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In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. [citation needed] Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. [1] Divided differences is a recursive division process.
The simplest method is to use finite difference approximations. A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). [1] Choosing a small number h, h represents a small change in x, and it can be either positive or negative.
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 evaluates this polynomial. Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences .
Let be the Lagrange interpolation polynomial for f at x 0, ..., x n.Then it follows from the Newton form of that the highest order term of is [, …,].. Let be the remainder of the interpolation, defined by =.
There, the function is a divided difference. In the generalized form here, the operator G {\displaystyle \ G\ } is the analogue of a divided difference for use in the Banach space . The operator G {\displaystyle \ G\ } is roughly equivalent to a matrix whose entries are all functions of vector arguments u {\displaystyle \ u\ } and v ...
Stanley is recalling 2.6 million mugs sold in the U.S. after the company received dozens of consumer complaints, including some users who reported getting burned and requiring medical attention ...
Thanks to the many holidays that take place during December, the month is almost synonymous with gift-giving. For some, receiving presents can occur beyond these seasonal holidays, especially if ...
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.