naginterfaces.library.lapackeig.dgeesx¶

naginterfaces.library.lapackeig.dgeesx(jobvs, sort, sense, a, select=None, data=None)[source]

dgeesx computes the eigenvalues, the real Schur form , and, optionally, the matrix of Schur vectors for an real nonsymmetric matrix .

For full information please refer to the NAG Library document for f08pb

https://www.nag.com/numeric/nl/nagdoc_28.7/flhtml/f08/f08pbf.html

Parameters
jobvsstr, length 1

If , Schur vectors are not computed.

If , Schur vectors are computed.

sortstr, length 1

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

Eigenvalues are not ordered.

Eigenvalues are ordered (see ).

sensestr, length 1

Determines which reciprocal condition numbers are computed.

None are computed.

Computed for average of selected eigenvalues only.

Computed for selected right invariant subspace only.

Computed for both.

If , or , .

afloat, array-like, shape

The matrix .

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

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

If , is used to select eigenvalues to sort to the top left of the Schur form.

An eigenvalue is selected if is .

If either one of a complex conjugate pair of eigenvalues is selected, then both are.

Note that a selected complex eigenvalue may no longer satisfy after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case is set to .

Parameters
wrfloat

The real and imaginary parts of the eigenvalue.

wifloat

The real and imaginary parts of the eigenvalue.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns
retvalbool

Must be if the eigenvalue is to be selected.

dataarbitrary, optional

User-communication data for callback functions.

Returns
afloat, ndarray, shape

is overwritten by its real Schur form .

sdimint

If , .

If , number of eigenvalues (after sorting) for which is . (Complex conjugate pairs for which is for either eigenvalue count as .)

wrfloat, ndarray, shape

See the description of .

wifloat, ndarray, shape

and contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form . Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.

vsfloat, ndarray, shape

If , contains the orthogonal matrix of Schur vectors.

If , is not referenced.

rcondefloat

If or , contains the reciprocal condition number for the average of the selected eigenvalues.

If or , is not referenced.

rcondvfloat

If or , contains the reciprocal condition number for the selected right invariant subspace.

If or , is not referenced.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: or .

(errno )

On entry, error in parameter .

Constraint: , , or .

(errno )

On entry, error in parameter .

Constraint: .

(errno )

The algorithm failed to compute all the eigenvalues.

Warns
NagAlgorithmicWarning
(errno )

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

(errno )

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

Notes

The real Schur factorization of is given by

where , the matrix of Schur vectors, is orthogonal and is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with and blocks. blocks will be standardized in the form

where . The eigenvalues of such a block are .

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

For further explanation of the reciprocal condition numbers and , see Section 4.8 of Anderson et al. (1999) (where these quantities are called and 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