tgsy2(3)

NAME
tgsy2 - tgsy2: Sylvester equation panel (?)

SYNOPSIS

  Functions

subroutine ctgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info)
CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
subroutine dtgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info)
DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).
subroutine stgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, iwork, pq, info)
STGSY2 solves the generalized Sylvester equation (unblocked algorithm).
subroutine ztgsy2 (trans, ijob, m, n, a, lda, b, ldb, c, ldc, d, ldd, e, lde, f, ldf, scale, rdsum, rdscal, info)
ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Detailed Description

Function Documentation

  subroutine ctgsy2 (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 rdsum, real rdscal, integer info)

CTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

    !>
    !> CTGSY2 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. 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 solving equation (1) corresponds 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) ],
    !>
    !> Ik is the identity matrix of size k and X**H is the 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 is used to compute an estimate of Dif[(A, D), (B, E)] =
    !> = sigma_min(Z) using reverse communication with CLACON.
    !>
    !> CTGSY2 also (IJOB >= 1) contributes to the computation in CTGSYL
    !> of an upper bound on the separation between to matrix pairs. Then
    !> the input (A, D), (B, E) are sub-pencils of two matrix pairs in
    !> CTGSYL.
    !>

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

    M

    !>          M is INTEGER
    !>          On entry, M specifies the order of A and D, and the row
    !>          dimension of C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          On entry, N specifies the order of B and E, and the column
    !>          dimension of C, F, R and L.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA, M)
    !>          On entry, A contains an upper triangular matrix.
    !>

    LDA

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

    B

    !>          B is COMPLEX array, dimension (LDB, N)
    !>          On entry, B contains an upper triangular matrix.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, C has been overwritten by the solution
    !>          R.
    !>

    LDC

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

    D

    !>          D is COMPLEX array, dimension (LDD, M)
    !>          On entry, D contains an upper triangular matrix.
    !>

    LDD

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

    E

    !>          E is COMPLEX array, dimension (LDE, N)
    !>          On entry, E contains an upper triangular matrix.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, F has been overwritten by the solution
    !>          L.
    !>

    LDF

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

    SCALE

    !>          SCALE is REAL
    !>          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

    !>          RDSUM is REAL
    !>          On entry, the sum of squares of computed contributions to
    !>          the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by
    !>          CTGSYL.
    !>

    RDSCAL

    !>          RDSCAL is REAL
    !>          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 CTGSY2 is called by
    !>          CTGSYL.
    !>

    INFO

    !>          INFO is INTEGER
    !>          On exit, if INFO is set to
    !>            =0: Successful exit
    !>            <0: If INFO = -i, input argument number i is illegal.
    !>            >0: The matrix pairs (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.

  subroutine dtgsy2 (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 rdsum, double precision rdscal, integer, dimension( * ) iwork, integer pq, integer info)

DTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

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**T, Im) ]             (2)
    !>            [ kron(In, D)  -kron(E**T, Im) ],
    !>
    !> 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.
    !> 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**T*y = scale*b for y,
    !> 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 is used to compute an estimate of Dif[(A, D), (B, E)] =
    !> sigma_min(Z) using reverse communication with DLACON.
    !>
    !> DTGSY2 also (IJOB >= 1) contributes to the computation in DTGSYL
    !> 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 DTGSYL for details.
    !>

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: 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

    !>          M is INTEGER
    !>          On entry, M specifies the order of A and D, and the row
    !>          dimension of C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          On entry, N specifies the order of B and E, and the column
    !>          dimension of C, F, R and L.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension (LDA, M)
    !>          On entry, A contains an upper quasi triangular matrix.
    !>

    LDA

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

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB, N)
    !>          On entry, B contains an upper quasi triangular matrix.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, C has been overwritten by the
    !>          solution R.
    !>

    LDC

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

    D

    !>          D is DOUBLE PRECISION array, dimension (LDD, M)
    !>          On entry, D contains an upper triangular matrix.
    !>

    LDD

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

    E

    !>          E is DOUBLE PRECISION array, dimension (LDE, N)
    !>          On entry, E contains an upper triangular matrix.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, F has been overwritten by the
    !>          solution L.
    !>

    LDF

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

    SCALE

    !>          SCALE is 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

    !>          RDSUM is 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 DTGSYL.
    !>

    RDSCAL

    !>          RDSCAL is 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

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

    PQ

    !>          PQ is INTEGER
    !>          On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
    !>          8-by-8) solved by this routine.
    !>

    INFO

    !>          INFO is 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.
    !>

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.

  subroutine stgsy2 (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 rdsum, real rdscal, integer, dimension( * ) iwork, integer pq, integer info)

STGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

    !>
    !> STGSY2 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**T, Im) ]             (2)
    !>            [ kron(In, D)  -kron(E**T, Im) ],
    !>
    !> 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.
    !> 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**T*y = scale*b for y,
    !> 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 is used to compute an estimate of Dif[(A, D), (B, E)] =
    !> sigma_min(Z) using reverse communication with SLACON.
    !>
    !> STGSY2 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
    !> STGSYL. See STGSYL for details.
    !>

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

    M

    !>          M is INTEGER
    !>          On entry, M specifies the order of A and D, and the row
    !>          dimension of C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          On entry, N specifies the order of B and E, and the column
    !>          dimension of C, F, R and L.
    !>

    A

    !>          A is REAL array, dimension (LDA, M)
    !>          On entry, A contains an upper quasi triangular matrix.
    !>

    LDA

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

    B

    !>          B is REAL array, dimension (LDB, N)
    !>          On entry, B contains an upper quasi triangular matrix.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, C has been overwritten by the
    !>          solution R.
    !>

    LDC

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

    D

    !>          D is REAL array, dimension (LDD, M)
    !>          On entry, D contains an upper triangular matrix.
    !>

    LDD

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

    E

    !>          E is REAL array, dimension (LDE, N)
    !>          On entry, E contains an upper triangular matrix.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, F has been overwritten by the
    !>          solution L.
    !>

    LDF

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

    SCALE

    !>          SCALE is REAL
    !>          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

    !>          RDSUM is REAL
    !>          On entry, the sum of squares of computed contributions to
    !>          the Dif-estimate under computation by STGSYL, 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 STGSY2 is called by STGSYL.
    !>

    RDSCAL

    !>          RDSCAL is REAL
    !>          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 STGSY2 is called by
    !>                STGSYL.
    !>

    IWORK

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

    PQ

    !>          PQ is INTEGER
    !>          On exit, the number of subsystems (of size 2-by-2, 4-by-4 and
    !>          8-by-8) solved by this routine.
    !>

    INFO

    !>          INFO is 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.
    !>

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.

  subroutine ztgsy2 (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 rdsum, double precision rdscal, integer info)

ZTGSY2 solves the generalized Sylvester equation (unblocked algorithm).

Purpose:

    !>
    !> ZTGSY2 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. 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 solving equation (1) corresponds 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) ],
    !>
    !> Ik is the identity matrix of size k 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 is used to compute an estimate of Dif[(A, D), (B, E)] =
    !> = sigma_min(Z) using reverse communication with ZLACON.
    !>
    !> ZTGSY2 also (IJOB >= 1) contributes to the computation in ZTGSYL
    !> of an upper bound on the separation between to matrix pairs. Then
    !> the input (A, D), (B, E) are sub-pencils of two matrix pairs in
    !> ZTGSYL.
    !>

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: 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

    !>          M is INTEGER
    !>          On entry, M specifies the order of A and D, and the row
    !>          dimension of C, F, R and L.
    !>

    N

    !>          N is INTEGER
    !>          On entry, N specifies the order of B and E, and the column
    !>          dimension of C, F, R and L.
    !>

    A

    !>          A is COMPLEX*16 array, dimension (LDA, M)
    !>          On entry, A contains an upper triangular matrix.
    !>

    LDA

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

    B

    !>          B is COMPLEX*16 array, dimension (LDB, N)
    !>          On entry, B contains an upper triangular matrix.
    !>

    LDB

    !>          LDB is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, C has been overwritten by the solution
    !>          R.
    !>

    LDC

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

    D

    !>          D is COMPLEX*16 array, dimension (LDD, M)
    !>          On entry, D contains an upper triangular matrix.
    !>

    LDD

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

    E

    !>          E is COMPLEX*16 array, dimension (LDE, N)
    !>          On entry, E contains an upper triangular matrix.
    !>

    LDE

    !>          LDE is INTEGER
    !>          The leading dimension of the matrix 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).
    !>          On exit, if IJOB = 0, F has been overwritten by the solution
    !>          L.
    !>

    LDF

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

    SCALE

    !>          SCALE is 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

    !>          RDSUM is DOUBLE PRECISION
    !>          On entry, the sum of squares of computed contributions to
    !>          the Dif-estimate under computation by ZTGSYL, 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 ZTGSY2 is called by
    !>          ZTGSYL.
    !>

    RDSCAL

    !>          RDSCAL is 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 ZTGSY2 is called by
    !>          ZTGSYL.
    !>

    INFO

    !>          INFO is INTEGER
    !>          On exit, if INFO is set to
    !>            =0: Successful exit
    !>            <0: If INFO = -i, input argument number i is illegal.
    !>            >0: The matrix pairs (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.

Author

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