We present algorithms for computing eigenvalues and eigenvectors of quantum spin chains.  In particular, we will examine efficient algorithms for computing interior eigenvalues of a finite size disordered spin chain with short range interactions.  We will also examine algorithms for computing the ground state of a translational invariant spin chain with infinite number of spins.  One way to solve such a problem is to use an infinite translational invariant tensor ring to represent the approximation to the eigenvector to be computed and update the core of such a tensor ring through a flexible power iteration. We discuss approximations made in each step of the power iteration and how these approximations affect the convergence of the method.

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