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:

              gular.

              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 num­
              ber of sub-diagonals of the matrix A.  K must  sat­
              isfy  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 coeffi­
              cients, supplied column by column, with the leading
              diagonal  of  the  matrix  in  row ( k + 1 ) of the
              array, the first super-diagonal starting  at  posi­
              tion  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     CON­
              TINUE 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 coeffi­
              cients, 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

              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 ele­
              ments 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.


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