What is the most efficient way to calucate the dominant eigenvector of a real symmetric tridiagonal matrix, and what's the corresponding time complexity bound? Could someone give me a reference for this special structure? Thank you in advance.

The following paper proposes an $O(nk)$ algorithm called $MR^3$ (Multiple Relatively Robust Representations) to compute $k$ orthogonal eigenvectors of an $n \times n$ symmetric tridiagonal matrix. An implementation of this method is also available in LAPACK.
(The paper also includes details of an algorithm called "GetVec" for computing an eigenvector corresponding to an isolated eigenvalue). Additionally, searching for the MRRR algorithm will turn up other relevant results for you 

