ctgsy2.f

          SUBROUTINE CTGSY2( TRANS, IJOB, M, N, A, LDA, B, LDB, C, LDC, D,
         $                   LDD, E, LDE, F, LDF, SCALE, RDSUM, RDSCAL,
         $                   INFO )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          TRANS
          INTEGER            IJOB, INFO, LDA, LDB, LDC, LDD, LDE, LDF, M, N
          REAL               RDSCAL, RDSUM, SCALE
    *     ..
    *     .. Array Arguments ..
          COMPLEX            A( LDA, * ), B( LDB, * ), C( LDC, * ),
         $                   D( LDD, * ), E( LDE, * ), F( LDF, * )
    *     ..
    *
    *  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', Im) ]             (2)
    *             [ kron(In, D)  -kron(E', Im) ],
    *
    *  Ik is the identity matrix of size k and X' 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'y = scale*b
    *  is solved for, which is equivalent to solve for R and L in
    *
    *              A' * R  + D' * L   = scale *  C           (3)
    *              R  * B' + L  * E'  = 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) CHARACTER*1
    *          = 'N', solve the generalized Sylvester equation (1).
    *          = 'T': solve the 'transposed' system (3).
    *
    *  IJOB    (input) 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) INTEGER
    *          On entry, M specifies the order of A and D, and the row
    *          dimension of C, F, R and L.
    *
    *  N       (input) INTEGER
    *          On entry, N specifies the order of B and E, and the column
    *          dimension of C, F, R and L.
    *
    *  A       (input) COMPLEX array, dimension (LDA, M)
    *          On entry, A contains an upper triangular matrix.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the matrix A. LDA >= max(1, M).
    *
    *  B       (input) COMPLEX array, dimension (LDB, N)
    *          On entry, B contains an upper triangular matrix.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the matrix B. LDB >= max(1, N).
    *
    *  C       (input/output) 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) INTEGER
    *          The leading dimension of the matrix C. LDC >= max(1, M).
    *
    *  D       (input) COMPLEX array, dimension (LDD, M)
    *          On entry, D contains an upper triangular matrix.
    *
    *  LDD     (input) INTEGER
    *          The leading dimension of the matrix D. LDD >= max(1, M).
    *
    *  E       (input) COMPLEX array, dimension (LDE, N)
    *          On entry, E contains an upper triangular matrix.
    *
    *  LDE     (input) INTEGER
    *          The leading dimension of the matrix E. LDE >= max(1, N).
    *
    *  F       (input/output) 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) INTEGER
    *          The leading dimension of the matrix F. LDF >= max(1, M).
    *
    *  SCALE   (output) 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) 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) 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) 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.
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
    *     Umea University, S-901 87 Umea, Sweden.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          INTEGER            LDZ
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, LDZ = 2 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOTRAN
          INTEGER            I, IERR, J, K
          REAL               SCALOC
          COMPLEX            ALPHA
    *     ..
    *     .. Local Arrays ..
          INTEGER            IPIV( LDZ ), JPIV( LDZ )
          COMPLEX            RHS( LDZ ), Z( LDZ, LDZ )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          EXTERNAL           LSAME
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           CAXPY, CGESC2, CGETC2, CSCAL, CLATDF, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          CMPLX, CONJG, MAX
    *     ..
    *     .. Executable Statements ..
    *
    *     Decode and test input parameters
    *
          INFO = 0
          IERR = 0
          NOTRAN = LSAME( TRANS, 'N' )
          IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'C' ) ) THEN
             INFO = -1
          ELSE IF( NOTRAN ) THEN
             IF( ( IJOB.LT.0 ) .OR. ( IJOB.GT.2 ) ) THEN
                INFO = -2
             END IF
          END IF
          IF( INFO.EQ.0 ) THEN
             IF( M.LE.0 ) THEN
                INFO = -3
             ELSE IF( N.LE.0 ) THEN
                INFO = -4
             ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
                INFO = -5
             ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
                INFO = -8
             ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
                INFO = -10
             ELSE IF( LDD.LT.MAX( 1, M ) ) THEN
                INFO = -12
             ELSE IF( LDE.LT.MAX( 1, N ) ) THEN
                INFO = -14
             ELSE IF( LDF.LT.MAX( 1, M ) ) THEN
                INFO = -16
             END IF
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'CTGSY2', -INFO )
             RETURN
          END IF
    *
          IF( NOTRAN ) THEN
    *
    *        Solve (I, J) - system
    *           A(I, I) * R(I, J) - L(I, J) * B(J, J) = C(I, J)
    *           D(I, I) * R(I, J) - L(I, J) * E(J, J) = F(I, J)
    *        for I = M, M - 1, ..., 1; J = 1, 2, ..., N
    *
             SCALE = ONE
             SCALOC = ONE
             DO 30 J = 1, N
                DO 20 I = M, 1, -1
    *
    *              Build 2 by 2 system
    *
                   Z( 1, 1 ) = A( I, I )
                   Z( 2, 1 ) = D( I, I )
                   Z( 1, 2 ) = -B( J, J )
                   Z( 2, 2 ) = -E( J, J )
    *
    *              Set up right hand side(s)
    *
                   RHS( 1 ) = C( I, J )
                   RHS( 2 ) = F( I, J )
    *
    *              Solve Z * x = RHS
    *
                   CALL CGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
                   IF( IERR.GT.0 )
         $            INFO = IERR
                   IF( IJOB.EQ.0 ) THEN
                      CALL CGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
                      IF( SCALOC.NE.ONE ) THEN
                         DO 10 K = 1, N
                            CALL CSCAL( M, CMPLX( SCALOC, ZERO ), C( 1, K ),
         $                              1 )
                            CALL CSCAL( M, CMPLX( SCALOC, ZERO ), F( 1, K ),
         $                              1 )
       10                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                   ELSE
                      CALL CLATDF( IJOB, LDZ, Z, LDZ, RHS, RDSUM, RDSCAL,
         $                         IPIV, JPIV )
                   END IF
    *
    *              Unpack solution vector(s)
    *
                   C( I, J ) = RHS( 1 )
                   F( I, J ) = RHS( 2 )
    *
    *              Substitute R(I, J) and L(I, J) into remaining equation.
    *
                   IF( I.GT.1 ) THEN
                      ALPHA = -RHS( 1 )
                      CALL CAXPY( I-1, ALPHA, A( 1, I ), 1, C( 1, J ), 1 )
                      CALL CAXPY( I-1, ALPHA, D( 1, I ), 1, F( 1, J ), 1 )
                   END IF
                   IF( J.LT.N ) THEN
                      CALL CAXPY( N-J, RHS( 2 ), B( J, J+1 ), LDB,
         $                        C( I, J+1 ), LDC )
                      CALL CAXPY( N-J, RHS( 2 ), E( J, J+1 ), LDE,
         $                        F( I, J+1 ), LDF )
                   END IF
    *
       20       CONTINUE
       30    CONTINUE
          ELSE
    *
    *        Solve transposed (I, J) - system:
    *           A(I, I)' * R(I, J) + D(I, I)' * L(J, J) = C(I, J)
    *           R(I, I) * B(J, J) + L(I, J) * E(J, J)   = -F(I, J)
    *        for I = 1, 2, ..., M, J = N, N - 1, ..., 1
    *
             SCALE = ONE
             SCALOC = ONE
             DO 80 I = 1, M
                DO 70 J = N, 1, -1
    *
    *              Build 2 by 2 system Z'
    *
                   Z( 1, 1 ) = CONJG( A( I, I ) )
                   Z( 2, 1 ) = -CONJG( B( J, J ) )
                   Z( 1, 2 ) = CONJG( D( I, I ) )
                   Z( 2, 2 ) = -CONJG( E( J, J ) )
    *
    *
    *              Set up right hand side(s)
    *
                   RHS( 1 ) = C( I, J )
                   RHS( 2 ) = F( I, J )
    *
    *              Solve Z' * x = RHS
    *
                   CALL CGETC2( LDZ, Z, LDZ, IPIV, JPIV, IERR )
                   IF( IERR.GT.0 )
         $            INFO = IERR
                   CALL CGESC2( LDZ, Z, LDZ, RHS, IPIV, JPIV, SCALOC )
                   IF( SCALOC.NE.ONE ) THEN
                      DO 40 K = 1, N
                         CALL CSCAL( M, CMPLX( SCALOC, ZERO ), C( 1, K ),
         $                           1 )
                         CALL CSCAL( M, CMPLX( SCALOC, ZERO ), F( 1, K ),
         $                           1 )
       40             CONTINUE
                      SCALE = SCALE*SCALOC
                   END IF
    *
    *              Unpack solution vector(s)
    *
                   C( I, J ) = RHS( 1 )
                   F( I, J ) = RHS( 2 )
    *
    *              Substitute R(I, J) and L(I, J) into remaining equation.
    *
                   DO 50 K = 1, J - 1
                      F( I, K ) = F( I, K ) + RHS( 1 )*CONJG( B( K, J ) ) +
         $                        RHS( 2 )*CONJG( E( K, J ) )
       50          CONTINUE
                   DO 60 K = I + 1, M
                      C( K, J ) = C( K, J ) - CONJG( A( I, K ) )*RHS( 1 ) -
         $                        CONJG( D( I, K ) )*RHS( 2 )
       60          CONTINUE
    *
       70       CONTINUE
       80    CONTINUE
          END IF
          RETURN
    *
    *     End of CTGSY2
    *
          END