subroutine dtgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO)
DTGSYL
DTGSYL
Purpose:
DTGSYL solves the generalized Sylvester equation:
A * R - L * B = scale * C (1)
D * R - L * E = scale * F
where R and L are unknown m-by-n matrices, (A, D), (B, E) and
(C, F) are given matrix pairs of size m-by-m, n-by-n and m-by-n,
respectively, with real entries. (A, D) and (B, E) must be in
generalized (real) Schur canonical form, i.e. A, B are upper quasi
triangular and D, E are upper triangular.
The solution (R, L) overwrites (C, F). 0 <= SCALE <= 1 is an output
scaling factor chosen to avoid overflow.
In matrix notation (1) is equivalent to solve Zx = scale b, where
Z is defined as
Z = [ kron(In, A) -kron(B**T, Im) ] (2)
[ kron(In, D) -kron(E**T, Im) ].
Here Ik is the identity matrix of size k and X**T is the transpose of
X. kron(X, Y) is the Kronecker product between the matrices X and Y.
If TRANS = 'T', DTGSYL solves the transposed system Z**T*y = scale*b,
which is equivalent to solve for R and L in
A**T * R + D**T * L = scale * C (3)
R * B**T + L * E**T = scale * -F
This case (TRANS = 'T') is used to compute an one-norm-based estimate
of Dif[(A,D), (B,E)], the separation between the matrix pairs (A,D)
and (B,E), using DLACON.
If IJOB >= 1, DTGSYL computes a Frobenius norm-based estimate
of Dif[(A,D),(B,E)]. That is, the reciprocal of a lower bound on the
reciprocal of the smallest singular value of Z. See [1-2] for more
information.
This is a level 3 BLAS algorithm.
Parameters:
TRANS is CHARACTER*1
= 'N', solve the generalized Sylvester equation (1).
= 'T', solve the 'transposed' system (3).
IJOB
IJOB is INTEGER
Specifies what kind of functionality to be performed.
=0: solve (1) only.
=1: The functionality of 0 and 3.
=2: The functionality of 0 and 4.
=3: Only an estimate of Dif[(A,D), (B,E)] is computed.
(look ahead strategy IJOB = 1 is used).
=4: Only an estimate of Dif[(A,D), (B,E)] is computed.
( DGECON on sub-systems is used ).
Not referenced if TRANS = 'T'.
M
M is INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.
N
N is INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.
A
A is DOUBLE PRECISION array, dimension (LDA, M)
The upper quasi triangular matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B
B is DOUBLE PRECISION array, dimension (LDB, N)
The upper quasi triangular matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C
C is DOUBLE PRECISION array, dimension (LDC, N)
On entry, C contains the right-hand-side of the first matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, C has been overwritten by
the solution R. If IJOB = 3 or 4 and TRANS = 'N', C holds R,
the solution achieved during the computation of the
Dif-estimate.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D
D is DOUBLE PRECISION array, dimension (LDD, M)
The upper triangular matrix D.
LDD
LDD is INTEGER
The leading dimension of the array D. LDD >= max(1, M).
E
E is DOUBLE PRECISION array, dimension (LDE, N)
The upper triangular matrix E.
LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1, N).
F
F is DOUBLE PRECISION array, dimension (LDF, N)
On entry, F contains the right-hand-side of the second matrix
equation in (1) or (3).
On exit, if IJOB = 0, 1 or 2, F has been overwritten by
the solution L. If IJOB = 3 or 4 and TRANS = 'N', F holds L,
the solution achieved during the computation of the
Dif-estimate.
LDF
LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M).
DIF
DIF is DOUBLE PRECISION
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Dif-function, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigma_min(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'T', DIF is not touched.
SCALE
SCALE is DOUBLE PRECISION
On exit SCALE is the scaling factor in (1) or (3).
If 0 < SCALE < 1, C and F hold the solutions R and L, resp.,
to a slightly perturbed system but the input matrices 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. Normally, SCALE = 1.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = 'N', LWORK >= max(1,2*M*N).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal 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 by XERBLA.
IWORK
IWORK is INTEGER array, dimension (M+N+6)
INFO
INFO is INTEGER
=0: successful exit
<0: If INFO = -i, the i-th argument had an illegal value.
>0: (A, D) and (B, E) have common or close eigenvalues.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Contributors:
References:
[1] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
for Solving the Generalized Sylvester Equation and Estimating the
Separation between Regular Matrix Pairs, Report UMINF - 93.23,
Department of Computing Science, Umea University, S-901 87 Umea,
Sweden, December 1993, Revised April 1994, Also as LAPACK Working
Note 75. To appear in ACM Trans. on Math. Software, Vol 22,
No 1, 1996.
[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
Equation (AR - LB, DR - LE ) = (C, F), SIAM J. Matrix Anal.
Appl., 15(4):1045-1060, 1994
[3] B. Kagstrom and L. Westin, Generalized Schur Methods with
Condition Estimators for Solving the Generalized Sylvester
Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7,
July 1989, pp 745-751.
Definition at line 301 of file dtgsyl.f.
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