DLATBS  -  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


SYNOPSIS

       SUBROUTINE DLATBS( UPLO,  TRANS,  DIAG, NORMIN, N, KD, AB,
                          LDAB, X, SCALE, CNORM, INFO )

           CHARACTER      DIAG, NORMIN, TRANS, UPLO

           INTEGER        INFO, KD, LDAB, N

           DOUBLE         PRECISION SCALE

           DOUBLE         PRECISION AB( LDAB, * ), CNORM( * ), X(
                          * )


PURPOSE

       DLATBS  solves 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. 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 over­
       flow, the Level 2 BLAS routine DTBSV 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'* x = s*b  (Transpose)
               =  'C':  Solve A'* x = s*b  (Conjugate transpose =
               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


       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) DOUBLE PRECISION 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) DOUBLE PRECISION 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   or   A'*  x  = s*b.  If SCALE = 0, the
               matrix A is singular or badly scaled, and the vec­
               tor 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 ille­
               gal value

       A rough bound on x is computed; if that is less than over­
       flow, DTBSV 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
       DTBSV 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'*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] ) /

            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 DTBSV if 1/M(n) and 1/G(n) are both
       greater than max(underflow, 1/overflow).


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