subroutine sgemv ( trans, m, n, alpha, a, lda, x, incx,
     $                   beta, y, incy )
*     .. scalar arguments ..
      real               alpha, beta
      integer            incx, incy, lda, m, n
      character*1        trans
*     .. array arguments ..
      real               a( lda, * ), x( * ), y( * )
*     ..
*
*  purpose
*  =======
*
*  sgemv  performs one of the matrix-vector operations
*
*     y := alpha*a*x + beta*y,   or   y := alpha*a'*x + beta*y,
*
*  where alpha and beta are scalars, x and y are vectors and a is an
*  m by n matrix.
*
*  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.
*
*  alpha  - real            .
*           on entry, alpha specifies the scalar alpha.
*           unchanged on exit.
*
*  a      - real             array of dimension ( lda, n ).
*           before entry, the leading m by n part of the array a must
*           contain the matrix of coefficients.
*           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
*           max( 1, m ).
*           unchanged on exit.
*
*  x      - real             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   - real            .
*           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      - real             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 with beta non-zero, 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.
*