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Newton's form has the simplicity that the new points are always added at one end: Newton's forward formula can add new points to the right, and Newton's backward formula can add new points to the left. The accuracy of polynomial interpolation depends on how close the interpolated point is to the middle of the x values of the set of points used ...
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.
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.
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 .
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.
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 ...
The general relation gives the Newton series = () = (, +), [citation needed] where is the Hurwitz zeta function and () the Bernoulli polynomial. The series does not converge, the identity holds formally.