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DLATRS(l)					)					DLATRS(l)

NAME
       DLATRS  -  solve  one of the triangular systems	A *x = s*b or A'*x = s*b  with scaling to
       prevent overflow

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

	   CHARACTER	  DIAG, NORMIN, TRANS, UPLO

	   INTEGER	  INFO, LDA, N

	   DOUBLE	  PRECISION SCALE

	   DOUBLE	  PRECISION A( LDA, * ), CNORM( * ), X( * )

PURPOSE
       DLATRS solves one of the triangular systems A *x = s*b or A'*x = s*b with scaling to  pre-
       vent  overflow. Here A is an upper or lower triangular matrix, 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 DTRSV  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 trian-
	       gular
	       = '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 triangular.  = 'N':  Non-unit trian-
	       gular
	       = '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.

       A       (input) DOUBLE PRECISION array, dimension (LDA,N)
	       The triangular matrix A.  If UPLO = 'U', the leading n by n upper triangular  part
	       of the array A contains the upper triangular matrix, and the strictly lower trian-
	       gular part of A is not referenced.  If UPLO = 'L', the leading n by n lower trian-
	       gular  part  of the array A contains the lower triangular matrix, and the strictly
	       upper triangular part of A is not referenced.  If DIAG = 'U',  the  diagonal  ele-
	       ments 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) DOUBLE PRECISION 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) 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 vector 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 illegal value

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

       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]
	    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  DTRSV  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'*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 DTRSV if 1/M(n) and 1/G(n) are  both  greater  than  max(underflow,
       1/overflow).

LAPACK version 3.0			   15 June 2000 				DLATRS(l)
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