An Orthogonal Similarity Reduction of a Matrix into Semiseparable Form

An orthogonal similarity reduction of a matrix to semiseparable form. Abstract It is well known how any symmetric matrix can be reduced by an orthogonal similarity transformation into tridiagonal form. Once the tridiagonal matrix has been computed, several algorithms can be used to compute either the whole spectrum or part of it. In this paper, we propose an algorithm to reduce any symmetric matrix into a similar semiseparable one of semiseparability rank 1, by orthogonal similarity transformations. A remarkable feature of the algorithm is that, after few steps of it, the largest eigenvalues, in absolute value, are already computed with high precision. Once the semiseparable matrix has been computed, to compute the whole spectrum either the same algorithm can be iterated or algorithms for computing the eigendecomposition of diagonal plus semiseparable matrices, available in the literature, can be used. These features of the proposed algorithm are confirmed by some numerical experiments. An orthogonal similarity reduction of a matrix to semiseparable form. * Abstract. It is well known how any symmetric matrix can be reduced by an orthogonal similarity transformation into tridiagonal form. Once the tridiagonal matrix has been computed, several algorithms can be used to compute either the whole spectrum or part of it. In this paper, we propose an algorithm to reduce any symmetric matrix into a similar semiseparable one of semiseparability rank 1, by orthogonal similarity transformations. A remarkable feature of the algorithm is that, after few steps of it, the largest eigenvalues, in absolute value, are already computed with high precision. Once the semiseparable matrix has been computed, to compute the whole spectrum either the same algorithm can be iterated or algorithms for computing the eigendecomposition of diagonal plus semiseparable matrices, available in the literature, can be used. These features of the proposed algorithm are confirmed by some numerical experiments.