SUBROUTINE ZLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
     $                   SCALE, CNORM, INFO )
*
*  -- LAPACK auxiliary routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          DIAG, NORMIN, TRANS, UPLO
      INTEGER            INFO, KD, LDAB, N
      DOUBLE PRECISION   SCALE
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   CNORM( * )
      COMPLEX*16         AB( LDAB, * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  ZLATBS solves one of the triangular systems
*
*     A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,
*
*  with scaling to prevent overflow, where A is an upper or lower
*  triangular band matrix.  Here A' denotes the transpose of A, x and b
*  are n-element vectors, and s is a scaling factor, usually less than
*  or equal to 1, chosen so that the components of x will be less than
*  the overflow threshold.  If the unscaled problem will not cause
*  overflow, the Level 2 BLAS routine ZTBSV is called.  If the matrix A
*  is singular (A(j,j) = 0 for some j), then s is set to 0 and a
*  non-trivial solution to A*x = 0 is returned.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the matrix A is upper or lower triangular.
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  TRANS   (input) CHARACTER*1
*          Specifies the operation applied to A.
*          = 'N':  Solve A * x = s*b     (No transpose)
*          = 'T':  Solve A**T * x = s*b  (Transpose)
*          = 'C':  Solve A**H * x = s*b  (Conjugate transpose)
*
*  DIAG    (input) CHARACTER*1
*          Specifies whether or not the matrix A is unit triangular.
*          = 'N':  Non-unit triangular
*          = 'U':  Unit triangular
*
*  NORMIN  (input) CHARACTER*1
*          Specifies whether CNORM has been set or not.
*          = 'Y':  CNORM contains the column norms on entry
*          = 'N':  CNORM is not set on entry.  On exit, the norms will
*                  be computed and stored in CNORM.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  KD      (input) INTEGER
*          The number of subdiagonals or superdiagonals in the
*          triangular matrix A.  KD >= 0.
*
*  AB      (input) COMPLEX*16 array, dimension (LDAB,N)
*          The upper or lower triangular band matrix A, stored in the
*          first KD+1 rows of the array. The j-th column of A is stored
*          in the j-th column of the array AB as follows:
*          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*
*  LDAB    (input) INTEGER
*          The leading dimension of the array AB.  LDAB >= KD+1.
*
*  X       (input/output) COMPLEX*16 array, dimension (N)
*          On entry, the right hand side b of the triangular system.
*          On exit, X is overwritten by the solution vector x.
*
*  SCALE   (output) DOUBLE PRECISION
*          The scaling factor s for the triangular system
*             A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
*          If SCALE = 0, the matrix A is singular or badly scaled, and
*          the vector x is an exact or approximate solution to A*x = 0.
*
*  CNORM   (input or output) DOUBLE PRECISION array, dimension (N)
*
*          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
*          contains the norm of the off-diagonal part of the j-th column
*          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
*          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
*          must be greater than or equal to the 1-norm.
*
*          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
*          returns the 1-norm of the offdiagonal part of the j-th column
*          of A.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -k, the k-th argument had an illegal value
*
*  Further Details
*  ======= =======
*
*  A rough bound on x is computed; if that is less than overflow, ZTBSV
*  is called, otherwise, specific code is used which checks for possible
*  overflow or divide-by-zero at every operation.
*
*  A columnwise scheme is used for solving A*x = b.  The basic algorithm
*  if A is lower triangular is
*
*       x[1:n] := b[1:n]
*       for j = 1, ..., n
*            x(j) := x(j) / A(j,j)
*            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
*       end
*
*  Define bounds on the components of x after j iterations of the loop:
*     M(j) = bound on x[1:j]
*     G(j) = bound on x[j+1:n]
*  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*
*  Then for iteration j+1 we have
*     M(j+1) <= G(j) / | A(j+1,j+1) |
*     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
*            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*
*  where CNORM(j+1) is greater than or equal to the infinity-norm of
*  column j+1 of A, not counting the diagonal.  Hence
*
*     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
*                  1<=i<=j
*  and
*
*     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
*                                   1<=i< j
*
*  Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTBSV if the
*  reciprocal of the largest M(j), j=1,..,n, is larger than
*  max(underflow, 1/overflow).
*
*  The bound on x(j) is also used to determine when a step in the
*  columnwise method can be performed without fear of overflow.  If
*  the computed bound is greater than a large constant, x is scaled to
*  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
*  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*
*  Similarly, a row-wise scheme is used to solve A**T *x = b  or
*  A**H *x = b.  The basic algorithm for A upper triangular is
*
*       for j = 1, ..., n
*            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
*       end
*
*  We simultaneously compute two bounds
*       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
*       M(j) = bound on x(i), 1<=i<=j
*
*  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
*  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
*  Then the bound on x(j) is
*
*       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*
*            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
*                      1<=i<=j
*
*  and we can safely call ZTBSV if 1/M(n) and 1/G(n) are both greater
*  than max(underflow, 1/overflow).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, HALF, ONE, TWO
      PARAMETER          ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
     $                   TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN, NOUNIT, UPPER
      INTEGER            I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND
      DOUBLE PRECISION   BIGNUM, GROW, REC, SMLNUM, TJJ, TMAX, TSCAL,
     $                   XBND, XJ, XMAX
      COMPLEX*16         CSUMJ, TJJS, USCAL, ZDUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IDAMAX, IZAMAX
      DOUBLE PRECISION   DLAMCH, DZASUM
      COMPLEX*16         ZDOTC, ZDOTU, ZLADIV
      EXTERNAL           LSAME, IDAMAX, IZAMAX, DLAMCH, DZASUM, ZDOTC,
     $                   ZDOTU, ZLADIV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, XERBLA, ZAXPY, ZDSCAL, ZTBSV
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1, CABS2
*     ..
*     .. Statement Function definitions ..
      CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
      CABS2( ZDUM ) = ABS( DBLE( ZDUM ) / 2.D0 ) +
     $                ABS( DIMAG( ZDUM ) / 2.D0 )
*     ..
*     .. Executable Statements ..
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      NOTRAN = LSAME( TRANS, 'N' )
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Test the input parameters.
*
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
     $         LSAME( TRANS, 'C' ) ) THEN
         INFO = -2
      ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
         INFO = -3
      ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
     $         LSAME( NORMIN, 'N' ) ) THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( KD.LT.0 ) THEN
         INFO = -6
      ELSE IF( LDAB.LT.KD+1 ) THEN
         INFO = -8
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'ZLATBS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine machine dependent parameters to control overflow.
*
      SMLNUM = DLAMCH( 'Safe minimum' )
      BIGNUM = ONE / SMLNUM
      CALL DLABAD( SMLNUM, BIGNUM )
      SMLNUM = SMLNUM / DLAMCH( 'Precision' )
      BIGNUM = ONE / SMLNUM
      SCALE = ONE
*
      IF( LSAME( NORMIN, 'N' ) ) THEN
*
*        Compute the 1-norm of each column, not including the diagonal.
*
         IF( UPPER ) THEN
*
*           A is upper triangular.
*
            DO 10 J = 1, N
               JLEN = MIN( KD, J-1 )
               CNORM( J ) = DZASUM( JLEN, AB( KD+1-JLEN, J ), 1 )
   10       CONTINUE
         ELSE
*
*           A is lower triangular.
*
            DO 20 J = 1, N
               JLEN = MIN( KD, N-J )
               IF( JLEN.GT.0 ) THEN
                  CNORM( J ) = DZASUM( JLEN, AB( 2, J ), 1 )
               ELSE
                  CNORM( J ) = ZERO
               END IF
   20       CONTINUE
         END IF
      END IF
*
*     Scale the column norms by TSCAL if the maximum element in CNORM is
*     greater than BIGNUM/2.
*
      IMAX = IDAMAX( N, CNORM, 1 )
      TMAX = CNORM( IMAX )
      IF( TMAX.LE.BIGNUM*HALF ) THEN
         TSCAL = ONE
      ELSE
         TSCAL = HALF / ( SMLNUM*TMAX )
         CALL DSCAL( N, TSCAL, CNORM, 1 )
      END IF
*
*     Compute a bound on the computed solution vector to see if the
*     Level 2 BLAS routine ZTBSV can be used.
*
      XMAX = ZERO
      DO 30 J = 1, N
         XMAX = MAX( XMAX, CABS2( X( J ) ) )
   30 CONTINUE
      XBND = XMAX
      IF( NOTRAN ) THEN
*
*        Compute the growth in A * x = b.
*
         IF( UPPER ) THEN
            JFIRST = N
            JLAST = 1
            JINC = -1
            MAIND = KD + 1
         ELSE
            JFIRST = 1
            JLAST = N
            JINC = 1
            MAIND = 1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 60
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, G(0) = max{x(i), i=1,...,n}.
*
            GROW = HALF / MAX( XBND, SMLNUM )
            XBND = GROW
            DO 40 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 60
*
               TJJS = AB( MAIND, J )
               TJJ = CABS1( TJJS )
*
               IF( TJJ.GE.SMLNUM ) THEN
*
*                 M(j) = G(j-1) / abs(A(j,j))
*
                  XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
               ELSE
*
*                 M(j) could overflow, set XBND to 0.
*
                  XBND = ZERO
               END IF
*
               IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
*                 G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
                  GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
               ELSE
*
*                 G(j) could overflow, set GROW to 0.
*
                  GROW = ZERO
               END IF
   40       CONTINUE
            GROW = XBND
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
            DO 50 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 60
*
*              G(j) = G(j-1)*( 1 + CNORM(j) )
*
               GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
   50       CONTINUE
         END IF
   60    CONTINUE
*
      ELSE
*
*        Compute the growth in A**T * x = b  or  A**H * x = b.
*
         IF( UPPER ) THEN
            JFIRST = 1
            JLAST = N
            JINC = 1
            MAIND = KD + 1
         ELSE
            JFIRST = N
            JLAST = 1
            JINC = -1
            MAIND = 1
         END IF
*
         IF( TSCAL.NE.ONE ) THEN
            GROW = ZERO
            GO TO 90
         END IF
*
         IF( NOUNIT ) THEN
*
*           A is non-unit triangular.
*
*           Compute GROW = 1/G(j) and XBND = 1/M(j).
*           Initially, M(0) = max{x(i), i=1,...,n}.
*
            GROW = HALF / MAX( XBND, SMLNUM )
            XBND = GROW
            DO 70 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )
     $            GO TO 90
*
*              G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
               XJ = ONE + CNORM( J )
               GROW = MIN( GROW, XBND / XJ )
*
               TJJS = AB( MAIND, J )
               TJJ = CABS1( TJJS )
*
               IF( TJJ.GE.SMLNUM ) THEN
*
*                 M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
                  IF( XJ.GT.TJJ )
     $               XBND = XBND*( TJJ / XJ )
               ELSE
*
*                 M(j) could overflow, set XBND to 0.
*
                  XBND = ZERO
               END IF
   70       CONTINUE
            GROW = MIN( GROW, XBND )
         ELSE
*
*           A is unit triangular.
*
*           Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
            GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
            DO 80 J = JFIRST, JLAST, JINC
*
*              Exit the loop if the growth factor is too small.
*
               IF( GROW.LE.SMLNUM )