NAG FL Interface
f08ygf (dtgsen)
1
Purpose
f08ygf reorders the generalized Schur factorization of a matrix pair in real generalized Schur form, so that a selected cluster of eigenvalues appears in the leading elements, or blocks on the diagonal of the generalized Schur form. The routine also, optionally, computes the reciprocal condition numbers of the cluster of eigenvalues and/or corresponding deflating subspaces.
2
Specification
Fortran Interface
Subroutine f08ygf ( |
ijob, wantq, wantz, select, n, a, lda, b, ldb, alphar, alphai, beta, q, ldq, z, ldz, m, pl, pr, dif, work, lwork, iwork, liwork, info) |
Integer, Intent (In) |
:: |
ijob, n, lda, ldb, ldq, ldz, lwork, liwork |
Integer, Intent (Out) |
:: |
m, iwork(max(1,liwork)), info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), b(ldb,*), q(ldq,*), z(ldz,*), dif(*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
alphar(n), alphai(n), beta(n), pl, pr, work(max(1,lwork)) |
Logical, Intent (In) |
:: |
wantq, wantz, select(n) |
|
C Header Interface
#include <nag.h>
void |
f08ygf_ (const Integer *ijob, const logical *wantq, const logical *wantz, const logical sel[], const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, double alphar[], double alphai[], double beta[], double q[], const Integer *ldq, double z[], const Integer *ldz, Integer *m, double *pl, double *pr, double dif[], double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, Integer *info) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08ygf_ (const Integer &ijob, const logical &wantq, const logical &wantz, const logical sel[], const Integer &n, double a[], const Integer &lda, double b[], const Integer &ldb, double alphar[], double alphai[], double beta[], double q[], const Integer &ldq, double z[], const Integer &ldz, Integer &m, double &pl, double &pr, double dif[], double work[], const Integer &lwork, Integer iwork[], const Integer &liwork, Integer &info) |
}
|
The routine may be called by the names f08ygf, nagf_lapackeig_dtgsen or its LAPACK name dtgsen.
3
Description
f08ygf factorizes the generalized real
by
matrix pair
in real generalized Schur form, using an orthogonal equivalence transformation as
where
are also in real generalized Schur form and have the selected eigenvalues as the leading diagonal elements, or diagonal blocks. The leading columns of
and
are the generalized Schur vectors corresponding to the selected eigenvalues and form orthonormal subspaces for the left and right eigenspaces (deflating subspaces) of the pair
.
The pair
are in real generalized Schur form if
is block upper triangular with
by
and
by
diagonal blocks and
is upper triangular as returned, for example, by
f08xaf, or
f08xef with
. The diagonal elements, or blocks, define the generalized eigenvalues
, for
, of the pair
. The eigenvalues are given by
but are returned as the pair
in order to avoid possible overflow in computing
. Optionally, the routine returns reciprocals of condition number estimates for the selected eigenvalue cluster,
and
, the right and left projection norms, and of deflating subspaces,
and
. For more information see Sections 2.4.8 and 4.11 of
Anderson et al. (1999).
If
and
are the result of a generalized Schur factorization of a matrix pair
then, optionally, the matrices
and
can be updated as
and
. Note that the condition numbers of the pair
are the same as those of the pair
.
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
5
Arguments
-
1:
– Integer
Input
-
On entry: specifies whether condition numbers are required for the cluster of eigenvalues (
and
) or the deflating subspaces (
and
).
- Only reorder with respect to select. No extras.
- Reciprocal of norms of ‘projections’ onto left and right eigenspaces with respect to the selected cluster ( and ).
- The upper bounds on and . -norm-based estimate ().
- Estimate of and . -norm-based estimate (). About five times as expensive as .
- Compute pl, pr and dif as in , and . Economic version to get it all.
- Compute pl, pr and dif as in , and .
Constraint:
.
-
2:
– Logical
Input
-
On entry: if
, update the left transformation matrix
.
If , do not update .
-
3:
– Logical
Input
-
On entry: if
, update the right transformation matrix
.
If , do not update .
-
4:
– Logical array
Input
-
On entry: specifies the eigenvalues in the selected cluster. To select a real eigenvalue
,
must be set to .TRUE..
To select a complex conjugate pair of eigenvalues and , corresponding to a by diagonal block, either or or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.
-
5:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
6:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
.
On entry: the matrix in the pair .
On exit: the updated matrix .
-
7:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08ygf is called.
Constraint:
.
-
8:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
b
must be at least
.
On entry: the matrix , in the pair .
On exit: the updated matrix
-
9:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08ygf is called.
Constraint:
.
-
10:
– Real (Kind=nag_wp) array
Output
-
On exit: see the description of
beta.
-
11:
– Real (Kind=nag_wp) array
Output
-
On exit: see the description of
beta.
-
12:
– Real (Kind=nag_wp) array
Output
-
On exit:
and
are the real and imaginary parts respectively of the
th eigenvalue, for
.
If is zero, then the th eigenvalue is real; if positive then is negative, and the th and st eigenvalues are a complex conjugate pair.
Conjugate pairs of eigenvalues correspond to the
by
diagonal blocks of
. These
by
blocks can be reduced by applying complex unitary transformations to
to obtain the complex Schur form
, where
is triangular (and complex). In this form
and
beta are the diagonals of
and
respectively.
-
13:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
q
must be at least
if
, and at least
otherwise.
On entry: if , the by matrix .
On exit: if
, the updated matrix
.
If
,
q is not referenced.
-
14:
– Integer
Input
-
On entry: the first dimension of the array
q as declared in the (sub)program from which
f08ygf is called.
Constraints:
- if , ;
- otherwise .
-
15:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
z
must be at least
if
, and at least
otherwise.
On entry: if , the by matrix .
On exit: if
, the updated matrix
.
If
,
z is not referenced.
-
16:
– Integer
Input
-
On entry: the first dimension of the array
z as declared in the (sub)program from which
f08ygf is called.
Constraints:
- if , ;
- otherwise .
-
17:
– Integer
Output
-
On exit: the dimension of the specified pair of left and right eigenspaces (deflating subspaces).
-
18:
– Real (Kind=nag_wp)
Output
-
19:
– Real (Kind=nag_wp)
Output
-
On exit: if
,
or
,
pl and
pr are lower bounds on the reciprocal of the norm of ‘projections’
and
onto left and right eigenspaces with respect to the selected cluster.
,
.
If or , .
If
,
or
,
pl and
pr are not referenced.
-
20:
– Real (Kind=nag_wp) array
Output
-
Note: the dimension of the array
dif
must be at least
.
On exit: if
,
store the estimates of
and
.
If or , are -norm-based upper bounds on and .
If or , are -norm-based estimates of and .
If or , .
If
or
,
dif is not referenced.
-
21:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
returns the minimum
lwork.
If
,
work is not referenced.
-
22:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08ygf is called.
If
, a workspace query is assumed; the routine only calculates the minimum sizes of the
work and
iwork arrays, 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 , ;
- if , or , ;
- if or , ;
- otherwise .
-
23:
– Integer array
Workspace
-
On exit: if
,
returns the minimum
liwork.
If
,
iwork is not referenced.
-
24:
– Integer
Input
-
On entry: the dimension of the array
iwork as declared in the (sub)program from which
f08ygf is called.
If
, a workspace query is assumed; the routine only calculates the minimum sizes of the
work and
iwork arrays, 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 , ;
- if or , ;
- otherwise .
-
25:
– 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.
-
Reordering of
failed because the transformed matrix pair would be too far from generalized Schur form; the problem is very ill-conditioned.
may have been partially reordered. If requested,
is returned in
and
,
pl and
pr.
7
Accuracy
The computed generalized Schur form is nearly the exact generalized Schur form for nearby matrices
and
, where
and
is the
machine precision. See Section 4.11 of
Anderson et al. (1999) for further details of error bounds for the generalized nonsymmetric eigenproblem, and for information on the condition numbers returned.
8
Parallelism and Performance
f08ygf 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 complex analogue of this routine is
f08yuf.
10
Example
This example reorders the generalized Schur factors
and
and update the matrices
and
given by
selecting the first and fourth generalized eigenvalues to be moved to the leading positions. Bases for the left and right deflating subspaces, and estimates of the condition numbers for the eigenvalues and Frobenius norm based bounds on the condition numbers for the deflating subspaces are also output.
10.1
Program Text
10.2
Program Data
10.3
Program Results