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 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 trans­
               pose)

       DIAG    (input) CHARACTER*1
               Specifies whether or not the matrix A is unit tri­
               angular.  = '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


       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 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*16 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,  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 approxi­
               mate 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 ille­
               gal value


FURTHER DETAILS

       A  rough  bound  on  x  is  computed; if that is less than

       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
       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
       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**T *x =  b
       or  A**H *x = b.  The basic algorithm for A upper triangu­
       lar 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)|
       )
                           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).


Man(1) output converted with man2html