Public Types | Public Member Functions | Static Public Attributes

RealSchur< _MatrixType > Class Template Reference

Performs a real Schur decomposition of a square matrix. More...

#include <RealSchur.h>

List of all members.

Public Types

enum  {
  RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
}
typedef _MatrixType MatrixType
typedef MatrixType::Scalar Scalar
typedef std::complex< typename
NumTraits< Scalar >::Real > 		ComplexScalar
typedef MatrixType::Index Index
typedef Matrix< ComplexScalar,
ColsAtCompileTime, 1, Options
&~RowMajor,
MaxColsAtCompileTime, 1 > 		EigenvalueType
typedef Matrix< Scalar,
ColsAtCompileTime, 1, Options
&~RowMajor,
MaxColsAtCompileTime, 1 > 		ColumnVectorType

Public Member Functions

 RealSchur (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
 Default constructor.
 RealSchur (const MatrixType &matrix, bool computeU=true)
 Constructor; computes real Schur decomposition of given matrix.
const MatrixType & matrixU () const
 Returns the orthogonal matrix in the Schur decomposition.
const MatrixType & matrixT () const
 Returns the quasi-triangular matrix in the Schur decomposition.
RealSchur & compute (const MatrixType &matrix, bool computeU=true)
 Computes Schur decomposition of given matrix.
ComputationInfo info () const
 Reports whether previous computation was successful.

Static Public Attributes

static const int m_maxIterations = 40
 Maximum number of iterations.

Detailed Description

template<typename _MatrixType>
class RealSchur< _MatrixType >

Performs a real Schur decomposition of a square matrix.

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

Given a real square matrix A, this class computes the real Schur decomposition: $ A = U T U^T $ where U is a real orthogonal matrix and T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose inverse is equal to its transpose, $ U^{-1} = U^T $. A quasi-triangular matrix is a block-triangular matrix whose diagonal consists of 1-by-1 blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the blocks on the diagonal of T are the same as the eigenvalues of the matrix A, and thus the real Schur decomposition is used in EigenSolver to compute the eigendecomposition of a matrix.

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

The documentation of RealSchur(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:
class ComplexSchur, class EigenSolver, class ComplexEigenSolver

Member Typedef Documentation

template<typename _MatrixType>
typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> RealSchur< _MatrixType >::ColumnVectorType
template<typename _MatrixType>
typedef std::complex<typename NumTraits<Scalar>::Real> RealSchur< _MatrixType >::ComplexScalar
template<typename _MatrixType>
typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> RealSchur< _MatrixType >::EigenvalueType
template<typename _MatrixType>
typedef MatrixType::Index RealSchur< _MatrixType >::Index
template<typename _MatrixType>
typedef _MatrixType RealSchur< _MatrixType >::MatrixType
template<typename _MatrixType>
typedef MatrixType::Scalar RealSchur< _MatrixType >::Scalar

Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator:
RowsAtCompileTime 
ColsAtCompileTime 
Options 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Constructor & Destructor Documentation

template<typename _MatrixType>
RealSchur< _MatrixType >::RealSchur ( Index  size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime  )  [inline]

Default constructor.

Parameters:
[in] size Positive 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.
template<typename _MatrixType>
RealSchur< _MatrixType >::RealSchur ( const MatrixType &  matrix,
bool  computeU = true 
) [inline]

Constructor; computes real Schur decomposition of given matrix.

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

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

Example: 

 Output: 


 





Member Function Documentation








template<typename _MatrixType> 

      

        

          RealSchur& RealSchur< _MatrixType >::compute 
          (
          const MatrixType & 
           matrix, 
        

        

          
          
          bool 
           computeU = true 
        

        

          
          )
          
        

      







Computes Schur decomposition of given matrix. 


Parameters:

  

    
[in] matrix Square matrix whose Schur decomposition is to be computed. 

    
[in] computeU If 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 Francis QR iterations with implicit double shift. The cost of computing the Schur decomposition depends on the number of iterations; as a rough guide, it may be taken to be $25n^3$ flops if computeU is true and $10n^3$ flops if computeU is false.


Example: 


 Output: 


 











template<typename _MatrixType> 

      

        

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

      







Reports whether previous computation was successful. 


Returns:
Success if computation was succesful, NoConvergence otherwise. 













template<typename _MatrixType> 

      

        

          const MatrixType& RealSchur< _MatrixType >::matrixT 
          (
          
           ) 
           const [inline]
        

      







Returns the quasi-triangular matrix in the Schur decomposition. 


Returns:
A const reference to the matrix T.


Precondition:
Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix.


See also:
RealSchur(const MatrixType&, bool) for an example 













template<typename _MatrixType> 

      

        

          const MatrixType& RealSchur< _MatrixType >::matrixU 
          (
          
           ) 
           const [inline]
        

      







Returns the orthogonal matrix in the Schur decomposition. 


Returns:
A const reference to the matrix U.


Precondition:
Either the constructor RealSchur(const MatrixType&, bool) or the member function compute(const MatrixType&, bool) has been called before to compute the Schur decomposition of a matrix, and computeU was set to true (the default value).


See also:
RealSchur(const MatrixType&, bool) for an example 







Member Data Documentation








template<typename _MatrixType> 

      

        

          const int RealSchur< _MatrixType >::m_maxIterations = 40 [static]
        

      







Maximum number of iterations. 


Maximum number of iterations allowed for an eigenvalue to converge. 







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



Generated on Thu Dec 1 2011 12:54:37 for HUMIM Tracker by 

doxygen 1.7.1