DLATPS  -  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 matrix stored in packed
       form


SYNOPSIS

       SUBROUTINE DLATPS( UPLO, TRANS, DIAG, NORMIN,  N,  AP,  X,
                          SCALE, CNORM, INFO )

           CHARACTER      DIAG, NORMIN, TRANS, UPLO

           INTEGER        INFO, N

           DOUBLE         PRECISION SCALE

           DOUBLE         PRECISION AP( * ), CNORM( * ), X( * )


PURPOSE

       DLATPS  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 matrix stored in packed form.
       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 DTPSV 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.

       AP      (input) DOUBLE PRECISION array, dimension
               (N*(N+1)/2)
               The upper or lower  triangular  matrix  A,  packed
               columnwise  in a linear array.  The j-th column of
               A is stored in the array AP as follows: if UPLO  =
               'U',  AP(i  +  (j-1)*j/2) = A(i,j) for 1<=i<=j; if
               UPLO = 'L', AP(i + (j-1)*(2n-j)/2)  =  A(i,j)  for
               j<=i<=n.

       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


FURTHER DETAILS

       A rough bound on x is computed; if that is less than over­
       flow, DTPSV 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

            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
       DTPSV 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] ) /
       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),

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


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