interpolation polynomiale exo7

Figure 1: Interpolating polynomial for data at three abscissas (x0; x1; x2) and two possible functions f(x) Given three points p(x) may not be a good estimate of interpolant cannot know what f does between the f (right) - the data points but the nature of the function f(x) may be unknown

We will show that there exists a unique interpolation polynomial Depending on how we represent the interpolation polynomial it can be computed more or less e ciently Notation: We denote the interpolating polynomial by P(fjx1; : : : ; xn)(x) Theorem (Fundamental Theorem of Algebra)

Spring 2020 Overview The point: Here we introduce polynomial interpolation - a critical tool used throughout computational math for building approximations to functions Some properties of the im-portant error formula are considered Related reading: Quarteroni Section 8 1 1 and 8 2

To motivate the need for interpolation of data and of functions; to derive three(!) di erent methods for computing a polynomial interpolant each particularly suitable for certain circumstances; to derive error expressions for the polynomial interpolation process; to construct Chebyshev interpolants which can provide very accurate approximation

Exo7 : Cours et exercices de mathématiques

DRAFT: NA/LA Section 1: Polynomial interpolation f(x0) If we wish to add a point (x1;f(x1)) to our interpolation we may do so by using the form p1(x) = p0(x)+a1(x x0) to ensure we do not a ect our previous success The value of a1 is uniquely determined by our requirement p1(x1) = p0(x1)+a1(x1 x0) = f(x1) as long as x1 6= x0 a1 = f(x1) p0