CGBMV - perform one of the matrix-vector operations   y :=
       alpha*A*x + beta*y, or y := alpha*A'*x + beta*y, or   y :=
       alpha*conjg( A' )*x + beta*y,


SYNOPSIS

       SUBROUTINE CGBMV (  TRANS, M, N, KL, KU, ALPHA, A, LDA, X,
                        INCX, BETA, Y, INCY )

           COMPLEX      ALPHA, BETA

           INTEGER      INCX, INCY, KL, KU, LDA, M, N

           CHARACTER*1  TRANS

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


PURPOSE

       CGBMV  performs one of the matrix-vector operations

       where alpha and beta are scalars, x and y are vectors  and
       A  is  an m by n band matrix, with kl sub-diagonals and ku
       super-diagonals.


PARAMETERS

       TRANS  - CHARACTER*1.
              On entry, TRANS specifies the operation to be  per­
              formed as follows:

              TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.

              TRANS = 'T' or 't'   y := alpha*A'*x + beta*y.

              TRANS  =  'C'  or  'c'   y := alpha*conjg( A' )*x +
              beta*y.

              Unchanged on exit.

       M      - INTEGER.
              On entry, M specifies the number  of  rows  of  the
              matrix  A.   M must be at least zero.  Unchanged on
              exit.

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

       KL     - INTEGER.
              On entry, KL specifies the number of  sub-diagonals
              of  the  matrix  A.  KL  must  satisfy   0 .le. KL.
              Unchanged on exit.

              On entry, KU specifies the number  of  super-diago­
              nals  of  the matrix A. KU must satisfy  0 .le. KU.
              Unchanged on exit.

       ALPHA  - COMPLEX         .
              On  entry,  ALPHA  specifies  the   scalar   alpha.
              Unchanged on exit.

       A      - COMPLEX          array of DIMENSION ( LDA, n ).
              Before entry, the leading ( kl + ku + 1 ) by n part
              of the array A must contain the matrix  of  coeffi­
              cients, supplied column by column, with the leading
              diagonal of the matrix in row ( ku +  1  )  of  the
              array,  the  first super-diagonal starting at posi­
              tion 2 in row ku, the first  sub-diagonal  starting
              at  position  1 in row ( ku + 2 ), and so on.  Ele­
              ments in the array A that do not correspond to ele­
              ments  in  the band matrix (such as the top left ku
              by ku triangle) are not referenced.  The  following
              program  segment  will  transfer a band matrix from
              conventional full matrix storage to band storage:

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

              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 ( kl + ku + 1 ).  Unchanged on exit.

       X      - COMPLEX          array of DIMENSION at least
              ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS =  'N'  or
              'n' and at least ( 1 + ( m - 1 )*abs( INCX ) ) oth­
              erwise.  Before entry, the incremented array X must
              contain the vector x.  Unchanged on exit.

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

       BETA   - COMPLEX         .
              On entry, BETA specifies the scalar beta. When BETA
              is supplied as zero then  Y  need  not  be  set  on
              input.  Unchanged on exit.

       Y      - COMPLEX          array of DIMENSION at least
              (  1  + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or
              'n' and at least ( 1 + ( n - 1 )*abs( INCY ) ) oth­
              erwise.  Before entry, the incremented array Y must

              the updated vector y.

       INCY   - INTEGER.
              On entry, INCY specifies the increment for the ele­
              ments of Y. INCY 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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