Contents
dtgsen - 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 diag-
onal blocks of the upper quasi-triangular matrix A and the
upper triangular B
SUBROUTINE DTGSEN(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)
INTEGER IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
INTEGER IWORK(*)
LOGICAL WANTQ, WANTZ
LOGICAL SELECT(*)
DOUBLE PRECISION PL, PR
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*),
BETA(*), Q(LDQ,*), Z(LDZ,*), DIF(*), WORK(*)
SUBROUTINE DTGSEN_64(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)
INTEGER*8 IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK,
INFO
INTEGER*8 IWORK(*)
LOGICAL*8 WANTQ, WANTZ
LOGICAL*8 SELECT(*)
DOUBLE PRECISION PL, PR
DOUBLE PRECISION A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*),
BETA(*), Q(LDQ,*), Z(LDZ,*), DIF(*), WORK(*)
F95 INTERFACE
SUBROUTINE TGSEN(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])
INTEGER :: IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK,
INFO
INTEGER, DIMENSION(:) :: IWORK
LOGICAL :: WANTQ, WANTZ
LOGICAL, DIMENSION(:) :: SELECT
REAL(8) :: PL, PR
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
REAL(8), DIMENSION(:,:) :: A, B, Q, Z
SUBROUTINE TGSEN_64(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])
INTEGER(8) :: IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK,
INFO
INTEGER(8), DIMENSION(:) :: IWORK
LOGICAL(8) :: WANTQ, WANTZ
LOGICAL(8), DIMENSION(:) :: SELECT
REAL(8) :: PL, PR
REAL(8), DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
REAL(8), DIMENSION(:,:) :: A, B, Q, Z
C INTERFACE
#include <sunperf.h>
void dtgsen(int ijob, int wantq, int wantz, int *select, int
n, double *a, int lda, double *b, int ldb, double
*alphar, double *alphai, double *beta, double *q,
int ldq, double *z, int ldz, int *m, double *pl,
double *pr, double *dif, int *info);
void dtgsen_64(long ijob, long wantq, long wantz, long
*select, long n, double *a, long lda, double *b,
long ldb, double *alphar, double *alphai, double
*beta, double *q, long ldq, double *z, long ldz,
long *m, double *pl, double *pr, double *dif, long
*info);
dtgsen 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 diag-
onal 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.
DTGSEN also computes the generalized eigenvalues
w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)
of the reordered matrix pair (A, B).
Optionally, DTGSEN 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, resp., and norms of
"projections" onto left and right eigenspaces w.r.t. the
selected cluster in the (1,1)-block.
IJOB (input)
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)
WANTZ (input)
SELECT (input)
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),
corresponding 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) The order of the matrices A and B. N >= 0.
A (input/output)
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)
The leading dimension of the array A. LDA >=
max(1,N).
B (input/output)
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.
LDB (input)
The leading dimension of the array B. LDB >=
max(1,N).
ALPHAR (output)
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 diagonals 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.
ALPHAI (output)
See the description of ALPHAR.
BETA (output)
See the description of ALPHAR.
Q (input/output)
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)
The leading dimension of the array Q. LDQ >= 1;
and if WANTQ = .TRUE., LDQ >= N.
Z (input/output)
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)
The leading dimension of the array Z. LDZ >= 1; If
WANTZ = .TRUE., LDZ >= N.
M (output)
The dimension of the specified pair of left and
right eigen- spaces (deflating subspaces). 0 <= M
<= N.
PL (output)
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. If M = 0 or M
= N, PL = PR = 1. If IJOB = 0, 2 or 3, PL and PR
are not referenced.
PR (output)
See the description of PL.
DIF (output)
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)
If IJOB = 0, WORK is not referenced. Otherwise,
on exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
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)
If IJOB = 0, IWORK is not referenced. Otherwise,
on exit, if INFO = 0, IWORK(1) returns the optimal
LIWORK.
LIWORK (input)
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)
=0: Successful exit.
<0: If INFO = -i, the i-th 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
ill-conditioned. (A, B) may have been partially
reordered. If requested, 0 is returned in DIF(*),
PL and PR.
DTGSEN first collects the selected eigenvalues by computing
orthogonal 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
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 correspond-
ing 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:
ifu[(A11, B11), (A22, B22)] = sigma-min( Zu )
and
where sigma-min(Zu) is the smallest singular value of the
(2*n1*n2)-by-(2*n1*n2) matrix
u = [ 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 com-
puted deflating subspaces is PS * 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
= ( 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 11*R - L*A22 = -A12
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
PS * norm((A, B)) / PL.
There are also global error bounds which valid for perturba-
tions 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 per-
turbed (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 refer-
ences for more information.
Note that if the default method for computing the
Frobenius-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 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
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.