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Eigen::RealSchur< _MatrixType > Class Template Reference

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

#include <RealSchur.h>

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. More...
 
 RealSchur (const MatrixType &matrix, bool computeU=true)
 Constructor; computes real Schur decomposition of given matrix. More...
 
const MatrixType & matrixU () const
 Returns the orthogonal matrix in the Schur decomposition. More...
 
const MatrixType & matrixT () const
 Returns the quasi-triangular matrix in the Schur decomposition. More...
 
RealSchur & compute (const MatrixType &matrix, bool computeU=true)
 Computes Schur decomposition of given matrix. More...
 
template<typename HessMatrixType , typename OrthMatrixType >
RealSchur & computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)
 Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T. More...
 
ComputationInfo info () const
 Reports whether previous computation was successful. More...
 
RealSchur & setMaxIterations (Index maxIters)
 Sets the maximum number of iterations allowed. More...
 
Index getMaxIterations ()
 Returns the maximum number of iterations.
 
template<typename HessMatrixType , typename OrthMatrixType >
RealSchur< MatrixType > & computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)
 

Static Public Attributes

static const int m_maxIterationsPerRow = 40
 Maximum number of iterations per row. More...
 

Detailed Description

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

Performs a real Schur decomposition of a square matrix.

Template Parameters
_MatrixTypethe 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

Definition at line 54 of file RealSchur.h.

Constructor & Destructor Documentation

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

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 83 of file RealSchur.h.

83  : RowsAtCompileTime)
84  : m_matT(size, size),
85  m_matU(size, size),
86  m_workspaceVector(size),
87  m_hess(size),
88  m_isInitialized(false),
89  m_matUisUptodate(false),
90  m_maxIters(-1)
91  { }
template<typename _MatrixType>
Eigen::RealSchur< _MatrixType >::RealSchur ( const MatrixType &  matrix,
bool  computeU = true 
)
inline

Constructor; computes real 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.

Example:

Output: 

 

Definition at line 103 of file RealSchur.h.


  104             : m_matT(matrix.rows(),matrix.cols()),
  105               m_matU(matrix.rows(),matrix.cols()),
  106               m_workspaceVector(matrix.rows()),
  107               m_hess(matrix.rows()),
  108               m_isInitialized(false),
  109               m_matUisUptodate(false),
  110               m_maxIters(-1)
  111     {
  112       compute(matrix, computeU);
  113     }
matrix
Definition: math3d.h:219


Eigen::RealSchur::compute
RealSchur & compute(const MatrixType &matrix, bool computeU=true)
Computes Schur decomposition of given matrix. 
Definition: RealSchur.h:246








Member Function Documentation








template<typename MatrixType > 

      

        

          RealSchur< MatrixType > & Eigen::RealSchur< 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 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: 
See also
compute(const MatrixType&, bool, Index) 



Definition at line 246 of file RealSchur.h.


  247 {
  248   eigen_assert(matrix.cols() == matrix.rows());
  249   Index maxIters = m_maxIters;
  250   if (maxIters == -1)
  251     maxIters = m_maxIterationsPerRow * matrix.rows();
  252 
  253   // Step 1. Reduce to Hessenberg form
  254   m_hess.compute(matrix);
  255 
  256   // Step 2. Reduce to real Schur form  
  257   computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
  258   
  259   return *this;
  260 }
Eigen::HessenbergDecomposition::matrixQ
HouseholderSequenceType matrixQ() const 
Reconstructs the orthogonal matrix Q in the decomposition. 
Definition: HessenbergDecomposition.h:232


Eigen::RealSchur::m_maxIterationsPerRow
static const int m_maxIterationsPerRow
Maximum number of iterations per row. 
Definition: RealSchur.h:221


matrix
Definition: math3d.h:219


Eigen::HessenbergDecomposition::matrixH
MatrixHReturnType matrixH() const 
Constructs the Hessenberg matrix H in the decomposition. 
Definition: HessenbergDecomposition.h:260


Eigen::HessenbergDecomposition::compute
HessenbergDecomposition & compute(const MatrixType &matrix)
Computes Hessenberg decomposition of given matrix. 
Definition: HessenbergDecomposition.h:150


Eigen::RealSchur::computeFromHessenberg
RealSchur & computeFromHessenberg(const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU)
Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T. 














template<typename _MatrixType> 



template<typename HessMatrixType , typename OrthMatrixType > 

      

        

          RealSchur& Eigen::RealSchur< _MatrixType >::computeFromHessenberg 
          (
          const HessMatrixType & 
          matrixH, 
        

        

          
          
          const OrthMatrixType & 
          matrixQ, 
        

        

          
          
          bool 
          computeU 
        

        

          
          )
          
        

      





Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T. 


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) 













template<typename _MatrixType> 



  

  
      

        

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

      

  		
  
inline  
  







Reports whether previous computation was successful. 


Returns
Success if computation was succesful, NoConvergence otherwise. 



Definition at line 193 of file RealSchur.h.


  194     {
  195       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
  196       return m_info;
  197     }












template<typename _MatrixType> 



  

  
      

        

          const MatrixType& Eigen::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 



Definition at line 143 of file RealSchur.h.


  144     {
  145       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
  146       return m_matT;
  147     }












template<typename _MatrixType> 



  

  
      

        

          const MatrixType& Eigen::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 



Definition at line 126 of file RealSchur.h.


  127     {
  128       eigen_assert(m_isInitialized && "RealSchur is not initialized.");
  129       eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
  130       return m_matU;
  131     }












template<typename _MatrixType> 



  

  
      

        

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

      

  		
  
inline  
  







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 204 of file RealSchur.h.


  205     {
  206       m_maxIters = maxIters;
  207       return *this;
  208     }






Member Data Documentation








template<typename _MatrixType> 



  

  
      

        

          const int Eigen::RealSchur< _MatrixType >::m_maxIterationsPerRow = 40
        

      

  		
  
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 40. 



Definition at line 221 of file RealSchur.h.







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





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