
DTGSYL(l) ) DTGSYL(l)
NAME
DTGSYL  solve the generalized Sylvester equation
SYNOPSIS
SUBROUTINE DTGSYL( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF,
SCALE, DIF, WORK, LWORK, IWORK, INFO )
CHARACTER TRANS
INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, M, N
DOUBLE PRECISION DIF, SCALE
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, *
), F( LDF, * ), WORK( * )
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 mbyn matrices, (A, D), (B, E) and (C, F) are given matrix
pairs of size mbym, nbyn and mbyn, respectively, with real entries. (A, D) and (B,
E) must be in generalized (real) Schur canonical form, i.e. A, B are upper quasi triangu
lar 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', Im) ] (2)
[ kron(In, D) kron(E', Im) ].
Here Ik is the identity matrix of size k and X' 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'*y = scale*b, which is equivalent to
solve for R and L in
A' * R + D' * L = scale * C (3)
R * B' + L * E' = scale * (F)
This case (TRANS = 'T') is used to compute an onenormbased 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 normbased 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 [12] for more information.
This is a level 3 BLAS algorithm.
ARGUMENTS
TRANS (input) CHARACTER*1
= 'N', solve the generalized Sylvester equation (1). = 'T', solve the 'trans
posed' system (3).
IJOB (input) 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
subsystems is used ). Not referenced if TRANS = 'T'.
M (input) INTEGER
The order of the matrices A and D, and the row dimension of the matrices C, F, R
and L.
N (input) INTEGER
The order of the matrices B and E, and the column dimension of the matrices C, F,
R and L.
A (input) DOUBLE PRECISION array, dimension (LDA, M)
The upper quasi triangular matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B (input) DOUBLE PRECISION array, dimension (LDB, N)
The upper quasi triangular matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C (input/output) DOUBLE PRECISION array, dimension (LDC, N)
On entry, C contains the righthandside 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 compu
tation of the Difestimate.
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D (input) DOUBLE PRECISION array, dimension (LDD, M)
The upper triangular matrix D.
LDD (input) INTEGER
The leading dimension of the array D. LDD >= max(1, M).
E (input) DOUBLE PRECISION array, dimension (LDE, N)
The upper triangular matrix E.
LDE (input) INTEGER
The leading dimension of the array E. LDE >= max(1, N).
F (input/output) DOUBLE PRECISION array, dimension (LDF, N)
On entry, F contains the righthandside 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 compu
tation of the Difestimate.
LDF (input) INTEGER
The leading dimension of the array F. LDF >= max(1, M).
DIF (output) DOUBLE PRECISION
On exit DIF is the reciprocal of a lower bound of the reciprocal of the Diffunc
tion, 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 (output) 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 matri
ces 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 (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
If IJOB = 0, WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1)
returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK > = 1. If IJOB = 1 or 2 and TRANS = 'N',
LWORK >= 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 (workspace) INTEGER array, dimension (M+N+6)
INFO (output) INTEGER
=0: successful exit
<0: If INFO = i, the ith argument had an illegal value.
>0: (A, D) and (B, E) have common or close eigenvalues.
FURTHER DETAILS
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.
[1] B. Kagstrom and P. Poromaa, LAPACKStyle 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, S901 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):10451060, 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 745751.
LAPACK version 3.0 15 June 2000 DTGSYL(l) 
