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 nxnxn term in the computational complexity. Finally we illustrate that special choices of the diagonal create a specific convergence behavior.

In this paper the definition of semiseparable matrices is investigated. Properties of the frequently used definition and the corresponding representation by generators are deduced. Corresponding to the class of tridiagonal matrices another definition of semiseparable matrices is introduced preserving the nice properties dual to the class of tridiagonal matrices. Several theorems and properties are included showing the viability of this alternative definition. Because of the alternative definition, the standard representation of semiseparable matrices is not satisfying anymore. The concept of a representation is explicitly formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation.

An algorithm to reduce a symmetric matrix to a similar semiseparable one of semiseparability rank 1, using orthogonal similarity transformations, is proposed in this paper. It is shown that, while running to completion, the proposed algorithm gives information on the spectrum of the similar initial matrix. In fact, the proposed algorithm shares the same properties of the Lanczos method and the Householder reduction to tridiagonal form. Furthermore, at each iteration, the proposed algorithm performs a step of the QR method without shift to a principal submatrix to retrieve the semiseparable structure. The latter step can be considered a kind of subspace-like iteration method, where the size of the subspace increases by one dimension at each step of the algorithm. Hence, when during the execution of the algorithm the Ritz values approximate the dominant eigenvalues closely enough, diagonal blocks will appear in the semiseparable part where the norm of the corresponding subdiagonal blocks goes to zero in the subsequent iteration steps, depending on the corresponding gap between the eigenvalues. A numerical experiment is included, illustrating the properties of the new algorithm.

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n x n matrix, this algorithm requires O(n^3) operations per iteration step. To reduce this complexity for a sytmmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable fortri, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method.

The linear space of all proper rational functions with prescribed poles is considered. Given a set of points in the complex plane and the weights, we define the discrete inner product. In this paper we derive a method to compute the coefficients of a recurrence relation generating a set of orthonormal rational basis functions with respect to the discrete inner product. We will show that these coefficients can be computed by solving an inverse eigenvalue problem for a matrix having a specific structure. In the case where all the points lie on the real line or on the unit circle, the computational complexity is reduced by an order of magnitude.

Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper. In particular, the solutions of the inverse eigenvalue problems for Toeplitz-plus-Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form. Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices. Numerical testing results show a clear advantage of such an application.

In this paper we discuss the structure of the factors of a QR- and a URV-factorization of a diagonal-plus -semiseparable matrix. The Q-factor of a QR-factorization has the diagonal-plus-semiseparable structure. The U- and V-factor of a URV-factorization are semiseparable lower Hessenberg orthogonal matrices. The strictly upper triangular part of the R-factor of a QR- and of a URV-factorization is the strictly upper triangular part of a rank-2 matrix. This latter fact provides a tool to construct a fast QR-solver and a fast URV-solver for linear systems of the form (D + S)x = b. c 2003 Elsevier B.V. All rights reserved.