STGSEN(3) LAPACK routine of NEC Numeric Library Collection STGSEN(3)
NAME
STGSEN
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)
PURPOSE
STGSEN reorders the generalized real Schur decomposition of a real
matrix pair (A, B) (in terms of an orthonormal equivalence trans-
formation Q**T * (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, resp.,
and norms of "projections" onto left and right eigenspaces w.r.t.
the selected cluster in the (1,1)-block.
ARGUMENTS
IJOB (input)
IJOB is 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 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
version to get it all.
=5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
WANTQ (input)
WANTQ is LOGICAL
.TRUE. : update the left transformation matrix Q;
.FALSE.: do not update Q.
WANTZ (input)
WANTZ is LOGICAL
.TRUE. : update the right transformation matrix Z;
.FALSE.: do not update Z.
SELECT (input)
SELECT is 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 complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.
N (input)
N is INTEGER
The order of the matrices A and B. N >= 0.
A (input/output)
A is 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)
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output)
B is 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.
LDB (input)
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
ALPHAR (output)
ALPHAR is REAL array, dimension (N)
ALPHAI (output)
ALPHAI is REAL array, dimension (N)
BETA (output)
BETA is 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 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 conjugate pair, with ALPHAI(j+1) negative.
Q (input/output)
Q is 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)
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= 1;
and if WANTQ = .TRUE., LDQ >= N.
Z (input/output)
Z is 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)
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1;
If WANTZ = .TRUE., LDZ >= N.
M (output)
M is INTEGER
The dimension of the specified pair of left and right eigen-
spaces (deflating subspaces). 0 <= M <= N.
PL (output)
PL is REAL
PR (output)
PR is 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.
If M = 0 or M = N, PL = PR = 1.
If IJOB = 0, 2 or 3, PL and PR are not referenced.
DIF (output)
DIF is 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 (output)
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
LWORK is 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 (output)
IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input)
LIWORK is 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)
INFO is 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 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**T*(A, B)*W = (A11 A12) (B11 B12) n1
( 0 A22),( 0 B22) n2
n1 n2 n1 n2
where N = n1+n2 and U**T 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**T, then the
reordered generalized real Schur form of (C, D) is given by
(C, D) = (Q*U)*(U**T*(A, B)*W)*(Z*W)**T,
and the first n1 columns of Q*U and Z*W span the corresponding
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**T, In1) ]
[ kron(In2, B11) -kron(B22**T, In1) ].
Here, Inx is the identity matrix of size nx and A22**T 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 computed 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 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))
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.
LAPACK routine 31 October 2017 STGSEN(3)