We present a unification and generalization of what is known in the literature as sequentially and hierarchically semiseparable (SSS and HSS) representations for matrices. These socalled tree quasi-separable (TQS) matrices contain sparse matrices with tree-structured adjacency graphs as an important sub case. TQS matrices inherit all the favorable algebraic properties of SSS and HSS under addition, products, and inversion. To arrive at these properties, we prove a key result that characterizes the conversion of any dense matrix into a TQS representation. Here, we specifically show through an explicit construction that the size of the representation is dictated by the ranks of certain Hankel blocks of the matrix. Analogous to SSS and HSS, TQS matrices admit fast matrix-vector products and direct solvers. A sketch of the associated algorithms is provided.

Matrix theory is the lingua franca of everyone who deals with dynamically evolving systems, and familiarity with efficient matrix computations is an essential part of the modern curriculum in dynamical systems and associated computation. This is a master's-level textbook on dynamical systems and computational matrix algebra. It is based on the remarkable identity of these two disciplines in the context of linear, time-variant, discrete-time systems and their algebraic equivalent, quasi-separable systems. The authors' approach provides a single, transparent framework that yields simple derivations of basic notions, as well as new and fundamental results such as constrained model reduction, matrix interpolation theory and scattering theory. This book outlines all the fundamental concepts that allow readers to develop the resulting recursive computational schemes needed to solve practical problems. An ideal treatment for graduate students and academics in electrical and computer engineering, computer science and applied mathematics.

This presentation addresses the modeling of a non-Gaussian stochastic process based on a fully ordered sequence of measurements of moments or correlations. To do so, it uses recent results in the parametrization of all the moment generating functions that interpolate the given or measured correlation data. While the parametrization problem appears to be adequately solved, the crucial numerical determination of the resulting cumulative probability function (cpf) or probability density function (pdf) appears to be a new interesting but unsolved problem, even in the single variable case. The talk introduces the problem and its background, hoping to provide motivation for further research.

Orthogonal filtering is presented as a generic technique in signal processing, that is capable of solving many classical signal processing problems in a streamlined fashion, with emphasis on a time variant setting. Orthogonal filtering consists in efficient recursive orthogonalization of data, either original signal data or model data. Historically, it goes back to the notion of 'inner-outer factorization' in Hardy space theory, a notion that engineers refer to as 'decomposition of a transfer function in a lossless phase factor and a minimal phase, hence invertible, factor.' In contrast to the historical setting, the paper adopts a fully numerical, time variant approach. It starts out by developing the method on a 4 by 4 (block) example, which is then generalized to arbitrary discrete time, time variant systems. Next, the paper illustrates the utilization of the method in some detail on two classical problems: Kalman filtering and optimal quadratic control of a linear (time variant) system (a la Bellman). A number of further applications are mentioned and briefly discussed.

The paper addresses the question of parametrization with independent parameters for a multi-variable, non-Gaussian set of stochastic variables, based on higher order moments. The issue is particularly relevant for the construction of low-complexity models that meet measured correlation data between powers of the variables. It turns out that such a parametrization exists in the case where the correlation data is stricty ordered by increasing degree, and the paper shows in outline how it can be constructed.

What do basic notions in systems and ecology mean? Several papers in this book propose specific ways of viewing systems (in particular ecological systems) and offer definitions and notions related to their proposed views. These often very compact “conceptual models” aim at providing a means to understand the behavior and evolution of real-world systems, be they economic, social, or ecological. The way real-world systems are viewed by people and politicians influences considerably how humanity deals with their natural surroundings and how they may decide to act in an ecologically favorable direction, given the fact that humanity’s actions obviously have major significance for the global earth’s well-being. The proposed conceptual models used by various authors in this book differ considerably from each other, making a discussion of their respective merits and shortcomings very meaningful. Six authors joined in the discussion, proposing, supporting, or criticizing points of view and aiming at clarifying the notions they use, while putting them in perspective. The discussion has been ordered as a question and answers session around the main themes. We hope readers will enjoy the clash of ideas and develop further insights motivated by them.

Linear time invariant (LTI) systems are often represented by rational forms or rational matrix functions. Such forms exhibit important properties of a system, but these properties are often thought to be due to time invariance. Are there such forms for linear time-variant (LTV) systems with comparable properties, generalizing the LTI case? One major obstacle to derive them has been the lack of a divisibility theory for general block lower-triangular matrices, in particular the lack of a Euclidean algorithm or, in its wake, Smith forms and Smith-McMillan forms. But do rational forms for matrices, generalizing the time-invariant properties, exist at all? It turns out they do. The theory presented here produces representations for block lower-triangular matrices as ratios of (block) staircase (or echelon) matrices, and shows how central properties such as co-prime factorization or Bezout identities hold. Instead of the Euclidean algorithm, the theory uses a technique derived from a paper of Paul Van Dooren, namely ‘dead-beat control’. The theory finds good applications in various domains (computational efficiency, determining pre-conditioners, control theory and model reduction via generalized forms of interpolation).

The paper considers interpolating models for non-linear, non-Gauss stochastic variables and processes, given a well-ordered set of moments of increasing order. The proposed models use a characterization with independent parameters, much in the style of the SchurLevinson parametrization for the linear, Gaussian case (a topic to which Tom Kailath made seminal contributions), but very different from it, given the different kind of structured matrices involved (Hankel-like instead of Toeplitz). The paper starts out with a review of the classical Hamburger-Akhiezer-Jacobi parametrization for one stochastic variable, using a (non-classical) dynamical system theory approach. Next, the paper generalizes these results to the multivariable case, and presents a detailed generalized Jacobilike (independent) parametrization for two variables. Like in the Schur-Levinson case, such parametrizations succeed in characterizing models that interpolate the moment data (given the complexity of the issue, only the 2D case is treated in this paper, but using a method that generalizes to more variables).

This issue of the CAS magazine presents papers that could not be accommodated in the 1st part of the Alfred Fettweis memorial special issue that appeared as the December 2018 issue. While the 1st part provided extensive views of Alfred Fettweis' personal life, scientific contributions, and several papers dealing with areas that were influenced by his scientific and technological contributions at l...

This paper explores similarities and differences of different types of stochastic modeling, namely the traditional covariance modeling based on Schur-Levinson theory vs. partial moment matching. The first case leads to positive definite Toplitz matrices, while the second case handles positive definite Hankel matrices (or generalized versions ot those). In both cases a special type of interpolation problem is solved, from which general solutions can be derived. However, the differences between the two problems and their solution methods soon appear. In the multi-dimensional Hankel case, a joint pdf has to be determined, while in the (generalized) Schur-Levinson case, only second order data is handled. In contrast to the traditional Schur-Levinson approach, the non-linear, non-Gaussian estimation filter is a derivative of the model filter and not vice versa. Some results presented here for the multivariate Hankel case are believed to be new.

Orthogonal filtering is a method to extract essential information from digital data using orthogonal transformations. It belongs to the category of Wave Digital Filters (WDF?s) as originally defined and considered by the late Alfred Fettweis, one of the principal founders of modern digital filter theory. In the original WDF theory, filtering is done using adders and an algebraically minimal number of multipliers exclusively. When, instead, the arithmetic is based (also exclusively) on purely orthogonal transformations (Jacobi/Givens rotations), a much larger category of lossless digital filters is obtained. In this paper, it is shown how central classical problems with many engineering applications, namely quadratically optimal tracking (Bellman), linear least squares estimation (Kalman) and spectral factorization (Wiener), among many other types of filters, produce natural orthogonal filters and can be obtained and designed using nothing more than orthogonal transformations. Simple proofs based on these insights are provided, together with a streamlined realization theory for the resulting data processors and filters. The paper uses nothing more than elementary matrix theory, and should be accessible to students with no other background, although it does at some point make the connection with the Beurling-Lax theory on inner-outer factorization and the Wiener theory on spectral factorization, putting these theories in a purely matrix algebra context.

The paper presents a new algorithm to compute the LU-factorization of a matrix represented in a quasiseparable or semiseparable form (i.e., using generators). It obtains the quasiseparable representations of the factors L and U of an N×N block matrix via O(N) arithmetic operations on the block entries. The algorithm uses recursions based exclusively on unitary transformations which provide numerical stability even in singular cases. The method of the paper is based on the theory developed in [1] and provides an alternative to the approach proposed in [7] for strongly regular matrices. The algorithm presented here works also for some matrices with possibly singular principle submatrices. The results of numerical tests show that also for strongly regular matrices the new algorithm is comparable with the previous methods.

The main objects of this chapter are “semi-separable systems,” sometimes called “quasi-separable systems.” These are systems of equations, in which the operator has a special structure, called “semi-separable” in this chapter. By this is meant that the operator, although typically infinite dimensional, has a recursive structure determined by sequences of finite matrices, called transition matrices. This type of operator occurs commonly in Dynamical System Theory for systems with a finite dimensional state space and/or in systems that arise from discretization of continuous time and space. They form a natural generalization of finite matrices and a complete theory based on sequences of finite matrices is available for them. The chapter concentrates on the invertibility of such systems: either the computation of inverses when they exist, or the computation of approximate inverses of the Moore–Penrose type when not. Semi-separable systems depend on a single principal variable (often identified with time or a single dimension in space). Although there are several types of semi-separable systems depending on the continuity of that principal variable, the present chapter concentrates on indexed systems (so-called discrete-time systems). This is the most straightforward and most appealing type for an introductory text. The main workhorse is “inner–outer factorization,” a technique that goes back to Hardy space theory and generalizes to any context of nest algebras, as is the one considered here. It is based on the definition of appropriate invariant subspaces in the range and co-range of the operator. It translates to attractive numerical algorithms, such as the celebrated “square-root algorithm,” which uses proven numerically stable operations such as QR-factorization and singular value decomposition (SVD).