The third major technique for polynomial interpolation is interpolation using Lagrange interpolating polynomials. Given a set of distinct x-values x0, x1, ... xn, define the n + 1 Lagrange interpolating polynomials for these values by (for i = 0, 1, ... n)

Li (x) is a polynomial of exact degree n and that Li(xj) = 0 if i ‰  j, and Li(xi) = 1. It follows that we can write the polynomial interpolant to (x0, y0), ... , (xn, yn) in the form
p(x) = cˆ© Lˆ©(x) + c1 L1(x) + ... cn Ln(x)
Where ci = yi, i = 0, 1, ... , n.
(a) Verify that p(x) = y0L0(x) + y1L1(x) + ... + ynLn(x) is the unique interpolating polynomial for this data.
(b) What is the linear system for the coefficients c0, c1, ... , cn, corresponding to 1 for the Vandermonde approach and to 4 for the Newton approach?
(c) Compare the three approaches to polynomial interpolation that we have seen. Which is most efficient with respect to finding the coefficients? Which is most efficient with respect to evaluating the interpolant somewhere between data points?

L,(x)