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A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or Newton polynomials. Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function.
The interpolation polynomial in the Lagrange form is the linear combination ():= ... For example, 4 equally spaced data points ,,, ...
Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than n such that the polynomial and its first few derivatives have the same values at m (fewer than n) given points as the given function ...
In the mathematical field of numerical analysis, Runge's phenomenon (German:) is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points.
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 range of the samples, unlike linear interpolation. For example, the interpolant above has a local maximum at x ≈ 1.566, f(x) ≈ 1. ...
One can use Lagrange polynomial interpolation to find an expression for this polynomial, = ) () + () ... show Example implementation in Python:
The scheme exploits the Lagrange interpolation theorem, specifically that points on the polynomial uniquely determines a polynomial of degree less than or equal to . For instance, 2 points are sufficient to define a line , 3 points are sufficient to define a parabola , 4 points to define a cubic curve and so forth.
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 ...