Linear Algebra Techniques Research Articles

Linear variational Diophantine techniques in mass balance of chemical reactions

Linear algebraic techniques based on minimization of thermodynamic functional and/or other constraints are illustrated for mass balance of chemical reactions that do not exhibit stoichiometrically unique solutions in the linear algebraic vector space. The techniques demonstrate elegant casting of chemical equations in terms of generalized linear Diophantine matrices and generalized elimination and variational schemes.

On computing a separating transcendence basis

Let k ( x 1 ,…, x n )/ k be a finitely generated field extension, and g 1 ,…, g r ε k (x). For k ( x 1 ,…, x n )/ k ( g 1 ,…, g r ) being separably generated (which in particular includes the case char( k ) = 0) we give a method to compute the transcendence degree and a separating transcendence basis of this extension by means of simple linear algebra techniques.

The role of the implicit function in the Lagrange multiplier rule

In this note the necessary condition for a constrained extremum point and the criteria for its classification using the implicit function, the chain rule and elementary techniques of linear algebra are presented.

Some computational problems arising in adaptive optics imaging systems

Recently there has been growing interest and progress in using numerical linear algebra techniques in adaptive optics imaging control computations. Real-time adaptive optics is a means for enhancing the resolution of ground based, optical telescopes beyond the limits previously imposed by the turbulent atmosphere. An adaptive optics system automatically corrects for light distortions caused by the medium of transmission. The system measures the characteristics of the phase of the arriving wavefront and corrects for the degradations by means of one or more deformable mirrors controlled by special purpose computers. No attempt is made in this paper to give a comprehensive survey of recent numerical linear applications in optical imaging. Rather, two fairly representative applications are discussed in some detail. The following research topics in the area of adaptive optics control systems, each involving the formulation and numerical solution of difficult problems in numerical linear algebra, are described: (1) Jacobi-like eigenvalue computations for multiple bandwidth deformable mirror control methods, and (2) covariance matrix computations for performance modeling of adaptive optics systems using fast Hankel transforms.

Letters to the Editor on the Paper ‘Fault Detection and Isolation for State Affine Systems’ by H. Hammouri, M. Kinnaert and E. H. El Yaagoubi

In this note we show how to test whether or not certain algebraic equations derived in the paper 'Fault Detection and Isolation for State Affine Systems', by H. Hammouri, M. Kinnaert, E.H. El Yaagoubi, are solvable and to describe a solution method. Test and solution rely upon elementary linear algebraic techniques.

Linear Differential Algorithm for Motion Recovery: AGeometric Approach

The aim of this paper is to explore a linear geometric algorithm for recovering the three dimensional motion of a moving camera from image velocities. Generic similarities and differences between the discrete approach and the differential approach are clearly revealed through a parallel development of an analogous motion estimation theory previously explored in Vieville, T. and Faugeras, O.D. 1995. In Proceedings of Fifth International Conference on Computer Vision, pp. 750–756s Zhuang, X. and Haralick, R.M. 1984. In Proceedings of the First International Conference on Artificial Intelligence Applications, pp. 366–375. We present a precise characterization of the space of differential essential matrices, which gives rise to a novel eigenvalue-decomposition-based 3D velocity estimation algorithm from the optical flow measurements. This algorithm gives a unique solution to the motion estimation problem and serves as a differential counterpart of the well-known SVD-based 3D displacement estimation algorithm for the discrete case. Since the proposed algorithm only involves linear algebra techniques, it may be used to provide a fast initial guess for more sophisticated nonlinear algorithms (Ma et al., 1998c. Electronic Research Laboratory Memorandum, UC Berkeley, UCB/ERL(M98/37)). Extensive simulation results are presented for evaluating the performance of our algorithm in terms of bias and sensitivity of the estimates with respect to different noise levels in image velocity measurements.

The Searchlight Problem for Radiative Transfer in a Finite Slab

A version of the discrete-ordinates method recently developed for radiative-transfer calculations is used along with numerical linear-algebra techniques and two-dimensional Fourier-transform procedures to establish the radiation flux and the z component of the radiation current at all locations in a finite plane-parallel layer irradiated by a beam incident only at one point on one surface. In addition to a general formulation basic to a beam that is incident at an oblique angle, for which the flux and current depend on three spatial variables, the Fourier transforms of the flux and current are inverted numerically for the two-dimensional case relevant to a normally incident beam. The reported numerical procedures, while computationally intensive, are thought to yield, for the considered test case, the radiation flux and the normal component of the radiation current with five figures of accuracy.

A fast numerical homogenization algorithm for finite element analysis

A numerical homogenization method is presented here to solve problems governed by partial differential equations with coefficients that are generic functions in R2. It consists of a recursive finite elements discretization and an algebraic homogenization. This method takes advantages of speed and memory occupation from the hierarchy of elements and nodes defined by the recursive discretization. It turns out that using the state-of-the-art general linear algebra techniques, all non-numerical data manipulations that are typically done before real computations, can be avoided. Copyright © 1999 John Wiley & Sons, Ltd.

A fast numerical homogenization algorithm for finite element analysis

A numerical homogenization method is presented here to solve problems governed by partial differential equations with coefficients that are generic functions in R2. It consists of a recursive finite elements discretization and an algebraic homogenization. This method takes advantages of speed and memory occupation from the hierarchy of elements and nodes defined by the recursive discretization. It turns out that using the state-of-the-art general linear algebra techniques, all non-numerical data manipulations that are typically done before real computations, can be avoided. Copyright © 1999 John Wiley & Sons, Ltd.

Being stable and discrete

Many discrete lattice systems possess solutions that take the form of localized, stationary structures. In this communication we introduce the discrete version of the Evans function, an analytic function whose zeros correspond to the eigenvalues of the linear stability problem for a spatially localized equilibrium solution. This function provides a convenient and useful tool for investigating the linear eigenvalue spectrum. Notably, it allows us to construct sufficient stability conditions and detect “internal modes” (neutral oscillatory modes that correspond to localized oscillations about the static structure). We illustrate with the discrete sine-Gordon equation, also known as the Frenkel–Kontorova model. A complementary approach suitable for systems with nearest neighbour coupling and based upon techniques of linear algebra (the bisection method) is also described.

A discrete-ordinates solution for multigroup transport theory with upscattering

A recently developed version of the discrete-ordinates method is used along with elementary numerical linear-algebra techniques to establish an efficient and especially accurate solution to a class of multigroup transport problems for which upscattering is an important aspect of the model. The problems considered are defined for finite plane-parallel media, and anisotropic scattering from any group to any group is included in the formulation. Computational details of the solution are discussed, and accurate numerical results for two previously defined test problems are established.

A discrete-ordinates solution for radiative-transfer models that include polarization effects

A recently developed version of the discrete-ordinates method is used along with elementary numerical linear-algebra techniques to establish an accurate solution for all components in a Fourier representation of the Stokes vector basic to the scattering of polarized light. Computational aspects of the solution are discussed, and numerical results for each of the four Stokes parameters are given for a test case based on an atmosphere of randomly orientated oblate spheroids.

A concise and accurate solution to Chandrasekhar’s basic problem in radiative transfer

A recently developed version of the discrete-ordinates method is used along with elementary numerical linear-algebra techniques to establish an efficient and especially accurate solution to what can be called Chandrasekhar’s basic problem in radiative transfer, namely the problem of computing the radiation intensity in a finite plane-parallel layer illuminated by an incident beam of radiation and in which scattering can be described by a (rather) general scattering law.

Wave-number solution for a string with distributed mass properties

A structure with spatially variable properties when excited at a single frequency will excite many wave numbers. In this paper the vibration of a string with a mass distribution m(x) given by a Fourier series expansion is analyzed. The equation governing the wave-number solution w(k) for the string displacement is obtained via Fourier transform. The final result is a finite-difference equation. The finite-difference equation relates all of the displacement wave numbers to each other. The solution to this equation is obtained by rewriting it as a system of simultaneous equations that are then solved by standard linear algebra techniques. Wave-number spectral solutions are obtained for different mass distribution. In one case the mass distribution is assumed to vary in a discontinuous manner between adjacent bays, in another case the mass is assumed to vary in a ‘‘triangular’’ manner. The wave-number solutions are presented in 3-D graphs so that the magnitude of the displacement can be seen as a function of the wave number and either the magnitude of the additional mass, the area over which the mass is distributed, or the length of the bay. In the limiting case as the mass distribution becomes very concentrated, the classical pass/stop band, familiar with periodic structures, is recovered.

A new efficient algorithm for computing Gröbner bases ( F4)

This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much intermediate computation as possible, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo p computations.

Some subclasses of Petri nets and the analysis of their structural properties: a new approach

The purpose of the paper is to consider some special types of Petri nets, introduced by Lien (1976), and to propose a complete and unified approach for the study of their structural properties by using techniques of linear algebra of matrices. We distinguish four subclasses: forward-conflict-free, backward-conflict-free, forward-concurrent-free, and backward-concurrent-free Petri nets. A modification of the classical incidence matrix results in a square matrix, called a modified incidence matrix, with nonpositive (nonnegative) off-diagonal elements when backward-(forward-) conflict-free or concurrent-free Petri nets are considered. The modified incidence matrix eigenvalues are computed and theorems on matrices of this type are used to prove several sufficient and/or necessary conditions for structural boundedness, liveness, repetitiveness, conservativeness, and consistency of these four subclasses of Petri nets.

Education

The Education section is intended to provide modules that teachers and students of applied mathematics and scientific computation can use directly in studying these fields, in courses and beyond. These modules are likely to fall into one or more of four broad categories: applications, special topics, software, and historical modules. The first issue of this section contained one paper describing a Web-based software package and a second discussing a modern special topic in applied mathematics. In this, the second issue, there are two different types of papers. Michael Berry (no relation to the Sir Michael Berry of the article with Keating earlier in this issue), Zlatko Drmac, and Liz Jessup describe the application of linear algebra to problems in information retrieval. In particular, they highlight an important recent technique in information retrieval called latent semantic indexing that itself draws heavily upon linear algebra techniques including the singular value decomposition. The early parts of the paper can serve as a very nice motivating example for students and teachers in both linear algebra and numerical computation courses, while the more detailed and advanced parts of the paper can constitute a unit in a graduate or advanced undergraduate course in numerical linear algebra that combines real applications with serious numerical mathematics. P. Broadbridge, G.R. Fulford, N.D. Fowkes, D.T.C. Chan, and C. Lassig provide a simple yet compelling use of mathematical modeling to understand a very practical problem, the appearance of bubbles in gummed wine labels. Using techniques that are understandable by undergraduates and a delightful writing style, they demonstrate the insight provided by mathematical modeling on a problem that is simple to describe yet perplexing to explain. The paper will make an excellent module in differential equation and mathematical modeling courses and can even be used as a motivating example in earlier courses. Both articles are written in the style that is meant to be the standard for this section: informal, easy to understand, and written to students. Unsolicited submissions form the primary source of potential papers for the Education section, although the editorial board also will recruit submissions when it is aware of good prospects. The goals and style of this section are somewhat different than is standard for academic journals and even for educationally-oriented papers in the previous version of SIAM Review. For these reasons, prospective authors are strongly encouraged to consult the section guidelines, which were published in this preface in the previous issue of the journal and are available on the web at http://www.siam.org/journals/sirev/revguide.htm. Prospective authors are also welcome to contact the Section Editor or Associate Editors with ideas and questions about planned submissions before making a formal submission.

Regularized symmetric indefinite systems in interior point methods for linear and quadratic optimization*

This paper presents linear algebra techniques used in the implementation of an interior point method for solving linear programs and convex quadratic programs with linear constraints. New regularization techniques for Newton systems applicable to both symmetric positive definite and symmetric indefinite systems are described. They transform the latter to quasidef-inite systems known to be strongly factorizable to a form of Cholesky-like factorization.Two different regularization techniques,primal; and dual, are very well suited to the (infeasible) primal-dual interior point algorithm. This particular algorithm, with an extension of multiple centrality correctors, is implemented in our solver HOPDM. Computational results are given to illustrate the potential advantages of the approach when applied to the solution of very large linear and convex quadratic programs.

Constrained Maximum-Entropy Sampling

A fundamental experimental design problem is to select a most informative subset, having prespecified size, from a set of correlated random variables. Instances of this problem arise in many applied domains such as meteorology, environmental statistics, and statistical geology. In these applications, observations can be collected at different locations and, possibly, at different times. Information is measured by “entropy.” Practical situations have further restrictions on the design space. For example, budgetary limits, geographical considerations, as well as legislative and political considerations may restrict the design space in a complicated manner. Using techniques of linear algebra, combinatorial optimization, and convex optimization, we develop upper and lower bounds on the optimal value for the Gaussian case. We describe how these bounds can be integrated into a branch-and-bound algorithm for the exact solution of these design problems. Finally, we describe how we have implemented this algorithm, and we present computational results for estimated covariance matrices corresponding to sets of environmental monitoring stations in the Ohio Valley of the United States.

Eigenvalue Locations of Generalized Companion Predictor Matrices

Generalized predictor companion matrices arise in the linear prediction approach for the fit of a weighted sum of $n$ exponentials to a given set of data points. They are special solutions of matrix equations of the type ${\sf H}(l+p)\, {\sf S}={\sf H}(l)$, where for each $l\ge 0$ ${\sf H}(l)$ is an $M\times N$ Hankel matrix obtained from this data ($M\ge N>n$). We discuss in this paper results about the eigenvalue locations of this class of solutions by means of linear algebra techniques. An application of these results in the case that all the exponents have either negative or positive real parts is that the n exponentials can correspond to eigenvalues which are outside the unit circle depending on the choice of generalized predictor companion matrices. The other (N-n) eigenvalues of these matrices always lie inside the unit circle and approach zero when p increases. This separation can facilitate their numerical calculation.