ZTGSEN - reorder the generalized Schur decomposition of a
complex matrix pair (A, B) (in terms of an unitary equiva
lence 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 equiva
lence 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 leading 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 "projections" onto left and right
(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.
F-norm-based estimate
(DIF(1:2)).
=3: Estimate of Difu and Difl. 1-norm-based esti
mate
(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: Com
pute 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 gener
alized Schur canonical form. On exit, A is over
written 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 gener
alized Schur canonical form. On exit, B is over
written by the reordered matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >=
max(1,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 general
ized 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 N-by-N
matrix. On exit, Q has been postmultiplied by the
left unitary transformation matrix which reorder
(A, B); The leading M columns of Q form orthonor
mal 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 N-by-N
matrix. On exit, Z has been postmultiplied by the
left unitary transformation matrix which reorder
(A, B); The leading M columns of Z form orthonor
mal 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 subspaces) 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 F-norm-based upper
bounds on
Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are
1-norm-based estimates of Difu and Difl, computed
using reversed communication with ZLACON. If M =
0 or N, DIF(1:2) = F-norm([A, B]). If IJOB = 0 or
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*(N-M) If IJOB = 3
or 5, LWORK >= 4*M*(N-M)
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*(N-M));
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 i-th argument had an illegal
value.
=1: Reordering of (A, B) failed because the trans
formed matrix pair (A, B) would be too far from
generalized Schur form; the problem is very ill-
conditioned. (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 comput
ing unitary U and W that move them to the top left corner
of (A, B). In other words, the selected eigenvalues are
the eigenvalues of (A11, B11) in
U'*(A, B)*W = (A11 A12) (B11 B12) n1
( 0 A22),( 0 B22) 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 corre
sponding deflating subspaces of (C, D) (Q and Z store Q*U
and Z*W, resp.).
Note that if the selected eigenvalue is sufficiently ill-
conditioned, 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)] = sigma-min( Zu )
and
Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11,
B11)],
where sigma-min(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
returned in PL and PR. They are computed as follows.
First we compute L and R so that P*(A, B)*Q is block diag
onal, 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 = (F-norm(L)**2+1)**(-1/2) and PR = (F-
norm(R)**2+1)**(-1/2). An approximate (asymptotic) 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 pertur
bations up to a certain restriction: A lower bound (x) on
the smallest F-norm(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 = ( F-norm(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
max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL *
PL)**(1/2))
max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR *
PR)**(1/2))
See LAPACK User's Guide section 4.11 or the following ref
erences for more information.
Note that if the default method for computing the Frobe
nius-norm- 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
Science,
Umea University, S-901 87 Umea, Sweden.
References
==========
[1] B. Kagstrom; A Direct Method for Reordering Eigenval
ues 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
Real-Time Applications, Kluwer Academic Publ. 1993, pp
195-218.
[2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with
Specified
Eigenvalues of a Regular Matrix Pair (A, B) and Condi
tion
Estimation: Theory, Algorithms and Software, Report
UMINF - 94.04, Department of Computing Science, Umea
University,
S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
Note 87.
To appear in Numerical Algorithms, 1996.
[3] B. Kagstrom and P. Poromaa, LAPACK-Style 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,
S-901 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.
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