A New O(n^2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem

A New O(n) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem by Inderjit Singh Dhillon Doctor of Philosophy in Computer Science University of California, Berkeley Professor James W. Demmel, Chair Computing the eigenvalues and orthogonal eigenvectors of an n n symmetric tridiagonal matrix is an important task that arises while solving any symmetric eigenproblem. All practical software requires O(n) time to compute all the eigenvectors and ensure their orthogonality when eigenvalues are close. In the rst part of this thesis we review earlier work and show how some existing implementations of inverse iteration can fail in surprising ways. The main contribution of this thesis is a new O(n), easily parallelizable algorithm for solving the tridiagonal eigenproblem. Three main advances lead to our new algorithm. A tridiagonal matrix is traditionally represented by its diagonal and o -diagonal elements. Our most important advance is in recognizing that its bidiagonal factors are \better" for computational purposes. The use of bidiagonals enables us to invoke a relative criterion to judge when eigenvalues are \close". The second advance comes by using multiple bidiagonal factorizations in order to compute di erent eigenvectors independently of each other. Thirdly, we use carefully chosen dqds-like transformations as inner loops to compute eigenpairs that are highly accurate and \faithful" to the various bidiagonal representations. Orthogonality of the eigenvectors is a consequence of this accuracy. Only O(n) work per eigenpair is needed by our new algorithm. Conventional wisdom is that there is usually a trade-o between speed and accuracy in numerical procedures, i.e., higher accuracy can be achieved only at the expense of greater computing time. An interesting aspect of our work is that increased accuracy in