Linear Algebra

A method is presented for the recovery of optical flow. The key idea is that the local spatial structure of optical flow, with the exception of surface boundaries, is usually rather coherent and can thus be appropriately approximated by a linear vector field. According to the proposed method, the optical flow components and their first order spatial derivatives are computed

A simple model order reduction technique is proposed for z-transfer functions. This method is based on approximate model matching in the frequency domain. The entire procedure is carried out in the z-domain and the resultant linear algebraic equations are solved to find the unknown parameters of the reduced-order model. An example favorably compares this method with some prevalent techniques.

The frenetic development of the current architectures places a strain on the current state-of-the-art programming environments. Harnessing the full potential of such architectures has been a tremendous task for the whole scientific computing community.

In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50-90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.

We present an introduction to some aspects of digital signal processing and time series analysis which are not always covered in classical textbooks. One of the objectives is to illustrate how mathematics and engineering can be combined in a fruitful interplay, resulting in many new techniques and methods in many different fields. We shall illustrate how the prediction problem is related to linear algebra, orthogonal polynomials, classical interpolation problems, inverse scattering theory, Toeplitz operators, network theory etc.

Problems of constructing a sediment budget for an inlet is often a challenge due to constraints on data collection, quality of data, and assumptions one is willing to consider. Existing literature does not provide a systematic foundation for equations considered and often engineers and geologists typically do not properly consider the mathematical constraints placed on the problem of sediment budget construction. The present technical note presents a simple methodology for assessing sediment budget, i.e. what is possible, and what is not possible via a matrix equation system and linear algebra. It is hoped that engineers and geologists will not impose unrealistic expectations on the sediment budget system via using such an approach. Some simple examples are given of both pitfalls and correct approaches to sediment budget development.

Hill cipher's susceptibility to cryptanalysis has rendered it unusable in practice, it still serves an important pedagogical role in both cryptology and linear algebra. Hill cipher requires inverse of the key matrix while decryption. In fact that not all the matrices have an inverse and therefore they will not be eligible as key matrices in the Hill cipher scheme. Furthermore, due to its linear nature, the basic Hill cipher succumbs to known-plaintext attacks. In order to repair these flaws of the original Hill cipher, in this paper we proposed Invertible, Involutory and Permutation Matrix Generation Methods for Hill Cipher System.

P ARPACK is a parallel version of the ARPACK software. ARPACK is a package of Fortran 77 subroutines which implement the Implicitly Restarted Arnoldi Method used for solving large sparse eigenvalue problems. A parallel implementation of ARPACK is presented which is portable across a wide range of distributed memory platforms and requires minimal changes to the serial code. The communication layers used for message passing are the Basic Linear Algebra Communication Subprograms (BLACS) developed for the ScaLAPACK project and Message Passing Interface(MPI).

We present and compare three constructive methods for realizing non-real spectra with three nonzero elements in the nonnegative inverse eigenvalue problem. We also provide some necessary conditions for realizability and numerical examples. In particular, we utilise the companion matrix.

Biochemical networks typically exhibit intricate topologies that hinder their analysis with control-theoretic tools. In this work we present a systematic methodology for the identification of the control structure of a reaction network. The method is based on a bandwidth reduction technique applied to the incidence matrix of the network's graph. In addition, in the case of mass-action and stable networks we show that it is possible to identify linear algebraic dependencies between the time-domain integrals of some species' concentrations. We consider the extrinsic apoptosis pathway and an activationinhibition mechanism to illustrate the application of our results.

Given the high relevance of visual input to human behavior, it is often important to precisely monitor the spatial orientation of the visual axis. One popular and accurate technique for measuring gaze orientation is based on the dual search coil. This technique does not allow for very large displacements of the subject, however, and is not robust with respect to translations of the head. More recently, less invasive procedures have been developed that record eye movements with camera-based systems attached to a helmet worn by the subject. Computational algorithms have also been developed that can calibrate eye orientation when the head's position is fixed. Given that camera-based systems measure the eye's position in its orbit, however, the reconstruction of gaze orientation is not as straightforward when the head is allowed to move. In this paper, we propose a new algorithm and calibration method to compute gaze orientation under unrestrained head conditions. Our method requires only the accurate measurement of orbital eye position (for instance, with a camera-based system), and the position of three points on the head. The calculations are expressed in terms of linear algebra, so can easily be interpreted and related to the geometry of the human body. Our calibration method has been tested experimentally and validated against independent data, proving that is it robust even under large translations, rotations, and torsions of the head.

This work presents an approximate solution method for the inÿnite-time nonlinear quadratic optimal control problem. The method is applicable to a large class of nonlinear systems and involves solving a Riccati equation and a series of algebraic equations. The conditions for uniqueness and stability of the resulting feedback policy are established. It is shown that the proposed approximation method is useful in determining the region in which the constrained and unconstrained optimal control problems are identical. A reactor control problem is used to illustrate the method.

Information security is an important issue. Hill Cipher is one of the most famous symmetric cryptosystem that can be used to protect information from unauthorized access.Hill cipher is a polygraph substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraph cipher which was practical to operate on more than three symbols at once, Hill Cipher has many advantages in data encryption. First, it is resistant to the frequency letter analysis. It's also very simple since it uses matrix multiplication. Finally, it has high speed and high throughput [1,13,14], However, noninvertible key matrix over is the main disadvantage of Hill Cipher, because few of the matrices have inverses over. This means that the encrypted text can't be decrypted [2]. This paper suggests a new technique in Hill Cipher algorithm to overcome its major problemnoninvertible key matrix. also our paper suggest enhancement the security of Hill Cipher against known plaintext attack because all steps in Hill Cipher depend on linear algebra calculation This is possible by using public key idea and key generating depending on various options and without linear algebra steps. Therefore, it will be difficult for the attacker to get the key. Finally, our paper will present a proper solution for the problem of sending secret key () for the first time. Our idea is using a public key public () that doesn't sent but generated in the encryption and decryption sides.

Many interesting problems in graph theory can be reduced to solving an eigenproblem of the adjacency matrix or Laplacian of a graph. Given the availability of high-quality linear algebra and graph libraries, one might expect that one could merely use a graph data structure within a eigensolver. However, conventional libraries are rigidly constructed, requiring conversion to library-specific data structures or using heavyweight abstraction methods that prevent efficient composition. The Generic Programming methodology addresses the problems of reusability and composability by careful factorization of a domain into efficient library abstractions. We describe the composition process that makes the data structures from a library supporting one domain usable with the algorithms of another library for a disjoint domain without conversion or heavyweight abstractions. To illustrate the process, we compose two separately-developed libraries, one for solving eigenproblems sequentially and the other for solving graph problems in parallel, effecting an efficient, scalable parallel graph eigensolver.

All the important parameters of a truncated dodecahedron (having 20 congruent equilateral triangular & 12 congruent regular decagonal faces each of equal edge length) such as normal distances & solid angles subtended by the faces, inner radius, outer radius, mean radius, surface area & volume have been calculated by using HCR's formula for regular polyhedron. This formula is a generalized dimensional formula which is applied on any of the five platonic solids i.e. reguler tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron & regular icosahedron to calculate their important parameters. It can be used in analysis, designing & modelling of regular n-polyhedrons.

The exponential of a matrix and the spectral decomposition of a matrix can be computed knowing nothing more than the eigenvalues of the matrix and the Cayley-Hamilton theorem. The arrangement of the ideas in this paper is simple enough to be taught to beginning students of ODEs.

Abstract—The diffraction of E-polarized plane waves by a tandem impedance,slit waveguide is investigated rigorously by using the Fourier transform technique in conjunction with the Mode Matching method. This mixed method,of formulation gives rise to a scalar Wiener-Hopf equation of the second kind,the solution of which contains infinitely many constants satisfying an infinite system of linear algebraic equations. A numerical solution

Batch laboratory scale experiments were conducted to investigate the removal of phosphate from aqueous and municipal wastewater samples by addition of FeCls.6HsO. The effect of pH, Fe-dose and initial phosphate concentration were assessed. Optimum phosphate removal, 63% for 1:l molar addition of Fe(II1) was observed at pH 4.5. However, a 155% excess of Fe-dose was necessary for complete phosphorus removal. Phosphorus removal from municipal wastewater was slightly higher than that observed for the aqueous solutions. A chemical precipitation mathematical model was developed and tested with the available experimental data. The model included a total of 15 chemical reactions and 4 solid phases with the option of single-phase precipitation or two-phase co-precipitation. The resulting system of non-linear algebraic equations was solved numerically, using the Wijngaarden-Dekker-Brent method.

The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.

It was observed long ago that the obstruction to the accurate computation of eigenvalues of large non-self-adjoint matrices is inherent in the problem. The basic idea is that the resolvent of a highly non-normal operator can be very large far away from the spectrum. This leads to an easily observable fact that algorithms for locating eigenvalues will typically find some "false eigenvalues". These false eigenvalues also explain one of the most surprising phenomena in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, in Berkeley) that one cannot always locally solve the PDE P u = f. Almost immediately after that discovery, Hörmander provided an explanation of Lewy's example showing that almost all operators with non-constanct complex valued coefficients are not locally solvable. In modern language, that was done by considering the essentially dual problem of existence of non-propagating singularities. The purpose of this article is to review this work in the context of "almost eigenvalues" and from the point of view of semi-classical analysis.

Many beginning university students struggle with the new approaches to mathematics that they find in their courses due to a shift in presentation of mathematical ideas, from a procedural approach to concept definitions and deductive derivations, with ideas building upon each other in quick succession. This paper highlights this situation by considering some conceptual processes and difficulties students find in learning about eigenvalues and eigenvectors. We use the theoretical framework of Tall's three worlds of mathematics, along with perspectives from process-object and representational theory. The results of the study describe the thinking about these concepts of groups by first and second year university students, and in particular the obstacles they faced, and the emerging links some were constructing between parts of their concept images formed from the embodied, symbolic and formal worlds. We also identify some fundamental problems with student understanding of the definition of eigenvectors that lead to problems using it, and some of the concepts underlying the difficulties.

In cryptography, we know the symmetric key algorithms and asymmetric key algorithms. We know that the asymmetric key algorithm is more secure than the symmetric key algorithm. Affine Cipher uses a symmetric key algorithm. In this paper, we introduce the Affine Cipher using asymmetric keys. Asymmetrical keys are formed from rectangular matrices.

Most methods for calculating the SVD (singular value decomposition) require to ®rst bidiagonalize the matrix. The blocked reduction of a general, dense matrix to bidiagonal form, as implemented in ScaLAPACK, does about one half of the operations with BLAS3. By subdividing the reduction into two stages dense 3 banded and banded 3 bidiagonal with cubic and quadratic arithmetic costs, respectively, we are able to carry out a much higher portion of the calculations in matrix±matrix multiplications. Thus, higher performance can be expected. This paper presents and compares three parallel techniques for reducing a full matrix to banded form. (The second reduction stage is described in another paper [B. Lang, Parallel Comput. 22 (1996) 1±18]). Numerical experiments on the Intel Paragon and IBM SP/1 distributed memory parallel computers demonstrate that the two-stage reduction approach can be signi®cantly superior if only the singular values are required. Ó . This work was partially funded by Deutsche Forschungsgemeinschaft, Gesch aftszeichen Fr 755/6-1 and Fr 755/6-2. 0167-8191/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 -8 1 9 1 ( 9 9 ) 0 0 0 4 1 -1 parallel computers [1,2,4] and to novel accuracy issues, do most of the work on a full or triangular matrix.

T I. INTRODUCTION HIS paper deals with some mathematical aspects of the discrete Fourier transform (DFT), studied with linear algebra and matrix theory methods. The vast majority of papers on the DFT have concerned computational issues, most notably the extensive literature ...

We describe a new algorithm for Gaussian Elimination suitable for general (unsymmetric and possibly singular) sparse matrices, of any entry type, which has a natural parallel and distributed-memory formulation but degrades gracefully to sequential execution. We present a sample MPI implementation of a program computing the rank of a sparse integer matrix using the proposed algorithm. Some preliminary performance measurements are presented and discussed, and the performance of the algorithm is compared to corresponding state-of-the-art algorithms for floating-point and integer matrices.

We present modelling software developed under MATLAB in which parameter estimations are obtained by using non-linear regression techniques. The different parameters appear in a set of non-linear algebraic and differential equations representing the model of the process. From experimental data obtained in discontinuous cultures a representative mathematical model (unstructured kinetic model) of the macroscopic behaviour of Lactobacillus acidophilus has been developed. An unstructured model expressed the specific rates of cell growth, lactic acid production and glucose consumption for batch fermentation. The model is formulated by considering the inhibition of growth under sub-optimal culture conditions during Lactobacillus acidophilus fermentation, which is accompanied by an increase of the maintenance energy. This study permits to predict the cellular behaviour at low growth temperatures and enables to define the response of the strain to sub-optimal temperature stress.

The aim of this study is to present a way of visualization to teaching and learning linear algebra at a level suitable for freshman and sophomore students. In this study, it has been shown how to use visualization approach through vector geometry in teaching and learning linear algebra concepts. This study concentrates basic linear algebra concepts such as determinant, vector subspace, linear dependence and linear independence, basis and dimension, linear transformation, Eigen value, dot product and projection.

Writing performance-critical programs can be frustrating because optimizing compilers for imperative languages tend to be unpredictable. For a subset of optimizations-those that simplify rather than reorder code-it would be useful to prove that a compiler reliably performs optimizations. We show that adopting a "superanalysis" approach to optimization enables such a proof. By analogy with linear algebra, we define the nullspace of an optimizer as those programs it reduces to the empty program. To span the nullspace, we define rewrite rules that de-optimize programs by introducing abstraction. For a model compiler we prove that any sequence of de-optimizing rewrite rule applications is undone by the optimizer. Thus, we are able to give programmers a clear mental model of what simplifications the compiler is guaranteed to perform, and make progress on the problem of "abstraction penalty" in imperative languages.

The Paper describes estimation and control strategies for models with bounded data uncertainties. We shall refer to them as BDU estimation and BDU control methods, for brevity. They are based on constrained game-type formulations that allow the designer to explicitly incorporate into the Problem Statement a priori information about bounds on the sizes of the uncertainties. In this way, the effect of uncertainties is not unnecessarily overemphasized beyond what is implied by the a priori bounds; consequently, overly conservative designs, as well as overly sensitive designs, are avoided. A feature of these new formulations is that geometric insights and recursive techniques, which are widely known and appreciated for classical quadratic-tost designs, tan also be pursued in this new framework. Also, algorithms for computing the optimal solutions with the same computational effort as Standard least-squares solutions exist, thus making the new formulations attractive for practical use. Moreover, the framework is broad enough to encompass applications across several disciplines, not just estimation and control. Examples will be given of a quadratic control design, an So0 control design, a total

We propose a novel spectral decomposition of a 4th-order covariance tensor, S. Just as the variability of vector (i.e., a 1st-order tensor)-valued random variable is characterized by a covariance matrix (i.e., a 2nd-order tensor), S, the variability of a 2nd-order tensor-valued random variable, D, is characterized by a 4th-order covariance tensor, S. Accordingly, just as the spectral decomposition of S is a linear combination of its eigenvalues and the outer product of its corresponding (1st-order tensors) eigenvectors, the spectral decomposition of S is a linear combination of its eigenvalues and the outer product of its corresponding 2nd-order eigentensors. Analogously, these eigenvalues and 2nd-order eigentensors can be used as features with which to represent and visualize variability in tensor-valued data. Here we suggest a framework to visualize the angular structure of S, and then use it to assess and characterize the variability of synthetic diffusion tensor magnetic resonance imaging (DTI) data. The spectral decomposition suggests a hierarchy of symmetries with which to classify the statistical anisotropy inherent in tensor data. We also present maximum likelihood estimates of the sample mean and covariance tensors associated with D, and derive formulae for the expected value of the mean and variance of the projection of D along a particular direction (i.e., the apparent diffusion coefficient or ADC). These findings would be difficult, if not impossible, to glean if we treated 2nd-order tensor random variables as vector-valued random variables, which is conventionally done in multi-variate statistical analysis. r

The assignment – selection problem used to find one-to- one match of given “Users” to “Laptops”, the main objective is to minimize the cost as per user requirement. This paper presents satisfactory solution for real assignment – Laptop selection problem using MATLAB coding.

With the latest developments in the area of advanced computer architectures, we are already seeing large scale machines at petascale level and we are faced with the exascale computing challenge. All these require scalability at system, algorithmic and mathematical model level. In particular, efficient scalable algorithms are required to bridge the performance gap. In this paper, examples of various approaches of designing scalable algorithms for such advanced architectures will be given. We will briefly present our approach to Monte Carlo scalable algorithms for Linear Algebra and explain how these approaches are extended to the field of Computational Finance. Implementation examples will be presented using Linear Algebra Problems and problems from Computational Finance. Furthermore, the corresponding properties of these algorithms will be outlined and discussed.

Tiling is one of the more important transformations for en-hancing locality of reference in programs. Intuitively, tiling a set of loops achieves the effect of interleaving iterations of these loops. Tiling of perfectly-nested loop nests (which are loop nests in which all assignment ...

The exponent of a primitive digraph is the smallest integer k such that for each ordered pair of (not necessarily distinct) vertices x and y there is a walk of length k from x to y. As a generalization of exponent, Brualdi and Liu (Linear Algebra Appl. 14 (1990) 483499) introduced three types of generalized exponents for primitive digraphs in 1990. In this Paper we extend their definitions of generalized exponents from primitive digraphs to general digraphs which are not necessarily primitive. We give necessary and sufficient conditions for the finiteness of these generalized exponents for graphs (undirected, corresponding to symmetric digraphs) and completely determine the largest finite values and the exponent sets of generalized exponents for the class of non-primitive graphs of Order n, the class of connected bipartite graphs of Order n and the class of trees of Order n.

In this work the magic squares of order n are addressed from the perspective of linear algebra, vector space, "Magic Space" and some of its basis is presented.

The case of sparse systems of equations that having more than one qualitatively different row echelon form.