naginterfaces.library.lapackeig.dtgsen

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

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://support.nag.com/numeric/nl/nagdoc_30/flhtml/f08/f08ygf.html

Parameters

ijobint

Specifies whether condition numbers are required for the cluster of eigenvalues ($p$ and $q$) or the deflating subspaces (${Dif}_u$ and ${Dif}_l$).

$ijob=0$

Only reorder with respect to $select$. No extras.

$ijob=1$

Reciprocal of norms of ‘projections’ onto left and right eigenspaces with respect to the selected cluster ($p$ and $q$).

$ijob=2$

The upper bounds on ${Dif}_u$ and ${Dif}_l$. $F$-norm-based estimate ($dif[0:2]$).

$ijob=3$

Estimate of ${Dif}_u$ and ${Dif}_l$. $1$-norm-based estimate ($dif[0:2]$). About five times as expensive as $ijob=2$.

$ijob=4$

Compute $pl$, $pr$ and $dif$ as in $ijob=0$, $1$ or $2$. Economic version to get it all.

$ijob=5$

Compute $pl$, $pr$ and $dif$ as in $ijob=0$, $1$ or $3$.

wantqbool

If $wantq=True$, update the left transformation matrix $Q$.

If $wantq=False$, do not update $Q$.

wantzbool

If $wantz=True$, update the right transformation matrix $Z$.

If $wantz=False$, do not update $Z$.

selectbool, array-like, shape $\left(n\right)$

Specifies the eigenvalues in the selected cluster. To select a real eigenvalue ${\lambda }_j$, $select[j-1]$ must be set to $True$.

To select a complex conjugate pair of eigenvalues ${\lambda }_j$ and ${\lambda }_{j+1}$, corresponding to a $2\times 2$ diagonal block, either $select[j-1]$ or $select\left[j\right]$ or both must be set to $True$; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.

afloat, array-like, shape $(n,n)$

The matrix $S$ in the pair $(S,T)$.

bfloat, array-like, shape $(n,n)$

The matrix $T$, in the pair $(S,T)$.

qfloat, array-like, shape $(:,:)$

Note: the required extent for this argument in dimension 1 is determined as follows: if $wantq=True$: $n$; otherwise: $1$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $wantq=True$: $n$; otherwise: $1$.

If $wantq=True$, the $n\times n$ matrix $Q$.

zfloat, array-like, shape $(:,:)$

Note: the required extent for this argument in dimension 1 is determined as follows: if $wantz=True$: $n$; otherwise: $1$.

Note: the required extent for this argument in dimension 2 is determined as follows: if $wantz=True$: $n$; otherwise: $1$.

If $wantz=True$, the $n\times n$ matrix $Z$.

Returns

afloat, ndarray, shape $(n,n)$

The updated matrix $\hat{S}$.

bfloat, ndarray, shape $(n,n)$

The updated matrix $\hat{T}$

alpharfloat, ndarray, shape $\left(n\right)$

See the description of $beta$.

alphaifloat, ndarray, shape $\left(n\right)$

See the description of $beta$.

betafloat, ndarray, shape $\left(n\right)$

$alphar[\text{j}-1]/beta[\text{j}-1]$ and $alphai[\text{j}-1]/beta[\text{j}-1]$ are the real and imaginary parts respectively of the $\text{j}$th eigenvalue, for $\text{j}=1,2,\dots ,n$.

If $alphai[j-1]$ is zero, then the $j$th eigenvalue is real; if positive then $alphai\left[j\right]$ is negative, and the $j$th and $(j+1)$st eigenvalues are a complex conjugate pair.

Conjugate pairs of eigenvalues correspond to the $2\times 2$ diagonal blocks of $\hat{S}$.

These $2\times 2$ blocks can be reduced by applying complex unitary transformations to $(\hat{S},\hat{T})$ to obtain the complex Schur form $(\tilde{S},\tilde{T})$, where $\tilde{S}$ is triangular (and complex).

In this form $alphar+ialphai$ and $beta$ are the diagonals of $\tilde{S}$ and $\tilde{T}$ respectively.

qfloat, ndarray, shape $(:,:)$

If $wantq=True$, the updated matrix $Q\hat{Q}$.

If $wantq=False$, $q$ is not referenced.

zfloat, ndarray, shape $(:,:)$

If $wantz=True$, the updated matrix $Z\hat{Z}$.

If $wantz=False$, $z$ is not referenced.

mint

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

plfloat

If $ijob=1$, $4$ or $5$, $pl$ and $pr$ are lower bounds on the reciprocal of the norm of ‘projections’ $p$ and $q$ onto left and right eigenspaces with respect to the selected cluster. $0<pl$, $pr\le 1$.

If $m=0$ or $m=n$, $pl=pr=1$.

If $ijob=0$, $2$ or $3$, $pl$ and $pr$ are not referenced.

prfloat

If $ijob=1$, $4$ or $5$, $pl$ and $pr$ are lower bounds on the reciprocal of the norm of ‘projections’ $p$ and $q$ onto left and right eigenspaces with respect to the selected cluster. $0<pl$, $pr\le 1$.

If $m=0$ or $m=n$, $pl=pr=1$.

If $ijob=0$, $2$ or $3$, $pl$ and $pr$ are not referenced.

diffloat, ndarray, shape $\left(2\right)$

If $ijob\ge 2$, $dif[0:2]$ store the estimates of ${Dif}_u$ and ${Dif}_l$.

If $ijob=2$ or $4$, $dif[0:2]$ are $F$-norm-based upper bounds on ${Dif}_u$ and ${Dif}_l$.

If $ijob=3$ or $5$, $dif[0:2]$ are $1$-norm-based estimates of ${Dif}_u$ and ${Dif}_l$.

If $m=0$ or $n$, $dif[0:2]$ $={\parallel (A,B)\parallel }_F$.

If $ijob=0$ or $1$, $dif$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $ijob$.

Constraint: $0\le ijob\le 5$.

(errno $-5$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $1$)

Reordering of $(S,T)$ failed because the transformed matrix pair would be too far from generalized Schur form; the problem is very ill-conditioned. $(S,T)$ may have been partially reordered. If requested, $0$ is returned in $dif\left[0\right]$ and $dif\left[1\right]$, $pl$ and $pr$.

Notes

dtgsen factorizes the generalized real $n\times n$ matrix pair $(S,T)$ in real generalized Schur form, using an orthogonal equivalence transformation as

$$S=\hat{Q}\hat{S}{\hat{Z}}^T\text{,}T=\hat{Q}\hat{T}{\hat{Z}}^T\text{,}$$

where $(\hat{S},\hat{T})$ are also in real generalized Schur form and have the selected eigenvalues as the leading diagonal elements, or diagonal blocks. The leading columns of $Q$ and $Z$ 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 $(S,T)$.

The pair $(S,T)$ are in real generalized Schur form if $S$ is block upper triangular with $1\times 1$ and $2\times 2$ diagonal blocks and $T$ is upper triangular as returned, for example, by dgges3(), or dhgeqz() with $\text{job}=\text{'S'}$. The diagonal elements, or blocks, define the generalized eigenvalues $({\alpha }_{\text{i}},{\beta }_{\text{i}})$, for $\text{i}=1,2,\dots ,n$, of the pair $(S,T)$. The eigenvalues are given by

$${\lambda }_i={\alpha }_i/{\beta }_i\text{,}$$

but are returned as the pair $({\alpha }_i,{\beta }_i)$ in order to avoid possible overflow in computing ${\lambda }_i$. Optionally, the function returns reciprocals of condition number estimates for the selected eigenvalue cluster, $p$ and $q$, the right and left projection norms, and of deflating subspaces, ${Dif}_u$ and ${Dif}_l$. For more information see Sections 2.4.8 and 4.11 of Anderson et al. (1999).

If $S$ and $T$ are the result of a generalized Schur factorization of a matrix pair $(A,B)$

$$A=QS{Z}^T\text{,}B=QT{Z}^T$$

then, optionally, the matrices $Q$ and $Z$ can be updated as $Q\hat{Q}$ and $Z\hat{Z}$. Note that the condition numbers of the pair $(S,T)$ are the same as those of the pair $(A,B)$.

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