NAG Library Routine Document
F08PNF (ZGEES)
1 Purpose
F08PNF (ZGEES) computes the eigenvalues, the Schur form , and, optionally, the matrix of Schur vectors for an by complex nonsymmetric matrix .
2 Specification
SUBROUTINE F08PNF ( |
JOBVS, SORT, SELECT, N, A, LDA, SDIM, W, VS, LDVS, WORK, LWORK, RWORK, BWORK, INFO) |
INTEGER |
N, LDA, SDIM, LDVS, LWORK, INFO |
REAL (KIND=nag_wp) |
RWORK(*) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), W(*), VS(LDVS,*), WORK(max(1,LWORK)) |
LOGICAL |
SELECT, BWORK(*) |
CHARACTER(1) |
JOBVS, SORT |
EXTERNAL |
SELECT |
|
The routine may be called by its
LAPACK
name zgees.
3 Description
The Schur factorization of
is given by
where
, the matrix of Schur vectors, is unitary and
is the Schur form. A complex matrix is in Schur form if it is upper triangular.
Optionally, F08PNF (ZGEES) also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left. The leading columns of form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.
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: JOBVS – CHARACTER(1)Input
On entry: if
, Schur vectors are not computed.
If , Schur vectors are computed.
Constraint:
or .
- 2: SORT – CHARACTER(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
- Eigenvalues are not ordered.
- Eigenvalues are ordered (see SELECT).
Constraint:
or .
- 3: SELECT – LOGICAL FUNCTION, supplied by the user.External Procedure
If
,
SELECT is used to select eigenvalues to sort to the top left of the Schur form.
If
,
SELECT is not referenced and F08PNF (ZGEES) may be called with the dummy function F08PNZ.
An eigenvalue is selected if is .TRUE..
The specification of
SELECT is:
- 1: W – COMPLEX (KIND=nag_wp)Input
On entry: the real and imaginary parts of the eigenvalue.
SELECT must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F08PNF (ZGEES) is called. Parameters denoted as
Input must
not be changed by this procedure.
- 4: N – INTEGERInput
On entry: , the order of the matrix .
Constraint:
.
- 5: A(LDA,) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
.
On entry: the by matrix .
On exit:
A is overwritten by its Schur form
.
- 6: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08PNF (ZGEES) is called.
Constraint:
.
- 7: SDIM – INTEGEROutput
On exit: if
,
.
If
,
number of eigenvalues for which
SELECT is .TRUE..
- 8: W() – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
W
must be at least
.
On exit: contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form .
- 9: VS(LDVS,) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VS
must be at least
if
, and at least
otherwise.
On exit: if
,
VS contains the unitary matrix
of Schur vectors.
If
,
VS is not referenced.
- 10: LDVS – INTEGERInput
On entry: the first dimension of the array
VS as declared in the (sub)program from which F08PNF (ZGEES) is called.
Constraints:
- if , ;
- otherwise .
- 11: WORK() – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
, the real part of
contains the minimum value of
LWORK required for optimal performance.
- 12: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08PNF (ZGEES) is called.
If
, 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
, where
is the optimal
block size for
F08NSF (ZGEHRD).
Constraint:
.
- 13: RWORK() – REAL (KIND=nag_wp) arrayWorkspace
-
Note: the dimension of the array
RWORK
must be at least
.
- 14: BWORK() – LOGICAL arrayWorkspace
-
Note: the dimension of the array
BWORK
must be at least
if
, and at least
otherwise.
If
,
BWORK is not referenced.
- 15: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
If and , the algorithm failed to compute all the eigenvalues.
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy . This could also be caused by underflow due to scaling.
7 Accuracy
The computed Schur factorization satisfies
where
and
is the
machine precision. See Section 4.8 of
Anderson et al. (1999) for further details.
The total number of floating point operations is proportional to .
The real analogue of this routine is
F08PAF (DGEES).
9 Example
This example finds the Schur factorization of the matrix
Note that the block size (NB) of 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 (f08pnfe.f90)
9.2 Program Data
Program Data (f08pnfe.d)
9.3 Program Results
Program Results (f08pnfe.r)