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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 is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences.
Of course, only a divided-difference method can be used for such a determination. For that purpose, the divided-difference formula and/or its x 0 point should be chosen so that the formula will use, for its linear term, the two data points between which the linear interpolation of interest would be done.
The difference quotient as a derivative needs no explanation, other than to point out that, since P 0 essentially equals P 1 = P 2 = ... = P ń (as the differences are infinitesimal), the Leibniz notation and derivative expressions do not distinguish P to P 0 or P ń:
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 =.
For arbitrary stencil points and any derivative of order < up to one less than the number of stencil points, the finite difference coefficients can be obtained by solving the linear equations [6] ( s 1 0 ⋯ s N 0 ⋮ ⋱ ⋮ s 1 N − 1 ⋯ s N N − 1 ) ( a 1 ⋮ a N ) = d !
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.