NAG Library Routine Document
F08XNF (ZGGES)
1 Purpose
F08XNF (ZGGES) 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
.
2 Specification
| SUBROUTINE F08XNF ( |
JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO) |
| INTEGER |
N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO |
| REAL (KIND=nag_wp) |
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 |
| EXTERNAL |
SELCTG |
|
The routine may be called by its
LAPACK
name zgges.
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, F08XNF (ZGGES) 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.
F08XNF (ZGGES) computes T to have real non-negative diagonal entries. The generalized Schur factorization, before reordering, is computed by the QZ algorithm.
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/lugGolub 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 and F08XNF (ZGGES) 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 is set to
N+2 in this case. (See
INFO below).
SELCTG must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F08XNF (ZGGES) is called. Parameters denoted as
Input must
not be changed by this procedure.
- 5: N – INTEGERInput
-
On entry:
n, the order of the matrices A and B.
- 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 first of the pair of matrices, A.
On exit:
A has been overwritten by its generalized Schur form
S.
- 7: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08XNF (ZGGES) 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 second of the pair of matrices, B.
On exit:
B has been overwritten by its generalized Schur form
T.
- 9: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F08XNF (ZGGES) is called.
Constraint:
LDB≥max1,N.
- 10: SDIM – INTEGEROutput
-
On exit: if
SORT='N',
SDIM=0.
If
SORT='S',
SDIM= number of eigenvalues (after sorting) for which
SELCTG is .TRUE..
- 11: ALPHA(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: see the description of
BETA.
- 12: BETA(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit:
ALPHAj/BETAj, for
j=1,2,…,N, will be the generalized eigenvalues.
ALPHAj, for
j=1,2,…,N and
BETAj, for
j=1,2,…,N, are the diagonals of the complex Schur form
A,B output by F08XNF (ZGGES). The
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.
- 13: 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.
- 14: LDVSL – INTEGERInput
-
On entry: the first dimension of the array
VSL as declared in the (sub)program from which F08XNF (ZGGES) is called.
Constraints:
- if JOBVSL='V', LDVSL≥max1,N;
- otherwise LDVSL≥1.
- 15: 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.
- 16: LDVSR – INTEGERInput
-
On entry: the first dimension of the array
VSR as declared in the (sub)program from which F08XNF (ZGGES) is called.
Constraints:
- if JOBVSR='V', LDVSR≥max1,N;
- otherwise LDVSR≥1.
- 17: 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.
- 18: LWORK – INTEGERInput
-
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08XNF (ZGGES) is called.
If
LWORK=-1, 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.
Suggested value:
for optimal performance,
LWORK must generally be larger than the minimum, say
2×N+nb×N
, where
nb
is the optimal
block size for
F08NSF (ZGEHRD).
Constraint:
LWORK≥max1,2×N.
- 19: RWORK(max1,8×N) – REAL (KIND=nag_wp) arrayWorkspace
- 20: 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.
- 21: 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
F08XAF (DGGES).
9 Example
This example finds the generalized Schur factorization of the matrix pair
A,B, where
and
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 (f08xnfe.f90)
9.2 Program Data
Program Data (f08xnfe.d)
9.3 Program Results
Program Results (f08xnfe.r)