SLATBS(l)								 )								 SLATBS(l)

NAME

       SLATBS - solve one of the triangular systems  A *x = s*b or A'*x = s*b  with scaling to prevent overflow, where A is an upper or lower tri-
       angular band matrix

SYNOPSIS

       SUBROUTINE SLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO )

	   CHARACTER	  DIAG, NORMIN, TRANS, UPLO

	   INTEGER	  INFO, KD, LDAB, N

	   REAL 	  SCALE

	   REAL 	  AB( LDAB, * ), CNORM( * ), X( * )

PURPOSE

       SLATBS solves one of the triangular systems A *x = s*b or A'*x = s*b with scaling to prevent overflow, where A is an upper or lower  trian-
       gular 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 STBSV 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'* 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.

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

       AB      (input) REAL 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) REAL 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  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) 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

FURTHER DETAILS

       A  rough bound on x is computed; if that is less than overflow, STBSV 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]
	    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 STBSV 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  com-
       puted  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'*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 STBSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

LAPACK version 3.0						   15 June 2000 							 SLATBS(l)