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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. Neville's algorithm evaluates this polynomial.
This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x) (dashed, black), which is the sum of the scaled basis polynomials y 0 ℓ 0 (x), y 1 ℓ 1 (x), y 2 ℓ 2 (x) and y 3 ℓ 3 (x). The interpolation polynomial passes through all four control points, and each scaled basis ...
Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to ...
The Hermite interpolation problem is a problem of linear algebra that has the coefficients of the interpolation polynomial as unknown variables and a confluent Vandermonde matrix as its matrix. [3] The general methods of linear algebra, and specific methods for confluent Vandermonde matrices are often used for computing the interpolation ...
Smith, Simon J. (2006), "Lebesgue constants in polynomial interpolation" (PDF), Annales Mathematicae et Informaticae, 33: 109– 123, ISSN 1787-5021 Ibrahimoglu, Bayram Ali (2016), "Lebesgue functions and Lebesgue constants in polynomial interpolation", Journal of Inequalities and Applications , 2016 : 2016:93, doi : 10.1186/s13660-016-1030-3 ...
Bernstein polynomials can be generalized to k dimensions – the resulting polynomials have the form B i 1 (x 1) B i 2 (x 2) ... B i k ( x k ) . [ 1 ] In the simplest case only products of the unit interval [0,1] are considered; but, using affine transformations of the line, Bernstein polynomials can also be defined for products [ a 1 , b 1 ...
This product is a monic polynomial of degree n. It may be shown that the maximum absolute value (maximum norm) of any such polynomial is bounded from below by 2 1−n. This bound is attained by the scaled Chebyshev polynomials 2 1−n T n, which are also monic. (Recall that |T n (x)| ≤ 1 for x ∈ [−1, 1]. [5])