Ellen Van Camp

Orthogonal similarity transformation of a symmetric matrix into a diagonal-plus-semiseparable one with free choice of the diagonal

In this paper we describe an orthogonal similarity transformation for transforming arbitrary symmetric matrices into a diagonal-plus-semiseparable matrix, where we can freely choose the diagonal. Very recently an algorithm was proposed for transforming arbitrary symmetric matrices into similar semiseparable ones. This reduction is strongly connected to the reduction to tridiagonal form. The class of semiseparable matrices can be considered as a subclass of the diagonalplus- semiseparable matrices. Therefore we can interpret the proposed algorithm here as an extension of the reduction to semiseparable form.A numerical experiment is performed comparing thereby the accuracy of this reduction algorithm with respect to the accuracy of the traditional reduction to tridiagonal form, and the reduction to semiseparable form. The experiment indicates that all three reduction algorithms are equally accurate. Moreover it is shown in the experiments that asymptotically all the three approaches have the same complexity, i.e. that they have the same factor preceding the n3 term in the computational complexity. Finally we illustrate that special choices of the diagonal create a specific convergence behavior.

On Computing the Spectral Decomposition of Symmetric Arrowhead Matrices

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.