dtgsyl (l)  Linux Manuals
dtgsyl: solves the generalized Sylvester equation
NAME
DTGSYL  solves the generalized Sylvester equationSYNOPSIS
 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: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 triangular 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
Here Ik is the identity matrix of size k and Xaq is the transpose of X. kron(X, Y) is the Kronecker product between the matrices X and Y. If TRANS = aqTaq, DTGSYL solves the transposed system Zaq*y = scale*b, which is equivalent to solve for R and L in
This case (TRANS = aqTaq) 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
 = aqNaq, solve the generalized Sylvester equation (1). = aqTaq, solve the aqtransposedaq 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 = aqTaq.  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 = aqNaq, C holds R, the solution achieved during the computation 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 = aqNaq, F holds L, the solution achieved during the computation 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 Diffunction, 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 = aqTaq, 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 matrices 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 (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 = aqNaq, 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+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 byBo Kagstrom and Peter Poromaa, Department of Computing Science,
Umea University, S901 87 Umea, Sweden.
[1] B. Kagstrom and P. Poromaa, LAPACKStyle Algorithms and Software
[2] B. Kagstrom, A Perturbation Analysis of the Generalized Sylvester
[3] B. Kagstrom and L. Westin, Generalized Schur Methods with