NAG Library Routine Document
F08XAF (DGGES)
1 Purpose
F08XAF (DGGES) computes the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right generalized Schur vectors for a pair of by real nonsymmetric matrices .
2 Specification
SUBROUTINE F08XAF ( |
JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO) |
INTEGER |
N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), B(LDB,*), ALPHAR(N), ALPHAI(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 dgges.
3 Description
The generalized Schur factorization for a pair of real matrices
is given by
where
and
are orthogonal,
is upper triangular and
is upper quasi-triangular with
by
and
by
diagonal blocks. The generalized eigenvalues,
, of
are computed from the diagonals of
and
and satisfy
where
is the corresponding generalized eigenvector.
is actually returned as the pair
such that
since
, or even both
and
can be zero. The columns of
and
are the left and right generalized Schur vectors of
.
Optionally, F08XAF (DGGES) can order the generalized eigenvalues on the diagonals of so that selected eigenvalues are at the top left. The leading columns of and then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
F08XAF (DGGES) computes to have non-negative diagonal elements, and the by blocks of correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the 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/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
, do not compute the left Schur vectors.
If , compute the left Schur vectors.
Constraint:
or .
- 2: JOBVSR – CHARACTER(1)Input
On entry: if
, do not compute the right Schur vectors.
If , compute the right Schur vectors.
Constraint:
or .
- 3: SORT – CHARACTER(1)Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
- Eigenvalues are not ordered.
- Eigenvalues are ordered (see SELCTG).
Constraint:
or .
- 4: SELCTG – LOGICAL FUNCTION, supplied by the user.External Procedure
If
,
SELCTG is used to select generalized eigenvalues to the top left of the generalized Schur form.
If
,
SELCTG is not referenced by F08XAF (DGGES), and may be called with the dummy function F08XAZ.
The specification of
SELCTG is:
FUNCTION SELCTG ( |
AR, AI, B) |
REAL (KIND=nag_wp) |
AR, AI, B |
|
- 1: AR – REAL (KIND=nag_wp)Input
- 2: AI – REAL (KIND=nag_wp)Input
- 3: B – REAL (KIND=nag_wp)Input
On entry: an eigenvalue
is selected if
. If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy after ordering. in this case.
SELCTG must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which F08XAF (DGGES) is called. Parameters denoted as
Input must
not be changed by this procedure.
- 5: N – INTEGERInput
On entry: , the order of the matrices and .
Constraint:
.
- 6: A(LDA,) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
.
On entry: the first of the pair of matrices, .
On exit:
A has been overwritten by its generalized Schur form
.
- 7: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08XAF (DGGES) is called.
Constraint:
.
- 8: B(LDB,) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
.
On entry: the second of the pair of matrices, .
On exit:
B has been overwritten by its generalized Schur form
.
- 9: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08XAF (DGGES) is called.
Constraint:
.
- 10: SDIM – INTEGEROutput
On exit: if
,
.
If
,
number of eigenvalues (after sorting) for which
SELCTG is .TRUE.. (Complex conjugate pairs for which
SELCTG is .TRUE. for either eigenvalue count as
.)
- 11: ALPHAR(N) – REAL (KIND=nag_wp) arrayOutput
On exit: see the description of
BETA.
- 12: ALPHAI(N) – REAL (KIND=nag_wp) arrayOutput
On exit: see the description of
BETA.
- 13: BETA(N) – REAL (KIND=nag_wp) arrayOutput
On exit:
, for
, will be the generalized eigenvalues.
, and
, for
, are the diagonals of the complex Schur form
that would result if the
by
diagonal blocks of the real Schur form of
were further reduced to triangular form using
by
complex unitary transformations.
If is zero, then the th eigenvalue is real; if positive, then the th and st eigenvalues are a complex conjugate pair, with negative.
Note: the quotients
and
may easily overflow or underflow, and
may even be zero. Thus, you should avoid naively computing the ratio
. However,
ALPHAR and
ALPHAI will always be less than and usually comparable with
in magnitude, and
BETA will always be less than and usually comparable with
.
- 14: VSL(LDVSL,) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VSL
must be at least
if
, and at least
otherwise.
On exit: if
,
VSL will contain the left Schur vectors,
.
If
,
VSL is not referenced.
- 15: LDVSL – INTEGERInput
On entry: the first dimension of the array
VSL as declared in the (sub)program from which F08XAF (DGGES) is called.
Constraints:
- if , ;
- otherwise .
- 16: VSR(LDVSR,) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VSR
must be at least
if
, and at least
otherwise.
On exit: if
,
VSR will contain the right Schur vectors,
.
If
,
VSR is not referenced.
- 17: LDVSR – INTEGERInput
On entry: the first dimension of the array
VSR as declared in the (sub)program from which F08XAF (DGGES) is called.
Constraints:
- if , ;
- otherwise .
- 18: WORK() – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
,
contains the minimum value of
LWORK required for optimal performance.
- 19: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08XAF (DGGES) 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 mimimum; add, say
, where
is the optimal
block size.
Constraints:
- if , ;
- otherwise .
- 20: BWORK() – LOGICAL arrayWorkspace
-
Note: the dimension of the array
BWORK
must be at least
if
, and at least
otherwise.
If
,
BWORK is not referenced.
- 21: 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.
The iteration failed. are not in Schur form, but , , and should be correct for .
Unexpected error returned from
F08XEF (DHGEQZ).
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy . This could also be caused by underflow due to scaling.
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
is the
machine precision. See Section 4.11 of
Anderson et al. (1999) for further details.
The total number of floating point operations is proportional to .
The complex analogue of this routine is
F08XNF (ZGGES).
9 Example
This example finds the generalized Schur factorization of the matrix pair
, where
such that the real eigenvalues of
correspond to the top left diagonal elements of the generalized Schur form,
.
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 (f08xafe.f90)
9.2 Program Data
Program Data (f08xafe.d)
9.3 Program Results
Program Results (f08xafe.r)