naginterfaces.library.lapackeig.zgeesx

naginterfaces.library.lapackeig.zgeesx(jobvs, sort, sense, a, select=None, data=None)

zgeesx computes the eigenvalues, the Schur form $T$, and, optionally, the matrix of Schur vectors $Z$ for an $n\times n$ complex nonsymmetric matrix $A$.

For full information please refer to the NAG Library document for f08pp

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f08/f08ppf.html

Parameters

jobvsstr, length 1

If $jobvs=\text{'N'}$, Schur vectors are not computed.

If $jobvs=\text{'V'}$, Schur vectors are computed.

sortstr, length 1

Specifies whether or not to order the eigenvalues on the diagonal of the Schur form.

$sort=\text{'N'}$

Eigenvalues are not ordered.

$sort=\text{'S'}$

Eigenvalues are ordered (see $select$).

sensestr, length 1

Determines which reciprocal condition numbers are computed.

$sense=\text{'N'}$

None are computed.

$sense=\text{'E'}$

Computed for average of selected eigenvalues only.

$sense=\text{'V'}$

Computed for selected right invariant subspace only.

$sense=\text{'B'}$

Computed for both.

If $sense=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$, $sort=\text{'S'}$.

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

The $n\times n$ matrix $A$.

selectNone or callable retval = select(w, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

If $sort=\text{'S'}$, $select$ is used to select eigenvalues to sort to the top left of the Schur form.

An eigenvalue $w[j-1]$ is selected if $select\left(w\right[j-1\left]\right)$ is $True$.

Parameters

wcomplex

The real and imaginary parts of the eigenvalue.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalbool

Must be $True$ if the eigenvalue is to be selected.

dataarbitrary, optional

User-communication data for callback functions.

Returns

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

$a$ is overwritten by its Schur form $T$.

sdimint

If $sort=\text{'N'}$, $sdim=0$.

If $sort=\text{'S'}$, $sdim=$ number of eigenvalues for which $select$ is $True$.

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

Contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form $T$.

vscomplex, ndarray, shape $(:,:)$

If $jobvs=\text{'V'}$, $vs$ contains the unitary matrix $Z$ of Schur vectors.

If $jobvs=\text{'N'}$, $vs$ is not referenced.

rcondefloat

If $sense=\text{'E'}$ or $\text{'B'}$, contains the reciprocal condition number for the average of the selected eigenvalues.

If $sense=\text{'N'}$ or $\text{'V'}$, $rconde$ is not referenced.

rcondvfloat

If $sense=\text{'V'}$ or $\text{'B'}$, $rcondv$ contains the reciprocal condition number for the selected right invariant subspace.

If $sense=\text{'N'}$ or $\text{'E'}$, $rcondv$ is not referenced.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $jobvs$.

Constraint: $jobvs=\text{'N'}$ or $\text{'V'}$.

(errno $-2$)

On entry, error in parameter $sort$.

Constraint: $sort=\text{'N'}$ or $\text{'S'}$.

(errno $-4$)

On entry, error in parameter $sense$.

Constraint: $sense=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.

(errno $-5$)

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

Constraint: $n\ge 0$.

(errno $(i>0)\text{and}(i\le n)$)

The $QR$ algorithm failed to compute all the eigenvalues.

Warns

NagAlgorithmicWarning

(errno $n+1$)

The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).

(errno $n+2$)

After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy $select=True$. This could also be caused by underflow due to scaling.

Notes

The Schur factorization of $A$ is given by

$$A=ZT{Z}^H\text{,}$$

where $Z$, the matrix of Schur vectors, is unitary and $T$ is the Schur form. A complex matrix is in Schur form if it is upper triangular.

Optionally, zgeesx also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues ($rconde$); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues ($rcondv$). The leading columns of $Z$ form an orthonormal basis for this invariant subspace.

For further explanation of the reciprocal condition numbers $rconde$ and $rcondv$, see Section 4.8 of Anderson et al. (1999) (where these quantities are called $s$ and $\text{sep}$ respectively).

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

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore