.TH "SRC/stgsyl.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/stgsyl.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBstgsyl\fP (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, \fBlde\fP, f, ldf, scale, dif, work, lwork, iwork, info)" .br .RI "\fBSTGSYL\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine stgsyl (character trans, integer ijob, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldc, * ) c, integer ldc, real, dimension( ldd, * ) d, integer ldd, real, dimension( \fBlde\fP, * ) e, integer lde, real, dimension( ldf, * ) f, integer ldf, real scale, real dif, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)" .PP \fBSTGSYL\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> STGSYL 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 real entries\&. (A, D) and (B, E) must be in !> generalized (real) 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 (1) is equivalent to solve Zx = scale b, where !> Z is defined as !> !> Z = [ kron(In, A) -kron(B**T, Im) ] (2) !> [ kron(In, D) -kron(E**T, Im) ]\&. !> !> Here Ik is the identity matrix of size k and X**T is the transpose of !> X\&. kron(X, Y) is the Kronecker product between the matrices X and Y\&. !> !> If TRANS = 'T', STGSYL solves the transposed system Z**T*y = scale*b, !> which is equivalent to solve for R and L in !> !> A**T * R + D**T * L = scale * C (3) !> R * B**T + L * E**T = scale * -F !> !> This case (TRANS = 'T') 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 SLACON\&. !> !> If IJOB >= 1, STGSYL 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\&. See [1-2] for more !> information\&. !> !> This is a level 3 BLAS algorithm\&. !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANS\fP .PP .nf !> TRANS is CHARACTER*1 !> = 'N': solve the generalized Sylvester equation (1)\&. !> = 'T': solve the 'transposed' system (3)\&. !> .fi .PP .br \fIIJOB\fP .PP .nf !> IJOB 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 IJOB = 1 is used)\&. !> = 4: Only an estimate of Dif[(A,D), (B,E)] is computed\&. !> ( SGECON on sub-systems is used )\&. !> Not referenced if TRANS = 'T'\&. !> .fi .PP .br \fIM\fP .PP .nf !> M is INTEGER !> The order of the matrices A and D, and the row dimension of !> the matrices C, F, R and L\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The order of the matrices B and E, and the column dimension !> of the matrices C, F, R and L\&. !> .fi .PP .br \fIA\fP .PP .nf !> A is REAL array, dimension (LDA, M) !> The upper quasi triangular matrix A\&. !> .fi .PP .br \fILDA\fP .PP .nf !> LDA is INTEGER !> The leading dimension of the array A\&. LDA >= max(1, M)\&. !> .fi .PP .br \fIB\fP .PP .nf !> B is REAL array, dimension (LDB, N) !> The upper quasi triangular matrix B\&. !> .fi .PP .br \fILDB\fP .PP .nf !> LDB is INTEGER !> The leading dimension of the array B\&. LDB >= max(1, N)\&. !> .fi .PP .br \fIC\fP .PP .nf !> C is REAL 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\&. !> .fi .PP .br \fILDC\fP .PP .nf !> LDC is INTEGER !> The leading dimension of the array C\&. LDC >= max(1, M)\&. !> .fi .PP .br \fID\fP .PP .nf !> D is REAL array, dimension (LDD, M) !> The upper triangular matrix D\&. !> .fi .PP .br \fILDD\fP .PP .nf !> LDD is INTEGER !> The leading dimension of the array D\&. LDD >= max(1, M)\&. !> .fi .PP .br \fIE\fP .PP .nf !> E is REAL array, dimension (LDE, N) !> The upper triangular matrix E\&. !> .fi .PP .br \fILDE\fP .PP .nf !> LDE is INTEGER !> The leading dimension of the array E\&. LDE >= max(1, N)\&. !> .fi .PP .br \fIF\fP .PP .nf !> F is REAL 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\&. !> .fi .PP .br \fILDF\fP .PP .nf !> LDF is INTEGER !> The leading dimension of the array F\&. LDF >= max(1, M)\&. !> .fi .PP .br \fIDIF\fP .PP .nf !> DIF is REAL !> 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 = 'T', DIF is not touched\&. !> .fi .PP .br \fISCALE\fP .PP .nf !> SCALE 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, C and F hold the !> solutions R and L, respectively, to the homogeneous system !> with C = F = 0\&. Normally, SCALE = 1\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is REAL array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. !> .fi .PP .br \fILWORK\fP .PP .nf !> LWORK 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\&. !> .fi .PP .br \fIIWORK\fP .PP .nf !> IWORK is INTEGER array, dimension (M+N+6) !> .fi .PP .br \fIINFO\fP .PP .nf !> INFO is 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 close eigenvalues\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBContributors:\fP .RS 4 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. .RE .PP \fBReferences:\fP .RS 4 .PP .nf !> !> [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\&. !> .fi .PP .RE .PP .PP Definition at line \fB296\fP of file \fBstgsyl\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.