stgsen(3P) Sun Performance Library stgsen(3P)NAMEstgsen - reorder the generalized real Schur decomposition of a real
matrix pair (A, B) (in terms of an orthonormal equivalence trans- for‐
mation 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)
INTEGER IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
INTEGER IWORK(*)
LOGICAL WANTQ, WANTZ
LOGICAL SELECT(*)
REAL PL, PR
REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), Q(LDQ,*),
Z(LDZ,*), DIF(*), WORK(*)
SUBROUTINE STGSEN_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(*)
REAL PL, PR
REAL 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 :: PL, PR
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
REAL, 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 :: PL, PR
REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
REAL, DIMENSION(:,:) :: A, B, Q, Z
C INTERFACE
#include <sunperf.h>
void stgsen(int ijob, int wantq, int wantz, int *select, int n, float
*a, int lda, float *b, int ldb, float *alphar, float *alphai,
float *beta, float *q, int ldq, float *z, int ldz, int *m,
float *pl, float *pr, float *dif, int *info);
void stgsen_64(long ijob, long wantq, long wantz, long *select, long n,
float *a, long lda, float *b, long ldb, float *alphar, float
*alphai, float *beta, float *q, long ldq, float *z, long ldz,
long *m, float *pl, float *pr, float *dif, long *info);
PURPOSEstgsen reorders the generalized real Schur decomposition of a real
matrix pair (A, B) (in terms of an orthonormal equivalence trans- for‐
mation 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- spa‐
ces (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 num‐
bers 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.
ARGUMENTS
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 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 ver‐
sion to get it all. =5: Compute PL, PR and DIF (i.e. 0, 1
and 3 above)
WANTQ (input)
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ (input)
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
SELECT (input)
SELECT specifies the eigenvalues in the selected cluster. To
select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
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
.TRUE.; a complex conjugate pair of eigenvalues must be
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 over‐
written 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 gen‐
eralized real Schur canonical form. On exit, B is overwrit‐
ten 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 posi‐
tive, then the j-th and (j+1)-st eigenvalues are a complex
conjugate 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 transforma‐
tion 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 transforma‐
tion 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 recipro‐
cal 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 rou‐
tine 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.
FURTHER DETAILS
STGSEN 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 decomposi‐
tion of a matrix pair (C, D) = Q*(A, B)*Z', then the reordered general‐
ized 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 corresponding deflat‐
ing 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 reorder‐
ing.
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 computed 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 com‐
puted 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 perturbations 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 ei‐
genvalue 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 associ‐
ated 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 references 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 Condition
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.
6 Mar 2009 stgsen(3P)
