Polynomial interpolation - Wikipedia
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 , with no two the same, a polynomial function …
5.1: Polynomial Interpolation - Mathematics LibreTexts
2022年5月31日 · The matrix is called the Vandermonde matrix, and can be constructed using the MATLAB function vander.m. The system of linear equations can be solved in MATLAB using the \operator, and the MATLAB function polyval.m can used to interpolate using the \(c\) coefficients. I will illustrate this in class and place the code on the website.
Figure 1: Interpolating polynomial for data at three nodes (x0; x1; x2) and two possible functions f(x). Given three points, p(x) may not be a good estimate of f (right) - the interpolant cannot know what f does between the data points. One approach to approximation is called interpolation. Suppose we have the data.
Polynomial Interpolation. I Given data x 1 x 2 x n f 1 f 2 f n (think of f i = f(x i)) we want to compute a polynomial p n 1 of degree at most n 1 such that p n 1(x i) = f i; i = 1;:::;n: I A polynomial that satis es these conditions is called interpolating polynomial. The points x i are called interpolation points or interpolation nodes.
Polynomial interpolation is one the most fundamental problems in numerical methods.
3.1 The Interpolating Polynomial We all know that two points determine a straight line. More precisely, any two points in the plane, (x1,y1) and (x2,y2), with x1 ̸= x2, determine a unique first-degree polynomial in x whose graph passes through the two points. There are many different formulas for the polynomial, but they all lead to the same ...
polynomial interpolation problem is particularly important: Given x1,...,xn (distinct) and y1,...,yn, find a polynomial pn−1(x) of degree n−1 (or less) such that pn−1(xi) = yi for i = 1:n. Thus, p2(x) = 1+4x−2x2 interpolates the points (−2,−15), (3,−5), and (1,3).
In this chapter, we will immediately put interpolation to use to formulate high-order quadrature and di erentiation rules. In other words the graph of the polynomial should pass through the points (xj; y j). A degree-N polynomial can be written as pN(x) = P N n …
such that p(xi) = yi for i = 1; 2; : : : ; n. The polynomial p(x) is called the interpolating polynomial for the data. We will prove that interpolating polynomials exist and are unique in the next few slides. To nd p(x), set up a system of n linear equations in the n variables r0; r1; r2; : : …
When f j(x)g are polynomials, we say polynomial interpolation. When f j(x)g are trigonometric, we say trigonometric interpolation. There is also piecewise interpolation. We will only consider polynomial interpolation in this course.