On computing the spectral decomposition of symmetric arrowhead matrices

Abstract

The computation of the spectral decomposition of a symmetric arrowhead matrix is an important problem in applied mathematics [10]. It is also the kernel of divide and conquer algorithms for computing the Schur decomposition of symmetric tridiagonal matrices [2,7,8] and diagonal-plus-semiseparable matrices [3,9]. The eigenvalues of symmetric arrowhead matrices are the zeros of a secular equation [5] and some iterative algorithms have been proposed for their computation [2,7,8]. An important issue of these algorithms is the choice of the initial guess. Let α1 ≤ α2 ≤ ... ≤ αn-1 be the entries of the main diagonal of a symmetric arrowhead matrix of order n. Denoted by λi, i = 1,... ,n, the corresponding eigenvalues, it is well know that αi ≤ λi,+1 ≤ αi+1,i = 1,... ,n -2. An algorithm for computing each eigenvalue λi, i = 1,... , n, of a symmetric arrowhead matrix with monotonic quadratic convergence, independent of the choice of the initial guess in the interval ]αi-1, αi[ is proposed in this paper. Although the eigenvalues of a symmetric arrowhead matrix can be computed efficiently, a loss of orthogonality can occur in the computed matrix of eigenvectors [2,7,8].In this paper we propose also a simple, stable and efficient way to compute the eigenvectors of arrowhead matrices. © Springer-Verlag 2004.