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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 ...
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.
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.
Composed in 1669, [4] during the mid-part of that year probably, [5] from ideas Newton had acquired during the period 1665–1666. [4] Newton wrote And whatever the common Analysis performs by Means of Equations of a finite number of Terms (provided that can be done) this new method can always perform the same by means of infinite Equations.
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 ...
In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. [ 1 ] Statement of the theorem