Yoji KURODA / Eigen

Eigne Matrix Class Library

Dependents:   Eigen_test Odometry_test AttitudeEstimation_usingTicker MPU9250_Quaternion_Binary_Serial ... more

Embed: (wiki syntax)

« Back to documentation index

ComplexSchur< _MatrixType > Class Template Reference
Public Types | Public Member Functions | Static Public Attributes

ComplexSchur< _MatrixType > Class Template Reference
[Eigenvalues module]

More...

#include <ComplexSchur.h>

Public Types

typedef MatrixType::Scalar Scalar
 Scalar type for matrices of type _MatrixType.
typedef std::complex< RealScalar > ComplexScalar
 Complex scalar type for _MatrixType.
typedef Matrix< ComplexScalar,
RowsAtCompileTime,
ColsAtCompileTime, Options,
MaxRowsAtCompileTime,
MaxColsAtCompileTime > 		ComplexMatrixType
 Type for the matrices in the Schur decomposition.

Public Member Functions

 ComplexSchur (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
 Default constructor.
 ComplexSchur (const MatrixType &matrix, bool computeU=true)
 Constructor; computes Schur decomposition of given matrix.
const ComplexMatrixType & matrixU () const
 Returns the unitary matrix in the Schur decomposition.
const ComplexMatrixType & matrixT () const
 Returns the triangular matrix in the Schur decomposition.
ComplexSchur & compute (const MatrixType &matrix, bool computeU=true)
 Computes Schur decomposition of given matrix.
template<typename HessMatrixType , typename OrthMatrixType >
ComplexSchur & computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU=true)
 Compute Schur decomposition from a given Hessenberg matrix.
ComputationInfo info () const
 Reports whether previous computation was successful.
ComplexSchur & setMaxIterations (Index maxIters)
 Sets the maximum number of iterations allowed.
Index getMaxIterations ()
 Returns the maximum number of iterations.

Static Public Attributes

static const int m_maxIterationsPerRow = 30
 Maximum number of iterations per row.

Detailed Description

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

Performs a complex Schur decomposition of a real or complex square matrix

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

Given a real or complex square matrix A, this class computes the Schur decomposition: $ A = U T U^*$ where U is a unitary complex matrix, and T is a complex upper triangular matrix. The diagonal of the matrix T corresponds to the eigenvalues of the matrix A.

Call the function compute() to compute the Schur decomposition of a given matrix. Alternatively, you can use the ComplexSchur(const MatrixType&, bool) constructor which computes the Schur decomposition at construction time. Once the decomposition is computed, you can use the matrixU() and matrixT() functions to retrieve the matrices U and V in the decomposition.

Note:
This code is inspired from Jampack
See also:
class RealSchur, class EigenSolver, class ComplexEigenSolver

Definition at line 51 of file ComplexSchur.h.

Member Typedef Documentation

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

Type for the matrices in the Schur decomposition.

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

Definition at line 81 of file ComplexSchur.h.

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 74 of file ComplexSchur.h.

typedef MatrixType::Scalar Scalar

Scalar type for matrices of type _MatrixType.

Definition at line 64 of file ComplexSchur.h.

Constructor & Destructor Documentation

ComplexSchur ( Index  size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime )

Default constructor.

Parameters:
[in]sizePositive integer, size of the matrix whose Schur decomposition will be computed.

The default constructor is useful in cases in which 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 94 of file ComplexSchur.h.

ComplexSchur ( const MatrixType &  matrix,
bool  computeU = true 
)

Constructor; computes Schur decomposition of given matrix.

Parameters:
[in]matrixSquare matrix whose Schur decomposition is to be computed.
[in]computeUIf true, both T and U are computed; if false, only T is computed.

This constructor calls compute() to compute the Schur decomposition.

See also:
matrixT() and matrixU() for examples.

Definition at line 112 of file ComplexSchur.h.

Member Function Documentation

ComplexSchur< MatrixType > & compute ( const MatrixType &  matrix,
bool  computeU = true 
)

Computes Schur decomposition of given matrix.

Parameters:
[in]matrixSquare matrix whose Schur decomposition is to be computed.
[in]computeUIf true, both T and U are computed; if false, only T is computed.
Returns:
Reference to *this

The Schur decomposition is computed by first reducing the matrix to Hessenberg form using the class HessenbergDecomposition. The Hessenberg matrix is then reduced to triangular form by performing QR iterations with a single shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken on the number of iterations; as a rough guide, it may be taken to be $25n^3$ complex flops, or $10n^3$ complex flops if computeU is false.

Example: 

 Output: 


See also:
compute(const MatrixType&, bool, Index) 



Definition at line 316 of file ComplexSchur.h.











      

        

          ComplexSchur< MatrixType > & computeFromHessenberg 
          (
          const HessMatrixType & 
           matrixH, 
        

        

          
          
          const OrthMatrixType & 
           matrixQ, 
        

        

          
          
          bool 
           computeU = true 
        

        

          
          )
          
        

      







Compute Schur decomposition from a given Hessenberg matrix. 


Parameters:

  

    
[in]matrixHMatrix in Hessenberg form H 

    
[in]matrixQorthogonal matrix Q that transform a matrix A to H : A = Q H Q^T 

    
computeUComputes the matriX U of the Schur vectors 

  

  




Returns:
Reference to *this 


This routine assumes that the matrix is already reduced in Hessenberg form matrixH using either the class HessenbergDecomposition or another mean. It computes the upper quasi-triangular matrix T of the Schur decomposition of H When computeU is true, this routine computes the matrix U such that A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix


NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix is not available, the user should give an identity matrix (Q.setIdentity())


See also:
compute(const MatrixType&, bool) 



Definition at line 338 of file ComplexSchur.h.











      

        

          Index getMaxIterations 
          (
           )
          
        

      







Returns the maximum number of iterations. 



Definition at line 233 of file ComplexSchur.h.











      

        

          ComputationInfo info 
          (
           )
           const
        

      







Reports whether previous computation was successful. 


Returns:
Success if computation was succesful, NoConvergence otherwise. 



Definition at line 215 of file ComplexSchur.h.











      

        

          const ComplexMatrixType& matrixT 
          (
           )
           const
        

      







Returns the triangular matrix in the Schur decomposition. 


Returns:
A const reference to the matrix T.


It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix.


Note that this function returns a plain square matrix. If you want to reference only the upper triangular part, use: 


 schur.matrixT().triangularView<Upper>() 

Example: 


 Output: 


 

Definition at line 161 of file ComplexSchur.h.











      

        

          const ComplexMatrixType& matrixU 
          (
           )
           const
        

      







Returns the unitary matrix in the Schur decomposition. 


Returns:
A const reference to the matrix U.


It is assumed that either the constructor ComplexSchur(const MatrixType& matrix, bool computeU) or the member function compute(const MatrixType& matrix, bool computeU) has been called before to compute the Schur decomposition of a matrix, and that computeU was set to true (the default value).


Example: 


 Output: 


 

Definition at line 137 of file ComplexSchur.h.











      

        

          ComplexSchur& setMaxIterations 
          (
          Index 
           maxIters )
          
        

      







Sets the maximum number of iterations allowed. 


If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size of the matrix. 



Definition at line 226 of file ComplexSchur.h.







Field Documentation






      

        

          const int m_maxIterationsPerRow = 30 [static]
        

      







Maximum number of iterations per row. 


If not otherwise specified, the maximum number of iterations is this number times the size of the matrix. It is currently set to 30. 



Definition at line 243 of file ComplexSchur.h.









Generated on Tue Jul 12 2022 17:47:05 by 

doxygen 1.7.2