Question

(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1 On ...

Answer 1

2.

Given function values I 20, 21, ... en f(x) fo, fi, ... fn with Ii + I; for i + j. There exists a unique polynomial Pn of degree less or equal n which interpolates these values, i.e. Ph(26) = fi, i = 0, 1,..., n. Several representations of Pn are known, we will present in this paper the basis transformations among four of them. (1)

A second representation is by means of the Lagrange polynomials 1:1) = (10(2), 11(2),..., In (2))? with 4() = 11.6.-", i = 0,1,...,1. lic ; Oi-lj i = 0,1,...,n. Pa(z) = ; 1;(x) = f*l(a). Since lí(Ti) = 1 and li(Lj) = ( for i # j the interpolation polynomial can be written as linear combination (2) j=0 Interpolating with the Lagrange formula (2) is not very efficient, since for every new value x we have to perform O(na) operations. There exists a variant called the Barycentric Formula [5] which requires only O(n) operations per interpolation point. We define the coefficients i i=0,...,n. (r; – xo)... (Li - 1-1)(x1 - Xi+1) ...(; – xn)' Then for new interpolation points r we compute the weights pi= li/(– Xi) and evaluate so Pn() = Lizo Hifi (3) Li=o Mix) with only O(n) operations. Thus in this form the Lagrange polynomials are computed by li(x) = _Hi(x) , i = 0,1,...,n.

The coefficients a = (ao, a1,..., an)" are determined by the interpolation condition (1) as solution of the linear system Va= f with the Vandermonde matrix 71 20... 26 | 1 x1 ... 21-1 V= 1 In ... mal ve her
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