ztgsyl(3S)
ZTGSYL - solve the generalized Sylvester equation
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NAME ZTGSYL - solve the generalized Sylvester equation SYNOPSIS SUBROUTINE ZTGSYL( 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( * ) COMPLEX*16 A( LDA, * ), B( LDB, * ), C( LDC, * ), D( LDD, * ), E( LDE, * ), F( LDF, * ), WORK( * ) 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 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. ARGUMENTS 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. FURTHER DETAILS 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. SEE ALSO INTRO_LAPACK(3S), INTRO_SCSL(3S) This man page is available only online.