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Integer Relation Algorithm
An integer relation between a set of real numbers ''x''1, ''x''2, ..., ''x''''n'' is a set of integers ''a''1, ''a''2, ..., ''a''''n'', not all 0, such that :a_1x_1 + a_2x_2 + \cdots + a_nx_n = 0.\, An integer relation algorithm is an algorithm for finding integer relations. Specifically, given a set of real numbers known to a given precision, an integer relation algorithm will either find an integer relation between them, or will determine that no integer relation exists with coefficients whose magnitudes are less than a certain upper bound. History For the case ''n'' = 2, an extension of the Euclidean algorithm can find any integer relation that exists between any two real numbers ''x''1 and ''x''2. The algorithm generates successive terms of the continued fraction expansion of ''x''1/''x''2; if there is an integer relation between the numbers, then their ratio is rational and the algorithm eventually terminates. *The Ferguson–Forcade algorithm was published in 1979 by Hela ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbitrary Precision Arithmetic
In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision. Several modern programming languages have built-in support for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of bits related to the size of the processor register, these implementations typically use variable-length arrays of digits. Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. It should not be co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon Plouffe
Simon Plouffe (born June 11, 1956) is a mathematician who discovered the Bailey–Borwein–Plouffe formula (BBP algorithm) which permits the computation of the ''n''th binary digit of π, in 1995. His other 2022 formula allows extracting the ''n''th digit of in decimal. He was born in Saint-Jovite, Quebec. He co-authored ''The Encyclopedia of Integer Sequences'', made into the web site On-Line Encyclopedia of Integer Sequences dedicated to integer sequences later in 1995. In 1975, Plouffe broke the world record for memorizing digits of π by reciting 4096 digits, a record which stood until 1977. See also *Fabrice Bellard, who discovered in 1997 a faster formula to compute pi. *PiHex PiHex was a distributed computing project organized by Colin Percival to calculate specific bits of . 1,246 contributors used idle time slices on almost two thousand computers to make its calculations. The software used for the project made use o ... Notes External links * * Plouffe websit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorization Of Polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain. Polynomial factorization is one of the fundamental components of computer algebra systems. The first polynomial factorization algorithm was published by Theodor von Schubert in 1793. Leopold Kronecker rediscovered Schubert's algorithm in 1882 and extended it to multivariate polynomials and coefficients in an algebraic extension. But most of the knowledge on this topic is not older than circa 1965 and the first computer algebra systems: When the long-known finite step algorithms were first put on computers, they turned out to be highly inefficient. The fact that almost any uni- or multivariate polynomial of degree up to 100 and with coefficients of a moderate size (up to 100 bits) can be factored by modern algorithms in a few minut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Symbolic Calculator
__NOTOC__ The Inverse Symbolic Calculator is an online number checker established July 18, 1995 by Peter Benjamin Borwein, Jonathan Michael Borwein and Simon Plouffe of the Canadian Centre for Experimental and Constructive Mathematics (Burnaby, Canada). A user will input a number and the Calculator will use an algorithm to search for and calculate closed-form expressions or suitable functions that have roots near this number. Hence, the calculator is of great importance for those working in numerical areas of experimental mathematics. The ISC contains 54 million mathematical constants. Plouffe's Inverter (opened in 1998) contains 214 million. A newer version of the tables with 3.702 billion entries (as of June 19, 2010) exists. In 2016, Plouffe released a portable version of Plouffe's Inverter containing 3 billion entries. Literature * John Conway, Richard K. Guy: ''Zahlenzauber'' (The Book of Numbers), End of Chapter 1 about Numbers in languages. Birkhäuser, 1997, . See a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Mathematically, the logistic map is written where is a number between zero and one, that represents the ratio of existing population to the maximum possible population. This nonlinear difference equation is intended to capture two effects: * ''reproduction'' where the population will increase at a rate proportional to the current population when the population size is small. * ''starvation'' (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory in quantum mechanics. History Quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century. Its deve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Zeta Function
In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by :\zeta(s_1,\ldots,s_k) = \sum_\ \frac = \sum_\ \prod_^k \frac,\! and converge when Re(''s''1) + ... + Re(''s''''i'') > ''i'' for all ''i''. Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When ''s''1, ..., ''s''''k'' are all positive integers (with ''s''1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the multiple polylogarithms. The ''k'' in the above definition is named the "depth" of a MZV, and the ''n'' = ''s''1 + ... + ''s''''k'' is known as the "weight". The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bailey–Borwein–Plouffe Formula
The Bailey–Borwein–Plouffe formula (BBP formula) is a formula for . It was discovered in 1995 by Simon Plouffe and is named after the authors of the article in which it was published, David H. Bailey, Peter Borwein, and Plouffe. Before that, it had been published by Plouffe on his own site. The formula is : \pi = \sum_^\left frac \left(\frac-\frac-\frac-\frac\right)\right/math> The BBP formula gives rise to a spigot algorithm for computing the ''n''th base-16 (hexadecimal) digit of (and therefore also the ''4n''th binary digit of ) without computing the preceding digits. This does ''not'' compute the ''n''th decimal of (i.e., in base 10). But another formula discovered by Plouffe in 2022 allows extracting the ''n''th digit of in decimal. BBP and BBP-inspired algorithms have been used in projects such as PiHex for calculating many digits of using distributed computing. The existence of this formula came as a surprise. It had been widely believed that computing the ''n''t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Integer
In recreational mathematics, an almost integer (or near-integer) is any number that is not an integer but is very close to one. Almost integers are considered interesting when they arise in some context in which they are unexpected. Almost integers relating to the golden ratio and Fibonacci numbers Well-known examples of almost integers are high powers of the golden ratio \phi=\frac\approx 1.618, for example: : \begin \phi^ & =\frac\approx 3571.00028 \\ pt\phi^ & =2889+1292\sqrt5 \approx 5777.999827 \\ pt\phi^ & =\frac\approx 9349.000107 \end The fact that these powers approach integers is non-coincidental, because the golden ratio is a Pisot–Vijayaraghavan number. The ratios of Fibonacci or Lucas numbers can also make almost integers, for instance: * \operatorname(360)/\operatorname(216) \approx 1242282009792667284144565908481.999999999999999999999999999999195 * \operatorname(361)/\operatorname(216) \approx 2010054515457065378082322433761.000000000000000000000000000 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-form Expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit, differentiation, or integration. The set of operations and functions may vary with author and context. Example: roots of polynomials The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation :ax^2+bx+c=0, is tractable since its solutions can be expressed as a closed-form expression, i.e. in terms of elementary functions: :x=\frac. Similarly, solutions of cubic and quartic (third and fourth degree) equations can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
