NAME

       ctgsy2.f -

SYNOPSIS

   Functions/Subroutines
       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).

Function/Subroutine Documentation

   subroutine ctgsy2 (characterTRANS, integerIJOB, integerM, integerN, complex, dimension( lda, *
       )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldc, * )C,
       integerLDC, complex, dimension( ldd, * )D, integerLDD, complex, dimension( lde, * )E,
       integerLDE, complex, dimension( ldf, * )F, integerLDF, realSCALE, realRDSUM, realRDSCAL,
       integerINFO)
       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 communicaton 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.

       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 ctgsy2.f.

Author

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