naginterfaces.library.lapackeig.ztgsen¶
- naginterfaces.library.lapackeig.ztgsen(ijob, wantq, wantz, select, a, b, q, z)[source]¶
ztgsen
reorders the generalized Schur factorization of a complex matrix pair in generalized Schur form, so that a selected cluster of eigenvalues appears in the leading elements on the diagonal of the generalized Schur form. The function also, optionally, computes the reciprocal condition numbers of the cluster of eigenvalues and/or corresponding deflating subspaces.For full information please refer to the NAG Library document for f08yu
https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08yuf.html
- Parameters
- ijobint
Specifies whether condition numbers are required for the cluster of eigenvalues ( and ) or the deflating subspaces ( and ).
Only reorder with respect to . No extras.
Reciprocal of norms of ‘projections’ onto left and right eigenspaces with respect to the selected cluster ( and ).
The upper bounds on and . -norm-based estimate ().
Estimate of and . -norm-based estimate (). About five times as expensive as .
Compute , and as in , or . Economic version to get it all.
Compute , and as in , or .
- wantqbool
If , update the left transformation matrix .
If , do not update .
- wantzbool
If , update the right transformation matrix .
If , do not update .
- selectbool, array-like, shape
Specifies the eigenvalues in the selected cluster. To select an eigenvalue , must be set to .
- acomplex, array-like, shape
The matrix in the pair .
- bcomplex, array-like, shape
The matrix , in the pair .
- qcomplex, array-like, shape
Note: the required extent for this argument in dimension 1 is determined as follows: if : ; otherwise: .
Note: the required extent for this argument in dimension 2 is determined as follows: if : ; otherwise: .
If , the matrix .
- zcomplex, array-like, shape
Note: the required extent for this argument in dimension 1 is determined as follows: if : ; otherwise: .
Note: the required extent for this argument in dimension 2 is determined as follows: if : ; otherwise: .
If , the matrix .
- Returns
- acomplex, ndarray, shape
The updated matrix .
- bcomplex, ndarray, shape
The updated matrix
- alphacomplex, ndarray, shape
and contain diagonal elements of and , respectively, when the pair has been reduced to generalized Schur form. , for , are the eigenvalues.
- betacomplex, ndarray, shape
and contain diagonal elements of and , respectively, when the pair has been reduced to generalized Schur form. , for , are the eigenvalues.
- qcomplex, ndarray, shape
If , the updated matrix .
If , is not referenced.
- zcomplex, ndarray, shape
If , the updated matrix .
If , is not referenced.
- mint
The dimension of the specified pair of left and right eigenspaces (deflating subspaces), where will be in the range .
- plfloat
If , or , and are lower bounds on the reciprocal of the norm of ‘projections’ and onto left and right eigenspace with respect to the selected cluster. , .
If or , .
If , or , and are not referenced.
- prfloat
If , or , and are lower bounds on the reciprocal of the norm of ‘projections’ and onto left and right eigenspace with respect to the selected cluster. , .
If or , .
If , or , and are not referenced.
- diffloat, ndarray, shape
If , store the estimates of and .
If or , are -norm-based upper bounds on and .
If or , are -norm-based estimates of and .
If or , .
If or , is not referenced.
- Raises
- NagValueError
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
On entry, error in parameter .
Constraint: .
- (errno )
Reordering of failed because the transformed matrix pair would be too far from generalized Schur form; the problem is very ill-conditioned. may have been partially reordered. If requested, is returned in and , and .
- Notes
ztgsen
factorizes the generalized complex matrix pair in generalized Schur form, using a unitary equivalence transformation aswhere are also in generalized Schur form and have the selected eigenvalues as the leading diagonal elements. The leading columns of and are the generalized Schur vectors corresponding to the selected eigenvalues and form orthonormal subspaces for the left and right eigenspaces (deflating subspaces) of the pair .
The pair are in generalized Schur form if and are upper triangular as returned, for example, by
zgges3()
, orzhgeqz()
with . The diagonal elements define the generalized eigenvalues , for , of the pair . The eigenvalues are given bybut are returned as the pair in order to avoid possible overflow in computing . Optionally, the function returns reciprocals of condition number estimates for the selected eigenvalue cluster, and , the right and left projection norms, and of deflating subspaces, and . For more information see Sections 2.4.8 and 4.11 of Anderson et al. (1999).
If and are the result of a generalized Schur factorization of a matrix pair
then, optionally, the matrices and can be updated as and . Note that the condition numbers of the pair are the same as those of the pair .
- References
Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug