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Taking = for some unknown function in Newton divided difference formulas, if the representation of x in the previous sections was instead taken to be = +, in terms of forward differences, the Newton forward interpolation formula is expressed as: () = (+) = = () whereas for the same in terms of backward differences, the Newton backward ...
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.
This expression is Newton's difference quotient (also known as a first-order divided difference). The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to h. As h approaches zero, the slope of the secant line approaches the slope of the tangent line.
Since the relationship between divided differences and backward differences is given as: [citation needed] [,, …,] =! (), taking = (), if the representation of x in the previous sections was instead taken to be = +, the Newton backward interpolation formula is expressed as: () = (+) = = () (). which is the interpolation of all points before .
Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences. It is similar to Aitken's algorithm (named after Alexander Aitken ), which is nowadays not used.
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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 =.
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