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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.
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.
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.
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 ...
Solving an interpolation problem leads to a problem in linear algebra where we have to solve a system of linear equations. Using a standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix .
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A. [1] [2] It states that [3]
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation.Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. [1]