Two-dimensional linear algebra

We introduce two-dimensional linear algebra, by which we do not mean two-dimensional vector spaces but rather the systematic replacement in linear algebra of sets by categories. This entails the study of categories that are simultaneously categories of algebras for a monad and categories of coalgebras for comonad on a category such as SymMon_s, the category of small symmetric monoidal categories. We outline relevant notions such as that of pseudo-closed 2-category, symmetric monoidal Lawvere theory, and commutativity of a symmetric monoidal Lawvere theory, and we explain the role of coalgebra, explaining its precedence over algebra in this setting. We outline salient results and perspectives given by the dual approach of algebra and coalgebra, extending to two dimensions the study of linear algebra.     

Using desktop computers to solve large-scale dense linear algebra problems

We provide experimental evidence that current desktop computers feature enough computational power to solve large-scale dense linear algebra problems. While the high computational cost of the numerical methods for solving these problems can be tackled by the multiple cores of current processors, we propose to use the disk to store the large data structures associated with these applications. Our results also show that the limited amount of RAM and the comparatively slow disk of the system pose no problem for the solution of very large dense linear systems and linear least-squares problems. Thus, current desktop computers are revealed as an appealing, cost-effective platform for research groups that have to deal with large dense linear algebra problems but have no direct access to large computing facilities.     

Deriving dense linear algebra libraries

Starting in the late 1960s computer scientists including Dijkstra and Hoare advocated goal-oriented programming and the formal derivation of algorithms. The chief impediment to realizing this for loop-based programs was that a priori determination of loop-invariants, a prerequisite for developing loops, was a task too complex for any but the simplest of operations. Around 2000, these techniques were for the first time successfully applied to the domain of high-performance dense linear algebra libraries. This has led to a multitude of papers (mostly published in the ACM Transactions for Mathematical Software), a system for the mechanical derivation of algorithms, and a high-performance linear algebra library, ${\tt libflame}$ , that includes more than a thousand variants of algorithms for more than a hundred linear algebra operations. To our knowledge, this success story has unfolded with limited awareness on the part the formal methods community. This paper reports on ten years of experience and is meant to raise that awareness.     

Parallel accurate linear algebra subroutines

In this paper we present a set of linear algebra subroutines which serve as building blocks for numerical software, and develop algorithms to implement these subroutines as a portable library for parallel computers. We consider these routines as a part of the standard arithmetic of a computer Therefore they have to deliver a validated result of high accuracy     

Towards a linear algebra of programming

The algebra of programming (AoP) is a discipline for programming from specifications using relation algebra. Specification vagueness and nondeterminism are captured by relations. (Final) implementations are functions. Probabilistic functions are half way between relations and functions: they express the propensity, or likelihood of ambiguous, multiple outputs. This paper puts forward a basis for a linear algebra of programming (LAoP) extending standard AoP towards probabilistic functions. Because of the quantitative essence of these functions, the allegory of binary relations which supports the AoP has to be extended. We show that, if one restricts to discrete probability spaces, categories of matrices provide adequate support for the extension, while preserving the pointfree reasoning style typical of the AoP.     

A linear algebra approach to OLAP

Inspired by the relational algebra of data processing, this paper addresses the foundations of data analytical processing from a linear algebra perspective. The paper investigates, in particular, how aggregation operations such as cross tabulations and data cubes essential to quantitative analysis of data can be expressed solely in terms of matrix multiplication, transposition and the Khatri–Rao variant of the Kronecker product. The approach offers a basis for deriving an algebraic theory of data consolidation, handling the quantitative as well as qualitative sides of data science in a natural, elegant and typed way. It also shows potential for parallel analytical processing, as the parallelization theory of such matrix operations is well acknowledged.     

Index transformation algorithms in a linear algebra framework

We present a linear algebraic formulation for a class of index transformations such as Gray code encoding and decoding, matrix transpose, bit reversal, vector reversal, shuffles, and other index or dimension permutations. This formulation unifies, simplifies, and can be used to derive algorithms for hypercube multiprocessors. We show how all the widely known properties of Gray codes, and some not so well-known properties as well, can be derived using this framework. Using this framework, we relate hypercube communications algorithms to Gauss-Jordan elimination on a matrix of 0's and 1's     

Verification of linear algebra programs in the spektr system

A subsystem for verification of linear algebra programs is described. The subsystem is implemented as part of the problem-oriented verification system SPEKTR.Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 136–144, September–October, 1992.     

An algebra for data flow diagram process decomposition

Data flow diagram process decomposition, as applied in the analysis phase of software engineering, is a top-down method that takes a process, and its input and output data flows, and logically implements the process as a network of smaller processes. The decomposition is generally performed in an ad hoc manner by an analyst applying heuristics, expertise, and knowledge to the problem. An algebra that formalizes process decomposition is presented using the De Marco representation scheme. In this algebra, the analyst relates the disjoint input and output sets of a single process by specifying the elements of an input/output connectivity matrix. A directed acyclic graph is constructed from the matrix and is the decomposition of the process. The graph basis, grammar matrix, and graph interpretations, and the operators of the algebra are discussed. A decomposition procedure for applying the algebra, prototype, and production tools and outlook are also discussed     

Looking back at dense linear algebra software

Over the years, computational physics and chemistry served as an ongoing source of problems that demanded the ever increasing performance from hardware as well as the software that ran on top of it. Most of these problems could be translated into solutions for systems of linear equations: the very topic of numerical linear algebra. Seemingly then, a set of efficient linear solvers could be solving important scientific problems for years to come. We argue that dramatic changes in hardware designs precipitated by the shifting nature of the marketplace of computer hardware had a continuous effect on the software for numerical linear algebra. The extraction of high percentages of peak performance continues to require adaptation of software. If the past history of this adaptive nature of linear algebra software is any guide then the future theme will feature changes as well–changes aimed at harnessing the incredible advances of the evolving hardware infrastructure.     

Usability levels for sparse linear algebra components

Sparse matrix computations are ubiquitous in high‐performance computing applications and often are their most computationally intensive part. In particular, efficient solution of large‐scale linear systems may drastically improve the overall application performance. Thus, the choice and implementation of the linear system solver are of paramount importance. It is difficult, however, to navigate through a multitude of available solver packages and to tune their performance to the problem at hand, mainly because of the plethora of interfaces, each requiring application adaptations to match the specifics of solver packages. For example, different ways of setting parameters and a variety of sparse matrix formats hinder smooth interactions of sparse matrix computations with user applications. In this paper, interfaces designed for components that encapsulate sparse matrix computations are discussed in the light of their matching with application usability requirements. Consequently, we distinguish three levels of interfaces, high, medium, and low, corresponding to the degree of user involvement in the linear system solution process and in sparse matrix manipulations. We demonstrate when each interface design choice is applicable and how it may be used to further users' scientific goals. Component computational overheads caused by various design choices are also examined, ranging from low level, for matrix manipulation components, to high level, in which a single component contains the entire linear system solver. Published in 2007 by John Wiley & Sons, Ltd.     

Unifying and optimizing parallel linear algebra algorithms

Two issues in linear algebra algorithms for multicomputers are addressed. First, how to unify parallel implementations of the same algorithm in a decomposition-independent way. Second, how to optimize naive parallel programs maintaining the decomposition independence. Several matrix decompositions are viewed as instances of a more general allocation function called subcube matrix decomposition. By this meta-decomposition, a programming environment characterized by general primitives that allow one to design meta-algorithms independently of a particular decomposition. The authors apply such a framework to the parallel solution of dense matrices. This demonstrates that most of the existing algorithms can be derived by suitably setting the primitives used in the meta-algorithm. A further application of this programming style concerns the optimization of parallel algorithms. The idea to overlap communication and computation has been extended from 1-D decompositions to 2-D decompositions. Thus, a first attempt towards a decomposition-independent definition of such optimization strategies is provided     

Typing linear algebra: A biproduct-oriented approach

Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts. This shifts the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming. The derivation of fusion, cancellation and abide laws from the biproduct equations makes it easy to calculate algorithms implementing matrix multiplication, the central operation of matrix algebra, ranging from its divide-and-conquer version to its vectorization implementation.     

Numerical linear algebra aspects of control design computations

The interplay between recent results and methodologies in numerical linear algebra and mathematical software and their application to problems arising in systems, control, and estimation theory is discussed. The impact of finite precision, finite range arithmetic [including the implications of the proposed IEEE floating point standard(s)] on control design computations is illustrated with numerous examples as are pertinent remarks concerning numerical stability and conditioning. Basic tools from numerical linear algebra such as linear equations, linear least squares, eigenproblems, generalized eigenproblems, and singular value decomposition are then outlined. A selected list of applications of the basic tools then follows including algorithms for solution of problems such as matrix exponentials, frequency response, system balancing, and matrix Riccati equations. The implementation of such algorithms as robust mathematical software is then discussed. A number of issues are addressed including characteristics of reliable mathematical software, availability and evaluation, language implications (Fortran, Ada, etc.), and the overall role of mathematical software as a component of computer-aided control system design.     

Deriving linear transformations in 3D using quaternion algebra

We propose a method for delivering linear transformation in the two parallel projections that works well with quaternion techniques. The article also presents a completed vector method to derive a linear transformation of the two kinds of parallel projections. Here, we propose a novel method that can achieve a unified framework based on quaternion techniques.     

Using computer algebra systems in mathematical classrooms

Abstract This paper describes a research whose main focus is the use of Computer Algebra Systems (CAS) in mathematical classrooms and the didactical possibilities linked with its use. The possibilities of integrating Self-Regulated Learning (SRL) within the CAS environment are brought into focus. Forty-three Israeli students (mean age 13.3) were assigned to two learning algebraic groups. The first group received explicit meta-cognitive SRL with CAS (CAS + SRL); the second group was exposed to CAS without SRL (CAS). Empirical results from the experimental and case study designs revealed that (CAS + SRL) students outperformed (CAS) students on algebraic thinking and that (CAS + SRL) students regulated their learning more effectively.