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Boole's Rule
In mathematics, Boole's rule, named after George Boole, is a method of numerical integration. Formula Simple Boole's Rule It approximates an integral: : \int_^ f(x)\,dx by using the values of at five equally spaced points: : \begin & x_0 = a\\ & x_1 = x_1 + h \\ & x_2 = x_1 + 2h \\ & x_3 = x_1 + 3h \\ & x_4 = x_1 +4h = b \end It is expressed thus in Abramowitz and Stegun: : \int_^ f(x)\,dx = \frac\bigl 7f(x_0) + 32 f(x_1) + 12 f(x_2) + 32 f(x_3) + 7f(x_4) \bigr+ \text where the error term is : -\,\frac for some number between and where . It is often known as Bode's rule, due to a typographical error that propagated from Abramowitz and Stegun. The following constitutes a very simple implementation of the method in Common Lisp which ignores the error term: (defun integrate-booles-rule (f x1 x5) "Calculates the Boole's rule numerical integral of the function F in the closed interval extending from inclusive X1 to inclusive X5 without error term inclusi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Titius–Bode Law
The Titius–Bode law (sometimes termed just Bode's law) is a formulaic prediction of spacing between planets in any given solar system. The formula suggests that, extending outward, each planet should be approximately twice as far from the Sun as the one before. The hypothesis correctly anticipated the orbits of Ceres (in the asteroid belt) and Uranus, but failed as a predictor of Neptune's orbit. It is named after Johann Daniel Titius and Johann Elert Bode. Later work by Blagg and Richardson significantly revised the original formula, and made predictions that were subsequently validated by new discoveries and observations. It is these re-formulations that offer "the best phenomenological representations of distances with which to investigate the theoretical significance of Titius–Bode type Laws". Original formulation The law relates the semi-major axis ~a_n~ of each planet outward from the Sun in units such that the Earth's semi-major axis is equal to 10: :~a = 4 + x~ where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic, and is best known as the author of '' The Laws of Thought'' (1854) which contains Boolean algebra. Boolean logic is credited with laying the foundations for the Information Age. Early life Boole was born in 1815 in Lincoln, Lincolnshire, England, the son of John Boole senior (1779–1848), a shoemaker and Mary Ann Joyce. He had a primary school education, and received lessons from his father, but due to a serious decline in business, he had little further formal and academic teaching. William Brooke, a bookseller in Lincoln, may have helped him with Latin, which he may also have learned at the school of Thomas Bainbridge. He was self-taught in modern la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to ''quadrature'') is more or less a synonym for ''numerical integration'', especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take ''quadrature'' to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abramowitz And Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Technology'' (NIST). Its full title is ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables''. A digital successor to the Handbook was released as the " Digital Library of Mathematical Functions" (DLMF) on 11 May 2010, along with a printed version, the '' NIST Handbook of Mathematical Functions'', published by Cambridge University Press. Overview Since it was first published in 1964, the 1046 page ''Handbook'' has been one of the most comprehensive sources of information on special functions, containing definitions, identities, approximations, plots, and tables of values of numerous functions used in virtually all fields of applied mathematics. The notation used in the ''Handbook'' is the ''de facto'' stand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Common Lisp
Common Lisp (CL) is a dialect of the Lisp programming language, published in ANSI standard document ''ANSI INCITS 226-1994 (S20018)'' (formerly ''X3.226-1994 (R1999)''). The Common Lisp HyperSpec, a hyperlinked HTML version, has been derived from the ANSI Common Lisp standard. The Common Lisp language was developed as a standardized and improved successor of Maclisp. By the early 1980s several groups were already at work on diverse successors to MacLisp: Lisp Machine Lisp (aka ZetaLisp), Spice Lisp, NIL and S-1 Lisp. Common Lisp sought to unify, standardise, and extend the features of these MacLisp dialects. Common Lisp is not an implementation, but rather a language specification. Several implementations of the Common Lisp standard are available, including free and open-source software and proprietary products. Common Lisp is a general-purpose, multi-paradigm programming language. It supports a combination of procedural, functional, and object-oriented programming para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton–Cotes Formulas
In numerical analysis, the Newton–Cotes formulas, also called the Newton–Cotes quadrature rules or simply Newton–Cotes rules, are a group of formulas for numerical integration (also called ''quadrature'') based on evaluating the integrand at equally spaced points. They are named after Isaac Newton and Roger Cotes. Newton–Cotes formulas can be useful if the value of the integrand at equally spaced points is given. If it is possible to change the points at which the integrand is evaluated, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are probably more suitable. Description It is assumed that the value of a function defined on , b/math> is known at n + 1 equally spaced points: a \leq x_0 < x_1 < \ldots < x_n \leq b. There are two classes of Newton–Cotes quadrature: they are called "closed" when and , i.e. they use the function values at the interval endpoints, and "open" when |
Simpson's Rule
In numerical integration, Simpson's rules are several approximations for definite integrals, named after Thomas Simpson (1710–1761). The most basic of these rules, called Simpson's 1/3 rule, or just Simpson's rule, reads \int_a^b f(x) \, dx \approx \frac \left (a) + 4f\left(\frac\right) + f(b)\right In German and some other languages, it is named after Johannes Kepler, who derived it in 1615 after seeing it used for wine barrels (barrel rule, ). The approximate equality in the rule becomes exact if is a polynomial up to and including 3rd degree. If the 1/3 rule is applied to ''n'' equal subdivisions of the integration range 'a'', ''b'' one obtains the composite Simpson's 1/3 rule. Points inside the integration range are given alternating weights 4/3 and 2/3. Simpson's 3/8 rule, also called Simpson's second rule, requires one more function evaluation inside the integration range and gives lower error bounds, but does not improve on order of the error. If the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Romberg's Method
In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. The method is named after Werner Romberg (1909–2003), who published the method in 1955. Method Using h_n = \frac, the method can be inductively defined by \begin R(0,0) &= h_1 (f(a) + f(b)) \\ R(n,0) &= \tfrac R(n-1,0) + h_n \sum_^ f(a + (2k-1)h_n) \\ R(n,m) &= R(n,m-1) + \tfrac (R(n,m-1) - R(n-1,m-1)) \\ &= \frac ( 4^m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macmillan Publishers
Macmillan Publishers (occasionally known as the Macmillan Group; formally Macmillan Publishers Ltd and Macmillan Publishing Group, LLC) is a British publishing company traditionally considered to be one of the 'Big Five' English language publishers. Founded in London in 1843 by Scottish brothers Daniel and Alexander MacMillan, the firm would soon establish itself as a leading publisher in Britain. It published two of the best-known works of Victorian era children’s literature, Lewis Carroll's ''Alice's Adventures in Wonderland'' (1865) and Rudyard Kipling's ''The Jungle Book'' (1894). Former Prime Minister of the United Kingdom, Harold Macmillan, grandson of co-founder Daniel, was chairman of the company from 1964 until his death in December 1986. Since 1999, Macmillan has been a wholly owned subsidiary of Holtzbrinck Publishing Group with offices in 41 countries worldwide and operations in more than thirty others. History Macmillan was founded in London in 1843 by Dani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
