CTGSYL(1) LAPACK routine (version 3.2) CTGSYL(1)NAME
CTGSYL - solves the generalized Sylvester equation
SYNOPSIS
SUBROUTINE CTGSYL( 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
REAL DIF, SCALE
INTEGER IWORK( * )
COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ),
E( LDE, * ), F( LDF, * ), WORK( * )
PURPOSE
CTGSYL 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 complex entries. A, B, D and E are upper triangular (i.e., (A,D)
and (B,E) in generalized Schur form). 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 Ix is the identity matrix of size x and X' is the conjugate trans‐
pose of X. Kron(X, Y) is the Kronecker product between the matrices X
and Y.
If TRANS = 'C', y in the conjugate transposed system Z'*y = scale*b is
solved for, 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 = 'C') 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 CLACON.
If IJOB >= 1, CTGSYL 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.
This is a level-3 BLAS algorithm.
ARGUMENTS
TRANS (input) CHARACTER*1
= 'N': solve the generalized sylvester equation (1).
= 'C': solve the "conjugate transposed" 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 is used). =4: Only an estimate of Dif[(A,D),
(B,E)] is computed. (CGECON on sub-systems is used). Not ref‐
erenced if TRANS = 'C'.
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) COMPLEX array, dimension (LDA, M)
The upper triangular matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B (input) COMPLEX array, dimension (LDB, N)
The upper triangular matrix B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C (input/output) COMPLEX 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 (input) INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D (input) COMPLEX 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) COMPLEX 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) COMPLEX 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 (input) INTEGER
The leading dimension of the array F. LDF >= max(1, M).
DIF (output) REAL
On exit DIF is the reciprocal of a lower bound of the recipro‐
cal 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 = 'C', DIF is not referenced.
SCALE (output) REAL
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, R and L will hold the
solutions to the homogenious system with C = F = 0.
WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
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 >= 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 (workspace) INTEGER array, dimension (M+N+2)
INFO (output) 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 very close eigenvalues.
FURTHER DETAILS
Based on contributions by
Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
[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.
LAPACK routine (version 3.2) November 2008 CTGSYL(1)
