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 $N \times N$ symmetric tridiagonal matrices is proposed. Unlike the standard eigensolvers that necessitate $O(N^3 )$ operations to compute the eigenvectors of such matrices, the proposed method computes both the eigenvalues and eigenvectors with only $O(N^2 )$ 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 $O(N^2 )$ Divide and Conquer operations one can compute the eigenvalues of an $N \times N$ 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.