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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.
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.
A difference list f is a single-argument function append L, which when given a linked list X as argument, returns a linked list containing L prepended to X. Concatenation of difference lists is implemented as function composition. The contents may be retrieved using f []. [1]
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if f {\displaystyle f} is a holomorphic function , real-valued on the real line, which can be evaluated at points in the complex plane near x {\displaystyle x} , then there are stable methods.
An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series. Another way of generalization is making coefficients μ k depend on point x: μ k = μ k (x), thus considering weighted finite difference. Also one may make the step h depend on point x: h = h(x).
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 ...
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.
In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. [1] Statement of the theorem