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      SLATBS - solve one of the triangular systems   A *x = s*b or
      A'*x = s*b  with scaling to prevent overflow, where A is an
      upper or lower triangular band matrix

                         LDAB, X, SCALE, CNORM, INFO )


          INTEGER        INFO, KD, LDAB, N

          REAL           SCALE

          REAL           AB( LDAB, * ), CNORM( * ), X( * )

      SLATBS solves one of the triangular systems 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 STBSV 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.

      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'* x = s*b  (Transpose)
              = 'C':  Solve A'* x = s*b  (Conjugate transpose =

      DIAG    (input) CHARACTER*1
              Specifies whether or not the matrix A is unit tri-
              angular.  = '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) REAL 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 >=

      X       (input/output) REAL 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) REAL
              The scaling factor s for the triangular system A * x
              = s*b  or  A'* 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) REAL array, dimension (N)

              If NORMIN = 'Y', CNORM is an input variable 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 variable 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

      A rough bound on x is computed; if that is less than over-
      flow, STBSV 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]

      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) | )

         |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 STBSV
      if the reciprocal of the largest M(j), j=1,..,n, is larger
      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 over-
      flow.  If the computed bound is greater than a large con-
      stant, 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'*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)

      We simultaneously compute two bounds
           G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ),
           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)| )

      and we can safely call STBSV if 1/M(n) and 1/G(n) are both
      greater than max(underflow, 1/overflow).