType&, bool).

See also: compute() for an example.

Definition at line 113 of file EigenSolver.h.

template<typename _MatrixType>

Eigen::EigenSolver< _MatrixType >::EigenSolver ( Index size )

inlineexplicit

Default constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also: EigenSolver()

Definition at line 121 of file EigenSolver.h.

template<typename _MatrixType>

template<typename InputType >

Eigen::EigenSolver< _MatrixType >::EigenSolver	(	const EigenBase< InputType > &	matrix,
		bool	computeEigenvectors = `true`
	)

inlineexplicit

Constructor; computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigenvalues and eigenvectors.

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
EigenSolver<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;
complex<double> lambda = es.eigenvalues()[0];
cout << "Consider the first eigenvalue, lambda = " << lambda << endl;
VectorXcd 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.cast<complex<double> >() * v << endl << endl;
MatrixXcd D = es.eigenvalues().asDiagonal();
MatrixXcd V = es.eigenvectors();
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

See also: compute()

Definition at line 147 of file EigenSolver.h.

Member Function Documentation

template<typename _MatrixType>

static void Eigen::EigenSolver< _MatrixType >::check_template_parameters ( )

inlinestaticprotected

Definition at line 305 of file EigenSolver.h.

template<typename _MatrixType>

template<typename InputType >

EigenSolver& Eigen::EigenSolver< _MatrixType >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeEigenvectors = `true`
	)

Computes eigendecomposition of given matrix.

Parameters

[in]	matrix	Square matrix whose eigendecomposition is to be computed.
[in]	computeEigenvectors	If true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

Returns: Reference to *this

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

The matrix is first reduced to real Schur form using the RealSchur class. The Schur decomposition is then used to compute the eigenvalues and eigenvectors.

The cost of the computation is dominated by the cost of the Schur decomposition, which is very approximately $ 25n^3 $ (where $ n $ is the size of the matrix) if computeEigenvectors is true, and $ 10n^3 $ if computeEigenvectors is false.

This method reuses of the allocated data in the EigenSolver object.

Example:

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

Output:

template<typename _MatrixType>

template<typename InputType >

EigenSolver<MatrixType>& Eigen::EigenSolver< _MatrixType >::compute	(	const EigenBase< InputType > &	matrix,
		bool	computeEigenvectors
	)

Definition at line 379 of file EigenSolver.h.

template<typename MatrixType >

void Eigen::EigenSolver< MatrixType >::doComputeEigenvectors ( )

private

Definition at line 458 of file EigenSolver.h.

template<typename _MatrixType>

const EigenvalueType& Eigen::EigenSolver< _MatrixType >::eigenvalues ( ) const

inline

Returns the eigenvalues of given matrix.

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

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before.

The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix. The eigenvalues are not sorted in any particular order.

Example:

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

Output:

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

Definition at line 244 of file EigenSolver.h.

template<typename MatrixType >

EigenSolver< MatrixType >::EigenvectorsType Eigen::EigenSolver< MatrixType >::eigenvectors ( ) const

Returns the eigenvectors of given matrix.

Returns: Matrix whose columns are the (possibly complex) eigenvectors.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).

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. The matrix returned by this function is the matrix $ V $ in the eigendecomposition $A = V D V^{-1}$ , if it exists.

Example:

MatrixXd ones = MatrixXd::Ones(3,3);
EigenSolver<MatrixXd> es(ones);
cout << "The first eigenvector of the 3x3 matrix of ones is:"
     << endl << es.eigenvectors().col(0) << endl;

Output:

See also: eigenvalues(), pseudoEigenvectors()

Definition at line 345 of file EigenSolver.h.

template<typename _MatrixType>

Index Eigen::EigenSolver< _MatrixType >::getMaxIterations ( )

inline

Returns the maximum number of iterations.

Definition at line 295 of file EigenSolver.h.

template<typename _MatrixType>

ComputationInfo Eigen::EigenSolver< _MatrixType >::info ( ) const

inline

Returns: NumericalIssue if the input contains INF or NaN values or overflow occured. Returns Success otherwise.

Definition at line 281 of file EigenSolver.h.

template<typename MatrixType >

MatrixType Eigen::EigenSolver< MatrixType >::pseudoEigenvalueMatrix ( ) const

Returns the block-diagonal matrix in the pseudo-eigendecomposition.

Returns: A block-diagonal matrix.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before.

The matrix $ D $ returned by this function is real and block-diagonal. The blocks on the diagonal are either 1-by-1 or 2-by-2 blocks of the form $\begin{bmatrix} u & v \\ -v & u \end{bmatrix}$ . These blocks are not sorted in any particular order. The matrix $ D $ and the matrix $ V $ returned by pseudoEigenvectors() satisfy $ AV = VD $ .

See also: pseudoEigenvectors() for an example, eigenvalues()

Definition at line 324 of file EigenSolver.h.

template<typename _MatrixType>

const MatrixType& Eigen::EigenSolver< _MatrixType >::pseudoEigenvectors ( ) const

inline

Returns the pseudo-eigenvectors of given matrix.

Returns: Const reference to matrix whose columns are the pseudo-eigenvectors.

Precondition: Either the constructor EigenSolver(const MatrixType&,bool) or the member function compute(const MatrixType&, bool) has been called before, and computeEigenvectors was set to true (the default).

The real matrix $ V $ returned by this function and the block-diagonal matrix $ D $ returned by pseudoEigenvalueMatrix() satisfy $ AV = VD $ .

Example:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
EigenSolver<MatrixXd> es(A);
MatrixXd D = es.pseudoEigenvalueMatrix();
MatrixXd V = es.pseudoEigenvectors();
cout << "The pseudo-eigenvalue matrix D is:" << endl << D << endl;
cout << "The pseudo-eigenvector matrix V is:" << endl << V << endl;
cout << "Finally, V * D * V^(-1) = " << endl << V * D * V.inverse() << endl;

Output:

See also: pseudoEigenvalueMatrix(), eigenvectors()

Definition at line 199 of file EigenSolver.h.

template<typename _MatrixType>

EigenSolver& Eigen::EigenSolver< _MatrixType >::setMaxIterations ( Index maxIters )

inline

Sets the maximum number of iterations allowed.

Definition at line 288 of file EigenSolver.h.

Member Data Documentation

template<typename _MatrixType>

bool Eigen::EigenSolver< _MatrixType >::m_eigenvectorsOk

protected

Definition at line 314 of file EigenSolver.h.

template<typename _MatrixType>

EigenvalueType Eigen::EigenSolver< _MatrixType >::m_eivalues

protected

Definition at line 312 of file EigenSolver.h.

template<typename _MatrixType>

MatrixType Eigen::EigenSolver< _MatrixType >::m_eivec

protected

Definition at line 311 of file EigenSolver.h.

template<typename _MatrixType>

ComputationInfo Eigen::EigenSolver< _MatrixType >::m_info

protected

Definition at line 315 of file EigenSolver.h.

template<typename _MatrixType>

bool Eigen::EigenSolver< _MatrixType >::m_isInitialized

protected

Definition at line 313 of file EigenSolver.h.

template<typename _MatrixType>

MatrixType Eigen::EigenSolver< _MatrixType >::m_matT

protected

Definition at line 317 of file EigenSolver.h.

template<typename _MatrixType>

RealSchur<MatrixType> Eigen::EigenSolver< _MatrixType >::m_realSchur

protected

Definition at line 316 of file EigenSolver.h.

template<typename _MatrixType>

ColumnVectorType Eigen::EigenSolver< _MatrixType >::m_tmp

protected

Definition at line 320 of file EigenSolver.h.

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

EigenSolver.h

Public Types

Public Member Functions

Protected Types

Static Protected Member Functions

Protected Attributes

Private Member Functions

Detailed Description

template<typename _MatrixType> class Eigen::EigenSolver< _MatrixType >

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename _MatrixType>
class Eigen::EigenSolver< _MatrixType >