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RedHat 9 (Linux i386) - man page for zlatbs (redhat section l)

ZLATBS(l)					)					ZLATBS(l)

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

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

	   CHARACTER	  DIAG, NORMIN, TRANS, UPLO

	   INTEGER	  INFO, KD, LDAB, N

	   DOUBLE	  PRECISION SCALE

	   DOUBLE	  PRECISION CNORM( * )

	   COMPLEX*16	  AB( LDAB, * ), X( * )

PURPOSE
       ZLATBS 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 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 ZTBSV 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 trans-
	       pose)
	       = '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-
	       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.

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

       AB      (input) COMPLEX*16 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  fol-
	       lows: 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*16 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,  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) 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, ZTBSV 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 ZTBSV 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)
	    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 ZTBSV if 1/M(n) and 1/G(n) are both greater than max(underflow,
       1/overflow).

LAPACK version 3.0			   15 June 2000 				ZLATBS(l)


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