Linear Algebra Techniques Research Articles

Detection Of Control Flow Errors In Parallel Programs At Compile Time

This paper describes a general technique to identify control flow errors in parallel programs, which can be automated into a compiler. The compiler builds a system of linear equations that describes the global control flow of the whole program. Solving these equations using standard techniques of linear algebra can locate a wide range of control flow bugs at compile time. This paper also describes an implementation of this control flow analysis technique in a prototype compiler for a well-known parallel programming language. In contrast to previous research in automated parallel program analysis, our technique is efficient for large programs, and does not limit the range of language features.

Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels

Using the graphical calculus and integration techniques introduced by the authors, we study the statistical properties of outputs of products of random quantum channels for entangled inputs. In particular, we revisit and generalize models of relevance for the recent counterexamples to the minimum output entropy additivity problems. Our main result is a classification of regimes for which the von Neumann entropy is lower on average than the elementary bounds that can be obtained with linear algebra techniques.

Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

Linear algebra technique in the study of linear representations of finite posets is developed in the paper. A concept of a quadratic wandering on a class of posets I is introduced and finite posets I are studied by means of the four integral bilinear forms b ˆ I , b I , b ¯ I , b I • : Z I × Z I → Z (1.1), the associated Coxeter transformations, and the Coxeter polynomials (in connection with bilinear forms of Dynkin diagrams, extended Dynkin diagrams and irreducible root systems are also studied). Bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of Dynkin diagrams and extended Dynkin diagrams are discussed. A homological interpretation of the bilinear forms (1.1) is given and Z -bilinear equivalences between them are discussed. By applying well-known results of Bongartz, Loupias, and Zavadskij-Shkabara, we give several characterisations of posets I, with the Euler form q ¯ I ( x ) = b ¯ I ( x , x ) weakly positive (resp. with the reduced Euler form q I • ( x ) = b I • ( x , x ) weakly positive), and posets I, with the Tits form q ˆ I ( x ) = b ¯ I ( x , x ) weakly positive.

Automatic Reproduction of a Genius Algorithm: Strassen's Algorithm Revisited by Genetic Search

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In 1968, Volker Strassen, a young German mathematician, announced a clever algorithm to reduce the asymptotic complexity of <formula formulatype="inline"><tex Notation="TeX">$n\times n$</tex> </formula> matrix multiplication from the order of <formula formulatype="inline"><tex Notation="TeX">$n^{3}$</tex></formula> to <formula formulatype="inline"><tex Notation="TeX">$n^{2.81}$</tex></formula>. It soon became one of the most famous scientific discoveries in the 20th century and provoked numerous studies by other mathematicians to improve upon it. Although a number of improvements have been made, Strassen's algorithm is still optimal in his original framework, the bilinear systems of 2<formula formulatype="inline"><tex Notation="TeX">$\,\times\,$</tex></formula>2 matrix multiplication, and people are still curious how Strassen developed his algorithm. We examined it to see if we could automatically reproduce Strassen's discovery using a search algorithm and find other algorithms of the same quality. In total, we found 608 algorithms that have the same quality as Strassen's, including Strassen's original algorithm. We partitioned the algorithms into nine different groups based on the way they are constructed. This paper was made possible by the combination of genetic search and linear-algebraic techniques. To the best of our knowledge, this is the first work that automatically reproduced Strassen's algorithm, and furthermore, discovered new algorithms with equivalent asymptotic complexity using a search algorithm. </para>

Costate Computation by a Chebyshev Pseudospectral Method

AMONG the various pseudospectral (PS) methods for optimal control [1], only the Legendre PS method has been mathematically proven to guarantee the feasibility, consistency, and convergence of the approximations [2–5]. As exemplified by its experimental andflight applications in national programs [6–10], it is not surprising that the Legendre PS method has become the method of choice [11–19] in both industry and academia for solving optimal control problems. Efforts to improve the Legendre PS methods by using either other polynomials [20–22] or point distributions [23,24] have not yet resulted in any rigorous framework for convergence of these approximations [24,25]. Compared to Legendre PS methods, Chebyshev PS methods [21,22] for optimal control are somewhat more attractive for a number of reasons. When a function is approximated, it is well known that a Chebyshev expansion is very close to the best polynomial approximation in the infinity norm [26,27]. In addition, Chebyshev polynomials have an attractive computational advantage in terms of the computation of Chebyshev–Gauss–Lobatto (CGL) nodes. Unlike the Legendre–Gauss–Lobatto (LGL) nodes, CGL nodes can be evaluated in closed form [26]. Thus, a Chebyshev PS method offers the possibility of rapid computation because it does not require the use of advanced numerical linear algebra techniques that are necessary for the calculation of LGL nodes [21]. A similar numerical advantage applies to the computation of the derivative via a fast Chebyshev differentiation scheme that is similar to a fastFourier-transform (FFT) computation. In the same spirit, integration is also fast because of the connection between the Clenshaw–Curtis integration and the FFT [27]. Despite these attractive properties, Chebyshev PS methods have not advanced beyond the works of [21,22]. This is, in part, due to the absence of a covector mapping theorem that is crucial for the computation of the costates and other covectors. The computation of costates and other covectors is important in solving practical optimal control problems as it provides ameans for verification and validation of the computed solution [25]. Beyond verification and validation, information about covectors can also be used to facilitate the design of guidance and control algorithms [28]. In this Note, we fill the key gap of costate computation for Chebyshev PSmethods by furthering themethod of Fahroo andRoss [21]. We do this by combining some recent results from Clenshaw– Curtis integration [27], the unification principles proposed by Fahroo and Ross [1,29,30], and the new results of Gong et al. [23].

Research on the Secure Multi-Party Computation of some Linear Algebra Problems

Considering constant-round protocols for generating random shared values, for secure multiplication and for addition of shared values, etc are available and can be met by known techniques in all standard models of communication. Protocols are presented allowing the players to securely solve standard computational problems in linear algebra. In particular, securely, efficiently and in constant-round compute determinant of matrices product, rank of a matrix, and determine similarity between matrices. If the basic protocols (addition and multiplication, etc) are unconditionally secure, then so are our protocols. Furthermore our protocols offer more efficient solutions than previous techniques for secure linear algebra.

Linear programming techniques for analysis and control of batches Petri nets

In this paper we consider Generalised Batches Petri nets (GBPN) and develop new linear algebraic techniques for the analysis of this model. Two main contributions are presented. The first contribution lies in the fact that although we consider the same GBPN model that has already be presented in the literature, we associate to this model a different semantics considering that the instantaneous firing flow of continuous and batch transitions are control variables that can take an arbitrary value provided they satisfy given constraints. The second contribution consists in the analysis of the steady state behavior of GBPN. We show that under the assumption that no discrete transition fires, a steady state can be characterized by solving a linear programming problem that takes into account the net structure and the initial marking.

Efficient computation in rational-valued P systems

In this paper, we describe a new representation for deterministic rational-valued P systems that allows us to form a bridge between membrane computing and linear algebra. On the one hand, we prove that an efficient computation for these P systems can be described using linear algebra techniques. In particular, we show that the computation for getting a configuration in such P systems can be carried out by multiplying appropriate matrices. On the other hand, we also show that membrane computing techniques can be used to get the nth power of a given matrix.

Periodic solutions of first-order autonomous quasi-linear systems

In this paper, using topological degree and linear algebra techniques, we prove that a certain class of quasi-linear systems of differential equations of the form x ˙ = Ax + μ f ( x , μ ) has at least one periodic solution, where μ is a small parameter and A is a constant n × n matrix. If μ is bounded away from zero and the components of f are polynomials in x 1 , … , x n , μ , then there exists at least one periodic solution under certain conditions. Finally, we consider several examples.

A method for decoding the neurophysiological spike-response transform

Many physiological responses elicited by neuronal spikes—intracellular calcium transients, synaptic potentials, muscle contractions—are built up of discrete, elementary responses to each spike. However, the spikes occur in trains of arbitrary temporal complexity, and each elementary response not only sums with previous ones, but can itself be modified by the previous history of the activity. A basic goal in system identification is to characterize the spike-response transform in terms of a small number of functions—the elementary response kernel and additional kernels or functions that describe the dependence on previous history—that will predict the response to any arbitrary spike train. Here we do this by developing further and generalizing the “synaptic decoding” approach of Sen et al. (1996). Given the spike times in a train and the observed overall response, we use least-squares minimization to construct the best estimated response and at the same time best estimates of the elementary response kernel and the other functions that characterize the spike-response transform. We avoid the need for any specific initial assumptions about these functions by using techniques of mathematical analysis and linear algebra that allow us to solve simultaneously for all of the numerical function values treated as independent parameters. The functions are such that they may be interpreted mechanistically. We examine the performance of the method as applied to synthetic data. We then use the method to decode real synaptic and muscle contraction transforms.

Matrix solution bounds of the continuous Lyapunov equation by using a bilinear transformation

By using a bilinear transformation and some linear algebraic techniques, new matrix bounds of the solution of the continuous algebraic Lyapunov equation (CALE) are derived in this paper. Comparing to existing works, these obtained matrix bounds are less restrictive and are easy to be calculated. A numerical example is also given to demonstrate the merits of the present results.

Simulation of electron spin resonance spectroscopy in diverse environments: An integrated approach

We discuss in this work a new software tool, named E-SpiReS (Electron Spin Resonance Simulations), aimed at the interpretation of dynamical properties of molecules in fluids from electron spin resonance (ESR) measurements. The code implements an integrated computational approach (ICA) for the calculation of relevant molecular properties that are needed in order to obtain spectral lines. The protocol encompasses information from atomistic level (quantum mechanical) to coarse grained level (hydrodynamical), and evaluates ESR spectra for rigid or flexible single or multi-labeled paramagnetic molecules in isotropic and ordered phases, based on a numerical solution of a stochastic Liouville equation. E-SpiReS automatically interfaces all the computational methodologies scheduled in the ICA in a way completely transparent for the user, who controls the whole calculation flow via a graphical interface. Parallelized algorithms are employed in order to allow running on calculation clusters, and a web applet Java has been developed with which it is possible to work from any operating system, avoiding the problems of recompilation. E-SpiReS has been used in the study of a number of different systems and two relevant cases are reported to underline the promising applicability of the ICA to complex systems and the importance of similar software tools in handling a laborious protocol. Program summary Program title: E-SpiReS Catalogue identifier: AEEM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEEM_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL v2.0 No. of lines in distributed program, including test data, etc.: 311 761 No. of bytes in distributed program, including test data, etc.: 10 039 531 Distribution format: tar.gz Programming language: C (core programs) and Java (graphical interface) Computer: PC and Macintosh Operating system: Unix and Windows Has the code been vectorized or parallelized?: Yes RAM: 2 048 000 000 Classification: 7.2 External routines: Babel-1.1, CLAPACK, BLAS, CBLAS, SPARSEBLAS, CQUADPACK, LEVMAR Nature of problem: Ab initio simulation of cw-ESR spectra of radicals in solution Solution method: E-SpiReS uses an hydrodynamic approach to calculate the diffusion tensor of the molecule, DFT methodologies to evaluate magnetic tensors and linear algebra techniques to solve numerically the stochastic Liouville equation to obtain an ESR spectrum. Running time: Variable depending on the task. It takes seconds for small molecules in the fast motional regime to hours for big molecules in viscous and/or ordered media.

Efficient Pricing of CPPI Using Markov Operators

Constant Proportion Portfolio Insurance (CPPI) is a strategy designed to give participation in a risky asset while protecting the invested capital. Some gap risk due to extreme events is often kept by the issuer of the product: a put option on the CPPI strategy is included in the product. In this paper we present a new method for the pricing of CPPIs and options on CPPIs, which is much faster and more accurate than the usual Monte-Carlo method. Provided the underlying follows a homogeneous process, the path-dependent CPPI strategy is reformulated into a Markov process in one variable, which allows to use efficient linear algebra techniques. Tail events, which are crucial in the pricing are handled smoothly. We incorporate in this framework linear thresholds, profit lock-in, performance coupons... The American exercise of open-ended CPPIs is handled naturally through backward propagation. Finally we use our pricing scheme to study the influence of various features on the gap risk of CPPI strategies.

An Asymptotic Approach for Testing P0-Matrices

A direct approach to the P-matrix or P0-matrix problem is to evaluate all the principal minors of matrix A using standard numerical linear algebra techniques with O(2nn3) computational time complexity. The computational time complexity of the P-matrix problem has been reduced from O(2nn3) to O(2n) by applying recursively a criterion for P-matrices based on Schur complementation. But this algorithm can be not directly applied to test the P0-matrices because the Schur complementation can be not computed when some zero diagonal elements appear.This paper proposes an asymptotic approach for testing P0-matrices with O(2n) computational time complexity. Some numerical examples show that the proposed algorithm is effective for testing P0-matrices.

Fixation at a locus with multiple alleles: Structure and solution of the Wright Fisher model

We consider the Wright Fisher model for a finite population of diploid sexual organisms where selection acts at a locus with multiple alleles. The mathematical description of a such a model requires vectors and matrices of a multidimensional nature, and hence has a considerable level of complexity. In the present work we avoid this complexity by introducing a simple mathematical transformation. This yields a description of the model in terms of ordinary vectors and ordinary matrices, thereby allowing standard linear algebra techniques to be directly employed. The new description yields a common mathematical representation of the Wright Fisher model that applies for arbitrary numbers of alleles. Within this framework, it is shown how the dynamics decomposes into component parts that are responsible for the different possible transitions of segregating and fixed populations, thereby allowing a clearer understanding of the population dynamics. This decomposition allows expressions to be directly derived for the mean time of fixation, the mean time of segregation (i.e., the sojourn time) and the probability of fixation. Numerical methods are discussed for the evaluation of these quantities.

Analyzing Reachability for Some Petri Nets With Fast Growing Markings

Using linear algebraic techniques, we analyse the computational complexity of testing reachability in Petri nets for which markings can grow very fast. This leads to two subclasses of Petri nets for which the reachability problem is PSPACE-complete. These subclasses are not contained in any other subclass for which complexity of the reachability problem was known, such as those given in Esparza and Nielsen's survey [Esparza, J. and M. Nielsen, Decidability issues for Petri nets — a survey, J. Inform. Process. Cybernet. 30 (1994), pp. 143–160]. We give an example where further extension of our subclasses fails to maintain the upper bound.

1+1+2 gravitational perturbations on LRS class II spacetimes: II. Decoupling gravito-electromagnetic 2-vector and scalar harmonic amplitudes

This is the second paper in a series that considers first-order, gauge-invariant and covariant, gravitational perturbations to locally rotationally symmetric (LRS) class II vacuum spacetimes. Focusing on the 1+1+2 gravito-electromagnetic (GEM) formalism, the first paper used linear algebra techniques to derive four decoupled equations that govern four specific combinations of the GEM 2-tensor harmonic amplitudes. This paper completes the decoupling of the 1+1+2 GEM system by showing how to derive seven new decoupled quantities. Four of these arise when considering the GEM 2-vector harmonic amplitudes and it is found that decoupling is achieved by combining these with the (2/3-sheet) shear 2-tensor harmonic amplitudes. The remaining three arise from the 1+1+2 GEM scalars. Two of which concern the 2-gradient of the gravito-electric scalar that must also be combined with shear 2-tensor amplitudes, whereas the other involves the gravito-magnetic scalar only.

Minimization of Frequency-Weighted $l_{2}$-Sensitivity Subject to $l_{2}$-Scaling Constraints for Two-Dimensional State-Space Digital Filters

This paper investigates the problem of frequency-weighted l2-sensitivity minimization subject to l2-scaling constraints for two-dimensional (2-D) state-space digital filters described by the Roesser model. It is shown that the Fornasini-Marchesini second model can be imbedded in the Roesser model. Two iterative methods are developed to solve the constrained optimization problem encountered. The first iterative method introduces a Lagrange function and optimizes it using some matrix-theoretic techniques and an efficient bisection method. The second iterative method converts the problem into an unconstrained optimization formulation by using linear-algebraic techniques and solves it by applying an efficient quasi-Newton algorithm. The optimal filter structure with minimum frequency-weighted l2-sensitivity and no overflow is then synthesized by an appropriate coordinate transformation. Case studies are presented to demonstrate the validity and effectiveness of the proposed techniques.

Optimal Synthesis of State-Estimate Feedback Controllers With Minimum $l_{2}$-Sensitivity

This paper investigates the problem of synthesizing the optimal structure of a state-estimate feedback controller with minimum l 2-sensitivity and no overflow. First, the l 2-sensitivity of a closed-loop transfer function with respect to the coefficients of a state-estimate feedback controller is analyzed. Next, two iterative techniques for obtaining the coordinate transformation matrix which constructs the optimal structure of a state-estimate feedback controller are developed so as to minimize an l 2-sensitivity measure subject to l 2-scaling constraints. One technique is based on a Lagrange function, some matrix-theoretic techniques, and an efficient bisection method. Another technique converts the problem into an unconstrained optimization formulation by using linear-algebraic techniques, and optimizes it by applying an efficient quasi-Newton method with closed-form formula for gradient evaluation. A numerical example is also presented to illustrate the utility of the proposed techniques.

On the defining polynomials of maximal real cyclotomic extensions

The aim of this paper is to show that the simplest techniques of linear algebra allow us to make explicit the defining equations of the maximal real cyclotomic extensions Q(³ + ³-1) of Q(³), where ³ stands for a primitive po-th rooot of unity with p a rational prime and o any positive integer.