
ZTGSEN(l) ) ZTGSEN(l)
NAME
ZTGSEN  reorder the generalized Schur decomposition of a complex matrix pair (A, B) (in
terms of an unitary equivalence trans formation Q' * (A, B) * Z), so that a selected
cluster of eigenvalues appears in the leading diagonal blocks of the pair (A,B)
SYNOPSIS
SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z,
LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO )
LOGICAL WANTQ, WANTZ
INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, M, N
DOUBLE PRECISION PL, PR
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION DIF( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), WORK( *
), Z( LDZ, * )
PURPOSE
ZTGSEN reorders the generalized Schur decomposition of a complex matrix pair (A, B) (in
terms of an unitary equivalence trans formation Q' * (A, B) * Z), so that a selected
cluster of eigenvalues appears in the leading diagonal blocks of the pair (A,B). The lead
ing columns of Q and Z form unitary bases of the corresponding left and right eigenspaces
(deflating subspaces). (A, B) must be in generalized Schur canonical form, that is, A and
B are both upper triangular.
ZTGSEN also computes the generalized eigenvalues
w(j)= ALPHA(j) / BETA(j)
of the reordered matrix pair (A, B).
Optionally, the routine computes estimates of reciprocal condition numbers for eigenvalues
and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e.
the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the
selected cluster and the eigenvalues outside the cluster, resp., and norms of "projec
tions" onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)block.
ARGUMENTS
IJOB (input) integer
Specifies whether condition numbers are required for the cluster of eigenvalues
(PL and PR) or the deflating subspaces (Difu and Difl):
=0: Only reorder w.r.t. SELECT. No extras.
=1: Reciprocal of norms of "projections" onto left and right eigenspaces w.r.t.
the selected cluster (PL and PR). =2: Upper bounds on Difu and Difl. Fnormbased
estimate
(DIF(1:2)).
=3: Estimate of Difu and Difl. 1normbased estimate
(DIF(1:2)). About 5 times as expensive as IJOB = 2. =4: Compute PL, PR and DIF
(i.e. 0, 1 and 2 above): Economic version to get it all. =5: Compute PL, PR and
DIF (i.e. 0, 1 and 3 above)
WANTQ (input) LOGICAL
WANTZ (input) LOGICAL
SELECT (input) LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To select an eigenvalue
w(j), SELECT(j) must be set to
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input/output) COMPLEX*16 array, dimension(LDA,N)
On entry, the upper triangular matrix A, in generalized Schur canonical form. On
exit, A is overwritten by the reordered matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) COMPLEX*16 array, dimension(LDB,N)
On entry, the upper triangular matrix B, in generalized Schur canonical form. On
exit, B is overwritten by the reordered matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N) The diagonal elements of A and B,
respectively, when the pair (A,B) has been reduced to generalized Schur form.
ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues.
Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
On entry, if WANTQ = .TRUE., Q is an NbyN matrix. On exit, Q has been postmul
tiplied by the left unitary transformation matrix which reorder (A, B); The lead
ing M columns of Q form orthonormal bases for the specified pair of left
eigenspaces (deflating subspaces). If WANTQ = .FALSE., Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1. If WANTQ = .TRUE., LDQ >= N.
Z (input/output) COMPLEX*16 array, dimension (LDZ,N)
On entry, if WANTZ = .TRUE., Z is an NbyN matrix. On exit, Z has been postmul
tiplied by the left unitary transformation matrix which reorder (A, B); The lead
ing M columns of Z form orthonormal bases for the specified pair of left
eigenspaces (deflating subspaces). If WANTZ = .FALSE., Z is not referenced.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= 1. If WANTZ = .TRUE., LDZ >= N.
M (output) INTEGER
The dimension of the specified pair of left and right eigenspaces, (deflating sub
spaces) 0 <= M <= N.
PL, PR (output) DOUBLE PRECISION If IJOB = 1, 4 or 5, PL, PR are lower bounds on
the reciprocal of the norm of "projections" onto left and right eigenspace with
respect to the selected cluster. 0 < PL, PR <= 1. If M = 0 or M = N, PL = PR =
1. If IJOB = 0, 2 or 3 PL, PR are not referenced.
DIF (output) DOUBLE PRECISION array, dimension (2).
If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
If IJOB = 2 or 4, DIF(1:2) are Fnormbased upper bounds on
Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1normbased estimates of Difu and
Difl, computed using reversed communication with ZLACON. If M = 0 or N, DIF(1:2)
= Fnorm([A, B]). If IJOB = 0 or 1, DIF is not referenced.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
IF IJOB = 0, WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1)
returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1 If IJOB = 1, 2 or 4, LWORK >=
2*M*(NM) If IJOB = 3 or 5, LWORK >= 4*M*(NM)
If LWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the WORK array, returns this value as the first entry of the WORK
array, and no error message related to LWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER, dimension (LIWORK)
IF IJOB = 0, IWORK is not referenced. Otherwise, on exit, if INFO = 0, IWORK(1)
returns the optimal LIWORK.
LIWORK (input) INTEGER
The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2 or 4, LIWORK >=
N+2; If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(NM));
If LIWORK = 1, then a workspace query is assumed; the routine only calculates the
optimal size of the IWORK array, returns this value as the first entry of the
IWORK array, and no error message related to LIWORK is issued by XERBLA.
INFO (output) INTEGER
=0: Successful exit.
<0: If INFO = i, the ith argument had an illegal value.
=1: Reordering of (A, B) failed because the transformed matrix pair (A, B) would
be too far from generalized Schur form; the problem is very illconditioned. (A,
B) may have been partially reordered. If requested, 0 is returned in DIF(*), PL
and PR.
FURTHER DETAILS
ZTGSEN first collects the selected eigenvalues by computing unitary U and W that move them
to the top left corner of (A, B). In other words, the selected eigenvalues are the eigen
values of (A11, B11) in
U'*(A, B)*W = (A11 A12) (B11 B12) n1
( 0 A22),( 0 B22) n2
n1 n2 n1 n2
where N = n1+n2 and U' means the conjugate transpose of U. The first n1 columns of U and W
span the specified pair of left and right eigenspaces (deflating subspaces) of (A, B).
If (A, B) has been obtained from the generalized real Schur decomposition of a matrix pair
(C, D) = Q*(A, B)*Z', then the reordered generalized Schur form of (C, D) is given by
(C, D) = (Q*U)*(U'*(A, B)*W)*(Z*W)',
and the first n1 columns of Q*U and Z*W span the corresponding deflating subspaces of (C,
D) (Q and Z store Q*U and Z*W, resp.).
Note that if the selected eigenvalue is sufficiently illconditioned, then its value may
differ significantly from its value before reordering.
The reciprocal condition numbers of the left and right eigenspaces spanned by the first n1
columns of U and W (or Q*U and Z*W) may be returned in DIF(1:2), corresponding to Difu and
Difl, resp.
The Difu and Difl are defined as:
Difu[(A11, B11), (A22, B22)] = sigmamin( Zu )
and
Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
where sigmamin(Zu) is the smallest singular value of the (2*n1*n2)by(2*n1*n2) matrix
Zu = [ kron(In2, A11) kron(A22', In1) ]
[ kron(In2, B11) kron(B22', In1) ].
Here, Inx is the identity matrix of size nx and A22' is the transpose of A22. kron(X, Y)
is the Kronecker product between the matrices X and Y.
When DIF(2) is small, small changes in (A, B) can cause large changes in the deflating
subspace. An approximate (asymptotic) bound on the maximum angular error in the computed
deflating subspaces is
EPS * norm((A, B)) / DIF(2),
where EPS is the machine precision.
The reciprocal norm of the projectors on the left and right eigenspaces associated with
(A11, B11) may be returned in PL and PR. They are computed as follows. First we compute L
and R so that P*(A, B)*Q is block diagonal, where
P = ( I L ) n1 Q = ( I R ) n1
( 0 I ) n2 and ( 0 I ) n2
n1 n2 n1 n2
and (L, R) is the solution to the generalized Sylvester equation
A11*R  L*A22 = A12
B11*R  L*B22 = B12
Then PL = (Fnorm(L)**2+1)**(1/2) and PR = (Fnorm(R)**2+1)**(1/2). An approximate (as
ymptotic) bound on the average absolute error of the selected eigenvalues is
EPS * norm((A, B)) / PL.
There are also global error bounds which valid for perturbations up to a certain restric
tion: A lower bound (x) on the smallest Fnorm(E,F) for which an eigenvalue of (A11, B11)
may move and coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), (i.e. (A
+ E, B + F), is
x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
An approximate bound on x can be computed from DIF(1:2), PL and PR.
If y = ( Fnorm(E,F) / x) <= 1, the angles between the perturbed (L', R') and unperturbed
(L, R) left and right deflating subspaces associated with the selected cluster in the
(1,1)blocks can be bounded as
maxangle(L, L') <= arctan( y * PL / (1  y * (1  PL * PL)**(1/2))
maxangle(R, R') <= arctan( y * PR / (1  y * (1  PR * PR)**(1/2))
See LAPACK User's Guide section 4.11 or the following references for more information.
Note that if the default method for computing the Frobeniusnorm based estimate DIF is
not wanted (see ZLATDF), then the parameter IDIFJB (see below) should be changed from 3 to
4 (routine ZLATDF (IJOB = 2 will be used)). See ZTGSYL for more details.
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.
References
==========
[1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
M.S. Moonen et al (eds), Linear Algebra for Large Scale and
RealTime Applications, Kluwer Academic Publ. 1993, pp 195218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condition
Estimation: Theory, Algorithms and Software, Report
UMINF  94.04, Department of Computing Science, Umea University,
S901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACKStyle Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF  93.23,
Department of Computing Science, Umea University, S901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
1996.
LAPACK version 3.0 15 June 2000 ZTGSEN(l) 
