  Let M be a matroid on ground set E. A subset l of E is called a `line' when
its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in
(M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a
system of linear constraints, one for each flat of M. Fractional matchings were
introduced by Vande Vate, who showed that the set of fractional matchings is a
half-integer relaxation of the matroid matching polytope.
  It was shown by Chang et al. that a maximum size fractional matching can be
found in polynomial time. In this paper we give a polynomial time algorithm to
find for any given weights on the lines in L, a maximum weight fractional
matching.
