naginterfaces.library.lapackeig.ztrsen¶
- naginterfaces.library.lapackeig.ztrsen(job, compq, select, t, q)[source]¶
ztrsen
reorders the Schur factorization of a complex general matrix so that a selected cluster of eigenvalues appears in the leading elements on the diagonal of the Schur form. The function also optionally computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.For full information please refer to the NAG Library document for f08qu
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f08/f08quf.html
- Parameters
- jobstr, length 1
Indicates whether condition numbers are required for the cluster of eigenvalues and/or the invariant subspace.
No condition numbers are required.
Only the condition number for the cluster of eigenvalues is computed.
Only the condition number for the invariant subspace is computed.
Condition numbers for both the cluster of eigenvalues and the invariant subspace are computed.
- compqstr, length 1
Indicates whether the matrix of Schur vectors is to be updated.
The matrix of Schur vectors is updated.
No Schur vectors are updated.
- selectbool, array-like, shape
Specifies the eigenvalues in the selected cluster. To select a complex eigenvalue , must be set .
- tcomplex, array-like, shape
The upper triangular matrix , as returned by
zhseqr()
.- qcomplex, array-like, shape
Note: the required extent for this argument in dimension 1 is determined as follows: if : ; if : ; otherwise: .
Note: the required extent for this argument in dimension 2 is determined as follows: if : ; if : ; otherwise: .
If , must contain the unitary matrix of Schur vectors, as returned by
zhseqr()
.
- Returns
- tcomplex, ndarray, shape
is overwritten by the updated matrix .
- qcomplex, ndarray, shape
If , contains the updated matrix of Schur vectors; the first columns of form an orthonormal basis for the specified invariant subspace.
If , is not referenced.
- wcomplex, ndarray, shape
The reordered eigenvalues of . The eigenvalues are stored in the same order as on the diagonal of .
- mint
, the dimension of the specified invariant subspace, which is the same as the number of selected eigenvalues (see ); .
- sfloat
If or , is a lower bound on the reciprocal condition number of the average of the selected cluster of eigenvalues. If or , .
If or , is not referenced.
- sepfloat
If or , is the estimated reciprocal condition number of the specified invariant subspace. If or , .
If or , is not referenced.
- Raises
- NagValueError
- (errno )
On entry, error in parameter .
Constraint: , , or .
- (errno )
On entry, error in parameter .
Constraint: or .
- (errno )
On entry, error in parameter .
Constraint: .
- Notes
ztrsen
reorders the Schur factorization of a complex general matrix , so that a selected cluster of eigenvalues appears in the leading diagonal elements of the Schur form.The reordered Schur form is computed by a unitary similarity transformation: . Optionally the updated matrix of Schur vectors is computed as , giving .
Let , where the selected eigenvalues are precisely the eigenvalues of the leading sub-matrix . Let be correspondingly partitioned as where consists of the first columns of . Then , and so the columns of form an orthonormal basis for the invariant subspace corresponding to the selected cluster of eigenvalues.
Optionally the function also computes estimates of the reciprocal condition numbers of the average of the cluster of eigenvalues and of the invariant subspace.
- References
Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore