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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.
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.
The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions.Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point.
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.
Python sets are very much like mathematical sets, and support operations like set intersection and union. Python also features a frozenset class for immutable sets, see Collection types. Dictionaries (class dict) are mutable mappings tying keys and corresponding values. Python has special syntax to create dictionaries ({key: value})
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 orange that has been sliced into two halves. In mathematics, division by two or halving has also been called mediation or dimidiation. [1] The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algorithm used division by two as one of its fundamental steps. [2]
The divided difference methods have the advantage that more data points can be added, for improved accuracy. The terms based on the previous data points can continue to be used. With the ordinary Lagrange formula, to do the problem with more data points would require re-doing the whole problem.