tgsyl(3)

NAME
tgsyl - tgsyl: Sylvester equation

SYNOPSIS

  Functions

subroutine ctgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
CTGSYL
subroutine dtgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
DTGSYL
subroutine stgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
STGSYL
subroutine ztgsyl (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, dif, work, lwork, iwork, info)
ZTGSYL

Detailed Description

Function Documentation

  subroutine ctgsyl (character trans, integer ijob, integer m, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldc, * ) c, integer ldc, complex, dimension( ldd, * ) d, integer ldd, complex, dimension( lde, * ) e, integer lde, complex, dimension( ldf, * ) f, integer ldf, real scale, real dif, complex, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

CTGSYL

Purpose:

    !>
    !> CTGSYL 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**H, Im) ]        (2)
    !>            [ kron(In, D)  -kron(E**H, Im) ],
    !>
    !> Here Ix is the identity matrix of size x and X**H 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**H *y = scale*b
    !> is solved for, which is equivalent to solve for R and L in
    !>
    !>             A**H * R + D**H * L = scale * C           (3)
    !>             R * B**H + L * E**H = 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 CLACON.
    !>
    !> If IJOB >= 1, CTGSYL 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': solve the generalized sylvester equation (1).
    !>          = 'C': solve the  system (3).
    !>

    IJOB

    !>          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 is used).
    !>          =4: Only an estimate of Dif[(A,D), (B,E)] is computed.
    !>              (CGECON on sub-systems is used).
    !>          Not referenced if TRANS = 'C'.
    !>

    M

    !>          M is INTEGER
    !>          The order of the matrices A and D, and the row dimension of
    !>          the matrices C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices B and E, and the column dimension
    !>          of the matrices C, F, R and L.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA, M)
    !>          The upper triangular matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1, M).
    !>

    B

    !>          B is COMPLEX array, dimension (LDB, N)
    !>          The upper triangular matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1, N).
    !>

    C

    !>          C is COMPLEX 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

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1, M).
    !>

    D

    !>          D is COMPLEX array, dimension (LDD, M)
    !>          The upper triangular matrix D.
    !>

    LDD

    !>          LDD is INTEGER
    !>          The leading dimension of the array D. LDD >= max(1, M).
    !>

    E

    !>          E is COMPLEX array, dimension (LDE, N)
    !>          The upper triangular matrix E.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the array E. LDE >= max(1, N).
    !>

    F

    !>          F is COMPLEX 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

    !>          LDF is INTEGER
    !>          The leading dimension of the array F. LDF >= max(1, M).
    !>

    DIF

    !>          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 = 'C', DIF is not referenced.
    !>

    SCALE

    !>          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, R and L will
    !>          hold the solutions to the homogeneous system with C = F = 0.
    !>

    WORK

    !>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          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.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (M+N+2)
    !>

    INFO

    !>          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 very close
    !>                eigenvalues.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:
    [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.

  subroutine dtgsyl (character trans, integer ijob, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldc, * ) c, integer ldc, double precision, dimension( ldd, * ) d, integer ldd, double precision, dimension( lde, * ) e, integer lde, double precision, dimension( ldf, * ) f, integer ldf, double precision scale, double precision dif, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

DTGSYL

Purpose:

    !>
    !> DTGSYL 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', DTGSYL 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 DLACON.
    !>
    !> If IJOB >= 1, DTGSYL 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': solve the generalized Sylvester equation (1).
    !>          = 'T': solve the 'transposed' system (3).
    !>

    IJOB

    !>          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.
    !>               ( DGECON on sub-systems is used ).
    !>          Not referenced if TRANS = 'T'.
    !>

    M

    !>          M is INTEGER
    !>          The order of the matrices A and D, and the row dimension of
    !>          the matrices C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices B and E, and the column dimension
    !>          of the matrices C, F, R and L.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA, M)
    !>          The upper quasi triangular matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1, M).
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB, N)
    !>          The upper quasi triangular matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1, N).
    !>

    C

    !>          C is DOUBLE PRECISION 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

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1, M).
    !>

    D

    !>          D is DOUBLE PRECISION array, dimension (LDD, M)
    !>          The upper triangular matrix D.
    !>

    LDD

    !>          LDD is INTEGER
    !>          The leading dimension of the array D. LDD >= max(1, M).
    !>

    E

    !>          E is DOUBLE PRECISION array, dimension (LDE, N)
    !>          The upper triangular matrix E.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the array E. LDE >= max(1, N).
    !>

    F

    !>          F is DOUBLE PRECISION 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

    !>          LDF is INTEGER
    !>          The leading dimension of the array F. LDF >= max(1, M).
    !>

    DIF

    !>          DIF is 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 = 'T', DIF is not touched.
    !>

    SCALE

    !>          SCALE is 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, C and F hold the
    !>          solutions R and L, respectively, to the homogeneous system
    !>          with C = F = 0. Normally, SCALE = 1.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          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.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (M+N+6)
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

    !>
    !>  [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.
    !>

  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( lde, * ) e, integer lde, real, dimension( ldf, * ) f, integer ldf, real scale, real dif, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

STGSYL

Purpose:

    !>
    !> 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': solve the generalized Sylvester equation (1).
    !>          = 'T': solve the 'transposed' system (3).
    !>

    IJOB

    !>          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'.
    !>

    M

    !>          M is INTEGER
    !>          The order of the matrices A and D, and the row dimension of
    !>          the matrices C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices B and E, and the column dimension
    !>          of the matrices C, F, R and L.
    !>

    A

    !>          A is REAL array, dimension (LDA, M)
    !>          The upper quasi triangular matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1, M).
    !>

    B

    !>          B is REAL array, dimension (LDB, N)
    !>          The upper quasi triangular matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1, N).
    !>

    C

    !>          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.
    !>

    LDC

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1, M).
    !>

    D

    !>          D is REAL array, dimension (LDD, M)
    !>          The upper triangular matrix D.
    !>

    LDD

    !>          LDD is INTEGER
    !>          The leading dimension of the array D. LDD >= max(1, M).
    !>

    E

    !>          E is REAL array, dimension (LDE, N)
    !>          The upper triangular matrix E.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the array E. LDE >= max(1, N).
    !>

    F

    !>          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.
    !>

    LDF

    !>          LDF is INTEGER
    !>          The leading dimension of the array F. LDF >= max(1, M).
    !>

    DIF

    !>          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.
    !>

    SCALE

    !>          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.
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          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.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (M+N+6)
    !>

    INFO

    !>          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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

    !>
    !>  [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.
    !>

  subroutine ztgsyl (character trans, integer ijob, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldc, * ) c, integer ldc, complex*16, dimension( ldd, * ) d, integer ldd, complex*16, dimension( lde, * ) e, integer lde, complex*16, dimension( ldf, * ) f, integer ldf, double precision scale, double precision dif, complex*16, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)

ZTGSYL

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**H, Im) ]        (2)
    !>            [ kron(In, D)  -kron(E**H, Im) ],
    !>
    !> Here Ix is the identity matrix of size x and X**H 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**H *y = scale*b
    !> is solved for, which is equivalent to solve for R and L in
    !>
    !>             A**H * R + D**H * L = scale * C           (3)
    !>             R * B**H + L * E**H = 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.
    !>

Parameters
    TRANS

    !>          TRANS is CHARACTER*1
    !>          = 'N': solve the generalized sylvester equation (1).
    !>          = 'C': solve the  system (3).
    !>

    IJOB

    !>          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 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

    !>          M is INTEGER
    !>          The order of the matrices A and D, and the row dimension of
    !>          the matrices C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrices B and E, and the column dimension
    !>          of the matrices C, F, R and L.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA, M)
    !>          The upper triangular matrix A.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A. LDA >= max(1, M).
    !>

    B

    !>          B is COMPLEX*16 array, dimension (LDB, N)
    !>          The upper triangular matrix B.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the array B. LDB >= max(1, N).
    !>

    C

    !>          C is 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

    !>          LDC is INTEGER
    !>          The leading dimension of the array C. LDC >= max(1, M).
    !>

    D

    !>          D is COMPLEX*16 array, dimension (LDD, M)
    !>          The upper triangular matrix D.
    !>

    LDD

    !>          LDD is INTEGER
    !>          The leading dimension of the array D. LDD >= max(1, M).
    !>

    E

    !>          E is COMPLEX*16 array, dimension (LDE, N)
    !>          The upper triangular matrix E.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the array E. LDE >= max(1, N).
    !>

    F

    !>          F is 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

    !>          LDF is INTEGER
    !>          The leading dimension of the array F. LDF >= max(1, M).
    !>

    DIF

    !>          DIF is 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

    !>          SCALE is 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 homogeneous system with C = F = 0.
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
    !>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    !>

    LWORK

    !>          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.
    !>

    IWORK

    !>          IWORK is INTEGER array, dimension (M+N+2)
    !>

    INFO

    !>          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 very close
    !>                eigenvalues.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Contributors:
    Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:
    [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.

Author

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