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Binary Splitting
In mathematics, binary splitting is a technique for speeding up numerical evaluation of many types of series with rational terms. In particular, it can be used to evaluate hypergeometric series at rational points. Method Given a series :S(a,b) = \sum_^b \frac where ''pn'' and ''qn'' are integers, the goal of binary splitting is to compute integers ''P''(''a'', ''b'') and ''Q''(''a'', ''b'') such that :S(a,b) = \frac{Q(a,b)}. The splitting consists of setting ''m'' = ''a'' + ''b'')/2and recursively computing ''P''(''a'', ''b'') and ''Q''(''a'', ''b'') from ''P''(''a'', ''m''), ''P''(''m'', ''b''), ''Q''(''a'', ''m''), and ''Q''(''m'', ''b''). When ''a'' and ''b'' are sufficiently close, ''P''(''a'', ''b'') and ''Q''(''a'', ''b'') can be computed directly from ''pa...pb'' and ''qa...qb''. Comparison with other methods Binary splitting requires more memory than direct term-by-term summation, but is asymptotically faster since the sizes of all occurring subproducts are re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Series (mathematics)
In mathematics, a series is, roughly speaking, an addition of Infinity, infinitely many Addition#Terms, terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greece, Ancient Greeks, the idea that a potential infinity, potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the Quadrature of the Parabola, quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hypergeometric Series
In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by and . There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic. History The term "hypergeometric series" was first used by John Wallis in his 1655 book ''Arithmetica Infinitor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Toom–Cook Multiplication
Toom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers. Given two large integers, ''a'' and ''b'', Toom–Cook splits up ''a'' and ''b'' into ''k'' smaller parts each of length ''l'', and performs operations on the parts. As ''k'' grows, one may combine many of the multiplication sub-operations, thus reducing the overall computational complexity of the algorithm. The multiplication sub-operations can then be computed recursively using Toom–Cook multiplication again, and so on. Although the terms "Toom-3" and "Toom–Cook" are sometimes incorrectly used interchangeably, Toom-3 is only a single instance of the Toom–Cook algorithm, where ''k'' = 3. Toom-3 reduces nine multiplications to five, and runs in Θ(''n''log(5)/log(3)) ≈ Θ(''n''1.46). In general, Toom-''k'' runs in , where , ''ne'' is the time spent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Schönhage–Strassen Algorithm
The Schönhage–Strassen algorithm is an asymptotically fast multiplication algorithm for large integers, published by Arnold Schönhage and Volker Strassen in 1971. It works by recursively applying fast Fourier transform (FFT) over the integers modulo 2^n + 1. The run-time bit complexity to multiply two -digit numbers using the algorithm is O(n \cdot \log n \cdot \log \log n) in big notation. The Schönhage–Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007. It is asymptotically faster than older methods such as Karatsuba and Toom–Cook multiplication, and starts to outperform them in practice for numbers beyond about 10,000 to 100,000 decimal digits. In 2007, Martin Fürer published an algorithm with faster asymptotic complexity. In 2019, David Harvey and Joris van der Hoeven demonstrated that multi-digit multiplication has theoretical O(n\log n) complexity; however, their algorithm has constant factors which make it im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Parallelization
Parallel computing is a type of computation in which many calculations or processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different forms of parallel computing: bit-level, instruction-level, data, and task parallelism. Parallelism has long been employed in high-performance computing, but has gained broader interest due to the physical constraints preventing frequency scaling.S.V. Adve ''et al.'' (November 2008)"Parallel Computing Research at Illinois: The UPCRC Agenda" (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits—increased clock frequency and smarter but increasingly complex architectures—are now hitting the so-called power wall. The computer industry has accepted that future performance increases must largely come from increasing the number of processors (or cores) on a die, rather than m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Checkpointing
Checkpointing is a technique that provides fault tolerance for computing systems. It involves saving a snapshot of an application's state, so that it can restart from that point in case of failure. This is particularly important for long-running applications that are executed in failure-prone computing systems. Checkpointing in distributed systems In the distributed computing environment, checkpointing is a technique that helps tolerate failures that would otherwise force a long-running application to restart from the beginning. The most basic way to implement checkpointing is to stop the application, copy all the required data from the memory to reliable storage (e.g., parallel file system), then continue with execution. In the case of failure, when the application restarts, it does not need to start from scratch. Rather, it will read the latest state ("the checkpoint") from the stable storage and execute from that point. While there is ongoing debate on whether checkpointing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Divide And Conquer Algorithm
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem. The divide-and-conquer technique is the basis of efficient algorithms for many problems, such as sorting (e.g., quicksort, merge sort), multiplying large numbers (e.g., the Karatsuba algorithm), finding the closest pair of points, syntactic analysis (e.g., top-down parsers), and computing the discrete Fourier transform ( FFT). Designing efficient divide-and-conquer algorithms can be difficult. As in mathematical induction, it is often necessary to generalize the problem to make it amenable to a recursive solution. The correctness of a divide-and-conquer algorithm is usually proved by mathematical induction, and its computational c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Class Library For Numbers
Class Library for Numbers (CLN) is a free library for arbitrary precision arithmetic. It operates on signed integers, rational numbers, floating point numbers, complex numbers, modular numbers, and univariate polynomials. Its implementation programming language is C++. Details CLN uses object oriented techniques and operator overloading to achieve a natural algebraic syntax: The sum ''x'' of two variables ''a'' and ''b'' is written as ''x'' = ''a'' + ''b'', as opposed to the function sum(&''x'', ''a'', ''b''). CLN uses class inheritance to model the natural subsets of the available number types: E.g. the integer class is a subtype of the rational class, just as the integer numbers are a subset of the rational numbers. The complex numbers and all its subtypes behave exactly like the types of numbers known to the Common Lisp language, giving CLN another meaning: it becomes an abbreviation of ''Common Lisp Numbers''. Due to this, CLN can be and is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
