clatbs(3F)

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

SYNOPSIS
     SUBROUTINE CLATBS( 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           CNORM( * )

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

PURPOSE
     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.


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