ctgsen(3P) Sun Performance Library ctgsen(3P)NAMEctgsen - 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 CTGSEN(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK,
LIWORK, INFO)
COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*),
WORK(*)
INTEGER IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
INTEGER IWORK(*)
LOGICAL WANTQ, WANTZ
LOGICAL SELECT(*)
REAL PL, PR
REAL DIF(*)
SUBROUTINE CTGSEN_64(IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK,
LIWORK, INFO)
COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*),
WORK(*)
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 DIF(*)
F95 INTERFACE
SUBROUTINE TGSEN(IJOB, WANTQ, WANTZ, SELECT, [N], A, [LDA], B, [LDB],
ALPHA, BETA, Q, [LDQ], Z, [LDZ], M, PL, PR, DIF, [WORK], [LWORK],
[IWORK], [LIWORK], [INFO])
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, Q, Z
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(:) :: DIF
SUBROUTINE TGSEN_64(IJOB, WANTQ, WANTZ, SELECT, [N], A, [LDA], B,
[LDB], ALPHA, BETA, Q, [LDQ], Z, [LDZ], M, PL, PR, DIF, [WORK],
[LWORK], [IWORK], [LIWORK], [INFO])
COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
COMPLEX, DIMENSION(:,:) :: A, B, Q, Z
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(:) :: DIF
C INTERFACE
#include <sunperf.h>
void ctgsen(int ijob, int wantq, int wantz, int *select, int n, complex
*a, int lda, complex *b, int ldb, complex *alpha, complex
*beta, complex *q, int ldq, complex *z, int ldz, int *m,
float *pl, float *pr, float *dif, int *info);
void ctgsen_64(long ijob, long wantq, long wantz, long *select, long n,
complex *a, long lda, complex *b, long ldb, complex *alpha,
complex *beta, complex *q, long ldq, complex *z, long ldz,
long *m, float *pl, float *pr, float *dif, long *info);
PURPOSEctgsen 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 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.
CTGSEN 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 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) LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ (input) LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
SELECT (input)
SELECT specifies the eigenvalues in the selected cluster. To
select an eigenvalue w(j), SELECT(j) must be set to .TRUE..
N (input) The order of the matrices A and B. N >= 0.
A (input/output)
On entry, the upper triangular matrix A, in generalized 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, in generalized 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).
ALPHA (output)
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.
BETA (output)
See the description of ALPHA.
Q (input/output)
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
orthonormal bases for the specified pair of left eigenspaces
(deflating subspaces). If WANTQ = .FALSE., Q is not refer‐
enced.
LDQ (input)
The leading dimension of the array Q. LDQ >= 1. 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 unitary 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 refer‐
enced.
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
eigenspaces, (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
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.
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, computed using reversed communi‐
cation with CLACON. 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 >= 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)
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+2; If IJOB = 3 or 5, LIWORK >= MAX(N+2,
2*M*(N-M));
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
CTGSEN 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 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 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 decomposi‐
tion of a matrix pair (C, D) = Q*(A, B)*Z', then the reordered general‐
ized 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
EPS * 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 PL.
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 CLATDF), then the parameter IDIFJB (see
below) should be changed from 3 to 4 (routine CLATDF (IJOB = 2 will be
used)). See CTGSYL 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 ctgsen(3P)
