NAG FL Interface
f08yhf (dtgsyl)
1
Purpose
f08yhf solves the generalized real quasi-triangular Sylvester equations.
2
Specification
Fortran Interface
Subroutine f08yhf ( |
trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info) |
Integer, Intent (In) |
:: |
ijob, m, n, lda, ldb, ldc, ldd, lde, ldf, lwork |
Integer, Intent (Out) |
:: |
iwork(max(1,m+n+6)), info |
Real (Kind=nag_wp), Intent (In) |
:: |
a(lda,*), b(ldb,*), d(ldd,*), e(lde,*) |
Real (Kind=nag_wp), Intent (Inout) |
:: |
c(ldc,*), f(ldf,*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
scale, dif, work(max(1,lwork)) |
Character (1), Intent (In) |
:: |
trans |
|
C Header Interface
#include <nag.h>
void |
f08yhf_ (const char *trans, const Integer *ijob, const Integer *m, const Integer *n, const double a[], const Integer *lda, const double b[], const Integer *ldb, double c[], const Integer *ldc, const double d[], const Integer *ldd, const double e[], const Integer *lde, double f[], const Integer *ldf, double *scal, double *dif, double work[], const Integer *lwork, Integer iwork[], Integer *info, const Charlen length_trans) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08yhf_ (const char *trans, const Integer &ijob, const Integer &m, const Integer &n, const double a[], const Integer &lda, const double b[], const Integer &ldb, double c[], const Integer &ldc, const double d[], const Integer &ldd, const double e[], const Integer &lde, double f[], const Integer &ldf, double &scal, double &dif, double work[], const Integer &lwork, Integer iwork[], Integer &info, const Charlen length_trans) |
}
|
The routine may be called by the names f08yhf, nagf_lapackeig_dtgsyl or its LAPACK name dtgsyl.
3
Description
f08yhf solves either the generalized real Sylvester equations
or the equations
where the pair
are given
by
matrices in real generalized Schur form,
are given
by
matrices in real generalized Schur form and
are given
by
matrices. The pair
are the
by
solution matrices, and
is an output scaling factor determined by the routine to avoid overflow in computing
.
Equations
(1) are equivalent to equations of the form
where
and
is the Kronecker product. Equations
(2) are then equivalent to
The pair
are in real generalized Schur form if
is block upper triangular with
by
and
by
diagonal blocks on the diagonal and
is upper triangular as returned, for example, by
f08xaf, or
f08xef with
.
Optionally, the routine estimates
, the separation between the matrix pairs
and
, which is the smallest singular value of
. The estimate can be based on either the Frobenius norm, or the
-norm. The
-norm estimate can be three to ten times more expensive than the Frobenius norm estimate, but makes the condition estimation uniform with the nonsymmetric eigenproblem. The Frobenius norm estimate provides a low cost, but equally reliable estimate. For more information see Sections 2.4.8.3 and 4.11.1.3 of
Anderson et al. (1999) and
Kågström and Poromaa (1996).
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
Kågström B (1994) A perturbation analysis of the generalized Sylvester equation SIAM J. Matrix Anal. Appl. 15 1045–1060
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103
5
Arguments
-
1:
– Character(1)
Input
-
On entry: if
, solve the generalized Sylvester equation
(1).
If
, solve the ‘transposed’ system
(2).
Constraint:
or .
-
2:
– Integer
Input
-
On entry: specifies what kind of functionality is to be performed when
.
- Solve (1) only.
- The functionality of and .
- The functionality of and .
- Only an estimate of is computed based on the Frobenius norm.
- Only an estimate of is computed based on the -norm.
If
,
ijob is not referenced.
Constraint:
if , .
-
3:
– Integer
Input
-
On entry: , the order of the matrices and , and the row dimension of the matrices , , and .
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the order of the matrices and , and the column dimension of the matrices , , and .
Constraint:
.
-
5:
– Real (Kind=nag_wp) array
Input
-
Note: the second dimension of the array
a
must be at least
.
On entry: the upper quasi-triangular matrix .
-
6:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08yhf is called.
Constraint:
.
-
7:
– Real (Kind=nag_wp) array
Input
-
Note: the second dimension of the array
b
must be at least
.
On entry: the upper quasi-triangular matrix .
-
8:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08yhf is called.
Constraint:
.
-
9:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
c
must be at least
.
On entry: contains the right-hand-side matrix .
On exit: if
,
or
,
c is overwritten by the solution matrix
.
If
and
or
,
c holds
, the solution achieved during the computation of the Dif estimate.
-
10:
– Integer
Input
-
On entry: the first dimension of the array
c as declared in the (sub)program from which
f08yhf is called.
Constraint:
.
-
11:
– Real (Kind=nag_wp) array
Input
-
Note: the second dimension of the array
d
must be at least
.
On entry: the upper triangular matrix .
-
12:
– Integer
Input
-
On entry: the first dimension of the array
d as declared in the (sub)program from which
f08yhf is called.
Constraint:
.
-
13:
– Real (Kind=nag_wp) array
Input
-
Note: the second dimension of the array
e
must be at least
.
On entry: the upper triangular matrix .
-
14:
– Integer
Input
-
On entry: the first dimension of the array
e as declared in the (sub)program from which
f08yhf is called.
Constraint:
.
-
15:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
f
must be at least
.
On entry: contains the right-hand side matrix .
On exit: if
,
or
,
f is overwritten by the solution matrix
.
If
and
or
,
f holds
, the solution achieved during the computation of the Dif estimate.
-
16:
– Integer
Input
-
On entry: the first dimension of the array
f as declared in the (sub)program from which
f08yhf is called.
Constraint:
.
-
17:
– Real (Kind=nag_wp)
Output
-
On exit:
, the scaling factor in
(1) or
(2).
If
,
c and
f hold the solutions
and
, respectively, to a slightly perturbed system but the input arrays
a,
b,
d and
e have not been changed.
If
,
c and
f hold the solutions
and
, respectively, to the homogeneous system with
. In this case
dif is not referenced.
Normally, .
-
18:
– Real (Kind=nag_wp)
Output
-
On exit: the estimate of
. If
,
dif is not referenced.
-
19:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
contains the minimum value of
lwork required for optimal performance.
-
20:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08yhf is called.
If
, a workspace query is assumed; the routine only calculates the minimum 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.
Constraints:
if
,
- if and or , ;
- otherwise .
-
21:
– Integer array
Workspace
-
-
22:
– 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.
-
and have common or close eigenvalues and so no solution could be computed.
7
Accuracy
See
Kågström (1994) for a perturbation analysis of the generalized Sylvester equation.
8
Parallelism and Performance
f08yhf 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 needed to solve the generalized Sylvester equations is approximately . The Frobenius norm estimate of does not require additional significant computation, but the -norm estimate is typically five times more expensive.
The complex analogue of this routine is
f08yvf.
10
Example
This example solves the generalized Sylvester equations
where
10.1
Program Text
10.2
Program Data
10.3
Program Results