dtrsyl.f

          SUBROUTINE DTRSYL( TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C,
         $                   LDC, SCALE, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          TRANA, TRANB
          INTEGER            INFO, ISGN, LDA, LDB, LDC, M, N
          DOUBLE PRECISION   SCALE
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DTRSYL solves the real Sylvester matrix equation:
    *
    *     op(A)*X + X*op(B) = scale*C or
    *     op(A)*X - X*op(B) = scale*C,
    *
    *  where op(A) = A or A**T, and  A and B are both upper quasi-
    *  triangular. A is M-by-M and B is N-by-N; the right hand side C and
    *  the solution X are M-by-N; and scale is an output scale factor, set
    *  <= 1 to avoid overflow in X.
    *
    *  A and B must be in Schur canonical form (as returned by DHSEQR), that
    *  is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks;
    *  each 2-by-2 diagonal block has its diagonal elements equal and its
    *  off-diagonal elements of opposite sign.
    *
    *  Arguments
    *  =========
    *
    *  TRANA   (input) CHARACTER*1
    *          Specifies the option op(A):
    *          = 'N': op(A) = A    (No transpose)
    *          = 'T': op(A) = A**T (Transpose)
    *          = 'C': op(A) = A**H (Conjugate transpose = Transpose)
    *
    *  TRANB   (input) CHARACTER*1
    *          Specifies the option op(B):
    *          = 'N': op(B) = B    (No transpose)
    *          = 'T': op(B) = B**T (Transpose)
    *          = 'C': op(B) = B**H (Conjugate transpose = Transpose)
    *
    *  ISGN    (input) INTEGER
    *          Specifies the sign in the equation:
    *          = +1: solve op(A)*X + X*op(B) = scale*C
    *          = -1: solve op(A)*X - X*op(B) = scale*C
    *
    *  M       (input) INTEGER
    *          The order of the matrix A, and the number of rows in the
    *          matrices X and C. M >= 0.
    *
    *  N       (input) INTEGER
    *          The order of the matrix B, and the number of columns in the
    *          matrices X and C. N >= 0.
    *
    *  A       (input) DOUBLE PRECISION array, dimension (LDA,M)
    *          The upper quasi-triangular matrix A, in Schur canonical form.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A. LDA >= max(1,M).
    *
    *  B       (input) DOUBLE PRECISION array, dimension (LDB,N)
    *          The upper quasi-triangular matrix B, in Schur canonical form.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B. LDB >= max(1,N).
    *
    *  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
    *          On entry, the M-by-N right hand side matrix C.
    *          On exit, C is overwritten by the solution matrix X.
    *
    *  LDC     (input) INTEGER
    *          The leading dimension of the array C. LDC >= max(1,M)
    *
    *  SCALE   (output) DOUBLE PRECISION
    *          The scale factor, scale, set <= 1 to avoid overflow in X.
    *
    *  INFO    (output) INTEGER
    *          = 0: successful exit
    *          < 0: if INFO = -i, the i-th argument had an illegal value
    *          = 1: A and B have common or very close eigenvalues; perturbed
    *               values were used to solve the equation (but the matrices
    *               A and B are unchanged).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NOTRNA, NOTRNB
          INTEGER            IERR, J, K, K1, K2, KNEXT, L, L1, L2, LNEXT
          DOUBLE PRECISION   A11, BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
         $                   SMLNUM, SUML, SUMR, XNORM
    *     ..
    *     .. Local Arrays ..
          DOUBLE PRECISION   DUM( 1 ), VEC( 2, 2 ), X( 2, 2 )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DDOT, DLAMCH, DLANGE
          EXTERNAL           LSAME, DDOT, DLAMCH, DLANGE
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLABAD, DLALN2, DLASY2, DSCAL, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Decode and Test input parameters
    *
          NOTRNA = LSAME( TRANA, 'N' )
          NOTRNB = LSAME( TRANB, 'N' )
    *
          INFO = 0
          IF( .NOT.NOTRNA .AND. .NOT.LSAME( TRANA, 'T' ) .AND. .NOT.
         $    LSAME( TRANA, 'C' ) ) THEN
             INFO = -1
          ELSE IF( .NOT.NOTRNB .AND. .NOT.LSAME( TRANB, 'T' ) .AND. .NOT.
         $         LSAME( TRANB, 'C' ) ) THEN
             INFO = -2
          ELSE IF( ISGN.NE.1 .AND. ISGN.NE.-1 ) THEN
             INFO = -3
          ELSE IF( M.LT.0 ) THEN
             INFO = -4
          ELSE IF( N.LT.0 ) THEN
             INFO = -5
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -7
          ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
             INFO = -9
          ELSE IF( LDC.LT.MAX( 1, M ) ) THEN
             INFO = -11
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DTRSYL', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( M.EQ.0 .OR. N.EQ.0 )
         $   RETURN
    *
    *     Set constants to control overflow
    *
          EPS = DLAMCH( 'P' )
          SMLNUM = DLAMCH( 'S' )
          BIGNUM = ONE / SMLNUM
          CALL DLABAD( SMLNUM, BIGNUM )
          SMLNUM = SMLNUM*DBLE( M*N ) / EPS
          BIGNUM = ONE / SMLNUM
    *
          SMIN = MAX( SMLNUM, EPS*DLANGE( 'M', M, M, A, LDA, DUM ),
         $       EPS*DLANGE( 'M', N, N, B, LDB, DUM ) )
    *
          SCALE = ONE
          SGN = ISGN
    *
          IF( NOTRNA .AND. NOTRNB ) THEN
    *
    *        Solve    A*X + ISGN*X*B = scale*C.
    *
    *        The (K,L)th block of X is determined starting from
    *        bottom-left corner column by column by
    *
    *         A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
    *
    *        Where
    *                  M                         L-1
    *        R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(J,L)].
    *                I=K+1                       J=1
    *
    *        Start column loop (index = L)
    *        L1 (L2) : column index of the first (first) row of X(K,L).
    *
             LNEXT = 1
             DO 60 L = 1, N
                IF( L.LT.LNEXT )
         $         GO TO 60
                IF( L.EQ.N ) THEN
                   L1 = L
                   L2 = L
                ELSE
                   IF( B( L+1, L ).NE.ZERO ) THEN
                      L1 = L
                      L2 = L + 1
                      LNEXT = L + 2
                   ELSE
                      L1 = L
                      L2 = L
                      LNEXT = L + 1
                   END IF
                END IF
    *
    *           Start row loop (index = K)
    *           K1 (K2): row index of the first (last) row of X(K,L).
    *
                KNEXT = M
                DO 50 K = M, 1, -1
                   IF( K.GT.KNEXT )
         $            GO TO 50
                   IF( K.EQ.1 ) THEN
                      K1 = K
                      K2 = K
                   ELSE
                      IF( A( K, K-1 ).NE.ZERO ) THEN
                         K1 = K - 1
                         K2 = K
                         KNEXT = K - 2
                      ELSE
                         K1 = K
                         K2 = K
                         KNEXT = K - 1
                      END IF
                   END IF
    *
                   IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                      SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
         $                   C( MIN( K1+1, M ), L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                      SCALOC = ONE
    *
                      A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                      DA11 = ABS( A11 )
                      IF( DA11.LE.SMIN ) THEN
                         A11 = SMIN
                         DA11 = SMIN
                         INFO = 1
                      END IF
                      DB = ABS( VEC( 1, 1 ) )
                      IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                         IF( DB.GT.BIGNUM*DA11 )
         $                  SCALOC = ONE / DB
                      END IF
                      X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 10 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
       10                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
    *
                   ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ),
         $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 20 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
       20                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K2, L1 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
    *
                      SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
         $                   C( MIN( K1+1, M ), L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
    *
                      SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
         $                   C( MIN( K1+1, M ), L2 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                      VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
    *
                      CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ),
         $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 30 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
       30                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L2 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                      VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L2 ), 1 )
                      SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 )
                      VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLASY2( .FALSE., .FALSE., ISGN, 2, 2,
         $                         A( K1, K1 ), LDA, B( L1, L1 ), LDB, VEC,
         $                         2, SCALOC, X, 2, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 40 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
       40                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 1, 2 )
                      C( K2, L1 ) = X( 2, 1 )
                      C( K2, L2 ) = X( 2, 2 )
                   END IF
    *
       50       CONTINUE
    *
       60    CONTINUE
    *
          ELSE IF( .NOT.NOTRNA .AND. NOTRNB ) THEN
    *
    *        Solve    A' *X + ISGN*X*B = scale*C.
    *
    *        The (K,L)th block of X is determined starting from
    *        upper-left corner column by column by
    *
    *          A(K,K)'*X(K,L) + ISGN*X(K,L)*B(L,L) = C(K,L) - R(K,L)
    *
    *        Where
    *                   K-1                        L-1
    *          R(K,L) = SUM [A(I,K)'*X(I,L)] +ISGN*SUM [X(K,J)*B(J,L)]
    *                   I=1                        J=1
    *
    *        Start column loop (index = L)
    *        L1 (L2): column index of the first (last) row of X(K,L)
    *
             LNEXT = 1
             DO 120 L = 1, N
                IF( L.LT.LNEXT )
         $         GO TO 120
                IF( L.EQ.N ) THEN
                   L1 = L
                   L2 = L
                ELSE
                   IF( B( L+1, L ).NE.ZERO ) THEN
                      L1 = L
                      L2 = L + 1
                      LNEXT = L + 2
                   ELSE
                      L1 = L
                      L2 = L
                      LNEXT = L + 1
                   END IF
                END IF
    *
    *           Start row loop (index = K)
    *           K1 (K2): row index of the first (last) row of X(K,L)
    *
                KNEXT = 1
                DO 110 K = 1, M
                   IF( K.LT.KNEXT )
         $            GO TO 110
                   IF( K.EQ.M ) THEN
                      K1 = K
                      K2 = K
                   ELSE
                      IF( A( K+1, K ).NE.ZERO ) THEN
                         K1 = K
                         K2 = K + 1
                         KNEXT = K + 2
                      ELSE
                         K1 = K
                         K2 = K
                         KNEXT = K + 1
                      END IF
                   END IF
    *
                   IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                      SCALOC = ONE
    *
                      A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                      DA11 = ABS( A11 )
                      IF( DA11.LE.SMIN ) THEN
                         A11 = SMIN
                         DA11 = SMIN
                         INFO = 1
                      END IF
                      DB = ABS( VEC( 1, 1 ) )
                      IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                         IF( DB.GT.BIGNUM*DA11 )
         $                  SCALOC = ONE / DB
                      END IF
                      X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 70 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
       70                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
    *
                   ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ),
         $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 80 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
       80                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K2, L1 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                      VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
    *
                      CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, B( L1, L1 ),
         $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 90 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
       90                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                      SUMR = DDOT( L1-1, C( K1, 1 ), LDC, B( 1, L2 ), 1 )
                      VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L1 ), 1 )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 )
                      SUMR = DDOT( L1-1, C( K2, 1 ), LDC, B( 1, L2 ), 1 )
                      VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLASY2( .TRUE., .FALSE., ISGN, 2, 2, A( K1, K1 ),
         $                         LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
         $                         2, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 100 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      100                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 1, 2 )
                      C( K2, L1 ) = X( 2, 1 )
                      C( K2, L2 ) = X( 2, 2 )
                   END IF
    *
      110       CONTINUE
      120    CONTINUE
    *
          ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN
    *
    *        Solve    A'*X + ISGN*X*B' = scale*C.
    *
    *        The (K,L)th block of X is determined starting from
    *        top-right corner column by column by
    *
    *           A(K,K)'*X(K,L) + ISGN*X(K,L)*B(L,L)' = C(K,L) - R(K,L)
    *
    *        Where
    *                     K-1                          N
    *            R(K,L) = SUM [A(I,K)'*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)'].
    *                     I=1                        J=L+1
    *
    *        Start column loop (index = L)
    *        L1 (L2): column index of the first (last) row of X(K,L)
    *
             LNEXT = N
             DO 180 L = N, 1, -1
                IF( L.GT.LNEXT )
         $         GO TO 180
                IF( L.EQ.1 ) THEN
                   L1 = L
                   L2 = L
                ELSE
                   IF( B( L, L-1 ).NE.ZERO ) THEN
                      L1 = L - 1
                      L2 = L
                      LNEXT = L - 2
                   ELSE
                      L1 = L
                      L2 = L
                      LNEXT = L - 1
                   END IF
                END IF
    *
    *           Start row loop (index = K)
    *           K1 (K2): row index of the first (last) row of X(K,L)
    *
                KNEXT = 1
                DO 170 K = 1, M
                   IF( K.LT.KNEXT )
         $            GO TO 170
                   IF( K.EQ.M ) THEN
                      K1 = K
                      K2 = K
                   ELSE
                      IF( A( K+1, K ).NE.ZERO ) THEN
                         K1 = K
                         K2 = K + 1
                         KNEXT = K + 2
                      ELSE
                         K1 = K
                         K2 = K
                         KNEXT = K + 1
                      END IF
                   END IF
    *
                   IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC,
         $                   B( L1, MIN( L1+1, N ) ), LDB )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                      SCALOC = ONE
    *
                      A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                      DA11 = ABS( A11 )
                      IF( DA11.LE.SMIN ) THEN
                         A11 = SMIN
                         DA11 = SMIN
                         INFO = 1
                      END IF
                      DB = ABS( VEC( 1, 1 ) )
                      IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                         IF( DB.GT.BIGNUM*DA11 )
         $                  SCALOC = ONE / DB
                      END IF
                      X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 130 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      130                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
    *
                   ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, A( K1, K1 ),
         $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 140 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      140                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K2, L1 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L2, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
    *
                      CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ),
         $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 150 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      150                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K1 ), 1, C( 1, L2 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L2, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( K1-1, A( 1, K2 ), 1, C( 1, L2 ), 1 )
                      SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
         $                   B( L2, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLASY2( .TRUE., .TRUE., ISGN, 2, 2, A( K1, K1 ),
         $                         LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
         $                         2, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 160 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      160                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 1, 2 )
                      C( K2, L1 ) = X( 2, 1 )
                      C( K2, L2 ) = X( 2, 2 )
                   END IF
    *
      170       CONTINUE
      180    CONTINUE
    *
          ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN
    *
    *        Solve    A*X + ISGN*X*B' = scale*C.
    *
    *        The (K,L)th block of X is determined starting from
    *        bottom-right corner column by column by
    *
    *            A(K,K)*X(K,L) + ISGN*X(K,L)*B(L,L)' = C(K,L) - R(K,L)
    *
    *        Where
    *                      M                          N
    *            R(K,L) = SUM [A(K,I)*X(I,L)] + ISGN*SUM [X(K,J)*B(L,J)'].
    *                    I=K+1                      J=L+1
    *
    *        Start column loop (index = L)
    *        L1 (L2): column index of the first (last) row of X(K,L)
    *
             LNEXT = N
             DO 240 L = N, 1, -1
                IF( L.GT.LNEXT )
         $         GO TO 240
                IF( L.EQ.1 ) THEN
                   L1 = L
                   L2 = L
                ELSE
                   IF( B( L, L-1 ).NE.ZERO ) THEN
                      L1 = L - 1
                      L2 = L
                      LNEXT = L - 2
                   ELSE
                      L1 = L
                      L2 = L
                      LNEXT = L - 1
                   END IF
                END IF
    *
    *           Start row loop (index = K)
    *           K1 (K2): row index of the first (last) row of X(K,L)
    *
                KNEXT = M
                DO 230 K = M, 1, -1
                   IF( K.GT.KNEXT )
         $            GO TO 230
                   IF( K.EQ.1 ) THEN
                      K1 = K
                      K2 = K
                   ELSE
                      IF( A( K, K-1 ).NE.ZERO ) THEN
                         K1 = K - 1
                         K2 = K
                         KNEXT = K - 2
                      ELSE
                         K1 = K
                         K2 = K
                         KNEXT = K - 1
                      END IF
                   END IF
    *
                   IF( L1.EQ.L2 .AND. K1.EQ.K2 ) THEN
                      SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
         $                   C( MIN( K1+1, M ), L1 ), 1 )
                      SUMR = DDOT( N-L1, C( K1, MIN( L1+1, N ) ), LDC,
         $                   B( L1, MIN( L1+1, N ) ), LDB )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
                      SCALOC = ONE
    *
                      A11 = A( K1, K1 ) + SGN*B( L1, L1 )
                      DA11 = ABS( A11 )
                      IF( DA11.LE.SMIN ) THEN
                         A11 = SMIN
                         DA11 = SMIN
                         INFO = 1
                      END IF
                      DB = ABS( VEC( 1, 1 ) )
                      IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
                         IF( DB.GT.BIGNUM*DA11 )
         $                  SCALOC = ONE / DB
                      END IF
                      X( 1, 1 ) = ( VEC( 1, 1 )*SCALOC ) / A11
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 190 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      190                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
    *
                   ELSE IF( L1.EQ.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, A( K1, K1 ),
         $                         LDA, ONE, ONE, VEC, 2, -SGN*B( L1, L1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 200 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      200                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K2, L1 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.EQ.K2 ) THEN
    *
                      SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
         $                   C( MIN( K1+1, M ), L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 1 ) = SGN*( C( K1, L1 )-( SUML+SGN*SUMR ) )
    *
                      SUML = DDOT( M-K1, A( K1, MIN( K1+1, M ) ), LDA,
         $                   C( MIN( K1+1, M ), L2 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L2, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 1 ) = SGN*( C( K1, L2 )-( SUML+SGN*SUMR ) )
    *
                      CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, B( L1, L1 ),
         $                         LDB, ONE, ONE, VEC, 2, -SGN*A( K1, K1 ),
         $                         ZERO, X, 2, SCALOC, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 210 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      210                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 2, 1 )
    *
                   ELSE IF( L1.NE.L2 .AND. K1.NE.K2 ) THEN
    *
                      SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 1 ) = C( K1, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K1, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L2 ), 1 )
                      SUMR = DDOT( N-L2, C( K1, MIN( L2+1, N ) ), LDC,
         $                   B( L2, MIN( L2+1, N ) ), LDB )
                      VEC( 1, 2 ) = C( K1, L2 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L1 ), 1 )
                      SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
         $                   B( L1, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 1 ) = C( K2, L1 ) - ( SUML+SGN*SUMR )
    *
                      SUML = DDOT( M-K2, A( K2, MIN( K2+1, M ) ), LDA,
         $                   C( MIN( K2+1, M ), L2 ), 1 )
                      SUMR = DDOT( N-L2, C( K2, MIN( L2+1, N ) ), LDC,
         $                   B( L2, MIN( L2+1, N ) ), LDB )
                      VEC( 2, 2 ) = C( K2, L2 ) - ( SUML+SGN*SUMR )
    *
                      CALL DLASY2( .FALSE., .TRUE., ISGN, 2, 2, A( K1, K1 ),
         $                         LDA, B( L1, L1 ), LDB, VEC, 2, SCALOC, X,
         $                         2, XNORM, IERR )
                      IF( IERR.NE.0 )
         $               INFO = 1
    *
                      IF( SCALOC.NE.ONE ) THEN
                         DO 220 J = 1, N
                            CALL DSCAL( M, SCALOC, C( 1, J ), 1 )
      220                CONTINUE
                         SCALE = SCALE*SCALOC
                      END IF
                      C( K1, L1 ) = X( 1, 1 )
                      C( K1, L2 ) = X( 1, 2 )
                      C( K2, L1 ) = X( 2, 1 )
                      C( K2, L2 ) = X( 2, 2 )
                   END IF
    *
      230       CONTINUE
      240    CONTINUE
    *
          END IF
    *
          RETURN
    *
    *     End of DTRSYL
    *
          END