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Eigen
3.2.5
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Performs a real QZ decomposition of a pair of square matrices. More...
Public Member Functions | |
RealQZ & | compute (const MatrixType &A, const MatrixType &B, bool computeQZ=true) |
Computes QZ decomposition of given matrix. | |
ComputationInfo | info () const |
Reports whether previous computation was successful. | |
Index | iterations () const |
Returns number of performed QR-like iterations. | |
const MatrixType & | matrixQ () const |
Returns matrix Q in the QZ decomposition. | |
const MatrixType & | matrixS () const |
Returns matrix S in the QZ decomposition. | |
const MatrixType & | matrixT () const |
Returns matrix S in the QZ decomposition. | |
const MatrixType & | matrixZ () const |
Returns matrix Z in the QZ decomposition. | |
RealQZ (const MatrixType &A, const MatrixType &B, bool computeQZ=true) | |
Constructor; computes real QZ decomposition of given matrices. | |
RealQZ (Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime) | |
Default constructor. | |
RealQZ & | setMaxIterations (Index maxIters) |
Performs a real QZ decomposition of a pair of square matrices.
This is defined in the Eigenvalues module.
#include <Eigen/Eigenvalues>
_MatrixType | the type of the matrix of which we are computing the real QZ decomposition; this is expected to be an instantiation of the Matrix class template. |
Given a real square matrices A and B, this class computes the real QZ decomposition: ,
where Q and Z are real orthogonal matrixes, T is upper-triangular matrix, and S is upper quasi-triangular matrix. An orthogonal matrix is a matrix whose inverse is equal to its transpose,
. A quasi-triangular matrix is a block-triangular matrix whose diagonal consists of 1-by-1 blocks and 2-by-2 blocks where further reduction is impossible due to complex eigenvalues.
The eigenvalues of the pencil can be obtained from 1x1 and 2x2 blocks on the diagonals of S and T.
Call the function compute() to compute the real QZ decomposition of a given pair of matrices. Alternatively, you can use the RealQZ(const MatrixType& B, const MatrixType& B, bool computeQZ) constructor which computes the real QZ decomposition at construction time. Once the decomposition is computed, you can use the matrixS(), matrixT(), matrixQ() and matrixZ() functions to retrieve the matrices S, T, Q and Z in the decomposition. If computeQZ==false, some time is saved by not computing matrices Q and Z.
Example:
MatrixXf A = MatrixXf::Random(4,4); MatrixXf B = MatrixXf::Random(4,4); RealQZ<MatrixXf> qz(4); // preallocate space for 4x4 matrices qz.compute(A,B); // A = Q S Z, B = Q T Z // print original matrices and result of decomposition cout << "A:\n" << A << "\n" << "B:\n" << B << "\n"; cout << "S:\n" << qz.matrixS() << "\n" << "T:\n" << qz.matrixT() << "\n"; cout << "Q:\n" << qz.matrixQ() << "\n" << "Z:\n" << qz.matrixZ() << "\n"; // verify precision cout << "\nErrors:" << "\n|A-QSZ|: " << (A-qz.matrixQ()*qz.matrixS()*qz.matrixZ()).norm() << ", |B-QTZ|: " << (B-qz.matrixQ()*qz.matrixT()*qz.matrixZ()).norm() << "\n|QQ* - I|: " << (qz.matrixQ()*qz.matrixQ().adjoint() - MatrixXf::Identity(4,4)).norm() << ", |ZZ* - I|: " << (qz.matrixZ()*qz.matrixZ().adjoint() - MatrixXf::Identity(4,4)).norm() << "\n";
Output:
A: 0.68 0.823 -0.444 -0.27 -0.211 -0.605 0.108 0.0268 0.566 -0.33 -0.0452 0.904 0.597 0.536 0.258 0.832 B: 0.271 -0.967 -0.687 0.998 0.435 -0.514 -0.198 -0.563 -0.717 -0.726 -0.74 0.0259 0.214 0.608 -0.782 0.678 S: 0.927 -0.928 0.643 -0.227 -0.594 0.36 0.146 -0.606 0 0 -0.398 -0.164 0 0 0 -1.12 T: 1.51 0.278 -0.238 0.501 0 -1.04 0.519 -0.239 0 0 -1.25 0.438 0 0 0 0.746 Q: 0.603 0.011 0.552 0.576 -0.142 0.243 0.761 -0.585 0.092 -0.958 0.152 -0.223 0.78 0.149 -0.306 -0.526 Z: 0.284 0.26 -0.696 0.606 -0.918 -0.108 -0.38 0.0406 -0.269 0.783 0.462 0.32 -0.0674 -0.555 0.398 0.727 Errors: |A-QSZ|: 8.79e-07, |B-QTZ|: 9.63e-07 |QQ* - I|: 6.22e-07, |ZZ* - I|: 4.22e-07
RealQZ | ( | Index | size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime |
) | [inline] |
Default constructor.
[in] | size | Positive integer, size of the matrix whose QZ 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.
RealQZ | ( | const MatrixType & | A, | |
const MatrixType & | B, | |||
bool | computeQZ = true | |||
) | [inline] |
RealQZ< MatrixType > & compute | ( | const MatrixType & | A, | |
const MatrixType & | B, | |||
bool | computeQZ = true | |||
) | [inline] |
Computes QZ decomposition of given matrix.
*this
References Eigen::NoConvergence, PlainObjectBase< Derived >::resize(), and Eigen::Success.
Referenced by GeneralizedEigenSolver< _MatrixType >::compute(), and RealQZ< MatrixType >::RealQZ().
ComputationInfo info | ( | ) | const [inline] |
Reports whether previous computation was successful.
Success
if computation was succesful, NoConvergence
otherwise. Referenced by GeneralizedEigenSolver< _MatrixType >::compute().
const MatrixType& matrixQ | ( | ) | const [inline] |
Returns matrix Q in the QZ decomposition.
const MatrixType& matrixS | ( | ) | const [inline] |
Returns matrix S in the QZ decomposition.
Referenced by GeneralizedEigenSolver< _MatrixType >::compute().
const MatrixType& matrixT | ( | ) | const [inline] |
Returns matrix S in the QZ decomposition.
Referenced by GeneralizedEigenSolver< _MatrixType >::compute().
const MatrixType& matrixZ | ( | ) | const [inline] |
Returns matrix Z in the QZ decomposition.
Referenced by GeneralizedEigenSolver< _MatrixType >::compute().
RealQZ& setMaxIterations | ( | Index | maxIters | ) | [inline] |
Sets the maximal number of iterations allowed to converge to one eigenvalue or decouple the problem.
Referenced by GeneralizedEigenSolver< _MatrixType >::setMaxIterations().