NAG FL Interface
f08xpf (zggesx)
1
Purpose
f08xpf computes the generalized eigenvalues, the generalized Schur form and, optionally, the left and/or right generalized Schur vectors for a pair of by complex nonsymmetric matrices .
Estimates of condition numbers for selected generalized eigenvalue clusters and Schur vectors are also computed.
2
Specification
Fortran Interface
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, Intent (In) |
:: |
n, lda, ldb, ldvsl, ldvsr, lwork, liwork |
Integer, Intent (Out) |
:: |
sdim, iwork(max(1,liwork)), info |
Real (Kind=nag_wp), Intent (Out) |
:: |
rconde(2), rcondv(2), rwork(max(1,8*n)) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), b(ldb,*), vsl(ldvsl,*), vsr(ldvsr,*) |
Complex (Kind=nag_wp), Intent (Out) |
:: |
alpha(n), beta(n), work(max(1,lwork)) |
Logical, External |
:: |
selctg |
Logical, Intent (Inout) |
:: |
bwork(*) |
Character (1), Intent (In) |
:: |
jobvsl, jobvsr, sort, sense |
|
C Header Interface
#include <nag.h>
void |
f08xpf_ (const char *jobvsl, const char *jobvsr, const char *sort, logical (NAG_CALL *selctg)(const Complex *a, const Complex *b), const char *sense, const Integer *n, Complex a[], const Integer *lda, Complex b[], const Integer *ldb, Integer *sdim, Complex alpha[], Complex beta[], Complex vsl[], const Integer *ldvsl, Complex vsr[], const Integer *ldvsr, double rconde[], double rcondv[], Complex work[], const Integer *lwork, double rwork[], Integer iwork[], const Integer *liwork, logical bwork[], Integer *info, const Charlen length_jobvsl, const Charlen length_jobvsr, const Charlen length_sort, const Charlen length_sense) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08xpf_ (const char *jobvsl, const char *jobvsr, const char *sort, logical (NAG_CALL *selctg)(const Complex &a, const Complex &b), const char *sense, const Integer &n, Complex a[], const Integer &lda, Complex b[], const Integer &ldb, Integer &sdim, Complex alpha[], Complex beta[], Complex vsl[], const Integer &ldvsl, Complex vsr[], const Integer &ldvsr, double rconde[], double rcondv[], Complex work[], const Integer &lwork, double rwork[], Integer iwork[], const Integer &liwork, logical bwork[], Integer &info, const Charlen length_jobvsl, const Charlen length_jobvsr, const Charlen length_sort, const Charlen length_sense) |
}
|
The routine may be called by the names f08xpf, nagf_lapackeig_zggesx or its LAPACK name zggesx.
3
Description
The generalized Schur factorization for a pair of complex matrices
is given by
where
and
are unitary,
and
are upper triangular. 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, f08xpf 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.
f08xpf computes to have real non-negative diagonal entries. The generalized Schur factorization, before reordering, is computed by the algorithm.
The reciprocals of the condition estimates, the reciprocal values of the left and right projection norms, are returned in
and
respectively, for the selected generalized eigenvalues, together with reciprocal condition estimates for the corresponding left and right deflating subspaces, in
and
. 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
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Character(1)
Input
-
On entry: if
, do not compute the left Schur vectors.
If , compute the left Schur vectors.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: if
, do not compute the right Schur vectors.
If , compute the right Schur vectors.
Constraint:
or .
-
3:
– 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:
– Logical Function, supplied by the user.
External Procedure
-
If
,
selctg is used to select generalized eigenvalues to be moved to the top left of the generalized Schur form.
If
,
selctg is not referenced by
f08xpf, and may be called with the dummy function
f08xnz.
The specification of
selctg is:
Fortran Interface
Logical |
:: |
selctg |
Complex (Kind=nag_wp), Intent (In) |
:: |
a, b |
|
C Header Interface
Nag_Boolean |
selctg_ (const Complex *a, const Complex *b) |
|
C++ Header Interface
#include <nag.h> extern "C" {
Nag_Boolean |
selctg_ (const Complex &a, const Complex &b) |
}
|
-
1:
– Complex (Kind=nag_wp)
Input
-
2:
– Complex (Kind=nag_wp)
Input
-
On entry: an eigenvalue
is selected if
is .TRUE..
Note that in the ill-conditioned case, a selected 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
f08xpf is called. Arguments denoted as
Input must
not be changed by this procedure.
-
5:
– Character(1)
Input
-
On entry: determines which reciprocal condition numbers are computed.
- None are computed.
- Computed for average of selected eigenvalues only.
- Computed for selected deflating subspaces only.
- Computed for both.
If , or , .
Constraint:
, , or .
-
6:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
7:
– Complex (Kind=nag_wp) array
Input/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
.
-
8:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08xpf is called.
Constraint:
.
-
9:
– Complex (Kind=nag_wp) array
Input/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
.
-
10:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08xpf is called.
Constraint:
.
-
11:
– Integer
Output
-
On exit: if
,
.
If
,
number of eigenvalues (after sorting) for which
selctg is .TRUE..
-
12:
– Complex (Kind=nag_wp) array
Output
-
On exit: see the description of
beta.
-
13:
– Complex (Kind=nag_wp) array
Output
-
On exit:
, for
, will be the generalized eigenvalues.
and
are the diagonals of the complex Schur form
.
will be non-negative real.
Note: the quotients
may easily overflow or underflow, and
may even be zero. Thus, you should avoid naively computing the ratio
. However,
alpha will always be less than and usually comparable with
in magnitude, and
beta will always be less than and usually comparable with
.
-
14:
– Complex (Kind=nag_wp) array
Output
-
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:
– Integer
Input
-
On entry: the first dimension of the array
vsl as declared in the (sub)program from which
f08xpf is called.
Constraints:
- if , ;
- otherwise .
-
16:
– Complex (Kind=nag_wp) array
Output
-
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:
– Integer
Input
-
On entry: the first dimension of the array
vsr as declared in the (sub)program from which
f08xpf is called.
Constraints:
- if , ;
- otherwise .
-
18:
– Real (Kind=nag_wp) array
Output
-
On exit: if
or
,
and
contain the reciprocal condition numbers for the average of the selected eigenvalues.
If
or
,
rconde is not referenced.
-
19:
– Real (Kind=nag_wp) array
Output
-
On exit: if
or
,
and
contain the reciprocal condition numbers for the selected deflating subspaces.
if
or
,
rcondv is not referenced.
-
20:
– Complex (Kind=nag_wp) array
Workspace
-
On exit: if
, the real part of
contains a bound on the value of
lwork required for optimal performance.
-
21:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08xpf is called.
If
, 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 , ;
- if , or , ;
- otherwise .
Note: . Note also that an error is only returned if
, but if
,
or
this may not be large enough. Consider increasing
lwork by
, where
is the optimal
block size.
-
22:
– Real (Kind=nag_wp) array
Workspace
-
Real workspace.
-
23:
– Integer array
Workspace
-
On exit: if
,
returns the minimum
liwork.
-
24:
– Integer
Input
-
On entry: the dimension of the array
iwork as declared in the (sub)program from which
f08xpf is called.
If
, 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 or , ;
- otherwise .
-
25:
– Logical array
Workspace
-
Note: the dimension of the array
bwork
must be at least
if
, and at least
otherwise.
If
,
bwork is not referenced.
-
26:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
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 from element .
-
The
iteration failed with an unexpected error, please contact
NAG.
-
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.
8
Parallelism and Performance
f08xpf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xpf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is proportional to .
The real analogue of this routine is
f08xbf.
10
Example
This example finds the generalized Schur factorization of the matrix pair
, where
and
such that the eigenvalues of
for which
correspond to the top left diagonal elements of the generalized Schur form,
. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding deflating subspaces are also returned.
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.
10.1
Program Text
10.2
Program Data
10.3
Program Results