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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.
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.
One method is to write the interpolation polynomial in the Newton form (i.e. using Newton basis) and use the method of divided differences to construct the coefficients, e.g. Neville's algorithm. The cost is O( n 2 ) operations.
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.
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.
Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both can be derived from the calculation of divided differences. However, there are other methods for computing a Hermite interpolating polynomial.
The principle of a difference engine is Newton's method of divided differences. If the initial value of a polynomial (and of its finite differences) is calculated by some means for some value of X, the difference engine can calculate any number of nearby values, using the method generally known as the method of finite differences.
Newton's method — based on linear approximation around the current iterate; quadratic convergence Kantorovich theorem — gives a region around solution such that Newton's method converges; Newton fractal — indicates which initial condition converges to which root under Newton iteration; Quasi-Newton method — uses an approximation of the ...