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 α ₁ ≤ α ₂ ≤... ≤ α _n − 1 be the entries of the main diagonal of a symmetric arrowhead matrix of order n. Denoted by λ _i , i=1, ..., n, the corresponding eigenvalues, it is well know that α _i ≤ λ _i + 1 ≤ α _i + 1, i=1,..., n-2. An algorithm for computing each eigenvalue λ _i , i=1, ..., n, of a symmetric arrowhead matrix with monotonic quadratic convergence, independent of the choice of the initial guess in the interval ]α _i − 1,α _i [ 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.