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_30.2/flhtml/f08/f08yuf.html

Parameters

ijobint

Specifies whether condition numbers are required for the cluster of eigenvalues ( $p$ and $q$ ) or the deflating subspaces ( ${D i f}_{u}$ and ${D i f}_{l}$ ).

$i j o b = 0$

Only reorder with respect to $s e l e c t$ . No extras.

$i j o b = 1$

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

$i j o b = 2$

The upper bounds on ${D i f}_{u}$ and ${D i f}_{l}$ . $F$ -norm-based estimate ( $d i f [0 : 2]$ ).

$i j o b = 3$

Estimate of ${D i f}_{u}$ and ${D i f}_{l}$ . $1$ -norm-based estimate ( $d i f [0 : 2]$ ). About five times as expensive as $i j o b = 2$ .

$i j o b = 4$

Compute $p l$ , $p r$ and $d i f$ as in $i j o b = 0$ , $1$ or $2$ . Economic version to get it all.

$i j o b = 5$

Compute $p l$ , $p r$ and $d i f$ as in $i j o b = 0$ , $1$ or $3$ .

wantqbool

If $w a n t q = T r u e$ , update the left transformation matrix $Q$ .

If $w a n t q = F a l s e$ , do not update $Q$ .

wantzbool

If $w a n t z = T r u e$ , update the right transformation matrix $Z$ .

If $w a n t z = F a l s e$ , do not update $Z$ .

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

Specifies the eigenvalues in the selected cluster. To select an eigenvalue $λ_{j}$ , $s e l e c t [j - 1]$ must be set to $T r u e$ .

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

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

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

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

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

Note: the required extent for this argument in dimension 1 is determined as follows: if $w a n t q = T r u e$ : $n$ ; otherwise: $1$ .

Note: the required extent for this argument in dimension 2 is determined as follows: if $w a n t q = T r u e$ : $n$ ; otherwise: $1$ .

If $w a n t q = T r u e$ , the $n \times n$ matrix $Q$ .

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

Note: the required extent for this argument in dimension 1 is determined as follows: if $w a n t z = T r u e$ : $n$ ; otherwise: $1$ .

Note: the required extent for this argument in dimension 2 is determined as follows: if $w a n t z = T r u e$ : $n$ ; otherwise: $1$ .

If $w a n t z = T r u e$ , the $n \times n$ matrix $Z$ .

Returns

acomplex, ndarray, shape $(n, n)$

The updated matrix $^S$ .

bcomplex, ndarray, shape $(n, n)$

The updated matrix $^T$

alphacomplex, ndarray, shape $(n)$

$a l p h a$ and $b e t a$ contain diagonal elements of $^S$ and $^T$ , respectively, when the pair $(S, T)$ has been reduced to generalized Schur form. $a l p h a [i - 1] / b e t a [i - 1]$ , for $i = 1, 2, \dots, n$ , are the eigenvalues.

betacomplex, ndarray, shape $(n)$

$a l p h a$ and $b e t a$ contain diagonal elements of $^S$ and $^T$ , respectively, when the pair $(S, T)$ has been reduced to generalized Schur form. $a l p h a [i - 1] / b e t a [i - 1]$ , for $i = 1, 2, \dots, n$ , are the eigenvalues.

qcomplex, ndarray, shape $(:, :)$

If $w a n t q = T r u e$ , the updated matrix $Q^Q$ .

If $w a n t q = F a l s e$ , $q$ is not referenced.

zcomplex, ndarray, shape $(:, :)$

If $w a n t z = T r u e$ , the updated matrix $Z^Z$ .

If $w a n t z = F a l s e$ , $z$ is not referenced.

mint

The dimension of the specified pair of left and right eigenspaces (deflating subspaces), where $m$ will be in the range $0 \leq m \leq n$ .

plfloat

If $i j o b = 1$ , $4$ or $5$ , $p l$ and $p r$ are lower bounds on the reciprocal of the norm of ‘projections’ $p$ and $q$ onto left and right eigenspace with respect to the selected cluster. $0 < p l$ , $p r \leq 1$ .

If $m = 0$ or $m = n$ , $p l = p r = 1$ .

If $i j o b = 0$ , $2$ or $3$ , $p l$ and $p r$ are not referenced.

prfloat

If $i j o b = 1$ , $4$ or $5$ , $p l$ and $p r$ are lower bounds on the reciprocal of the norm of ‘projections’ $p$ and $q$ onto left and right eigenspace with respect to the selected cluster. $0 < p l$ , $p r \leq 1$ .

If $m = 0$ or $m = n$ , $p l = p r = 1$ .

If $i j o b = 0$ , $2$ or $3$ , $p l$ and $p r$ are not referenced.

diffloat, ndarray, shape $(2)$

If $i j o b \geq 2$ , $d i f [0 : 2]$ store the estimates of ${D i f}_{u}$ and ${D i f}_{l}$ .

If $i j o b = 2$ or $4$ , $d i f [0 : 2]$ are $F$ -norm-based upper bounds on ${D i f}_{u}$ and ${D i f}_{l}$ .

If $i j o b = 3$ or $5$ , $d i f [0 : 2]$ are $1$ -norm-based estimates of ${D i f}_{u}$ and ${D i f}_{l}$ .

If $m = 0$ or $n$ , $d i f [0 : 2]$ $= {∥ (A, B) ∥}_{F}$ .

If $i j o b = 0$ or $1$ , $d i f$ is not referenced.

Raises

NagValueError

(errno $- 1$ )

On entry, error in parameter $i j o b$ .

Constraint: $0 \leq i j o b \leq 5$ .

(errno $- 5$ )

On entry, error in parameter $n$ .

Constraint: $n \geq 0$ .

(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 $d i f [0]$ and $d i f [1]$ , $p l$ and $p r$ .

Notes

ztgsen factorizes the generalized complex $n \times n$ matrix pair $(S, T)$ in generalized Schur form, using a unitary equivalence transformation as

S =^Q^S {^Z}^{H}, T =^Q^T {^Z}^{H},

where $(^S,^T)$ are also in generalized Schur form and have the selected eigenvalues as the leading diagonal elements. 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 generalized Schur form if $S$ and $T$ are upper triangular as returned, for example, by zgges3(), or zhgeqz() with $job ='S'$ . The diagonal elements define the generalized eigenvalues $(α_{i}, β_{i})$ , for $i = 1, 2, \dots, n$ , of the pair $(S, T)$ . The eigenvalues are given by

λ_{i} = α_{i} / β_{i},

but are returned as the pair $(α_{i}, β_{i})$ in order to avoid possible overflow in computing $λ_{i}$ . 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, ${D i f}_{u}$ and ${D i f}_{l}$ . 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 = Q S Z^{H}, B = Q T Z^{H}

then, optionally, the matrices $Q$ and $Z$ can be updated as $Q^Q$ and $Z^Z$ . 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

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naginterfaces.library.lapackeig.ztgsen¶