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NAME
     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

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

     CHARACTER DIAG, NORMIN, TRANS, UPLO

     INTEGER INFO, KD, LDAB, N

     REAL SCALE

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



     #include <sunperf.h>

     void slatbs(char uplo, char trans, char diag,  char  normin,
               int  n,  int  kd, float *sab, int ldab, float *sx,
               float *sscale, float *cnorm, int *info) ;

PURPOSE
     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.


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
               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  >=
               KD+1.

     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 vec-
               tor 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  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

FURTHER DETAILS
     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]
          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  STBSV
     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 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)
          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 STBSV if 1/M(n) and 1/G(n)  are  both
     greater than max(underflow, 1/overflow).

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