naginterfaces.library.lapackeig.dtgsen

naginterfaces.library.lapackeig.dtgsen(ijob, wantq, wantz, select, a, b, q, z)[source]

dtgsen reorders the generalized Schur factorization of a matrix pair in real generalized Schur form, so that a selected cluster of eigenvalues appears in the leading elements, or blocks 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 f08yg

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f08/f08ygf.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 a real eigenvalue , must be set to .

To select a complex conjugate pair of eigenvalues and , corresponding to a diagonal block, either or or both must be set to ; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.

afloat, array-like, shape

The matrix in the pair .

bfloat, array-like, shape

The matrix , in the pair .

qfloat, 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 .

zfloat, 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
afloat, ndarray, shape

The updated matrix .

bfloat, ndarray, shape

The updated matrix

alpharfloat, ndarray, shape

See the description of .

alphaifloat, ndarray, shape

See the description of .

betafloat, ndarray, shape

and are the real and imaginary parts respectively of the th eigenvalue, for .

If is zero, then the th eigenvalue is real; if positive then is negative, and the th and st eigenvalues are a complex conjugate pair.

Conjugate pairs of eigenvalues correspond to the diagonal blocks of .

These blocks can be reduced by applying complex unitary transformations to to obtain the complex Schur form , where is triangular (and complex).

In this form and are the diagonals of and respectively.

qfloat, ndarray, shape

If , the updated matrix .

If , is not referenced.

zfloat, ndarray, shape

If , the updated matrix .

If , is not referenced.

mint

The dimension of the specified pair of left and right eigenspaces (deflating subspaces).

plfloat

If , or , and are lower bounds on the reciprocal of the norm of ‘projections’ and onto left and right eigenspaces 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 eigenspaces 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

dtgsen factorizes the generalized real matrix pair in real generalized Schur form, using an orthogonal equivalence transformation as

where are also in real generalized Schur form and have the selected eigenvalues as the leading diagonal elements, or diagonal blocks. 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 real generalized Schur form if is block upper triangular with and diagonal blocks and is upper triangular as returned, for example, by dgges3(), or dhgeqz() with . The diagonal elements, or blocks, define the generalized eigenvalues , for , of the pair . The eigenvalues are given by

but 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