Computing the eigenvalues of symmetric tridiagonal matrices via a Cayley transformation

In this paper we present a new algorithm for solving the symmetric tridiagonal eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg's implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable. It is comparable to other algorithms in terms of accuracy. Although it is not the fastest algorithm, it is not conspicuously slow either. It is approximately as fast as the symmetric tridiagonal QR algorithm.