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Reciprocal Difference
In mathematics, the reciprocal difference of a finite sequence of numbers (x_0, x_1, ..., x_n) on a function f(x) is defined inductively by the following formulas: :\rho_1(x_1, x_2) = \frac :\rho_2(x_1, x_2, x_3) = \frac + f(x_2) :\rho_n(x_1,x_2,\ldots,x_)=\frac+\rho_(x_2,\ldots,x_) See also *Divided differences In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its ... References * *{{cite book, last=Abramowitz, first=Milton, author2=Irene A. Stegun, title= Handbook of Mathematical Functions, origyear=1964, year=1972, publisher=Dover, edition=ninth Dover printing, tenth GPO printing, isbn=0-486-61272-4, pag878} Finite differences ... [...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] |
Finite Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or ''infinite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divided Differences
In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. Divided differences is a recursive division process. Given a sequence of data points (x_0, y_0),\ldots,(x_, y_), the method calculates the coefficients of the interpolation polynomial of these points in the Newton form. Definition Given ''n'' + 1 data points :(x_0, y_0),\ldots,(x_, y_) where the x_k are assumed to be pairwise distinct, the forward divided differences are defined as: : _k:= y_k, \qquad k \in \ : _k,\ldots,y_:= \frac, \qquad k\in\,\ j\in\. To make the recursive process of computation clearer, the divided differences can be put in tabular form, where the columns correspond to the value of ''j'' above, and each entry in the table is computed from the difference of the entries to its immediate lower le ... [...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] |
