NAG Library Function Document
nag_ztgsyl (f08yvc)
1
Purpose
nag_ztgsyl (f08yvc) solves the generalized complex triangular Sylvester equations.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_ztgsyl (Nag_OrderType order,
Nag_TransType trans,
Integer ijob,
Integer m,
Integer n,
const Complex a[],
Integer pda,
const Complex b[],
Integer pdb,
Complex c[],
Integer pdc,
const Complex d[],
Integer pdd,
const Complex e[],
Integer pde,
Complex f[],
Integer pdf,
double *scale,
double *dif,
NagError *fail) |
|
3
Description
nag_ztgsyl (f08yvc) solves either the generalized complex Sylvester equations
or the equations
where the pair
are given
by
matrices in generalized Schur form,
are given
by
matrices in generalized Schur form and
are given
by
matrices. The pair
are the
by
solution matrices, and
is an output scaling factor determined by the function 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 generalized Schur form if
and
are upper triangular as returned, for example, by
nag_zgges (f08xnc), or
nag_zhgeqz (f08xsc) with
.
Optionally, the function 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
http://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:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_TransTypeInput
-
On entry: if
, solve the generalized Sylvester equation
(1).
If
, solve the ‘conjugate transposed’ system
(2).
Constraint:
or .
- 3:
– IntegerInput
-
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 , .
- 4:
– IntegerInput
-
On entry: , the order of the matrices and , and the row dimension of the matrices , , and .
Constraint:
.
- 5:
– IntegerInput
-
On entry: , the order of the matrices and , and the column dimension of the matrices , , and .
Constraint:
.
- 6:
– const ComplexInput
-
Note: the dimension,
dim, of the array
a
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the upper triangular matrix .
- 7:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 8:
– const ComplexInput
-
Note: the dimension,
dim, of the array
b
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the upper triangular matrix .
- 9:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraint:
.
- 10:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
c
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
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.
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
c.
Constraints:
- if ,
;
- if , .
- 12:
– const ComplexInput
-
Note: the dimension,
dim, of the array
d
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the upper triangular matrix .
- 13:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
d.
Constraint:
.
- 14:
– const ComplexInput
-
Note: the dimension,
dim, of the array
e
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the upper triangular matrix .
- 15:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
e.
Constraint:
.
- 16:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
f
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
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.
- 17:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
f.
Constraints:
- if ,
;
- if , .
- 18:
– double *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, .
- 19:
– double *Output
-
On exit: the estimate of
. If
,
dif is not referenced.
- 20:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_EIGENVALUES
-
and have common or close eigenvalues and so no solution could be computed.
- NE_ENUM_INT
-
On entry, and .
Constraint: if , .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
See
Kågström (1994) for a perturbation analysis of the generalized Sylvester equation.
8
Parallelism and Performance
nag_ztgsyl (f08yvc) 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 function. 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 real analogue of this function is
nag_dtgsyl (f08yhc).
10
Example
This example solves the generalized Sylvester equations
where
and
10.1
Program Text
Program Text (f08yvce.c)
10.2
Program Data
Program Data (f08yvce.d)
10.3
Program Results
Program Results (f08yvce.r)