DTGSEN(1) LAPACK routine (version 3.2) DTGSEN(1)NAME
DTGSEN - 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
SYNOPSIS
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 )
LOGICAL WANTQ, WANTZ
INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, M, N
DOUBLE PRECISION PL, PR
LOGICAL SELECT( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
LDB, * ), BETA( * ), DIF( * ), Q( LDQ, * ), WORK( *
), Z( LDZ, * )
PURPOSE
DTGSEN 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 DGGES), 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 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) INTEGER
Specifies whether condition numbers are required for the clus‐
ter 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: Com‐
pute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic version
to get it all. =5: Compute 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 .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 com‐
plex conjugate pair of eigenvalues must be 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) DOUBLE PRECISION 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 overwrit‐
ten by the reordered matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension(LDB,N)
On entry, the upper triangular matrix B, with (A, B) in gener‐
alized real Schur canonical form. On exit, B is overwritten by
the reordered matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA
(output) DOUBLE PRECISION array, dimension (N) On exit,
(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen‐
eralized 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 con‐
jugate pair, with ALPHAI(j+1) negative.
Q (input/output) DOUBLE PRECISION 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 refer‐
enced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= 1; and if WANTQ =
.TRUE., LDQ >= N.
Z (input/output) DOUBLE PRECISION 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 refer‐
enced.
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 (output) DOUBLE PRECISION
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 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.
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. If M = 0 or N, DIF(1:2) = F-
norm([A, B]). If IJOB = 0 or 1, DIF is not referenced.
WORK (workspace/output) DOUBLE PRECISION array,
dimension (MAX(1,LWORK)) 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 (MAX(1,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 transformed matrix
pair (A, B) would be too far from generalized Schur form; the
problem is very ill-conditioned. (A, B) may have been par‐
tially reordered. If requested, 0 is returned in DIF(*), PL
and PR.
FURTHER DETAILS
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 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 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 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
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 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 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 references for
more information.
Note that if the default method for computing the Frobenius-norm- based
estimate DIF is not wanted (see DLATDF), then the parameter IDIFJB (see
below) should be changed from 3 to 4 (routine DLATDF (IJOB = 2 will be
used)). See DTGSYL 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.
LAPACK routine (version 3.2) November 2008 DTGSEN(1)
