CTGSYL (3)  Linux Manuals
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).
IJOBIJOB 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'.
MM is INTEGER The order of the matrices A and D, and the row dimension of the matrices C, F, R and L.
NN is INTEGER The order of the matrices B and E, and the column dimension of the matrices C, F, R and L.
AA is COMPLEX array, dimension (LDA, M) The upper triangular matrix A.
LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1, M).
BB is COMPLEX array, dimension (LDB, N) The upper triangular matrix B.
LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1, N).
CC 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.
LDCLDC is INTEGER The leading dimension of the array C. LDC >= max(1, M).
DD is COMPLEX array, dimension (LDD, M) The upper triangular matrix D.
LDDLDD is INTEGER The leading dimension of the array D. LDD >= max(1, M).
EE is COMPLEX array, dimension (LDE, N) The upper triangular matrix E.
LDELDE is INTEGER The leading dimension of the array E. LDE >= max(1, N).
FF 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.
LDFLDF is INTEGER The leading dimension of the array F. LDF >= max(1, M).
DIFDIF 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.
SCALESCALE 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.
WORKWORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORKLWORK 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.
IWORKIWORK is INTEGER array, dimension (M+N+2)
INFOINFO 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., H(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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