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ComplexEigenSolver< _MatrixType > Class Template Reference

ComplexEigenSolver< _MatrixType > Class Template Reference
[Eigenvalues module]

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#include <ComplexEigenSolver.h>

Public Types

typedef _MatrixType MatrixType
 Synonym for the template parameter _MatrixType.
typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type MatrixType.
typedef std::complex< RealScalar > ComplexScalar
 Complex scalar type for MatrixType.
typedef Matrix< ComplexScalar,
ColsAtCompileTime, 1, Options
&(~RowMajor),
MaxColsAtCompileTime, 1 > 
EigenvalueType
 Type for vector of eigenvalues as returned by eigenvalues().
typedef Matrix< ComplexScalar,
RowsAtCompileTime,
ColsAtCompileTime, Options,
MaxRowsAtCompileTime,
MaxColsAtCompileTime > 
EigenvectorType
 Type for matrix of eigenvectors as returned by eigenvectors().

Public Member Functions

 ComplexEigenSolver ()
 Default constructor.
 ComplexEigenSolver (Index size)
 Default Constructor with memory preallocation.
 ComplexEigenSolver (const MatrixType &matrix, bool computeEigenvectors=true)
 Constructor; computes eigendecomposition of given matrix.
const EigenvectorTypeeigenvectors () const
 Returns the eigenvectors of given matrix.
const EigenvalueTypeeigenvalues () const
 Returns the eigenvalues of given matrix.
ComplexEigenSolvercompute (const MatrixType &matrix, bool computeEigenvectors=true)
 Computes eigendecomposition of given matrix.
ComputationInfo info () const
 Reports whether previous computation was successful.
ComplexEigenSolversetMaxIterations (Index maxIters)
 Sets the maximum number of iterations allowed.
Index getMaxIterations ()
 Returns the maximum number of iterations.

Detailed Description

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

Computes eigenvalues and eigenvectors of general complex matrices

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

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 main function in this class is compute(), which computes the eigenvalues and eigenvectors of a given function. The documentation for that function contains an example showing the main features of the class.

See also:
class EigenSolver, class SelfAdjointEigenSolver

Definition at line 45 of file ComplexEigenSolver.h.


Member Typedef Documentation

typedef std::complex<RealScalar> 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 71 of file ComplexEigenSolver.h.

typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options&(~RowMajor), MaxColsAtCompileTime, 1> 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 78 of file ComplexEigenSolver.h.

typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> EigenvectorType

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 85 of file ComplexEigenSolver.h.

typedef _MatrixType MatrixType

Synonym for the template parameter _MatrixType.

Definition at line 50 of file ComplexEigenSolver.h.

typedef MatrixType::Scalar Scalar

Scalar type for matrices of type MatrixType.

Definition at line 61 of file ComplexEigenSolver.h.


Constructor & Destructor Documentation

Default constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via compute().

Definition at line 92 of file ComplexEigenSolver.h.

ComplexEigenSolver ( Index  size )

Default Constructor with memory preallocation.

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

See also:
ComplexEigenSolver()

Definition at line 107 of file ComplexEigenSolver.h.

ComplexEigenSolver ( const MatrixType matrix,
bool  computeEigenvectors = true 
)

Constructor; computes eigendecomposition of given matrix.

Parameters:
[in]matrixSquare matrix whose eigendecomposition is to be computed.
[in]computeEigenvectorsIf true, both the eigenvectors and the eigenvalues are computed; if false, only the eigenvalues are computed.

This constructor calls compute() to compute the eigendecomposition.

Definition at line 125 of file ComplexEigenSolver.h.


Member Function Documentation

ComplexEigenSolver< MatrixType > & compute ( const MatrixType matrix,
bool  computeEigenvectors = true 
)

Computes eigendecomposition of given matrix.

Parameters:
[in]matrixSquare matrix whose eigendecomposition is to be computed.
[in]computeEigenvectorsIf 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 complex 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 Schur form using the ComplexSchur 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 $ O(n^3) $ where $ n $ is the size of the matrix.

Example:

Output:

Definition at line 258 of file ComplexEigenSolver.h.

const EigenvalueType& eigenvalues (  ) const

Returns the eigenvalues of given matrix.

Returns:
A const reference to the column vector containing the eigenvalues.
Precondition:
Either the constructor ComplexEigenSolver(const MatrixType& matrix, bool) or the member function compute(const MatrixType& matrix, bool) has been called before to compute the eigendecomposition of a matrix.

This function returns a column vector containing the eigenvalues. 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:

Output:

Definition at line 181 of file ComplexEigenSolver.h.

const EigenvectorType& eigenvectors (  ) const

Returns the eigenvectors of given matrix.

Returns:
A const reference to the matrix whose columns are the eigenvectors.
Precondition:
Either the constructor ComplexEigenSolver(const MatrixType& matrix, bool) or the member function compute(const MatrixType& matrix, bool) has been called before to compute the eigendecomposition of a matrix, and computeEigenvectors was set to true (the default).

This function returns a matrix whose columns are the eigenvectors. Column $ k $ 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:

Output:

Definition at line 156 of file ComplexEigenSolver.h.

Index getMaxIterations (  )

Returns the maximum number of iterations.

Definition at line 231 of file ComplexEigenSolver.h.

ComputationInfo info (  ) const

Reports whether previous computation was successful.

Returns:
Success if computation was succesful, NoConvergence otherwise.

Definition at line 217 of file ComplexEigenSolver.h.

ComplexEigenSolver& setMaxIterations ( Index  maxIters )

Sets the maximum number of iterations allowed.

Definition at line 224 of file ComplexEigenSolver.h.