STGSEN - reorder the generalized real Schur decomposition
of a real matrix pair (A, B) (in terms of an orthonormal
equivalence trans- formation Q' * (A, B) * Z), so that a
selected cluster of eigenvalues appears in the leading
diagonal blocks of the upper quasi-triangular matrix A and
the upper triangular B
SYNOPSIS
SUBROUTINE STGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA,
B, LDB, ALPHAR, ALPHAI, 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
REAL PL, PR
LOGICAL SELECT( * )
INTEGER IWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
B( LDB, * ), BETA( * ), DIF( * ), Q(
LDQ, * ), WORK( * ), Z( LDZ, * )
PURPOSE
STGSEN reorders the generalized real Schur decomposition
of a real matrix pair (A, B) (in terms of an orthonormal
equivalence trans- formation Q' * (A, B) * Z), so that a
selected cluster of eigenvalues appears in the leading
diagonal blocks of the upper quasi-triangular matrix A and
the upper triangular B. The leading columns of Q and Z
form orthonormal bases of the corresponding left and right
eigen- spaces (deflating subspaces). (A, B) must be in
generalized real Schur canonical form (as returned by
SGGES), i.e. A is block upper triangular with 1-by-1 and
2-by-2 diagonal blocks. B is upper triangular.
STGSEN also computes the generalized eigenvalues
w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
of the reordered matrix pair (A, B).
Optionally, STGSEN computes the 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,
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.
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 a real eigenvalue w(j),
SELECT(j) must be set to w(j) and w(j+1), corre
sponding to a 2-by-2 diagonal block, either
SELECT(j) or SELECT(j+1) or both must be set to
either both included in the cluster or both
excluded.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
A (input/output) REAL array, dimension(LDA,N)
On entry, the upper quasi-triangular matrix A,
with (A, B) in generalized real 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) REAL array, dimension(LDB,N)
On entry, the upper triangular matrix B, with (A,
B) in generalized real Schur canonical form. On
exit, B is overwritten by the reordered matrix B.
The leading dimension of the array B. LDB >=
max(1,N).
ALPHAR (output) REAL array, dimension (N)
ALPHAI (output) REAL array, dimension (N) BETA
(output) REAL array, dimension (N) On exit,
(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. ALPHAR(j) +
ALPHAI(j)*i and BETA(j),j=1,...,N are the diago
nals of the complex Schur form (S,T) that would
result if the 2-by-2 diagonal blocks of the real
generalized Schur form of (A,B) were further
reduced to triangular form using complex unitary
transformations. If ALPHAI(j) is zero, then the
j-th eigenvalue is real; if positive, then the j-
th and (j+1)-st eigenvalues are a complex conju
gate pair, with ALPHAI(j+1) negative.
Q (input/output) REAL 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 orthogonal transformation matrix which
reorder (A, B); The leading 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;
and if WANTQ = .TRUE., LDQ >= N.
Z (input/output) REAL 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 orthogonal transformation matrix which
reorder (A, B); The leading 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 eigen- spaces (deflating subspaces). 0 <= M
<= N.
PL, PR (output) REAL If IJOB = 1, 4 or 5, PL, PR
are lower bounds on the reciprocal of the norm of
"projections" onto left and right eigenspaces with
respect to the selected cluster. 0 < PL, PR <= 1.
or 3, PL and PR are not referenced.
DIF (output) REAL 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. If M = 0
or N, DIF(1:2) = F-norm([A, B]). If IJOB = 0 or
1, DIF is not referenced.
WORK (workspace/output) REAL 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 >= 4*N+16.
If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-
M)). If IJOB = 3 or 5, LWORK >= MAX(4*N+16,
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 array, 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+6. If IJOB = 3 or
5, LIWORK >= MAX(2*M*(N-M), N+6).
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-
reordered. If requested, 0 is returned in DIF(*),
PL and PR.
FURTHER DETAILS
STGSEN first collects the selected eigenvalues by comput
ing orthogonal U and W that move them to the top left cor
ner 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
n1 n2 n1 n2
where N = n1+n2 and U' means the 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 real 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.
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 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))
erences for more information.
Note that if the default method for computing the Frobe
nius-norm- based estimate DIF is not wanted (see SLATDF),
then the parameter IDIFJB (see below) should be changed
from 3 to 4 (routine SLATDF (IJOB = 2 will be used)). See
STGSYL for more details.
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing
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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