NAG Library Routine Document
F08XPF (ZGGESX)
1 Purpose
F08XPF (ZGGESX) computes the generalized eigenvalues, the generalized Schur form
S,T
and, optionally, the left and/or right generalized Schur vectors for a pair of n by n complex nonsymmetric matrices
A,B
.
Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.
2 Specification
SUBROUTINE F08XPF ( |
JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO) |
INTEGER |
N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
REAL (KIND=nag_wp) |
RCONDE(2), RCONDV(2), RWORK(max(1,8*N)) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), B(LDB,*), ALPHA(N), BETA(N), VSL(LDVSL,*), VSR(LDVSR,*), WORK(max(1,LWORK)) |
LOGICAL |
SELCTG, BWORK(*) |
CHARACTER(1) |
JOBVSL, JOBVSR, SORT, SENSE |
EXTERNAL |
SELCTG |
|
The routine may be called by its
LAPACK
name zggesx.
3 Description
The generalized Schur factorization for a pair of complex matrices
A,B
is given by
where
Q and
Z are unitary,
T and
S are upper triangular. The generalized eigenvalues,
λ
, of
A,B
are computed from the diagonals of
T and
S and satisfy
where
z is the corresponding generalized eigenvector.
λ
is actually returned as the pair
α,β
such that
since
β
, or even both
α
and
β
can be zero. The columns of
Q and
Z are the left and right generalized Schur vectors of
A,B
.
Optionally, F08XPF (ZGGESX) can order the generalized eigenvalues on the diagonals of
S,T
so that selected eigenvalues are at the top left. The leading columns of Q and Z then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
F08XPF (ZGGESX) computes T to have real non-negative diagonal entries. The generalized Schur factorization, before reordering, is computed by the QZ algorithm.
The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in
RCONDE1
and
RCONDE2
respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in
RCONDV1
and
RCONDV2
. See Section 4.11 of
Anderson et al. (1999) for further information.
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/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: JOBVSL – CHARACTER(1)Input
On entry: if
JOBVSL='N', do not compute the left Schur vectors.
If JOBVSL='V', compute the left Schur vectors.
Constraint:
JOBVSL='N' or 'V'.
- 2: JOBVSR – CHARACTER(1)Input
On entry: if
JOBVSR='N', do not compute the right Schur vectors.
If JOBVSR='V', compute the right Schur vectors.
Constraint:
JOBVSR='N' or 'V'.
- 3: SORT – CHARACTER(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
- SORT='N'
- Eigenvalues are not ordered.
- SORT='S'
- Eigenvalues are ordered (see SELCTG).
Constraint:
SORT='N' or 'S'.
- 4: SELCTG – LOGICAL FUNCTION, supplied by the user.External Procedure
If
SORT='S',
SELCTG is used to select generalized eigenvalues to the top left of the generalized Schur form.
If
SORT='N',
SELCTG is not referenced by F08XPF (ZGGESX), and may be called with the dummy function F08XNZ.
The specification of
SELCTG is:
COMPLEX (KIND=nag_wp) |
A, B |
|
- 1: A – COMPLEX (KIND=nag_wp)Input
- 2: B – COMPLEX (KIND=nag_wp)Input
On entry: an eigenvalue
Aj / Bj is selected if
SELCTG Aj,Bj is .TRUE..
Note that in the ill-conditioned case, a selected generalized eigenvalue may no longer satisfy SELCTG Aj,Bj=.TRUE. after ordering. INFO=N+2 in this case.
SELCTG must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F08XPF (ZGGESX) is called. Parameters denoted as
Input must
not be changed by this procedure.
- 5: SENSE – CHARACTER(1)Input
On entry: determines which reciprocal condition numbers are computed.
- SENSE='N'
- None are computed.
- SENSE='E'
- Computed for average of selected eigenvalues only.
- SENSE='V'
- Computed for selected deflating subspaces only.
- SENSE='B'
- Computed for both.
If SENSE='E', 'V' or 'B', SORT='S'.
Constraint:
SENSE='N', 'E', 'V' or 'B'.
- 6: N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint:
N≥0.
- 7: 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 first of the pair of matrices, A.
On exit:
A has been overwritten by its generalized Schur form
S.
- 8: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08XPF (ZGGESX) is called.
Constraint:
LDA≥max1,N.
- 9: 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 second of the pair of matrices, B.
On exit:
B has been overwritten by its generalized Schur form
T.
- 10: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08XPF (ZGGESX) is called.
Constraint:
LDB≥max1,N.
- 11: SDIM – INTEGEROutput
On exit: if
SORT='N',
SDIM=0.
If
SORT='S',
SDIM= number of eigenvalues (after sorting) for which
SELCTG is .TRUE..
- 12: ALPHA(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: see the description of
BETA.
- 13: BETA(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit:
ALPHAj/BETAj, for
j=1,2,…,N, will be the generalized eigenvalues.
ALPHAj and
BETAj,j=1,2,…,N are the diagonals of the complex Schur form
S,T.
BETAj will be non-negative real.
Note: the quotients
ALPHAj/BETAj may easily overflow or underflow, and
BETAj may even be zero. Thus, you should avoid naively computing the ratio
α/β. However,
ALPHA will always be less than and usually comparable with
A in magnitude, and
BETA will always be less than and usually comparable with
B.
- 14: VSL(LDVSL,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VSL
must be at least
max1,N if
JOBVSL='V', and at least
1 otherwise.
On exit: if
JOBVSL='V',
VSL will contain the left Schur vectors,
Q.
If
JOBVSL='N',
VSL is not referenced.
- 15: LDVSL – INTEGERInput
On entry: the first dimension of the array
VSL as declared in the (sub)program from which F08XPF (ZGGESX) is called.
Constraints:
- if JOBVSL='V', LDVSL≥ max1,N ;
- otherwise LDVSL≥1.
- 16: VSR(LDVSR,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VSR
must be at least
max1,N if
JOBVSR='V', and at least
1 otherwise.
On exit: if
JOBVSR='V',
VSR will contain the right Schur vectors,
Z.
If
JOBVSR='N',
VSR is not referenced.
- 17: LDVSR – INTEGERInput
On entry: the first dimension of the array
VSR as declared in the (sub)program from which F08XPF (ZGGESX) is called.
Constraints:
- if JOBVSR='V', LDVSR≥ max1,N ;
- otherwise LDVSR≥1.
- 18: RCONDE(2) – REAL (KIND=nag_wp) arrayOutput
On exit: if
SENSE='E' or
'B',
RCONDE1 and
RCONDE2 contain the reciprocal condition numbers for the average of the selected eigenvalues.
If
SENSE='N' or
'V',
RCONDE is not referenced.
- 19: RCONDV(2) – REAL (KIND=nag_wp) arrayOutput
On exit: if
SENSE='V' or
'B',
RCONDV1 and
RCONDV2 contain the reciprocal condition numbers for the selected deflating subspaces.
if
SENSE='N' or
'E',
RCONDV is not referenced.
- 20: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains a bound on the 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 F08XPF (ZGGESX) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the bound on the optimal size of the
WORK array and the minimum size of the
IWORK array, 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 N=0, LWORK≥1;
- if SENSE='E', 'V' or 'B', LWORK≥ max1,2×N,2×SDIM×N-SDIM ;
- otherwise LWORK≥ max1,2×N .
Note: 2×SDIM×N-SDIM≤N×N/2. Note also that an error is only returned if
LWORK < max1,2×N , but if
SENSE='E',
'V' or
'B' this may not be large enough. Consider increasing
LWORK by
nb, where
nb is the optimal
block size.
- 22: RWORK(max1,8×N) – REAL (KIND=nag_wp) arrayWorkspace
Real workspace.
- 23: IWORK(max1,LIWORK) – INTEGER arrayWorkspace
On exit: if
INFO=0,
IWORK1 returns the minimum
LIWORK.
- 24: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which F08XPF (ZGGESX) is called.
If
LIWORK=-1, a workspace query is assumed; the routine only calculates the bound on the optimal size of the
WORK array and the minimum size of the
IWORK array, 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 SENSE='N' or N=0, LIWORK≥1;
- otherwise LIWORK≥N+2.
- 25: BWORK(*) – LOGICAL arrayWorkspace
-
Note: the dimension of the array
BWORK
must be at least
1 if
SORT='N', and at least
max1,N otherwise.
If
SORT='N',
BWORK is not referenced.
- 26: 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 to N
The QZ iteration failed. A,B are not in Schur form, but ALPHAj and BETAj should be correct for j=INFO+1,…,N.
- INFO=N+1
Unexpected error returned from
F08XSF (ZHGEQZ).
- INFO=N+2
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy SELCTG=.TRUE.. This could also be caused by underflow due to scaling.
- INFO=N+3
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
7 Accuracy
The computed generalized Schur factorization satisfies
where
and
ε is the
machine precision. See Section 4.11 of
Anderson et al. (1999) for further details.
8 Further Comments
The total number of floating point operations is proportional to n3.
The real analogue of this routine is
F08XBF (DGGESX).
9 Example
This example finds the generalized Schur factorization of the matrix pair
A,B, where
and
such that the eigenvalues of
A,B for which
λ<6 correspond to the top left diagonal elements of the generalized Schur form,
S,T. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding deflating subspaces are also returned.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
9.1 Program Text
Program Text (f08xpfe.f90)
9.2 Program Data
Program Data (f08xpfe.d)
9.3 Program Results
Program Results (f08xpfe.r)