Public Member Functions | Static Public Attributes | List of all members
RealSchur< _MatrixType > Class Template Reference

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

#include <RealSchur.h>

Public Member Functions

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

Static Public Attributes

static const int m_maxIterations
 Maximum number of iterations.
 

Detailed Description

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

Performs a real Schur decomposition of a square matrix.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>
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

Constructor & Destructor Documentation

◆ RealSchur() [1/2]

template<typename _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.

Referenced by compute().

◆ RealSchur() [2/2]

template<typename _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:

MatrixXd A = MatrixXd::Random(6,6);
cout << "Here is a random 6x6 matrix, A:" << endl << A << endl << endl;
RealSchur<MatrixXd> schur(A);
cout << "The orthogonal matrix U is:" << endl << schur.matrixU() << endl;
cout << "The quasi-triangular matrix T is:" << endl << schur.matrixT() << endl << endl;
MatrixXd U = schur.matrixU();
MatrixXd T = schur.matrixT();
cout << "U * T * U^T = " << endl << U * T * U.transpose() << endl;
Eigen::MatrixBase< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > >::transpose
Eigen::Transpose< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > > transpose()
Definition Transpose.h:199
Eigen::Matrix< double, Dynamic, Dynamic >::Random
static const CwiseNullaryOp< internal::scalar_random_op< Scalar >, Matrix< double, _Rows, _Cols, _Options, _MaxRows, _MaxCols > > Random(Index rows, Index cols)
Definition Random.h:49
Eigen::RealSchur::RealSchur
RealSchur(Index size=RowsAtCompileTime==Dynamic ? 1 :RowsAtCompileTime)
Default constructor.
Definition RealSchur.h:83

Output: 

Here is a random 6x6 matrix, A:
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

The orthogonal matrix U is:
  0.348  -0.754 0.00435  -0.351  0.0145   0.432
  -0.16  -0.266  -0.747   0.457  -0.366  0.0571
  0.505  -0.157  0.0746   0.644   0.518  -0.177
  0.703   0.324  -0.409  -0.349  -0.187  -0.275
  0.296   0.372    0.24   0.324  -0.379   0.684
 -0.126   0.305   -0.46  -0.161   0.647   0.485
The quasi-triangular matrix T is:
   -0.2   -1.83   0.864   0.271    1.09    0.14
  0.647   0.298 -0.0536   0.676  -0.288   0.023
      0       0   0.967  -0.201  -0.429   0.847
      0       0       0   0.353   0.602   0.694
      0       0       0       0   0.572   -1.03
      0       0       0       0  0.0184   0.664

U * T * U^T = 
   0.68   -0.33   -0.27  -0.717  -0.687  0.0259
 -0.211   0.536  0.0268   0.214  -0.198   0.678
  0.566  -0.444   0.904  -0.967   -0.74   0.225
  0.597   0.108   0.832  -0.514  -0.782  -0.408
  0.823 -0.0452   0.271  -0.726   0.998   0.275
 -0.605   0.258   0.435   0.608  -0.563  0.0486

References compute().

Member Function Documentation

◆ compute()

template<typename MatrixType>
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:

MatrixXf A = MatrixXf::Random(4,4);
RealSchur<MatrixXf> schur(4);
schur.compute(A, /* computeU = */ false);
cout << "The matrix T in the decomposition of A is:" << endl << schur.matrixT() << endl;
schur.compute(A.inverse(), /* computeU = */ false);
cout << "The matrix T in the decomposition of A^(-1) is:" << endl << schur.matrixT() << endl;
Eigen::MatrixBase< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > >::inverse
const internal::inverse_impl< Matrix< _Scalar, _Rows, _Cols, _Options, _MaxRows, _MaxCols > > inverse() const
Definition Inverse.h:316

Output: 

The matrix T in the decomposition of A is:
 0.523 -0.698  0.148  0.742
 0.475  0.986 -0.793  0.721
     0      0  -0.28  -0.77
     0      0 0.0145 -0.367
The matrix T in the decomposition of A^(-1) is:
-3.06 -4.57 -6.05  5.39
0.168 -2.62 -3.33  3.86
    0     0 0.434  0.56
    0     0 -1.06  1.35

References m_maxIterations, Eigen::NoConvergence, RealSchur(), and Eigen::Success.

Referenced by RealSchur().

◆ info()

template<typename _MatrixType>
ComputationInfo info ( ) const
inline

Reports whether previous computation was successful.

Returns
Success if computation was succesful, NoConvergence otherwise.

◆ matrixT()

template<typename _MatrixType>
const 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

◆ matrixU()

template<typename _MatrixType>
const 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

◆ m_maxIterations

template<typename _MatrixType>
const int m_maxIterations
static

Maximum number of iterations.

Maximum number of iterations allowed for an eigenvalue to converge.

Referenced by compute().

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