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MP is a package of ANSI Standard Fortran (ANS X3.9-1966) subroutines for performing multiple-precision floating-point arithmetic and evaluating elementary and special functions. The subroutines are machine independent and the precision is arbitrary, subject to storage limitations. The User's Guide describes the routines and their calling sequences, example and test programs, use of the Augment precompiler, and gives installation instructions for the package.
The main goal of our work is to formally prove the correctness of the key commands of the SCHUR software, an interactive program for calculating with characters of Lie groups and symmetric functions. The core of the computations relies on enumeration and manipulation of combinatorial structures. As a first "proof of concept", we present a formal proof of the conjugate function, written in C. This function computes the conjugate of an integer partition. To formally prove this program, we use the Frama-C software. It allows us to annotate C functions and to generate proof obligations, which are proved using several automated theorem provers. In this paper, we also draw on methodology, discussing on how to formally prove this kind of program.
Electronic Geometry Textbook is a knowledge management system that manages geometric textbook knowledge to enable users to construct and share dynamic geometry textbooks interactively and efficiently. Based on a knowledge base organizing and storing the knowledge represented in specific languages, the system implements interfaces for maintaining the data representing that knowledge as well as relations among those data, for automatically generating readable documents for viewing or printing, and for automatically discovering the relations among knowledge data. An interface has been developed for users to create geometry textbooks with automatic checking, in real time, of the consistency of the structure of each resulting textbook. By integrating an external geometric theorem prover and an external dynamic geometry software package, the system offers the facilities for automatically proving theorems and generating dynamic figures in the created textbooks. This paper provides a comprehensive account of the current version of Electronic Geometry Textbook.
Implementation of an editing process for Content MathML formulas in common visual style is a real challenge for a software developer who does not really want the user to have to understand the structure of Content MathML in order to edit an expression, since it is expected that users are often not that technically minded. In this paper, we demonstrate how this aim is achieved in the context of the Formulator project and discuss features of this MathML editor, which provides a user with a WYSIWYG editing style while authoring MathML documents with Content or mixed markup. We also present the approach taken to enhance availability of the MathML editor to end-users, demonstrating an online version of the editor that runs inside a Web browser.
This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.
A pseudo-random number generator (RNG) might be used to generate w-bit random samples in d dimensions if the number of state bits is at least dw. Some RNGs perform better than others and the concept of equidistribution has been introduced in the literature in order to rank different RNGs. We define what it means for a RNG to be (d,w)-equidistributed, and then argue that (d,w)-equidistribution is not necessarily a desirable property.
The importance of stochasticity within biological systems has been shown repeatedly during the last years and has raised the need for efficient stochastic tools. We present SABRE, a tool for stochastic analysis of biochemical reaction networks. SABRE implements fast adaptive uniformization (FAU), a direct numerical approximation algorithm for computing transient solutions of biochemical reaction networks. Biochemical reactions networks represent biological systems studied at a molecular level and these reactions can be modeled as transitions of a Markov chain. SABRE accepts as input the formalism of guarded commands, which it interprets either as continuous-time or as discrete-time Markov chains. Besides operating in a stochastic mode, SABRE may also perform a deterministic analysis by directly computing a mean-field approximation of the system under study. We illustrate the different functionalities of SABRE by means of biological case studies.
This is an announcement of the first official release ver.1.0 of a Python system NZMATH for number theory. We overview all functions in NZMATH 1.0, show main properties after former report on NZMATH 0.5.0, and describe new features for stable development. The most important point of the release is that we can now treat number fields. The second big change is that new type of polynomial programs are provided. Elliptic curve primality proving and its related programs are also available, where we partly use a library outside NZMATH as an advantage of writing the system only by Python. On method of development, a new feature is that NZMATH is registered on SourceForge as an open source project to keep continuous development of the project. This is a unique attempt among existing systems for number theory.
In this paper we present a Java implementation of the algorithm that computes Buchbereger's and reduced Groebner's basis step by step. The Java application enables graphical representation of the intersection of two surfaces in 3-dimensional space and determines conditions of existence and planarity of the intersection.
Random numbers play a crucial role in science and industry. Many numerical methods require the use of random numbers, in particular the Monte Carlo method. Therefore it is of paramount importance to have efficient random number generators. The differences, advantages and disadvantages of true and pseudo random number generators are discussed with an emphasis on the intrinsic details of modern and fast pseudo random number generators. Furthermore, standard tests to verify the quality of the random numbers produced by a given generator are outlined. Finally, standard scientific libraries with built-in generators are presented, as well as different approaches to generate nonuniform random numbers. Potential problems that one might encounter when using large parallel machines are discussed.
We introduce a new OpenMath content dictionary, named tensor1, containing symbols for the expression of tensor formulas. These symbols support the expression of non-Cartesian coordinates and invariant, multilinear expressions in the context of coordinate transformations. While current OpenMath symbols support the expression of linear algebra formulas using matrices and vectors, we find that there is an underlying assumption of Cartesian, or standard, coordinates that makes the expression of general tensor formulas difficult, if not impossible. In introducing these new OpenMath symbols for the expression of tensor formulas, we attempt to maintain, as much as possible, consistency with prior OpenMath symbol definitions for linear algebra.
This article introduces yet another representation of rotations in 3-space. The rotations form a 3-dimensional projective space, which fact has not been exploited in Computer Science. We use the four affine patches of this projective space to parametrize the rotations. This affine patch representation is more compact than quaternions (which require 4 components for calculations), encompasses the entire rotation group without singularities (unlike the Euler angles and rotation vector approaches), and requires only ratios of linear or quadratic polynomials for basic computations (unlike the Euler angles and rotation vector approaches which require transcendental functions). As an example, we derive the differential equation for the integration of angular velocity using this affine patch representation of rotations. We remark that the complexity of this equation is the same as the corresponding quaternion equation, but has advantages over the quaternion approach e.g. renormalization to unit length is not required, and state space has no dead directions.
Mathematical learning environments help students in mastering mathematical knowledge. Mature environments typically offer thousands of interactive exercises. Providing feedback to students solving interactive exercises requires domain reasoners for doing the exercise-specific calculations. Since a domain reasoner has to solve an exercise in the same way a student should solve it, the structure of domain reasoners should follow the layered structure of the mathematical domains. Furthermore, learners, teachers, and environment builders have different requirements for adapting domain reasoners, such as providing more details, disallowing or enforcing certain solutions, and combining multiple mathematical domains in a new domain. In previous work we have shown how domain reasoners for solving interactive exercises can be expressed in terms of rewrite strategies, rewrite rules, and views. This paper shows how users can adapt and configure such domain reasoners to their own needs. This is achieved by enabling users to explicitly communicate the components that are used for solving an exercise.
This paper proposes a type of pseudorandom number generator, Mersenne Twister for Graphic Processor (MTGP), for efficient generation on graphic processessing units (GPUs). MTGP supports large state sizes such as 11213 bits, and uses the high parallelism of GPUs in computing many steps of the recursion in parallel. The second proposal is a parameter-set generator for MTGP, named MTGP Dynamic Creator (MTGPDC). MT- GPDC creates up to 2^32 distinct parameter sets which generate sequences with high-dimensional uniformity. This facility is suitable for a large grid of GPUs where each GPU requires separate random number streams. MTGP is based on linear recursion over the two-element field, and has better high-dimensional equidistribution than the Mersenne Twister pseudorandom number generator.
In this work we describe an efficient implementation of a hierarchy of algorithms for the decomposition of dense matrices over the field with two elements (GF(2)). Matrix decomposition is an essential building block for solving dense systems of linear and non-linear equations and thus much research has been devoted to improve the asymptotic complexity of such algorithms. In this work we discuss an implementation of both well-known and improved algorithms in the M4RI library. The focus of our discussion is on a new variant of the M4RI algorithm - denoted MMPF in this work -- which allows for considerable performance gains in practice when compared to the previously fastest implementation. We provide performance figures on x86_64 CPUs to demonstrate the viability of our approach.
Generalized sparse matrix-matrix multiplication is a key primitive for many high performance graph algorithms as well as some linear solvers such as multigrid. We present the first parallel algorithms that achieve increasing speedups for an unbounded number of processors. Our algorithms are based on two-dimensional block distribution of sparse matrices where serial sections use a novel hypersparse kernel for scalability. We give a state-of-the-art MPI implementation of one of our algorithms. Our experiments show scaling up to thousands of processors on a variety of test scenarios.
"The term 'Linked Data' refers to a set of best practices for publishing and connecting structured data on the web". Linked Data make the Semantic Web work practically, which means that information can be retrieved without complicated lookup mechanisms, that a lightweight semantics enables scalable reasoning, and that the decentral nature of the Web is respected. OpenMath Content Dictionaries (CDs) have the same characteristics - in principle, but not yet in practice. The Linking Open Data movement has made a considerable practical impact: Governments, broadcasting stations, scientific publishers, and many more actors are already contributing to the "Web of Data". Queries can be answered in a distributed way, and services aggregating data from different sources are replacing hard-coded mashups. However, these services are currently entirely lacking mathematical functionality. I will discuss real-world scenarios, where today's RDF-based Linked Data do not quite get their job done, but where an integration of OpenMath would help - were it not for certain conceptual and practical restrictions. I will point out conceptual shortcomings in the OpenMath 2 specification and common bad practices in publishing CDs and then propose concrete steps to overcome them and to contribute OpenMath CDs to the Web of Data.
Multi-dimensional arrays are among the most fundamental and most useful data structures of all. In C++, excellent template libraries exist for arrays whose dimension is fixed at runtime. Arrays whose dimension can change at runtime have been implemented in C. However, a generic object-oriented C++ implementation of runtime-flexible arrays has so far been missing. In this article, we discuss our new implementation called Marray, a package of class templates that fills this gap. Marray is based on views as an underlying concept. This concept brings some of the flexibility known from script languages such as R and MATLAB to C++. Marray is free both for commercial and non-commercial use and is publicly available from www.andres.sc/marray