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Baillie–PSW Primality Test
The Baillie–PSW primality test is a probabilistic primality testing algorithm that determines whether a number is composite or is a probable prime. It is named after Robert Baillie, Carl Pomerance, John Selfridge, and Samuel Wagstaff. The Baillie–PSW test is a combination of a strong Fermat probable prime test to base 2 and a strong Lucas probable prime test. The Fermat and Lucas test each have their own list of pseudoprimes, that is, composite numbers that pass the test. For example, the first ten strong pseudoprimes to base 2 are : 2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, and 52633 . The first ten strong Lucas pseudoprimes (with Lucas parameters (''P'', ''Q'') defined by Selfridge's Method A) are : 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and 58519 . There is no known overlap between these lists of strong Fermat pseudoprimes and strong Lucas pseudoprimes, and there is even evidence that the numbers in these lists tend to be different kin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomized Algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized algor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently ranks as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perl
Perl is a family of two high-level, general-purpose, interpreted, dynamic programming languages. "Perl" refers to Perl 5, but from 2000 to 2019 it also referred to its redesigned "sister language", Perl 6, before the latter's name was officially changed to Raku in October 2019. Though Perl is not officially an acronym, there are various backronyms in use, including "Practical Extraction and Reporting Language". Perl was developed by Larry Wall in 1987 as a general-purpose Unix scripting language to make report processing easier. Since then, it has undergone many changes and revisions. Raku, which began as a redesign of Perl 5 in 2000, eventually evolved into a separate language. Both languages continue to be developed independently by different development teams and liberally borrow ideas from each other. The Perl languages borrow features from other programming languages including C, sh, AWK, and sed; They provide text processing facilities without the arbitrary data-le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OpenJDK
OpenJDK (Open Java Development Kit) is a free and open-source implementation of the Java Platform, Standard Edition (Java SE). It is the result of an effort Sun Microsystems began in 2006. The implementation is licensed under the GPL-2.0-only with a linking exception. Were it not for the GPL linking exception, components that linked to the Java class library would be subject to the terms of the GPL license. OpenJDK is the official reference implementation of Java SE since version 7. Components The OpenJDK project produces a number of components: most importantly the virtual machine ( HotSpot), the Java Class Library and the Java compiler (javac). The web-browser plugin and Web Start, which form part of Oracle Java, are not included in OpenJDK. Sun previously indicated that they would try to open-source these components, but neither Sun nor Oracle have done so. The only currently available free plugin and Web Start implementations are those provided by IcedTea. OpenJDK 9+ su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java (software Platform)
Java is a set of computer software and specifications developed by James Gosling at Sun Microsystems, which was later acquired by the Oracle Corporation, that provides a system for developing application software and deploying it in a cross-platform computing environment. Java is used in a wide variety of computing platforms from embedded devices and mobile phones to enterprise servers and supercomputers. Java applets, which are less common than standalone Java applications, were commonly run in secure, sandboxed environments to provide many features of native applications through being embedded in HTML pages. Writing in the Java programming language is the primary way to produce code that will be deployed as byte code in a Java virtual machine (JVM); byte code compilers are also available for other languages, including Ada, JavaScript, Python, and Ruby. In addition, several languages have been designed to run natively on the JVM, including Clojure, Groovy, and Scala. J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Library For Number Theory
The Fast Library for Number Theory (FLINT) is a C library for number theory applications. The two major areas of functionality currently implemented in FLINT are polynomial arithmetic over the integers and a quadratic sieve. The library is designed to be compiled with the GNU Multi-Precision Library (GMP) and is released under the GNU General Public License. It is developed by William Hart of the University of Kaiserslautern (formerly University of Warwick) and David Harvey of University of New South Wales (formerly Harvard University) to address the speed limitations of the PARI and NTL NTL may refer to: Companies * NTL Incorporated and NTL Internet, later Virgin Media, communications media company ** NTL Ireland, later Virgin Media Ireland * Arqiva, UK company formerly ''NTL Broadcast'' and ''National Transcommunications L ... libraries.Page 10 of thProject DescriptionfoUW 2006 COMPMATH Proposalby William Stein Design Philosophy * Asymptotically Fast Algorithms * I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxima (software)
Maxima () is a computer algebra system (CAS) based on a 1982 version of Macsyma. It is written in Common Lisp and runs on all POSIX platforms such as macOS, Unix, BSD, and Linux, as well as under Microsoft Windows and Android. It is free software released under the terms of the GNU General Public License (GPL). History Maxima is based on a 1982 version of Macsyma, which was developed at Massachusetts Institute of Technology, MIT with funding from the United States Department of Energy and other government agencies. A version of Macsyma was maintained by Bill Schelter from 1982 until his death in 2001. In 1998, Schelter obtained permission from the Department of Energy to release his version under the GPL. That version, now called Maxima, is maintained by an independent group of users and developers. Maxima does not include any of the many modifications and enhancements made to the commercial version of Macsyma during 1982–1999. Though the core functionality remains similar, cod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, number theory, calculus and statistics. The first version of SageMath was released on 24 February 2005 as free and open-source software under the terms of the GNU General Public License version 2, with the initial goals of creating an "open source alternative to Magma, Maple, Mathematica, and MATLAB". The originator and leader of the SageMath project, William Stein, was a mathematician at the University of Washington. SageMath uses a syntax resembling Python's, supporting procedural, functional and object-oriented constructs. Development Stein realized when designing Sage that there were many open-source mathematics software packages already written in different languages, namely C, C++, Common Lisp, Fortran and Python. Rather tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PARI/GP
PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. Versions 2.1.0 and higher are distributed under the GNU General Public License. It runs on most common operating systems. System overview The PARI/GP system is a package that is capable of doing formal computations on recursive types at high speed; it is primarily aimed at number theorists. Its three main strengths are its speed, the possibility of directly using data types that are familiar to mathematicians, and its extensive algebraic number theory module. The PARI/GP system consists of the following standard components: * PARI is a C library, allowing for fast computations, and which can be called from a high-level language application (for instance, written in C, C++, Pascal, Fortran, Perl, or Python). * gp is an easy-to-use interactive command line interface giving access to the PARI functions. It functions as a sophisticated programmable calculator which contains most o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematica
Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimization, plotting functions and various types of data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other programming languages. It was conceived by Stephen Wolfram, and is developed by Wolfram Research of Champaign, Illinois. The Wolfram Language is the programming language used in ''Mathematica''. Mathematica 1.0 was released on June 23, 1988 in Champaign, Illinois and Santa Clara, California. __TOC__ Notebook interface Wolfram Mathematica (called ''Mathematica'' by some of its users) is split into two parts: the kernel and the front end. The kernel interprets expressions (Wolfram Language code) and returns result expressions, which can then be displayed by the front end. The origin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
