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.KeywordsInitial GuessNewton MethodSpectral DecompositionSecular EquationExact EigenvalueThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.