
ctgsyl.f(3) LAPACK ctgsyl.f(3)
NAME
ctgsyl.f 
SYNOPSIS
Functions/Subroutines
subroutine ctgsyl (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF,
SCALE, DIF, WORK, LWORK, IWORK, INFO)
CTGSYL
Function/Subroutine Documentation
subroutine ctgsyl (characterTRANS, integerIJOB, integerM, integerN, complex, dimension( lda, *
)A, integerLDA, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldc, * )C,
integerLDC, complex, dimension( ldd, * )D, integerLDD, complex, dimension( lde, * )E,
integerLDE, complex, dimension( ldf, * )F, integerLDF, realSCALE, realDIF, complex,
dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)
CTGSYL
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 mbyn matrices, (A, D), (B, E) and
(C, F) are given matrix pairs of size mbym, nbyn and mbyn,
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**H, Im) ] (2)
[ kron(In, D) kron(E**H, Im) ],
Here Ix is the identity matrix of size x and X**H is the conjugate
transpose 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**H *y = scale*b
is solved for, which is equivalent to solve for R and L in
A**H * R + D**H * L = scale * C (3)
R * B**H + L * E**H = scale * F
This case (TRANS = 'C') 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 CLACON.
If IJOB >= 1, CTGSYL 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.
This is a level3 BLAS algorithm.
Parameters:
TRANS
TRANS is CHARACTER*1
= 'N': solve the generalized sylvester equation (1).
= 'C': solve the "conjugate 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 is used).
=4: Only an estimate of Dif[(A,D), (B,E)] is computed.
(CGECON on subsystems is used).
Not referenced if TRANS = 'C'.
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 COMPLEX array, dimension (LDA, M)
The upper triangular matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1, M).
B
B is COMPLEX array, dimension (LDB, N)
The upper triangular matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1, N).
C
C is COMPLEX 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 computation of the
Difestimate.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1, M).
D
D is COMPLEX 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 COMPLEX 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 COMPLEX 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 computation of the
Difestimate.
LDF
LDF is INTEGER
The leading dimension of the array F. LDF >= max(1, M).
DIF
DIF is REAL
On exit DIF is the reciprocal of a lower bound of the
reciprocal of the Diffunction, i.e. DIF is an upper bound of
Dif[(A,D), (B,E)] = sigmamin(Z), where Z as in (2).
IF IJOB = 0 or TRANS = 'C', DIF is not referenced.
SCALE
SCALE is 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
WORK is COMPLEX 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+2)
INFO
INFO is INTEGER
=0: successful exit
<0: If INFO = i, the ith argument had an illegal value.
>0: (A, D) and (B, E) have common or very close
eigenvalues.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S901
87 Umea, Sweden.
References:
[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.
Definition at line 294 of file ctgsyl.f.
Author
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