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Livre :
Expressions régulières,
Syntaxe et mise en oeuvre :
GNU/Linux |
CentOS 4.8 |
i386 |
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ztgsyl(l) |
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ZTGSYL - solve the generalized Sylvester equation
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SUBROUTINE ZTGSYL( |
TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO ) | ||
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CHARACTER |
TRANS | ||
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INTEGER |
IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, M, N | ||
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DOUBLE |
PRECISION DIF, SCALE | ||
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INTEGER |
IWORK( * ) | ||
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COMPLEX*16 |
A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * ), WORK( * ) |
ZTGSYL 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 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’*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 ZLACON.
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.
This is a level-3 BLAS algorithm.
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. (ZGECON on sub-systems is used). Not
referenced 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*16 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*16 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*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 = ’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*16 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*16 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*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 = ’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) 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 = ’C’, DIF is not referenced.
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.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
IF IJOB = 0, WORK is not referenced. Otherwise,
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+2)
If IJOB = 0, IWORK is not referenced.
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.
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.
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ztgsyl(l) | |