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CLATPS(l)					)					CLATPS(l)

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





	   REAL 	  CNORM( * )

	   COMPLEX	  AP( * ), X( * )

       CLATPS 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, where A is an upper or lower triangular matrix
       stored in packed form.  Here A**T denotes the transpose of A, A**H denotes  the	conjugate
       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
       CTPSV 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.

       UPLO    (input) CHARACTER*1
	       Specifies whether the matrix A is upper or lower triangular.  = 'U':  Upper trian-
	       = 'L':  Lower triangular

       TRANS   (input) CHARACTER*1
	       Specifies the operation applied to A.  = 'N':  Solve A * x =  s*b      (No  trans-
	       = 'T':  Solve A**T * x = s*b  (Transpose)
	       = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

       DIAG    (input) CHARACTER*1
	       Specifies whether or not the matrix A is unit triangular.  = 'N':  Non-unit trian-
	       = '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.

       AP      (input) COMPLEX 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) COMPLEX array, dimension (N)
	       On entry, the right hand side b of the triangular system.  On exit, X is overwrit-
	       ten 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 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 illegal value

       A rough bound on x is computed; if that is less than overflow, CTPSV is called, otherwise,
       specific code is used which checks for possible overflow or divide-by-zero at every opera-

       A columnwise scheme is used for solving A*x = b.  The basic algorithm if A is lower trian-
       gular 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]

       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) | )

	  |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  CTPSV  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  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**T *x = b  or A**H  *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)

       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)| )

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

LAPACK version 3.0			   15 June 2000 				CLATPS(l)
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