CTGSY2(3)      LAPACK routine of NEC Numeric Library Collection      CTGSY2(3)



NAME
       CTGSY2

SYNOPSIS
       SUBROUTINE CTGSY2 (TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D, LDD,
           E, LDE, F, LDF, SCALE, RDSUM, RDSCAL, INFO)



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.




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

           IJOB      (input)
                     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         (input)
                     M is INTEGER
                     On entry, M specifies the order of A and D, and the row
                     dimension of C, F, R and L.

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

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

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

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

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

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

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

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

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

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

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

           SCALE     (output)
                     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     (input/output)
                     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    (input/output)
                     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      (output)
                     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.



LAPACK routine                  31 October 2017                      CTGSY2(3)