Performs a complex Schur decomposition of a real or complex square matrix. More...
#include <ComplexSchur.h>
Public Types | |
enum | { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, Options = MatrixType::Options, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime } |
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 Eigen::Index | Index |
typedef _MatrixType | MatrixType |
typedef NumTraits< Scalar >::Real | RealScalar |
typedef MatrixType::Scalar | Scalar |
Scalar type for matrices of type _MatrixType . More... | |
Public Member Functions | |
template<typename InputType > | |
ComplexSchur (const EigenBase< InputType > &matrix, bool computeU=true) | |
Constructor; computes Schur decomposition of given matrix. More... | |
ComplexSchur (Index size=RowsAtCompileTime==Dynamic ? 1 :RowsAtCompileTime) | |
Default constructor. More... | |
template<typename InputType > | |
ComplexSchur< MatrixType > & | compute (const EigenBase< InputType > &matrix, bool computeU) |
template<typename InputType > | |
ComplexSchur & | compute (const EigenBase< InputType > &matrix, bool computeU=true) |
Computes Schur decomposition of given matrix. More... | |
template<typename HessMatrixType , typename OrthMatrixType > | |
ComplexSchur< MatrixType > & | computeFromHessenberg (const HessMatrixType &matrixH, const OrthMatrixType &matrixQ, bool computeU) |
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. More... | |
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 = 30 |
Maximum number of iterations per row. More... | |
Protected Attributes | |
HessenbergDecomposition< MatrixType > | m_hess |
ComputationInfo | m_info |
bool | m_isInitialized |
ComplexMatrixType | m_matT |
ComplexMatrixType | m_matU |
bool | m_matUisUptodate |
Index | m_maxIters |
Private Member Functions | |
ComplexScalar | computeShift (Index iu, Index iter) |
void | reduceToTriangularForm (bool computeU) |
bool | subdiagonalEntryIsNeglegible (Index i) |
Friends | |
struct | internal::complex_schur_reduce_to_hessenberg< MatrixType, NumTraits< Scalar >::IsComplex > |
Performs a complex Schur decomposition of a real or complex square matrix.
\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.
Definition at line 51 of file ComplexSchur.h.
typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> Eigen::ComplexSchur< _MatrixType >::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
.
Definition at line 81 of file ComplexSchur.h.
typedef std::complex<RealScalar> Eigen::ComplexSchur< _MatrixType >::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.
Definition at line 74 of file ComplexSchur.h.
typedef Eigen::Index Eigen::ComplexSchur< _MatrixType >::Index |
Definition at line 66 of file ComplexSchur.h.
typedef _MatrixType Eigen::ComplexSchur< _MatrixType >::MatrixType |
Definition at line 54 of file ComplexSchur.h.
typedef NumTraits<Scalar>::Real Eigen::ComplexSchur< _MatrixType >::RealScalar |
Definition at line 65 of file ComplexSchur.h.
typedef MatrixType::Scalar Eigen::ComplexSchur< _MatrixType >::Scalar |
Scalar type for matrices of type _MatrixType
.
Definition at line 64 of file ComplexSchur.h.
anonymous enum |
Enumerator | |
---|---|
RowsAtCompileTime | |
ColsAtCompileTime | |
Options | |
MaxRowsAtCompileTime | |
MaxColsAtCompileTime |
Definition at line 55 of file ComplexSchur.h.
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inlineexplicit |
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.
Definition at line 94 of file ComplexSchur.h.
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inlineexplicit |
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.
Definition at line 113 of file ComplexSchur.h.
ComplexSchur<MatrixType>& Eigen::ComplexSchur< _MatrixType >::compute | ( | const EigenBase< InputType > & | matrix, |
bool | computeU | ||
) |
Definition at line 322 of file ComplexSchur.h.
ComplexSchur& Eigen::ComplexSchur< _MatrixType >::compute | ( | const EigenBase< InputType > & | 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:
ComplexSchur<MatrixType>& Eigen::ComplexSchur< _MatrixType >::computeFromHessenberg | ( | const HessMatrixType & | matrixH, |
const OrthMatrixType & | matrixQ, | ||
bool | computeU | ||
) |
Definition at line 344 of file ComplexSchur.h.
ComplexSchur& Eigen::ComplexSchur< _MatrixType >::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 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())
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private |
Compute the shift in the current QR iteration.
Definition at line 281 of file ComplexSchur.h.
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inline |
Returns the maximum number of iterations.
Definition at line 235 of file ComplexSchur.h.
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inline |
Reports whether previous computation was successful.
Success
if computation was successful, NoConvergence
otherwise. Definition at line 217 of file ComplexSchur.h.
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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:
Definition at line 162 of file ComplexSchur.h.
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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:
Definition at line 138 of file ComplexSchur.h.
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private |
Definition at line 390 of file ComplexSchur.h.
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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 228 of file ComplexSchur.h.
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inlineprivate |
If m_matT(i+1,i) is neglegible in floating point arithmetic compared to m_matT(i,i) and m_matT(j,j), then set it to zero and return true, else return false.
Definition at line 266 of file ComplexSchur.h.
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friend |
Definition at line 259 of file ComplexSchur.h.
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protected |
Definition at line 249 of file ComplexSchur.h.
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protected |
Definition at line 250 of file ComplexSchur.h.
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protected |
Definition at line 251 of file ComplexSchur.h.
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protected |
Definition at line 248 of file ComplexSchur.h.
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protected |
Definition at line 248 of file ComplexSchur.h.
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protected |
Definition at line 252 of file ComplexSchur.h.
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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.
Definition at line 245 of file ComplexSchur.h.
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protected |
Definition at line 253 of file ComplexSchur.h.