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In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. [ 1 ] Given a set of n + 1 data points (
Download QR code; Print/export ... it may be preferable to replace the polynomial interpolation of ... The result in the lower right corner of the triangular array is ...
A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial and the function.
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.
As explained above in Applications, the polynomial interpolation problem for () = + + + + satisfying () =, …, = is equivalent to the matrix equation =, which has the unique solution =. There are other known formulas which solve the interpolation problem, which must be equivalent to the unique a = V − 1 y {\displaystyle a=V^{-1}y} , so they ...
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])
This polynomial is further reduced to = + + which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing () and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were ...
The process of interpolation maps the function to a polynomial . This defines a mapping X {\displaystyle X} from the space C ([ a , b ]) of all continuous functions on [ a , b ] to itself. The map X is linear and it is a projection on the subspace Π n of polynomials of degree n or less.