SLATPS  -  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 SLATPS( UPLO, TRANS, DIAG, NORMIN,  N,  AP,  X,
                          SCALE, CNORM, INFO )

           CHARACTER      DIAG, NORMIN, TRANS, UPLO

           INTEGER        INFO, N

           REAL           SCALE

           REAL           AP( * ), CNORM( * ), X( * )


PURPOSE

       SLATPS  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 STPSV 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) REAL 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) 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,  STPSV  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+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
       STPSV  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),
       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(0) * product ( ( 1 + CNORM(i) ) /  |A(i,i)|
       )
                           1<=i<=j

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


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