CLATRS - solve one of the triangular systems  A * x = s*b,
       A**T * x = s*b, or A**H * x = s*b,


SYNOPSIS

       SUBROUTINE CLATRS( UPLO, TRANS, DIAG, NORMIN, N,  A,  LDA,
                          X, SCALE, CNORM, INFO )

           CHARACTER      DIAG, NORMIN, TRANS, UPLO

           INTEGER        INFO, LDA, N

           REAL           SCALE

           REAL           CNORM( * )

           COMPLEX        A( LDA, * ), X( * )


PURPOSE

       CLATRS  solves  one of the triangular systems A * x = s*b,
       A**T * x = s*b, or A**H * x = s*b, with scaling to prevent
       overflow.   Here A is an upper or lower triangular matrix,
       A**T denotes the transpose of A, A**H denotes  the  conju­
       gate  transpose of A, x and b are n-element vectors, and s
       is a scaling factor, usually less than or equal to 1, cho­
       sen  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 CTRSV 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**T * x = s*b  (Transpose)
               = 'C':  Solve A**H * x =  s*b   (Conjugate  trans­
               pose)

       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

               norms will be computed and stored in CNORM.

       N       (input) INTEGER
               The order of the matrix A.  N >= 0.

       A       (input) COMPLEX array, dimension (LDA,N)
               The triangular matrix A.  If UPLO = 'U', the lead­
               ing  n  by  n upper triangular part of the array A
               contains the  upper  triangular  matrix,  and  the
               strictly  lower triangular part of A is not refer­
               enced.  If UPLO = 'L', the leading n  by  n  lower
               triangular  part of the array A contains the lower
               triangular matrix, and the strictly upper triangu­
               lar  part  of A is not referenced.  If DIAG = 'U',
               the diagonal elements of A are also not referenced
               and are assumed to be 1.

       LDA     (input) INTEGER
               The  leading dimension of the array A.  LDA >= max
               (1,N).

       X       (input/output) COMPLEX 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,  A**T * x = s*b,  or  A**H * x = s*b.  If
               SCALE = 0, the  matrix  A  is  singular  or  badly
               scaled,  and  the vector x is an exact or approxi­
               mate 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

       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
       CTRSV  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**T *x = b
       or A**H *x = b.  The basic algorithm for A upper  triangu­
       lar 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]  ),
       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 CTRSV if 1/M(n) and 1/G(n) are both
       greater than max(underflow, 1/overflow).


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