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

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





	   REAL		  CNORM( * )

	   COMPLEX	  AB( LDAB, * ), X( * )

       CLATBS  solves  one  of	the triangular systems with scaling to prevent
       overflow, where A is an upper or lower triangular band matrix.  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
       CTBSV  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**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 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.

       KD      (input) INTEGER
	       The  number of subdiagonals or superdiagonals in the triangular
	       matrix A.  KD >= 0.

       AB      (input) COMPLEX array, dimension (LDAB,N)
	       The upper or lower triangular band  matrix  A,  stored  in  the
	       first KD+1 rows of the array. The j-th column of A is stored in
	       the j-th column of the array AB as  follows:  if	 UPLO  =  'U',
	       AB(kd+1+i-j,j)  =  A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L',
	       AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

       LDAB    (input) INTEGER
	       The leading dimension of the array AB.  LDAB >= KD+1.

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

 LAPACK auxiliary routine (versioNovember 2008			     CLATBS(1)

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