Eigen
3.2.9

Performs a complex Schur decomposition of a real or complex square matrix.
This is defined in the Eigenvalues module.
_MatrixType  the 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: 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.
Public Types  
typedef Matrix< ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime >  ComplexMatrixType 
Type for the matrices in the Schur decomposition. More...  
typedef std::complex< RealScalar >  ComplexScalar 
Complex scalar type for _MatrixType . More...  
typedef MatrixType::Scalar  Scalar 
Scalar type for matrices of type _MatrixType .  
Public Member Functions  
ComplexSchur (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)  
Default constructor. More...  
ComplexSchur (const MatrixType &matrix, bool computeU=true)  
Constructor; computes Schur decomposition of given matrix. More...  
ComplexSchur &  compute (const MatrixType &matrix, bool computeU=true) 
Computes Schur decomposition of given matrix. More...  
template<typename HessMatrixType , typename OrthMatrixType >  
ComplexSchur &  computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU=true) 
Compute Schur decomposition from a given Hessenberg matrix. More...  
Index  getMaxIterations () 
Returns the maximum number of iterations.  
ComputationInfo  info () const 
Reports whether previous computation was successful. More...  
const ComplexMatrixType &  matrixT () const 
Returns the triangular matrix in the Schur decomposition. More...  
const ComplexMatrixType &  matrixU () const 
Returns the unitary matrix in the Schur decomposition. More...  
ComplexSchur &  setMaxIterations (Index maxIters) 
Sets the maximum number of iterations allowed. More...  
Static Public Attributes  
static const int  m_maxIterationsPerRow 
Maximum number of iterations per row. More...  
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
.
typedef std::complex<RealScalar> ComplexScalar 

inline 
Default constructor.
[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.

inline 
Constructor; computes Schur decomposition of given matrix.
[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.
ComplexSchur< MatrixType > & compute  (  const MatrixType &  matrix, 
bool  computeU = true 

) 
Computes Schur decomposition of given matrix.
[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
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 complex flops, or complex flops if computeU is false.
Example:
Output:
The matrix T in the decomposition of A is: (0.691,1.63) (0.763,0.144) (0.104,0.836) (0.462,0.378) (0,0) (0.758,1.22) (0.65,0.772) (0.244,0.113) (0,0) (0,0) (0.137,0.505) (0.0687,0.404) (0,0) (0,0) (0,0) (1.52,0.402) The matrix T in the decomposition of A^(1) is: (0.501,1.84) (1.01,0.984) (0.636,1.3) (0.676,0.352) (0,0) (0.369,0.593) (0.0733,0.18) (0.0658,0.0263) (0,0) (0,0) (0.222,0.521) (0.191,0.121) (0,0) (0,0) (0,0) (0.614,0.162)
References ComplexSchur< _MatrixType >::computeFromHessenberg(), and Eigen::Success.
Referenced by ComplexSchur< MatrixType >::ComplexSchur().
ComplexSchur& computeFromHessenberg  (  const HessMatrixType &  matrixH, 
const OrthMatrixType &  matrixQ,  
bool  computeU = true 

) 
Compute Schur decomposition from a given Hessenberg matrix.
[in]  matrixH  Matrix in Hessenberg form H 
[in]  matrixQ  orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T 
computeU  Computes the matriX U of the Schur vectors 
*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 quasitriangular 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())
Referenced by ComplexSchur< _MatrixType >::compute().

inline 
Reports whether previous computation was successful.
Success
if computation was succesful, NoConvergence
otherwise. Referenced by ComplexEigenSolver< _MatrixType >::info().

inline 
Returns the triangular matrix in the Schur decomposition.
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:
Example:
Output:
Here is a random 4x4 matrix, A: (0.211,0.68) (0.108,0.444) (0.435,0.271) (0.198,0.687) (0.597,0.566) (0.258,0.0452) (0.214,0.717) (0.782,0.74) (0.605,0.823) (0.0268,0.27) (0.514,0.967) (0.563,0.998) (0.536,0.33) (0.832,0.904) (0.608,0.726) (0.678,0.0259) The triangular matrix T is: (0.691,1.63) (0.763,0.144) (0.104,0.836) (0.462,0.378) (0,0) (0.758,1.22) (0.65,0.772) (0.244,0.113) (0,0) (0,0) (0.137,0.505) (0.0687,0.404) (0,0) (0,0) (0,0) (1.52,0.402)

inline 
Returns the unitary matrix in the Schur decomposition.
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:
Here is a random 4x4 matrix, A: (0.211,0.68) (0.108,0.444) (0.435,0.271) (0.198,0.687) (0.597,0.566) (0.258,0.0452) (0.214,0.717) (0.782,0.74) (0.605,0.823) (0.0268,0.27) (0.514,0.967) (0.563,0.998) (0.536,0.33) (0.832,0.904) (0.608,0.726) (0.678,0.0259) The unitary matrix U is: (0.122,0.271) (0.354,0.255) (0.7,0.321) (0.0909,0.346) (0.247,0.23) (0.435,0.395) (0.184,0.38) (0.492,0.347) (0.859,0.0877) (0.00469,0.21) (0.256,0.0163) (0.133,0.355) (0.116,0.195) (0.484,0.432) (0.183,0.359) (0.559,0.231)

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.
Referenced by ComplexEigenSolver< _MatrixType >::setMaxIterations().

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.