Computes eigenvalues and eigenvectors of general matrices
_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 are scalars
and vectors
such that
. If
is a diagonal matrix with the eigenvalues on the diagonal, and
is a matrix with the eigenvectors as its columns, then
. The matrix
is almost always invertible, in which case we have
. 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 and
satisfying
, just like the eigendecomposition, if the matrix
is not required to be diagonal, but if it is allowed to have blocks of the form
(where and
are real numbers) on the diagonal. These blocks correspond to complex eigenvalue pairs
. 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.
Definition at line 78 of file EigenSolver.h.
#include <src/Eigenvalues/EigenSolver.h>
Public Types | |
enum | { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime } |
typedef _MatrixType | MatrixType |
Synonym for the template parameter _MatrixType . | |
typedef MatrixType::Scalar | Scalar |
Scalar type for matrices of type MatrixType. | |
typedef NumTraits< Scalar >::Real | RealScalar |
typedef MatrixType::Index | Index |
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 > | EigenvectorsType |
Type for matrix of eigenvectors as returned by eigenvectors(). | |
Public Member Functions | |
EigenSolver () | |
Default constructor. | |
EigenSolver (Index size) | |
Default constructor with memory preallocation. | |
EigenSolver (const MatrixType &matrix, bool computeEigenvectors=true) | |
Constructor; computes eigendecomposition of given matrix. | |
EigenvectorsType | eigenvectors () const |
Returns the eigenvectors of given matrix. | |
const MatrixType & | pseudoEigenvectors () const |
Returns the pseudo-eigenvectors of given matrix. | |
MatrixType | pseudoEigenvalueMatrix () const |
Returns the block-diagonal matrix in the pseudo-eigendecomposition. | |
const EigenvalueType & | eigenvalues () const |
Returns the eigenvalues of given matrix. | |
EigenSolver & | compute (const MatrixType &matrix, bool computeEigenvectors=true) |
Computes eigendecomposition of given matrix. | |
ComputationInfo | info () const |
Protected Types | |
typedef Matrix< Scalar, ColsAtCompileTime, 1, Options &~RowMajor, MaxColsAtCompileTime, 1 > | ColumnVectorType |
Protected Attributes | |
MatrixType | m_eivec |
EigenvalueType | m_eivalues |
bool | m_isInitialized |
bool | m_eigenvectorsOk |
RealSchur< MatrixType > | m_realSchur |
MatrixType | m_matT |
ColumnVectorType | m_tmp |
Private Member Functions | |
void | doComputeEigenvectors () |
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenSolver< _MatrixType >::ColumnVectorType [protected] |
Definition at line 307 of file EigenSolver.h.
typedef std::complex<RealScalar> 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 104 of file EigenSolver.h.
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> 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 111 of file EigenSolver.h.
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> 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 118 of file EigenSolver.h.
typedef MatrixType::Index EigenSolver< _MatrixType >::Index |
Definition at line 96 of file EigenSolver.h.
typedef _MatrixType EigenSolver< _MatrixType >::MatrixType |
Synonym for the template parameter _MatrixType
.
Definition at line 83 of file EigenSolver.h.
typedef NumTraits<Scalar>::Real EigenSolver< _MatrixType >::RealScalar |
Definition at line 95 of file EigenSolver.h.
typedef MatrixType::Scalar EigenSolver< _MatrixType >::Scalar |
Scalar type for matrices of type MatrixType.
Definition at line 94 of file EigenSolver.h.
anonymous enum |
Definition at line 85 of file EigenSolver.h.
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).
Definition at line 127 of file EigenSolver.h.
EigenSolver< _MatrixType >::EigenSolver | ( | Index | size ) | [inline] |
Default constructor with memory preallocation.
Like the default constructor but with preallocation of the internal data according to the specified problem size.
Definition at line 135 of file EigenSolver.h.
EigenSolver< _MatrixType >::EigenSolver | ( | const MatrixType & | matrix, |
bool | computeEigenvectors = true |
||
) | [inline] |
Constructor; computes eigendecomposition of given matrix.
[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:
Output:
Definition at line 160 of file EigenSolver.h.
References EigenSolver< _MatrixType >::compute().
EigenSolver< MatrixType > & EigenSolver< MatrixType >::compute | ( | const MatrixType & | matrix, |
bool | computeEigenvectors = true |
||
) |
Computes eigendecomposition of given matrix.
[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
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 (where
is the size of the matrix) if
computeEigenvectors
is true, and if
computeEigenvectors
is false.
This method reuses of the allocated data in the EigenSolver object.
Example:
Output:
Definition at line 362 of file EigenSolver.h.
References abs(), EigenSolver< _MatrixType >::compute(), internal::sqrt(), and Success.
Referenced by EigenSolver< _MatrixType >::compute(), and EigenSolver< _MatrixType >::EigenSolver().
void EigenSolver< MatrixType >::doComputeEigenvectors | ( | ) | [private] |
Definition at line 426 of file EigenSolver.h.
References abs(), cdiv(), imag(), mrpt::math::norm(), real(), mrpt::math::size(), t(), and internal::y.
const EigenvalueType& EigenSolver< _MatrixType >::eigenvalues | ( | ) | const [inline] |
Returns the eigenvalues of given matrix.
The eigenvalues are repeated according to their algebraic multiplicity, so there are as many eigenvalues as rows in the matrix.
Example:
Output:
Definition at line 255 of file EigenSolver.h.
References eigen_assert, EigenSolver< _MatrixType >::m_eivalues, and EigenSolver< _MatrixType >::m_isInitialized.
EigenSolver< MatrixType >::EigenvectorsType EigenSolver< MatrixType >::eigenvectors | ( | ) | const |
Returns the eigenvectors of given matrix.
computeEigenvectors
was set to true (the default).Column of the returned matrix is an eigenvector corresponding to eigenvalue number
as returned by eigenvalues(). The eigenvectors are normalized to have (Euclidean) norm equal to one. The matrix returned by this function is the matrix
in the eigendecomposition
, if it exists.
Example:
Output:
Definition at line 332 of file EigenSolver.h.
References PlainObjectBase< Derived >::coeffRef(), eigen_assert, imag(), internal::isMuchSmallerThan(), and real().
ComputationInfo EigenSolver< _MatrixType >::info | ( | ) | const [inline] |
Definition at line 290 of file EigenSolver.h.
References eigen_assert, RealSchur< _MatrixType >::info(), EigenSolver< _MatrixType >::m_isInitialized, and EigenSolver< _MatrixType >::m_realSchur.
MatrixType EigenSolver< MatrixType >::pseudoEigenvalueMatrix | ( | ) | const |
Returns the block-diagonal matrix in the pseudo-eigendecomposition.
The matrix 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
. The matrix
and the matrix
returned by pseudoEigenvectors() satisfy
.
Definition at line 312 of file EigenSolver.h.
References eigen_assert, imag(), internal::isMuchSmallerThan(), and real().
const MatrixType& EigenSolver< _MatrixType >::pseudoEigenvectors | ( | ) | const [inline] |
Returns the pseudo-eigenvectors of given matrix.
computeEigenvectors
was set to true (the default).The real matrix returned by this function and the block-diagonal matrix
returned by pseudoEigenvalueMatrix() satisfy
.
Example:
Output:
Definition at line 212 of file EigenSolver.h.
References eigen_assert, EigenSolver< _MatrixType >::m_eigenvectorsOk, EigenSolver< _MatrixType >::m_eivec, and EigenSolver< _MatrixType >::m_isInitialized.
bool EigenSolver< _MatrixType >::m_eigenvectorsOk [protected] |
Definition at line 303 of file EigenSolver.h.
Referenced by EigenSolver< _MatrixType >::pseudoEigenvectors().
EigenvalueType EigenSolver< _MatrixType >::m_eivalues [protected] |
Definition at line 301 of file EigenSolver.h.
Referenced by EigenSolver< _MatrixType >::eigenvalues().
MatrixType EigenSolver< _MatrixType >::m_eivec [protected] |
Definition at line 300 of file EigenSolver.h.
Referenced by EigenSolver< _MatrixType >::pseudoEigenvectors().
bool EigenSolver< _MatrixType >::m_isInitialized [protected] |
Definition at line 302 of file EigenSolver.h.
Referenced by EigenSolver< _MatrixType >::eigenvalues(), EigenSolver< _MatrixType >::info(), and EigenSolver< _MatrixType >::pseudoEigenvectors().
MatrixType EigenSolver< _MatrixType >::m_matT [protected] |
Definition at line 305 of file EigenSolver.h.
RealSchur<MatrixType> EigenSolver< _MatrixType >::m_realSchur [protected] |
Definition at line 304 of file EigenSolver.h.
Referenced by EigenSolver< _MatrixType >::info().
ColumnVectorType EigenSolver< _MatrixType >::m_tmp [protected] |
Definition at line 308 of file EigenSolver.h.
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