NAME

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

SYNOPSIS

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

           DOUBLE       PRECISION ALPHA, BETA

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

           CHARACTER*1  TRANS

           DOUBLE       PRECISION A( LDA, * ), X( * ), Y( * )

PURPOSE

       DGBMV  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 performed 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*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.

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

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

       A      - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
              Before entry, the leading ( kl + ku + 1 ) by n part of the array A must contain the
              matrix of coefficients, supplied column by column, with the leading diagonal of the
              matrix  in  row  (  ku  +  1  )  of the array, the first super-diagonal starting at
              position 2 in row ku, the first sub-diagonal starting at position 1 in row ( ku + 2
              ),  and  so  on.  Elements in the array A that do not correspond to elements 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      - DOUBLE PRECISION array of DIMENSION at least
              ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n' and at least ( 1 + (  m  -  1
              )*abs(  INCX ) ) otherwise.  Before entry, the incremented array X must contain the
              vector x.  Unchanged on exit.

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

       BETA   - DOUBLE PRECISION.
              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      - DOUBLE PRECISION array of DIMENSION at least
              ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n' and at least ( 1 + (  n  -  1
              )*abs(  INCY ) ) otherwise.  Before entry, the incremented array Y must contain the
              vector y. On exit, Y is overwritten by the updated vector y.

       INCY   - INTEGER.
              On entry, INCY specifies the increment for the elements 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.