SRC/dtgsy2.f(3)

NAME
SRC/dtgsy2.f

SYNOPSIS

  Functions/Subroutines
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).

Function/Subroutine Documentation

  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.

Definition at line 271 of file dtgsy2.f.

Author

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