ztgsy2.f(3)

NAME
ztgsy2.f -

SYNOPSIS

  Functions/Subroutines

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).

Function/Subroutine Documentation

  subroutine ztgsy2 (characterTRANS, integerIJOB, integerM, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( ldc, * )C, integerLDC, complex*16, dimension( ldd, * )D, integerLDD, complex*16, dimension( lde, * )E, integerLDE, complex*16, dimension( ldf, * )F, integerLDF, double precisionSCALE, double precisionRDSUM, double precisionRDSCAL, integerINFO)

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 conjuguate 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 communicaton 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.

Date:
    September 2012

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

Definition at line 258 of file ztgsy2.f.

Author

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