Nicola Mastronardi

A note on the representation and definition of semiseparable matrices

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. Copyright

Fast Structured Total Least Squares Algorithm for Solving the Basic Deconvolution Problem

In this paper we develop a fast algorithm for the basic deconvolution problem. First we show that the kernel problem to be solved in the basic deconvolution problem is a so-called structured total least squares problem. Due to the low displacement rank of the involved matrices and the sparsity of the generators, we are able to develop a fast algorithm. We apply the new algorithm on a deconvolution problem arising in a medical application in renography. By means of this example, we show the increased computational performance of our algorithm as compared to other algorithms for solving this type of structured total least squares problem. In addition, Monte-Carlo simulations indicate the superior statistical performance of the structured total least squares estimator compared to other estimators such as the ordinary total least squares estimator.

Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) problem when constraints on the matrix structure need to be imposed. Similar to the ordinary TLS approach, the STLS approach can be used to determine the parameter vector of a linear model, given some noisy measurements. In many signal processing applications, the imposition of this matrix structure constraint is necessary for obtaining Maximum Likelihood (ML) estimates of the parameter vector. In this paper we consider the Toeplitz (Hankel) STLS problem (i.e., an STLS problem in which the Toeplitz (Hankel) structure needs to be preserved). A fast implementation of an algorithm for solving this frequently occurring STLS problem is proposed. The increased efficiency is obtained by exploiting the low displacement rank of the involved matrices and the sparsity of the associated generators.The fast implementation is compared to two other implementations of algorithms for solving the Toeplitz (Hankel) STLS problem. The comparison is carried out on a recently proposed speech compression scheme. The numerical results confirm the high efficiency of the newly proposed fast implementation: the straightforward implementations have a complexity of O((m+n)3) and O(m3) whereas the proposed implementation has a complexity of O(mn+n2).

An implicit QR algorithm for symmetric semiseparable matrices

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n×nmatrix, this algorithm requires O(n3) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is £rst reduced to tridiagonal form using orthogonal similarity transformations. In the report [Van Barel, Vandebril, Mastronardi 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 form, 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 ∗The research of the £rst and second author was supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scienti£c Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), and G.0184.02 (CORFU: Constructive study of Orthogonal Functions), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, PrimeMinister’s Of£ce for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identi£cation & Modelling). The work of the third author was partially supported by MIUR, grant number 2002014121. The scienti£c responsibility rests with the authors. †Email: raf.vandebril@cs.kuleuven.ac.be ‡Email: marc.vanbarel@cs.kuleuven.ac.be §Email: nicola.mastronardi@cs.kuleuven.ac.be

A fast algorithm for subspace state-space system identification via exploitation of the displacement structure

Two recent approaches (Van Overschee, De Moor, N4SID, Automatica 30 (1) (1994) 75; Verhaegen, Int. J. Control 58(3) (1993) 555) in subspace identification problems require the computation of the R factor of the QR factorization of a block-Hankel matrix H, which, in general has a huge number of rows. Since the data are perturbed by noise, the involved matrix H is, in general, full rank. It is well known that, from a theoretical point of view. the R factor of the PR factorization of H is equivalent to the Cholesky factor of the correlation matrix HTH, apart from a multiplication by a sign matrix. In Sima (Proceedings Second NICONET Workshop, Paris-Versailles, December 3, 1999, p. 75), a fast Cholesky factorization of the correlation matrix, exploiting the block-Hankel structure of H, is described. In this paper we consider a fast algorithm to compute the R factor based on the generalized Schur algorithm. The proposed algorithm allows to handle the rank-deficient case

Orthogonal Rational Functions and Structured Matrices

The linear space of all proper rational functions with prescribed poles is considered. Given a set of points zi in the complex plane and the weights wi we define the discrete inner product

A QR –method for computing the singular values via semiseparable matrices

\langle \phi,\psi \rangle := \sum_{i=0}^n |w_i|^2 \overline{\phi(z_i)} \psi(z_i).

Some numerical algorithms to evaluate Hadamard finite-part integrals

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 zi lie on the real line or on the unit circle, the computational complexity is reduced by an order of magnitude.