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CTGSY2(l)) CTGSY2(l)NAMECTGSY2 - solve the generalized Sylvester equation A * R - L * B = scale * C (1) D * R - L * E = scale * F using Level 1 and 2 BLAS, where R and L are unknown M-by-N matrices,SYNOPSISSUBROUTINE CTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO ) CHARACTER TRANS INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N REAL RDSCAL, RDSUM, SCALE COMPLEX A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * )PURPOSECTGSY2 solves the generalized Sylvester equation A * R - L * B = scale * C (1) D * R - L * E = scale * F using Level 1 and 2 BLAS, 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. 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 solving equation (1) corresponds to solve Zx = scale * b, where Z is defined as Z = [ kron(In, A) -kron(B', Im) ] (2) [ kron(In, D) -kron(E', Im) ], Ik is the identity matrix of size k and X' is the transpose of X. kron(X, Y) is the Kro- necker 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 *This case is used to compute an estimate of Dif[(A, D), (B, E)] = = sigma_min(Z) using reverse communicaton with CLACON. CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL of an upper bound on the separation between to matrix pairs. Then the input (A, D), (B, E) are sub-pencils of two matrix pairs in CTGSYL.-FARGUMENTSTRANS (input) CHARACTER = 'N', solve the generalized Sylvester equation (1). = 'T': solve the 'trans- posed' system (3). IJOB (input) INTEGER Specifies what kind of functionality to be performed. =0: solve (1) only. =1: A contribution from this subsystem to a Frobenius norm-based estimate of the separation between two matrix pairs is computed. (look ahead strategy is used). =2: A contribution from this subsystem to a Frobenius norm-based estimate of the separation between two matrix pairs is computed. (SGECON on sub-systems is used.) Not referenced if TRANS = 'T'. M (input) INTEGER On entry, M specifies the order of A and D, and the row dimension of C, F, R and L. N (input) INTEGER On entry, N specifies the order of B and E, and the column dimension of C, F, R and L. A (input) COMPLEX array, dimension (LDA, M) On entry, A contains an upper triangular matrix. LDA (input) INTEGER The leading dimension of the matrix A. LDA >= max(1, M). B (input) COMPLEX array, dimension (LDB, N) On entry, B contains an upper triangular matrix. LDB (input) INTEGER The leading dimension of the matrix B. LDB >= max(1, N). C (input/ output) COMPLEX array, dimension (LDC, N) On entry, C contains the right-hand-side of the first matrix equation in (1). On exit, if IJOB = 0, C has been overwritten by the solution R. LDC (input) INTEGER The leading dimension of the matrix C. LDC >= max(1, M). D (input) COMPLEX array, dimension (LDD, M) On entry, D contains an upper triangular matrix. LDD (input) INTEGER The leading dimension of the matrix D. LDD >= max(1, M). E (input) COMPLEX array, dimension (LDE, N) On entry, E contains an upper triangular matrix. LDE (input) INTEGER The leading dimension of the matrix E. LDE >= max(1, N). F (input/ output) COMPLEX array, dimension (LDF, N) On entry, F contains the right-hand-side of the second matrix equation in (1). On exit, if IJOB = 0, F has been overwritten by the solution L. LDF (input) INTEGER The leading dimension of the matrix F. LDF >= max(1, M). SCALE (output) REAL On exit, 0 <= SCALE <= 1. If 0 < SCALE < 1, the solutions R and L (C and F on entry) will hold the solutions to a slightly perturbed system but the input matri- ces A, B, D and E have not been changed. If SCALE = 0, R and L will hold the solu- tions to the homogeneous system with C = F = 0. Normally, SCALE = 1. RDSUM (input/output) REAL On entry, the sum of squares of computed contributions to the Dif-estimate under computation by CTGSYL, where the scaling factor RDSCAL (see below) has been fac- tored out. On exit, the corresponding sum of squares updated with the contribu- tions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL. RDSCAL (input/output) REAL On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when CTGSY2 is called by CTGSYL. INFO (output) INTEGER On exit, if INFO is set to =0: Successful exit <0: If INFO =, input argument number i is illegal. >0: The matrix pairs (A, D) and (B, E) have common or very close eigenvalues.-iFURTHER DETAILSBased on contributions by Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.LAPACK version 3.015 June 2000 CTGSY2(l)

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