```      SUBROUTINE ZTRSYL( 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 ..
COMPLEX*16         A( LDA, * ), B( LDB, * ), C( LDC, * )
*     ..
*
*  Purpose
*  =======
*
*  ZTRSYL solves the complex 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**H, and A and B are both upper 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.
*
*  Arguments
*  =========
*
*  TRANA   (input) CHARACTER*1
*          Specifies the option op(A):
*          = 'N': op(A) = A    (No transpose)
*          = 'C': op(A) = A**H (Conjugate transpose)
*
*  TRANB   (input) CHARACTER*1
*          Specifies the option op(B):
*          = 'N': op(B) = B    (No transpose)
*          = 'C': op(B) = B**H (Conjugate transpose)
*
*  ISGN    (input) INTEGER
*          = +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) COMPLEX*16 array, dimension (LDA,M)
*          The upper triangular matrix A.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A. LDA >= max(1,M).
*
*  B       (input) COMPLEX*16 array, dimension (LDB,N)
*          The upper triangular matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B. LDB >= max(1,N).
*
*  C       (input/output) COMPLEX*16 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   ONE
PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            NOTRNA, NOTRNB
INTEGER            J, K, L
DOUBLE PRECISION   BIGNUM, DA11, DB, EPS, SCALOC, SGN, SMIN,
\$                   SMLNUM
COMPLEX*16         A11, SUML, SUMR, VEC, X11
*     ..
*     .. Local Arrays ..
DOUBLE PRECISION   DUM( 1 )
*     ..
*     .. External Functions ..
LOGICAL            LSAME
DOUBLE PRECISION   DLAMCH, ZLANGE
EXTERNAL           LSAME, DLAMCH, ZLANGE, ZDOTC, ZDOTU, ZLADIV
*     ..
*     .. External Subroutines ..
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, 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, 'C' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRNB .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( 'ZTRSYL', -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
SMLNUM = SMLNUM*DBLE( M*N ) / EPS
BIGNUM = ONE / SMLNUM
SMIN = MAX( SMLNUM, EPS*ZLANGE( 'M', M, M, A, LDA, DUM ),
\$       EPS*ZLANGE( '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
*
DO 30 L = 1, N
DO 20 K = M, 1, -1
*
SUML = ZDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
\$                C( MIN( K+1, M ), L ), 1 )
SUMR = ZDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
VEC = C( K, L ) - ( SUML+SGN*SUMR )
*
SCALOC = ONE
A11 = A( K, K ) + SGN*B( L, L )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
\$               SCALOC = ONE / DB
END IF
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 10 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
10             CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
20       CONTINUE
30    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
*
DO 60 L = 1, N
DO 50 K = 1, M
*
SUML = ZDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
SUMR = ZDOTU( L-1, C( K, 1 ), LDC, B( 1, L ), 1 )
VEC = C( K, L ) - ( SUML+SGN*SUMR )
*
SCALOC = ONE
A11 = DCONJG( A( K, K ) ) + SGN*B( L, L )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
\$               SCALOC = ONE / DB
END IF
*
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 40 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
40             CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
50       CONTINUE
60    CONTINUE
*
ELSE IF( .NOT.NOTRNA .AND. .NOT.NOTRNB ) THEN
*
*        Solve    A'*X + ISGN*X*B' = C.
*
*        The (K,L)th block of X is determined starting from
*        upper-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
*           R(K,L) = SUM [A'(I,K)*X(I,L)] +
*                    I=1
*                           N
*                     ISGN*SUM [X(K,J)*B'(L,J)].
*                          J=L+1
*
DO 90 L = N, 1, -1
DO 80 K = 1, M
*
SUML = ZDOTC( K-1, A( 1, K ), 1, C( 1, L ), 1 )
SUMR = ZDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
\$                B( L, MIN( L+1, N ) ), LDB )
VEC = C( K, L ) - ( SUML+SGN*DCONJG( SUMR ) )
*
SCALOC = ONE
A11 = DCONJG( A( K, K )+SGN*B( L, L ) )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
\$               SCALOC = ONE / DB
END IF
*
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 70 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
70             CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
80       CONTINUE
90    CONTINUE
*
ELSE IF( NOTRNA .AND. .NOT.NOTRNB ) THEN
*
*        Solve    A*X + ISGN*X*B' = 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                          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
*
DO 120 L = N, 1, -1
DO 110 K = M, 1, -1
*
SUML = ZDOTU( M-K, A( K, MIN( K+1, M ) ), LDA,
\$                C( MIN( K+1, M ), L ), 1 )
SUMR = ZDOTC( N-L, C( K, MIN( L+1, N ) ), LDC,
\$                B( L, MIN( L+1, N ) ), LDB )
VEC = C( K, L ) - ( SUML+SGN*DCONJG( SUMR ) )
*
SCALOC = ONE
A11 = A( K, K ) + SGN*DCONJG( B( L, L ) )
DA11 = ABS( DBLE( A11 ) ) + ABS( DIMAG( A11 ) )
IF( DA11.LE.SMIN ) THEN
A11 = SMIN
DA11 = SMIN
INFO = 1
END IF
DB = ABS( DBLE( VEC ) ) + ABS( DIMAG( VEC ) )
IF( DA11.LT.ONE .AND. DB.GT.ONE ) THEN
IF( DB.GT.BIGNUM*DA11 )
\$               SCALOC = ONE / DB
END IF
*
X11 = ZLADIV( VEC*DCMPLX( SCALOC ), A11 )
*
IF( SCALOC.NE.ONE ) THEN
DO 100 J = 1, N
CALL ZDSCAL( M, SCALOC, C( 1, J ), 1 )
100             CONTINUE
SCALE = SCALE*SCALOC
END IF
C( K, L ) = X11
*
110       CONTINUE
120    CONTINUE
*
END IF
*
RETURN
*
*     End of ZTRSYL
*
END

```