dtgsy2(3S)
DTGSY2 - solve the generalized Sylvester equation
As shipped in IRIX 6.5.22. Last changed in IRIX 6.5.19.
NAME DTGSY2 - solve the generalized Sylvester equation SYNOPSIS SUBROUTINE DTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, IWORK, PQ, INFO ) CHARACTER TRANS INTEGER IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N, PQ DOUBLE PRECISION RDSCAL, RDSUM, SCALE INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * ) IMPLEMENTATION These routines are part of the SCSL Scientific Library and can be loaded using either the -lscs or the -lscs_mp option. The -lscs_mp option directs the linker to use the multi-processor version of the library. When linking to SCSL with -lscs or -lscs_mp, the default integer size is 4 bytes (32 bits). Another version of SCSL is available in which integers are 8 bytes (64 bits). This version allows the user access to larger memory sizes and helps when porting legacy Cray codes. It can be loaded by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use only one of the two versions; 4-byte integer and 8-byte integer library calls cannot be mixed. PURPOSE DTGSY2 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, with real entries. (A, D) and (B, E) must be in generalized 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 solving equation (1) corresponds to solve Z*x = 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 Kronecker product between the matrices X and Y. In the process of solving (1), we solve a number of such systems where Dim(In), Dim(In) = 1 or 2. If TRANS = 'T', solve the transposed system Z'*y = scale*b for y, 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 is used to compute an estimate of Dif[(A, D), (B, E)] = sigma_min(Z) using reverse communicaton with DLACON. DTGSY2 also (IJOB >= 1) contributes to the computation in STGSYL of an upper bound on the separation between to matrix pairs. Then the input (A, D), (B, E) are sub-pencils of the matrix pair in DTGSYL. See STGSYL for details. ARGUMENTS TRANS (input) CHARACTER = 'N', solve the generalized Sylvester equation (1). = 'T': solve the 'transposed' 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. (DGECON 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) DOUBLE PRECISION array, dimension (LDA, M) On entry, A contains an upper quasi triangular matrix. LDA (input) INTEGER The leading dimension of the matrix A. LDA >= max(1, M). B (input) DOUBLE PRECISION array, dimension (LDB, N) On entry, B contains an upper quasi triangular matrix. LDB (input) INTEGER The leading dimension of the matrix B. LDB >= max(1, N). C (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 matrices A, B, D and E have not been changed. If SCALE = 0, R and L will hold the solutions to the homogeneous system with C = F = 0. Normally, SCALE = 1. RDSUM (input/output) DOUBLE PRECISION On entry, the sum of squares of computed contributions to the Dif-estimate under computation by DTGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL. RDSCAL (input/output) DOUBLE PRECISION 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 DTGSY2 is called by DTGSYL. IWORK (workspace) INTEGER array, dimension (M+N+2) PQ (output) INTEGER On exit, the number of subsystems (of size 2-by-2, 4-by-4 and 8- by-8) solved by this routine. INFO (output) INTEGER On exit, if INFO is set to =0: Successful exit <0: If INFO = -i, the i-th argument had an illegal value. >0: The matrix pairs (A, D) and (B, E) have common or very close eigenvalues. FURTHER DETAILS Based on contributions by Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. SEE ALSO INTRO_LAPACK(3S), INTRO_SCSL(3S) This man page is available only online.