4,909 research outputs found
Can Computer Algebra be Liberated from its Algebraic Yoke ?
So far, the scope of computer algebra has been needlessly restricted to exact
algebraic methods. Its possible extension to approximate analytical methods is
discussed. The entangled roles of functional analysis and symbolic programming,
especially the functional and transformational paradigms, are put forward. In
the future, algebraic algorithms could constitute the core of extended symbolic
manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure
Sparse Automatic Differentiation for Large-Scale Computations Using Abstract Elementary Algebra
Most numerical solvers and libraries nowadays are implemented to use
mathematical models created with language-specific built-in data types (e.g.
real in Fortran or double in C) and their respective elementary algebra
implementations. However, built-in elementary algebra typically has limited
functionality and often restricts flexibility of mathematical models and
analysis types that can be applied to those models. To overcome this
limitation, a number of domain-specific languages with more feature-rich
built-in data types have been proposed. In this paper, we argue that if
numerical libraries and solvers are designed to use abstract elementary algebra
rather than language-specific built-in algebra, modern mainstream languages can
be as effective as any domain-specific language. We illustrate our ideas using
the example of sparse Jacobian matrix computation. We implement an automatic
differentiation method that takes advantage of sparse system structures and is
straightforward to parallelize in MPI setting. Furthermore, we show that the
computational cost scales linearly with the size of the system.Comment: Submitted to ACM Transactions on Mathematical Softwar
*-Continuous Kleene -Algebras for Energy Problems
Energy problems are important in the formal analysis of embedded or
autonomous systems. Using recent results on star-continuous Kleene
omega-algebras, we show here that energy problems can be solved by algebraic
manipulations on the transition matrix of energy automata. To this end, we
prove general results about certain classes of finitely additive functions on
complete lattices which should be of a more general interest.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Quantum Computers and Quantum Computer Languages: Quantum Assembly Language and Quantum C
We show a representation of Quantum Computers defines Quantum Turing Machines with associated Quantum Grammars. We then create examples of Quantum Grammars. Lastly we develop an algebraic approach to high level Quantum Languages using Quantum Assembly language and Quantum C language as examples
User-friendly Support for Common Concepts in a Lightweight Verifier
Machine verification of formal arguments can only increase our confidence in the correctness of those arguments, but the costs of employing machine verification still outweigh the benefits for some common kinds of formal reasoning activities. As a result, usability is becoming increasingly important in the design of formal verification tools. We describe the "aartifact" lightweight verification system, designed for processing formal arguments involving basic, ubiquitous mathematical concepts. The system is a prototype for investigating potential techniques for improving the usability of formal verification systems. It leverages techniques drawn both from existing work and from our own efforts. In addition to a parser for a familiar concrete syntax and a mechanism for automated syntax lookup, the system integrates (1) a basic logical inference algorithm, (2) a database of propositions governing common mathematical concepts, and (3) a data structure that computes congruence closures of expressions involving relations found in this database. Together, these components allow the system to better accommodate the expectations of users interested in verifying formal arguments involving algebraic and logical manipulations of numbers, sets, vectors, and related operators and predicates. We demonstrate the reasonable performance of this system on typical formal arguments and briefly discuss how the system's design contributed to its usability in two case studies
New Algebraic Formulation of Density Functional Calculation
This article addresses a fundamental problem faced by the ab initio
community: the lack of an effective formalism for the rapid exploration and
exchange of new methods. To rectify this, we introduce a novel, basis-set
independent, matrix-based formulation of generalized density functional
theories which reduces the development, implementation, and dissemination of
new ab initio techniques to the derivation and transcription of a few lines of
algebra. This new framework enables us to concisely demystify the inner
workings of fully functional, highly efficient modern ab initio codes and to
give complete instructions for the construction of such for calculations
employing arbitrary basis sets. Within this framework, we also discuss in full
detail a variety of leading-edge ab initio techniques, minimization algorithms,
and highly efficient computational kernels for use with scalar as well as
shared and distributed-memory supercomputer architectures
Modelica - A Language for Physical System Modeling, Visualization and Interaction
Modelica is an object-oriented language for modeling of large, complex and heterogeneous physical systems. It is suited for multi-domain modeling, for example for modeling of mechatronics including cars, aircrafts and industrial robots which typically consist of mechanical, electrical and hydraulic subsystems as well as control systems. General equations are used for modeling of the physical phenomena, No particular variable needs to be solved for manually. A Modelica tool will have enough information to do that automatically. The language has been designed to allow tools to generate efficient code automatically. The modeling effort is thus reduced considerably since model components can be reused and tedious and error-prone manual manipulations are not needed. The principles of object-oriented modeling and the details of the Modelica language as well as several examples are presented
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