  By application of the theory for second-order linear differential equations
with two turning points developed in \cite{Olver1975}, uniform asymptotic
approximations are obtained for the Lam\'{e} and Mathieu functions with a large
real parameter. The approximations are expressed in terms of parabolic cylinder
functions, and are uniformly valid in their respective real open intervals. In
all cases explicit bounds are supplied for the error terms associated with the
approximations. Approximations are also obtained for their respective
eigenvalues.
