NAG Library Routine Document
F08YHF (DTGSYL)
1 Purpose
F08YHF (DTGSYL) solves the generalized real quasi-triangular Sylvester equations.
2 Specification
| 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 |
IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, IWORK(max(1,M+N+6)), INFO |
| REAL (KIND=nag_wp) |
A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), SCALE, DIF, WORK(max(1,LWORK)) |
| CHARACTER(1) |
TRANS |
|
The routine may be called by its
LAPACK
name dtgsyl.
3 Description
F08YHF (DTGSYL) solves either the generalized real Sylvester equations
or the equations
where the pair
A,D are given
m by
m matrices in real generalized Schur form,
B,E are given
n by
n matrices in real generalized Schur form and
C,F are given
m by
n matrices. The pair
R,L are the
m by
n solution matrices, and
α is an output scaling factor determined by the routine to avoid overflow in computing
R,L.
Equations
(1) are equivalent to equations of the form
where
and
⊗ is the Kronecker product. Equations
(2) are then equivalent to
The pair
S,T are in real generalized Schur form if
S is block upper triangular with
1 by
1 and
2 by
2 diagonal blocks on the diagonal and
T is upper triangular as returned, for example, by
F08XAF (DGGES), or
F08XEF (DHGEQZ) with
JOB='S'.
Optionally, the routine estimates
DifA,D,B,E, the separation between the matrix pairs
A,D and
B,E, which is the smallest singular value of
Z. The estimate can be based on either the Frobenius norm, or the
1-norm. The
1-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/lugKågström B (1994) A perturbation analysis of the generalized Sylvester equation
AR-LB,DR-LE=c,F 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 Parameters
- 1: TRANS – CHARACTER(1)Input
-
On entry: if
TRANS='N', solve the generalized Sylvester equation
(1).
If
TRANS='T', solve the ‘transposed’ system
(2).
Constraint:
TRANS='N' or 'T'.
- 2: IJOB – INTEGERInput
-
On entry: specifies what kind of functionality is to be performed when
TRANS='N'.
- IJOB=0
- Solve (1) only.
- IJOB=1
- The functionality of IJOB=0 and 3.
- IJOB=2
- The functionality of IJOB=0 and 4.
- IJOB=3
- Only an estimate of
Dif
A,D,B,E
is computed based on the Frobenius norm.
- IJOB=4
- Only an estimate of
Dif
A,D,B,E
is computed based on the 1-norm.
If
TRANS='T',
IJOB is not referenced.
- 3: M – INTEGERInput
-
On entry: m, the order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
Constraint:
M≥0.
- 4: N – INTEGERInput
-
On entry:
n, the order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
Constraint:
N≥0.
- 5: A(LDA,*) – REAL (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
A
must be at least
max1,M.
On entry: the upper
quasi-triangular matrix A.
- 6: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint:
LDA≥max1,M.
- 7: B(LDB,*) – REAL (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the upper
quasi-triangular
matrix B.
- 8: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint:
LDB≥max1,N.
- 9: C(LDC,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
C
must be at least
max1,N.
On entry: contains the right-hand-side matrix C.
On exit: if
TRANS='T' or
'N' and
IJOB=0,
1 or
2,
C is overwritten by the solution matrix
R.
If
TRANS='N' and
IJOB=3 or
4,
C holds
R, the solution achieved during the computation of the Dif estimate.
- 10: LDC – INTEGERInput
-
On entry: the first dimension of the array
C as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint:
LDC≥max1,M.
- 11: D(LDD,*) – REAL (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
D
must be at least
max1,M.
On entry: the upper triangular matrix D.
- 12: LDD – INTEGERInput
-
On entry: the first dimension of the array
D as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint:
LDD≥max1,M.
- 13: E(LDE,*) – REAL (KIND=nag_wp) arrayInput
-
Note: the second dimension of the array
E
must be at least
max1,N.
On entry: the upper triangular matrix E.
- 14: LDE – INTEGERInput
-
On entry: the first dimension of the array
E as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint:
LDE≥max1,N.
- 15: F(LDF,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
F
must be at least
max1,N.
On entry: contains the right-hand side matrix F.
On exit: if
TRANS='T' or
'N' and
IJOB=0,
1 or
2,
F is overwritten by the solution matrix
L.
If
TRANS='N' and
IJOB=3 or
4,
F holds
L, the solution achieved during the computation of the Dif estimate.
- 16: LDF – INTEGERInput
-
On entry: the first dimension of the array
F as declared in the (sub)program from which F08YHF (DTGSYL) is called.
Constraint:
LDF≥max1,M.
- 17: SCALE – REAL (KIND=nag_wp)Output
-
On exit:
α, the scaling factor in
(1) or
(2).
If
0<SCALE<1,
C and
F hold the solutions
R and
L, respectively, to a slightly perturbed system but the input arrays
A,
B,
D and
E have not been changed.
If
SCALE=0,
C and
F hold the solutions
R and
L, respectively, to the homogeneous system with
C=F=0. In this case
DIF is not referenced.
Normally, SCALE=1.
- 18: DIF – REAL (KIND=nag_wp)Output
-
On exit: the estimate of
Dif. If
TRANS='T' or
'N' and
IJOB=0,
DIF is not referenced.
- 19: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 20: LWORK – INTEGERInput
-
On entry:
the dimension of the array
WORK as declared in the (sub)program from which F08YHF (DTGSYL) is called.
If
LWORK=-1, 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
LWORK≠-1,
- if TRANS='N' and IJOB=1 or 2, LWORK≥max1,2×M×N;
- otherwise LWORK≥1.
.
- 21: IWORK(max1,M+N+6) – INTEGER arrayWorkspace
- 22: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
A,D and B,E 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 Further Comments
The total number of floating point operations needed to solve the generalized Sylvester equations is approximately 2mnn+m. The Frobenius norm estimate of Dif does not require additional significant computation, but the 1-norm estimate is typically five times more expensive.
The complex analogue of this routine is
F08YVF (ZTGSYL).
9 Example
This example solves the generalized Sylvester equations
where
9.1 Program Text
Program Text (f08yhfe.f90)
9.2 Program Data
Program Data (f08yhfe.d)
9.3 Program Results
Program Results (f08yhfe.r)