template<typename MatrixType_>
class Eigen::RealSchur< MatrixType_ >
Performs a real Schur decomposition of a square matrix.
This is defined in the Eigenvalues module.
- Template Parameters
-
| MatrixType_ | the 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
Definition at line 56 of file RealSchur.h.
template<typename MatrixType_ >
template<typename InputType >
Constructor; computes real 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.
Example:
cout <<
"Here is a random 6x6 matrix, A:" << endl <<
A << endl << endl;
cout <<
"The orthogonal matrix U is:" << endl <<
schur.matrixU() << endl;
cout <<
"The quasi-triangular matrix T is:" << endl <<
schur.matrixT() << endl << endl;
cout <<
"U * T * U^T = " << endl << U *
T * U.transpose() << endl;
ComplexSchur< MatrixXcf > schur(4)
static const RandomReturnType Random()
Matrix< double, Dynamic, Dynamic > MatrixXd
Dynamic×Dynamic matrix of type double.
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 0.0146 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 0.139
0.647 0.298 -0.0536 0.676 -0.288 0.0231
0 0 0.967 -0.201 -0.429 0.847
0 0 0 0.353 0.603 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
Definition at line 106 of file RealSchur.h.
107 :
m_matT(matrix.rows(),matrix.cols()),
108 m_matU(matrix.rows(),matrix.cols()),
115 compute(matrix.derived(), computeU);
RealSchur & compute(const EigenBase< InputType > &matrix, bool computeU=true)
Computes Schur decomposition of given matrix.
template<typename MatrixType_ >
template<typename InputType >
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 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:
RealSchur<MatrixXf>
schur(4);
cout <<
"The matrix T in the decomposition of A is:" << endl <<
schur.matrixT() << endl;
schur.compute(
A.inverse(),
false);
cout <<
"The matrix T in the decomposition of A^(-1) is:" << endl <<
schur.matrixT() << endl;
Matrix< float, Dynamic, Dynamic > MatrixXf
Dynamic×Dynamic matrix of type float.
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 -5.97 5.48
0.168 -2.62 -3.27 3.9
0 0 0.427 0.573
0 0 -1.05 1.35
- See also
- compute(const MatrixType&, bool, Index)
template<typename MatrixType_ >
template<typename HessMatrixType , typename OrthMatrixType >
| RealSchur& Eigen::RealSchur< MatrixType_ >::computeFromHessenberg |
( |
const HessMatrixType & |
matrixH, |
|
|
const OrthMatrixType & |
matrixQ, |
|
|
bool |
computeU |
|
) |
| |
Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T.
- 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)