Index appearing in the upper left corner.

See also schur, ordeig, ordqz.

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<ordqz> - Reordering of eigenvalues in QZ factorization.

ORDQZ Reorder eigenvalues in QZ factorization.

[AAS,BBS,QS,ZS] = ORDQZ(AA,BB,Q,Z,SELECT) reorders the QZ factorization

Q*A*Z = AA, Q*B*Z = BB of a matrix pair (A,B) so that a selected cluster

of eigenvalues appears in the leading (upper left) diagonal blocks of the

quasitriangular pair (AA,BB), and the corresponding invariant subspace is

spanned by the leading columns of Z. The logical vector SELECT specifies

the selected cluster as E(SELECT) where E is the vector of eigenvalues

as they appear along the diagonal of AA-t*BB. Use E = ORDEIG(AA,BB) to

extract E from AA and BB.

ORDQZ takes the matrices AA,BB,Q,Z produced by the QZ command and

returns the reordered pair (AAS,BBS) and the cumulative orthogonal

transformations QS and ZS such that QS*A*ZS = AAS, QS*B*ZS = BBS.

Set Q=[] or Z=[] to get the incremental QS,ZS transforming (AA,BB)

into (AAS,BBS).

[AAS,BBS,...] = ORDQZ(AA,BB,Q,Z,KEYWORD) sets the selected cluster to

include all eigenvalues in one of the following regions:

KEYWORD Selected Region

'lhp' left-half plane (real(E)<0)

'rhp' right-half plane (real(E)>0)

'udi' interior of unit disk (abs(E)<1)

'udo' exterior of unit disk (abs(E)>1)

ORDQZ can also reorder multiple clusters at once. Given a vector

CLUSTERS of cluster indices, commensurate with E = EIG(AA,BB), and

such that all eigenvalues with same CLUSTERS value form one cluster,

[...] = ORDQZ(AA,BB,Q,Z,CLUSTERS) will sort the specified clusters

in descending order along the diagonal of (AAS,BBS), the cluster with

highest index appearing in the upper left corner.

See also qz, ordeig, ordschur.

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<ordeig> - Eigenvalues of quasitriangular matrices.

ORDEIG Eigenvalues of quasitriangular matrices.

E = ORDEIG(T) takes a quasitriangular Schur matrix T (typically

produced by SCHUR) and returns the vector E of eigenvalues in

their order of appearance down the diagonal of T.

E = ORDEIG(AA,BB) takes a quasitriangular matrix pair AA, BB

(typically produced by QZ) and returns the generalized eigenvalues

In their order of appearance down the diagonal of aa-t*bb.

ORDEIG is an order-preserving version of EIG for use with ORDSCHUR

and ORDQZ. It is also faster than EIG for quasitriangular matrices.

Example:

a = rand(10);

[u,t] = schur(a);

e = ordeig(t)

% Move eigenvalues with magnitude < 0.5 to the

% upper-left corner of T

[u,t] = ordschur(u,t,abs(e)<0.5);

abs(ordeig(t))

See also ordschur, ordqz, schur, qz, eig.

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