An ODE-based method for computing the approximate greatest common divisor of polynomials

Efficient parallel factorization and solution of structured and unstructured linear systems

A matrix Euclidean algorithm induced by state space realization

Linear quantum systems: A tutorial

We give a probabilistic algorithm for computing the greatest common divisor (GCD) of two polynomials over an algebraic number field. We can compute the GCD using O(llog5(l)) expected binary operations where l is the size of the GCD given by standard estimations. Since we require time Ω(l) just to write down the GCD, the algorithm is close to optimal.

We propose an algorithm to compute a greatest common divisor (GCD) of univariate polynomials with large integer coefficients on Graphics Processing Units (GPUs). At the highest level, our algorithm relies on modular techniques to decompose the problem into subproblems that can be solved separately. Next, we employ resultant-based or matrix algebra methods to compute a GCD of each modular image in parallel. Our approach exhibits block structure to distribute the computation of a single modular GCD over several thread blocks, and thus to remove any hardware limitations on the maximal size of polynomials that can be handled. To "combine" all modular results, we have adopted Mixed-Radix Conversion (MRC) algorithm running on the GPU. Our approach shows a significant speed-up over host-based GCD algorithm from Maple 13.

The interaction between numerical linear algebra and control theory has crucially influenced the development of numerical algorithms for linear systems in the past. Since the performance of a control system can often be measured in terms of eigenvalues or singular values, matrix eigenvalue methods have become an important tool for the implementation of control algorithms. Standard numerical methods for eigenvalue or singular value computations are based on the QR-algorithm. However, a number of computational problems in control and signal processing are not amenable to standard numerical theory or cannot be easily solved using current numerical software packages. Various examples can be found in the digital filter design area. For instance, the task of finding sensitivity optimal realizations for finite word length implementations requires the solution of highly nonlinear optimization problems for which no standard numerical solution of algorithms exist.KeywordsGlobal MinimumHomogeneous SpaceGlobal ConvergenceCholesky FactorBalance RealizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this article, we review a number of iterative and analytical techniques to approximate structured singular values or µ-values which is a straightforward generalization of singular values for square and rectangular matrices. The µ-value is a well-known tool which acts as a strong link between numerical linear algebra and control theory. The computation of structured singular value provides a platform to study and discuss stability, performance, and robustness of the system. Furthermore, we review some very important literature that discusses the applications of structured singular values in different areas of engineering.

Finding the greatest common divisor (GCD) of two given polynomials is a basic problem in algebraic computing. The problem is usually stated as follows: given the (real or complex) coefficients of two polynomials, compute the coefficients of their greatest common divisor.

Book Review of 'Linear Network Theory'

This chapter provides a self-contained overview of modern linear feedback control theory. We focus on key methods for laminar and transitional flow stabilization, introducing concepts from optimal control theory and system identification to build intuition. Optimal regulators and filters are discussed, along with system identification.

It is common practice to use linear plant models and linear controllers in the control systems design. Such approach has simple explanation applying to plants with insignificant nonlinearity or to those, functioning closely to a working point. But linear controllers, indeed with some modification, are used even for plants with significant non-linearities. Because of several reasons the non-linear controllers have not broad application. First, the linear control theory is well developed; while the non-linear control methods are clear for few engineers in practice. Second, there are some technological and economical difficulties to get high quality study of the process to be controlled in order to build detailed (more precise) non-linear plant model. Third, new ideas in the field of the control theory are continuously realized, which expand the span of the linear control systems applications as an alternative to utilizing complicated models at the expense of troubles of theoretical and practical nature.

We consider regular and determinant forms for representing the index of an arbitrary rational function based on a proposed generalization of the Routh method. We propose forms for representing the index relative to the negative real axis, the positive real axis and the entire axis, as well as the greatest common divisor of polynomials both in terms of the elements of the Routh method and in terms of the determinant of the corresponding Hurwitz matrix.

The problem considered here is an integral part of computations for algebraic control problems. The paper introduces the notion of normal factorization of polynomials and then presents a new hybrid algorithm for the computation of this factorization. The advantage of such a factorization is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots. The presented algorithm has the ability to specify the roots of the polynomials without computing them explicitly. Also it may be used for investigating the clustering of the roots of the polynomials. The developed procedure is based on the use of algorithms determining the greatest common divisor of polynomials. The algorithm can be implemented symbolically for the specification of well separated roots and numerically for the specification of roots belonging in approximate clusters.

A new method is suggested for solving the polynomial equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(s)X(s)+B(s)Y(s)=C(s)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A (s), B (s), C (s)</tex> are given polynomials and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X (s), Y (s)</tex> are to be determined. The method is then used for the formulation of the partial-fraction expansion of transfer functions, and for computing the greatest common divisor of polynomials

Normal factorisation of polynomials and computational issues