Detailed Description

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

Performs a complex Schur decomposition of a real or complex square matrix.

This is defined in the Eigenvalues module.

#include <Eigen/Eigenvalues>

Template Parameters

_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: $ A = U T U^*$ 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.

Note: This code is inspired from Jampack

See Also: class RealSchur, class EigenSolver, class ComplexEigenSolver

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...

Member Typedef Documentation

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

Complex scalar type for _MatrixType.

This is std::complex<Scalar> if Scalar is real (e.g., float or double) and just Scalar if Scalar is complex.

Constructor & Destructor Documentation

ComplexSchur ( 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.

ComplexSchur	(	const MatrixType &	matrix,
		bool	computeU = `true`
	)

inline

Constructor; 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.

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

See Also: matrixT() and matrixU() for examples.

Member Function Documentation

ComplexSchur< 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 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 $25n^3$ complex flops, or $10n^3$ complex flops if computeU is false.

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
ComplexSchur<MatrixXcf> schur(4);
schur.compute(A);
cout << "The matrix T in the decomposition of A is:" << endl << schur.matrixT() << endl;
schur.compute(A.inverse());
cout << "The matrix T in the decomposition of A^(-1) is:" << endl << schur.matrixT() << endl;

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)

See Also: compute(const MatrixType&, bool, Index)

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.

Parameters

[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

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)

Referenced by ComplexSchur< _MatrixType >::compute().

ComputationInfo info ( ) const

inline

Reports whether previous computation was successful.

Returns: Success if computation was succesful, NoConvergence otherwise.

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

const ComplexMatrixType& matrixT ( ) const

inline

Returns the triangular matrix in the Schur decomposition.

Returns: A const reference to the matrix T.

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:

schur.matrixT().triangularView<Upper>()

Example:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
ComplexSchur<MatrixXcf> schurOfA(A, false); // false means do not compute U
cout << "The triangular matrix T is:" << endl << schurOfA.matrixT() << endl;

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)

const ComplexMatrixType& matrixU ( ) const

inline

Returns the unitary matrix in the Schur decomposition.

Returns: A const reference to the matrix U.

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:

MatrixXcf A = MatrixXcf::Random(4,4);
cout << "Here is a random 4x4 matrix, A:" << endl << A << endl << endl;
ComplexSchur<MatrixXcf> schurOfA(A);
cout << "The unitary matrix U is:" << endl << schurOfA.matrixU() << endl;

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)

ComplexSchur& 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.

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

Member Data Documentation

const int m_maxIterationsPerRow

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.

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

ComplexSchur.h

Detailed Description

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

Public Types

Public Member Functions

Static Public Attributes

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

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