ztgsyl - solve the generalized Sylvester equation
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*1 TRANS DOUBLE COMPLEX A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), WORK(*) INTEGER IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER IWORK(*) DOUBLE PRECISION SCALE, DIF SUBROUTINE ZTGSYL_64(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) CHARACTER*1 TRANS DOUBLE COMPLEX A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), WORK(*) INTEGER*8 IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER*8 IWORK(*) DOUBLE PRECISION SCALE, DIF F95 INTERFACE SUBROUTINE TGSYL(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: TRANS COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, B, C, D, E, F INTEGER :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER, DIMENSION(:) :: IWORK REAL(8) :: SCALE, DIF SUBROUTINE TGSYL_64(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: TRANS COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, B, C, D, E, F INTEGER(8) :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER(8), DIMENSION(:) :: IWORK REAL(8) :: SCALE, DIF C INTERFACE #include <sunperf.h> void ztgsyl(char trans, int ijob, int m, int n, doublecomplex *a, int lda, doublecomplex *b, int ldb, doublecomplex *c, int ldc, doublecomplex *d, int ldd, doublecomplex *e, int lde, double- complex *f, int ldf, double *scale, double *dif, int *info); void ztgsyl_64(char trans, long ijob, long m, long n, doublecomplex *a, long lda, doublecomplex *b, long ldb, doublecomplex *c, long ldc, doublecomplex *d, long ldd, doublecomplex *e, long lde, doublecomplex *f, long ldf, double *scale, double *dif, long *info);
Oracle Solaris Studio Performance Library ztgsyl(3P) 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*1 TRANS DOUBLE COMPLEX A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), WORK(*) INTEGER IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER IWORK(*) DOUBLE PRECISION SCALE, DIF SUBROUTINE ZTGSYL_64(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) CHARACTER*1 TRANS DOUBLE COMPLEX A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), E(LDE,*), F(LDF,*), WORK(*) INTEGER*8 IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER*8 IWORK(*) DOUBLE PRECISION SCALE, DIF F95 INTERFACE SUBROUTINE TGSYL(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: TRANS COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, B, C, D, E, F INTEGER :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER, DIMENSION(:) :: IWORK REAL(8) :: SCALE, DIF SUBROUTINE TGSYL_64(TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD, E, LDE, F, LDF, SCALE, DIF, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: TRANS COMPLEX(8), DIMENSION(:) :: WORK COMPLEX(8), DIMENSION(:,:) :: A, B, C, D, E, F INTEGER(8) :: IJOB, M, N, LDA, LDB, LDC, LDD, LDE, LDF, LWORK, INFO INTEGER(8), DIMENSION(:) :: IWORK REAL(8) :: SCALE, DIF C INTERFACE #include <sunperf.h> void ztgsyl(char trans, int ijob, int m, int n, doublecomplex *a, int lda, doublecomplex *b, int ldb, doublecomplex *c, int ldc, doublecomplex *d, int ldd, doublecomplex *e, int lde, double- complex *f, int ldf, double *scale, double *dif, int *info); void ztgsyl_64(char trans, long ijob, long m, long n, doublecomplex *a, long lda, doublecomplex *b, long ldb, doublecomplex *c, long ldc, doublecomplex *d, long ldd, doublecomplex *e, long lde, doublecomplex *f, long ldf, double *scale, double *dif, long *info); 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 trans- pose 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) = 'N': solve the generalized sylvester equation (1). = 'C': solve the "conjugate transposed" system (3). IJOB (input) 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) The order of the matrices A and D, and the row dimension of the matrices C, F, R and L. N (input) The order of the matrices B and E, and the column dimension of the matrices C, F, R and L. A (input) The upper triangular matrix A. LDA (input) The leading dimension of the array A. LDA >= max(1, M). B (input) The upper triangular matrix B. LDB (input) The leading dimension of the array B. LDB >= max(1, N). C (input/output) 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 com- putation of the Dif-estimate. LDC (input) The leading dimension of the array C. LDC >= max(1, M). D (input) The upper triangular matrix D. LDD (input) The leading dimension of the array D. LDD >= max(1, M). E (input) The upper triangular matrix E. LDE (input) The leading dimension of the array E. LDE >= max(1, N). F (input/output) 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 com- putation of the Dif-estimate. LDF (input) The leading dimension of the array F. LDF >= max(1, M). SCALE (output) 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. DIF (output) On exit DIF is the reciprocal of a lower bound of the recip- rocal 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. WORK (workspace) If IJOB = 0, WORK is not referenced. Otherwise, on exit, if INFO=0 then WORK(1) returns the optimal LWORK. LWORK (input) 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) INFO (output) =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. 7 Nov 2015 ztgsyl(3P)