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DLATRS(1)	    LAPACK auxiliary routine (version 3.2)	     DLATRS(1)

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

			  CNORM, INFO )




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

       DLATRS  solves  one  of	the  triangular	 systems 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.

       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 triangular.  =
	       '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 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 trian‐
	       gular matrix, and the strictly lower triangular 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 elements of A are also  not	refer‐
	       enced 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 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 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) con‐
	       tains 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  off‐
	       diagonal 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, DTRSV 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) := 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  col‐
       umn 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 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  column‐
       wise method can be performed without fear of overflow.  If the computed
       bound is greater than a large constant, x is scaled  to	prevent	 over‐
       flow,  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)
       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 DTRSV if 1/M(n) and 1/G(n) are both greater than
       max(underflow, 1/overflow).

 LAPACK auxiliary routine (versioNovember 2008			     DLATRS(1)

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