TITLE: Dominant eigenvector of a real symmetric tridiagonal matrix
QUESTION [6 upvotes]: What is the most efficient way to calculate the dominant eigenvector of a real symmetric tridiagonal matrix? What's the corresponding time complexity bound? 
Could someone give me a reference for this special structure? Thank you in advance.

REPLY [3 votes]: 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.

I. S. Dhillon and B. N. Parlett. Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices. (LAA 387, 2004).

(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