Is there a lapack routine which does not require the full matrix as input?
If your matrix is symmetric (or Hermitian) and with a narrow bandwidth we have some direct eigensolver for this.
Otherwise => nope.
You probably need to turn to iterative methods package.
If the matrix is symmetric or Hermitian, check out PRIMME or BLOPEX.
If the matrix is nonsymmetric, checkout ARPACK.
You have some larger software like Trilinos or Petsc that might be worth checking out.