.TH ZTGSYL 1 "February 2007" " LAPACK routine (version 3.1.1) " " LAPACK routine (version 3.1.1) "
.SH NAME
ZTGSYL - the generalized Sylvester equation
.SH SYNOPSIS
.TP 19
SUBROUTINE ZTGSYL(
TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK,
IWORK, INFO )
.TP 19
.ti +4
CHARACTER
TRANS
.TP 19
.ti +4
INTEGER
IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF,
LWORK, M, N
.TP 19
.ti +4
DOUBLE
PRECISION DIF, SCALE
.TP 19
.ti +4
INTEGER
IWORK( * )
.TP 19
.ti +4
COMPLEX*16
A( LDA, * ), B( LDB, * ), C( LDC, * ),
D( LDD, * ), E( LDE, * ), F( LDF, * ),
WORK( * )
.SH PURPOSE
ZTGSYL solves the generalized Sylvester equation:
A * R - L * B = scale * C (1)
.br
D * R - L * E = scale * F
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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
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is an output scaling factor chosen to avoid overflow.
.br
In matrix notation (1) is equivalent to solve Zx = scale*b, where Z
is defined as
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Z = [ kron(In, A) -kron(B\(aq, Im) ] (2)
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[ kron(In, D) -kron(E\(aq, Im) ],
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Here Ix is the identity matrix of size x and X\(aq is the conjugate
transpose of X. Kron(X, Y) is the Kronecker product between the
matrices X and Y.
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If TRANS = \(aqC\(aq, y in the conjugate transposed system Z\(aq*y = scale*b
is solved for, which is equivalent to solve for R and L in
A\(aq * R + D\(aq * L = scale * C (3)
.br
R * B\(aq + L * E\(aq = scale * -F
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This case (TRANS = \(aqC\(aq) 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 ZLACON.
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If IJOB >= 1, ZTGSYL 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.
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This is a level-3 BLAS algorithm.
.br
.SH ARGUMENTS
.TP 8
TRANS (input) CHARACTER*1
= \(aqN\(aq: solve the generalized sylvester equation (1).
.br
= \(aqC\(aq: solve the "conjugate transposed" system (3).
.TP 8
IJOB (input) INTEGER
Specifies what kind of functionality to be performed.
=0: solve (1) only.
.br
=1: The functionality of 0 and 3.
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=2: The functionality of 0 and 4.
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=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.
(ZGECON on sub-systems is used).
Not referenced if TRANS = \(aqC\(aq.
.TP 8
M (input) INTEGER
The order of the matrices A and D, and the row dimension of
the matrices C, F, R and L.
.TP 8
N (input) INTEGER
The order of the matrices B and E, and the column dimension
of the matrices C, F, R and L.
.TP 8
A (input) COMPLEX*16 array, dimension (LDA, M)
The upper triangular matrix A.
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1, M).
.TP 8
B (input) COMPLEX*16 array, dimension (LDB, N)
The upper triangular matrix B.
.TP 8
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1, N).
.TP 8
C (input/output) COMPLEX*16 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 = \(aqN\(aq, C holds R,
the solution achieved during the computation of the
Dif-estimate.
.TP 8
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1, M).
.TP 8
D (input) COMPLEX*16 array, dimension (LDD, M)
The upper triangular matrix D.
.TP 8
LDD (input) INTEGER
The leading dimension of the array D. LDD >= max(1, M).
.TP 8
E (input) COMPLEX*16 array, dimension (LDE, N)
The upper triangular matrix E.
.TP 8
LDE (input) INTEGER
The leading dimension of the array E. LDE >= max(1, N).
.TP 8
F (input/output) COMPLEX*16 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 = \(aqN\(aq, F holds L,
the solution achieved during the computation of the
Dif-estimate.
.TP 8
LDF (input) INTEGER
The leading dimension of the array F. LDF >= max(1, M).
.TP 8
DIF (output) 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 = \(aqC\(aq, DIF is not referenced.
.TP 8
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 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.
.TP 8
WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
.TP 8
LWORK (input) INTEGER
The dimension of the array WORK. LWORK > = 1.
If IJOB = 1 or 2 and TRANS = \(aqN\(aq, 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.
.TP 8
IWORK (workspace) INTEGER array, dimension (M+N+2)
.TP 8
INFO (output) INTEGER
=0: successful exit
.br
<0: If INFO = -i, the i-th argument had an illegal value.
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>0: (A, D) and (B, E) have common or very close
eigenvalues.
.SH FURTHER DETAILS
Based on contributions by
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Bo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S-901 87 Umea, Sweden.
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[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.
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[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.
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[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.
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