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NAME

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

SYNOPSIS

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

           INTEGER      INCX, K, LDA, N

           CHARACTER*1  DIAG, TRANS, UPLO

           COMPLEX*16   A( LDA, * ), X( * )

PURPOSE

       ZTBSV  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*16       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*16       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.