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

RealSchur< _MatrixType > Class Template Reference
[Eigenvalues module]

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#include <RealSchur.h>

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.
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.
ComputationInfo info () const
 Reports whether previous computation was successful.
RealSchur & 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 = 40
 Maximum number of iterations per row.

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

RealSchur ( 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 83 of file RealSchur.h.

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

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.







Member Function Documentation






      

        

          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.











      

        

          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) 



Definition at line 263 of file RealSchur.h.











      

        

          Index getMaxIterations 
          (
           )
          
        

      







Returns the maximum number of iterations. 



Definition at line 211 of file RealSchur.h.











      

        

          ComputationInfo info 
          (
           )
           const
        

      







Reports whether previous computation was successful. 


Returns:
Success if computation was succesful, NoConvergence otherwise. 



Definition at line 193 of file RealSchur.h.











      

        

          const MatrixType& matrixT 
          (
           )
           const
        

      







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.











      

        

          const MatrixType& matrixU 
          (
           )
           const
        

      







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.











      

        

          RealSchur& 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 204 of file RealSchur.h.







Field Documentation






      

        

          const int 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.









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