An O(N2 ) Method for Computing the Eigensystem of N ˟ N Symmetric Tridiagonal Matrices by the Divide and Conquer Approach

An efficient method to solve the eigenproblem of N x N symmetric tridiagonal matrices is proposed. Unlike the standard eigensolvers that necessitate O(N3) operations to compute the eigenvectors of such matrices, the proposed method computes both the eigenvalues and eigenvectors with only O(N2) operations. The method is based on serial implementation of the recently introduced Divide and Conquer algorithm [3], [1], [4]. It exploits the fact that by O(N2) Divide and Conquer operations one can compute the eigenvalues of an N x N symmetric tridiagonal matrix and a small number of pairs of successive rows of its eigenvector matrix. The rest of the eigenvectors (either all together or one at a time) are computed by linear three-term recurrence relations. The paper is concluded with numerical examples that demonstrate the superiority of the proposed method for a special class of symmetric tridiagonal matrices, by saving an order of magnitude in execution time at the expense of sacrificing a few orders of accuracy, although for symmetric tridiagonal matrices in general, the method appears to be unstable.