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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.
Calculating the interpolating polynomial is computationally expensive (see computational complexity) compared to linear interpolation. Furthermore, polynomial interpolation may exhibit oscillatory artifacts, especially at the end points (see Runge's phenomenon). Polynomial interpolation can estimate local maxima and minima that are outside the ...
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 ...
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.
Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation. In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, [1] is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's ...
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 ...
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])