NAG Library Routine Document
F08YUF (ZTGSEN)
1 Purpose
F08YUF (ZTGSEN) reorders the generalized Schur factorization of a complex matrix pair in generalized Schur form, so that a selected cluster of eigenvalues appears in the leading elements on the diagonal of the generalized Schur form. The routine also, optionally, computes the reciprocal condition numbers of the cluster of eigenvalues and/or corresponding deflating subspaces.
2 Specification
| SUBROUTINE F08YUF ( |
IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, LWORK, IWORK, LIWORK, INFO) |
| INTEGER |
IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
| REAL (KIND=nag_wp) |
PL, PR, DIF(*) |
| COMPLEX (KIND=nag_wp) |
A(LDA,*), B(LDB,*), ALPHA(N), BETA(N), Q(LDQ,*), Z(LDZ,*), WORK(max(1,LWORK)) |
| LOGICAL |
WANTQ, WANTZ, SELECT(N) |
|
The routine may be called by its
LAPACK
name ztgsen.
3 Description
F08YUF (ZTGSEN) factorizes the generalized complex
n by
n matrix pair
S,T in generalized Schur form, using a unitary equivalence transformation as
where
S^,T^ are also in generalized Schur form and have the selected eigenvalues as the leading diagonal elements. The leading columns of
Q and
Z are the generalized Schur vectors corresponding to the selected eigenvalues and form orthonormal subspaces for the left and right eigenspaces (deflating subspaces) of the pair
S,T.
The pair
S,T are in generalized Schur form if
S and
T are upper triangular as returned, for example, by
F08XNF (ZGGES), or
F08XSF (ZHGEQZ) with
JOB='S'. The diagonal elements define the generalized eigenvalues
αi,βi, for
i=1,2,…,n, of the pair
S,T. The eigenvalues are given by
but are returned as the pair
αi,βi in order to avoid possible overflow in computing
λi. Optionally, the routine returns reciprocals of condition number estimates for the selected eigenvalue cluster,
p and
q, the right and left projection norms, and of deflating subspaces,
Difu and
Difl. For more information see Sections 2.4.8 and 4.11 of
Anderson et al. (1999).
If
S and
T are the result of a generalized Schur factorization of a matrix pair
A,B
then, optionally, the matrices
Q and
Z can be updated as
QQ^ and
ZZ^. Note that the condition numbers of the pair
S,T are the same as those of the pair
A,B.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug5 Parameters
- 1: IJOB – INTEGERInput
-
On entry: specifies whether condition numbers are required for the cluster of eigenvalues (
p and
q) or the deflating subspaces (
Difu and
Difl).
- IJOB=0
- Only reorder with respect to SELECT. No extras.
- IJOB=1
- Reciprocal of norms of ‘projections’ onto left and right eigenspaces with respect to the selected cluster (p and q).
- IJOB=2
- The upper bounds on Difu and Difl. F-norm-based estimate
(DIF1:2).
- IJOB=3
- Estimate of Difu and Difl. 1-norm-based estimate
(DIF1:2).
About five times as expensive as IJOB=2.
- IJOB=4
- Compute PL, PR and DIF as in IJOB=0, 1 and 2. Economic version to get it all.
- IJOB=5
- Compute PL, PR and DIF as in IJOB=0, 1 and 3.
- 2: WANTQ – LOGICALInput
-
On entry: if
WANTQ=.TRUE., update the left transformation matrix
Q.
If WANTQ=.FALSE., do not update Q.
- 3: WANTZ – LOGICALInput
-
On entry: if
WANTZ=.TRUE., update the right transformation matrix
Z.
If WANTZ=.FALSE., do not update Z.
- 4: SELECT(N) – LOGICAL arrayInput
-
Note: the dimension of the array
SELECT
must be at least
max1,N.
On entry: specifies the eigenvalues in the selected cluster. To select an eigenvalue λj, SELECTj must be set to .TRUE..
- 5: N – INTEGERInput
-
On entry:
n, the order of the matrices
S and T.
Constraint:
N≥0.
- 6: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the matrix S in the pair S,T.
On exit: the updated matrix S^.
- 7: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08YUF (ZTGSEN) is called.
Constraint:
LDA≥max1,N.
- 8: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the matrix T, in the pair S,T.
On exit: the updated matrix T^
- 9: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F08YUF (ZTGSEN) is called.
Constraint:
LDB≥max1,N.
- 10: ALPHA(N) – COMPLEX (KIND=nag_wp) arrayOutput
- 11: BETA(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit:
ALPHA and
BETA contain diagonal elements of
S^ and
T^, respectively, when the pair
S,T has been reduced to generalized Schur form.
ALPHAi/BETAi, for
i=1,2,…,N, are the eigenvalues.
- 12: Q(LDQ,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
Q
must be at least
max1,N if
WANTQ=.TRUE., and at least
1 otherwise.
On entry: if WANTQ=.TRUE., the n by n matrix Q.
On exit: if
WANTQ=.TRUE., the updated matrix
QQ^.
If
WANTQ=.FALSE.,
Q is not referenced.
- 13: LDQ – INTEGERInput
-
On entry: the first dimension of the array
Q as declared in the (sub)program from which F08YUF (ZTGSEN) is called.
Constraints:
- if WANTQ=.TRUE., LDQ≥max1,N;
- otherwise LDQ≥1.
- 14: Z(LDZ,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
Z
must be at least
max1,N if
WANTZ=.TRUE., and at least
1 otherwise.
On entry: if WANTZ=.TRUE., the n by n matrix Z.
On exit: if
WANTZ=.TRUE., the updated matrix
ZZ^.
If
WANTZ=.FALSE.,
Z is not referenced.
- 15: LDZ – INTEGERInput
-
On entry: the first dimension of the array
Z as declared in the (sub)program from which F08YUF (ZTGSEN) is called.
Constraints:
- if WANTZ=.TRUE., LDZ≥max1,N;
- otherwise LDZ≥1.
- 16: M – INTEGEROutput
-
On exit: the dimension of the specified pair of left and right eigenspaces (deflating subspaces).
Constraint:
0≤M≤N.
- 17: PL – REAL (KIND=nag_wp)Output
- 18: PR – REAL (KIND=nag_wp)Output
-
On exit: if
IJOB=1,
4 or
5,
PL and
PR are lower bounds on the reciprocal of the norm of ‘projections’
p and
q 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 and
PR are not referenced.
- 19: DIF(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
DIF
must be at least
2.
On exit: if
IJOB≥2,
DIF1:2 store the estimates of
Difu and
Difl.
If IJOB=2 or 4, DIF1:2 are F-norm-based upper bounds on Difu and Difl.
If IJOB=3 or 5, DIF1:2 are 1-norm-based estimates of Difu and Difl.
If M=0 or n, DIF1:2=A,BF.
If
IJOB=0 or
1,
DIF is not referenced.
- 20: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 21: LWORK – INTEGERInput
-
On entry:
the dimension of the array
WORK as declared in the (sub)program from which F08YUF (ZTGSEN) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the minimum sizes of the
WORK and
IWORK arrays, returns these values as the first entries of the
WORK and
IWORK arrays, and no error message related to
LWORK or
LIWORK is issued.
Constraints:
if
LWORK≠-1,
- if IJOB=1, 2 or 4, LWORK≥max1,2×M×N-M;
- if IJOB=3 or 5, LWORK≥max1,4×M×N-M;
- otherwise LWORK≥1.
.
- 22: IWORK(max1,LIWORK) – INTEGER arrayWorkspace
On exit: if
INFO=0,
IWORK1 returns the minimum
LIWORK.
- 23: LIWORK – INTEGERInput
-
On entry: the dimension of the array
IWORK as declared in the (sub)program from which F08YUF (ZTGSEN) is called.
If
LIWORK=-1, a workspace query is assumed; the routine only calculates the minimum sizes of the
WORK and
IWORK arrays, returns these values as the first entries of the
WORK and
IWORK arrays, and no error message related to
LWORK or
LIWORK is issued.
Constraints:
if
LIWORK≠-1,
- if IJOB=1, 2 or 4, LIWORK≥N+2;
- if IJOB=3 or 5, LIWORK≥maxN+2,2×M×N-M;
- otherwise LIWORK≥1.
.
- 24: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO=1
Reordering of
S,T failed because the transformed matrix pair
S^,T^ would be too far from generalized Schur form; the problem is very ill-conditioned.
S,T may have been partially reordered. If requested,
0 is returned in
DIF1:2,
PL and
PR.
7 Accuracy
The computed generalized Schur form is nearly the exact generalized Schur form for nearby matrices
S+E and
T+F, where
and
ε is the
machine precision. See Section 4.11 of
Anderson et al. (1999) for further details of error bounds for the generalized nonsymmetric eigenproblem, and for information on the condition numbers returned.
8 Further Comments
The real analogue of this routine is
F08YGF (DTGSEN).
9 Example
This example reorders the generalized Schur factors
S and
T and update the matrices
Q and
Z given by
selecting the second and third generalized eigenvalues to be moved to the leading positions. Bases for the left and right deflating subspaces, and estimates of the condition numbers for the eigenvalues and Frobenius norm based bounds on the condition numbers for the deflating subspaces are also output.
9.1 Program Text
Program Text (f08yufe.f90)
9.2 Program Data
Program Data (f08yufe.d)
9.3 Program Results
Program Results (f08yufe.r)