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Neville's Schema
In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given ''n'' + 1 points, there is a unique polynomial of degree ''≤ n'' which goes through the given points. Neville's algorithm evaluates this polynomial. Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the 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 o .... It is similar to Aitken's algorithm (named after Alexander Aitken), which is nowadays not used. The algorithm Given a set of ''n''+1 data points (''x''''i'', ''y''''i'') where no two ''x''''i'' are the same, the interpolating polynomial is the polynomial ''p'' of degree at most ''n'' with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no two x_j the same, a polynomial function p(x) is said to interpolate the data if p(x_j)=y_j for each j\in\. Two common explicit formulas for this polynomial are the Lagrange polynomials and Newton polynomials. Applications Polynomials can be used to approximate complicated curves, for example, the shapes of letters in typography, given a few points. A relevant application is the evaluation of the natural logarithm and trigonometric functions: pick a few known data points, create a lookup table, and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation also forms the basis for algorithms in numerical quadrature and numerical ordinary differential equations and Secure Multi P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Harold Neville
Eric Harold Neville, known as E. H. Neville (1 January 1889 London, England – 22 August 1961 Reading, Berkshire, England) was an English mathematician. A heavily fictionalised portrayal of his life is rendered in the 2007 novel ''The Indian Clerk''. He is the one who convinced Srinivasa Ramanujan to come to England. Early life and education Eric Harold Neville was born in London on 1 January 1889. He attended the William Ellis School, where his mathematical abilities were recognised and encouraged by his mathematics teacher, T. P. Nunn. In 1907, he entered Trinity College, Cambridge. He graduated second wrangler two years later. He was elected to a Fellowship at Trinity College. While there he became acquainted with other Cambridge fellows, most notably Bertrand Russell and G. H. Hardy. In 1913 Neville married Alice Farnfield (1875-1956); they had a son Eric Russell Neville in 1914 who died before his first birthday. Neville remained married to Alice until her death. Ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Polynomial
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method. Definition Given a set of ''k'' + 1 data points :(x_0, y_0),\ldots,(x_j, y_j),\ldots,(x_k, y_k) where no two ''x''''j'' are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials :N(x) := \sum_^ a_ n_(x) with the Newton basis polynomials defined as :n_j(x) := \prod_^ (x - x_i) for ''j'' > 0 and n_0(x) \equiv 1. The coefficients are defined as :a_j := _0,\ldots,y_j/math> where : _0,\ldots,y_j/math> is the notation for divided differences. Thus the Newton polynomial can be written as :N(x) = _0+ _0,y_1x-x_0) + \cdots + _0,\ldots,y_kx-x_0)(x-x_1)\cdots( ... [...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 lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aitken Interpolation
Aitken interpolation is an algorithm used for polynomial interpolation that was derived by the mathematician Alexander Aitken. It is similar to Neville's algorithm. See also Aitken's delta-squared process In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method, used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926.Alexa ... or Aitken Extrapolation. References External links * Polynomials Interpolation {{mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Aitken
Alexander Craig "Alec" Aitken (1 April 1895 – 3 November 1967) was one of New Zealand's most eminent mathematicians. In a 1935 paper he introduced the concept of generalized least squares, along with now standard vector/matrix notation for the linear regression model. Another influential paper co-authored with his student Harold Silverstone established the lower bound on the variance of an estimator, now known as Cramér–Rao bound. He was elected to the Royal Society of Literature for his World War I memoir, ''Gallipoli to the Somme''. Life and work Aitken was born on 1 April 1895 in Dunedin, the eldest of the seven children of Elizabeth Towers and William Aitken. He was of Scottish descent, his grandfather having emigrated from Lanarkshire in 1868. His mother was from Wolverhampton. He was educated at Otago Boys' High School in Dunedin (1908–13) where he was school dux and won the Thomas Baker Calculus Scholarship in his last year at school. He saw active service ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for '' Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomials
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
