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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 ...
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.
[1] Divided differences is a recursive division process. Given a sequence of data points (,), …, (,), the method calculates the coefficients of the interpolation polynomial of these points in the Newton form.
Romberg's method is a Newton–Cotes formula ... it may be preferable to replace the polynomial interpolation of Richardson with the ... 1: 0 4×16 − 0 / 3 ...
If we consider as a variable in a topological space, and the function () mapping to a Banach space, then the problem is treated as "interpolation of operators". [11] The classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other subsequent results.
Given n + 1 points, there is a unique polynomial of degree ≤ n which goes through the given points. Neville's algorithm evaluates this polynomial. Neville's algorithm evaluates this polynomial. Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences .
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
In numerical analysis, multivariate interpolation or multidimensional interpolation is interpolation on multivariate functions, having more than one variable or defined over a multi-dimensional domain. [1] A common special case is bivariate interpolation or two-dimensional interpolation, based on two variables or two dimensions.