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NAME

       CTBSV  -  solve one of the systems of equations   A*x = b, or A'*x = b, or conjg( A' )*x =
       b,

SYNOPSIS

       SUBROUTINE CTBSV ( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )

           INTEGER      INCX, K, LDA, N

           CHARACTER*1  DIAG, TRANS, UPLO

           COMPLEX      A( LDA, * ), X( * )

PURPOSE

       CTBSV  solves one of the systems of equations

       where b and x are n element vectors and A is an n by n unit, or non-unit, upper  or  lower
       triangular band matrix, with ( k + 1 ) diagonals.

       No  test  for singularity or near-singularity is included in this routine. Such tests must
       be performed before calling this routine.

PARAMETERS

       UPLO   - CHARACTER*1.
              On entry, UPLO specifies whether the matrix is an upper or lower triangular  matrix
              as follows:

              UPLO = 'U' or 'u'   A is an upper triangular matrix.

              UPLO = 'L' or 'l'   A is a lower triangular matrix.

              Unchanged on exit.

       TRANS  - CHARACTER*1.
              On entry, TRANS specifies the equations to be solved as follows:

              TRANS = 'N' or 'n'   A*x = b.

              TRANS = 'T' or 't'   A'*x = b.

              TRANS = 'C' or 'c'   conjg( A' )*x = b.

              Unchanged on exit.

       DIAG   - CHARACTER*1.
              On entry, DIAG specifies whether or not A is unit triangular as follows:

              DIAG = 'U' or 'u'   A is assumed to be unit triangular.

              DIAG = 'N' or 'n'   A is not assumed to be unit triangular.

              Unchanged on exit.

       N      - INTEGER.
              On  entry,  N  specifies  the  order  of  the  matrix  A.  N must be at least zero.
              Unchanged on exit.

       K      - INTEGER.
              On entry with UPLO = 'U' or 'u', K specifies the number of super-diagonals  of  the
              matrix A.  On entry with UPLO = 'L' or 'l', K specifies the number of sub-diagonals
              of the matrix A.  K must satisfy  0 .le. K.  Unchanged on exit.

       A      - COMPLEX          array of DIMENSION ( LDA, n ).
              Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) by n part of the array A
              must contain the upper triangular band part of the matrix of coefficients, supplied
              column by column, with the leading diagonal of the matrix in row ( k + 1 )  of  the
              array, the first super-diagonal starting at position 2 in row k, and so on. The top
              left k by k triangle of the array A  is  not  referenced.   The  following  program
              segment will transfer an upper triangular band matrix from conventional full matrix
              storage to band storage:

              DO 20, J = 1, N M = K + 1 - J DO 10, I = MAX( 1, J - K ), J A( M + I, J ) = matrix(
              I, J ) 10    CONTINUE 20 CONTINUE

              Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) by n part of the array A
              must contain the lower triangular band part of the matrix of coefficients, supplied
              column  by  column,  with the leading diagonal of the matrix in row 1 of the array,
              the first sub-diagonal starting at position 1 in row 2, and so on. The bottom right
              k  by  k  triangle of the array A is not referenced.  The following program segment
              will transfer a lower triangular band matrix from conventional full matrix  storage
              to band storage:

              DO  20, J = 1, N M = 1 - J DO 10, I = J, MIN( N, J + K ) A( M + I, J ) = matrix( I,
              J ) 10    CONTINUE 20 CONTINUE

              Note that when DIAG = 'U' or 'u' the elements of the array A corresponding  to  the
              diagonal  elements  of  the matrix are not referenced, but are assumed to be unity.
              Unchanged on exit.

       LDA    - INTEGER.
              On entry, LDA specifies the first dimension of A as declared in the  calling  (sub)
              program. LDA must be at least ( k + 1 ).  Unchanged on exit.

       X      - COMPLEX          array of dimension at least
              (  1 + ( n - 1 )*abs( INCX ) ).  Before entry, the incremented array X must contain
              the n element right-hand side vector b. On exit, X is overwritten with the solution
              vector x.

       INCX   - INTEGER.
              On  entry,  INCX  specifies  the  increment for the elements of X. INCX must not be
              zero.  Unchanged on exit.

              Level 2 Blas routine.

              -- Written on 22-October-1986.  Jack Dongarra, Argonne  National  Lab.   Jeremy  Du
              Croz,  Nag  Central  Office.  Sven Hammarling, Nag Central Office.  Richard Hanson,
              Sandia National Labs.