AMATH242 Study Guide - Quiz Guide: Cubic Hermite Spline, Linear Combination, Lagrange Polynomial

• School: University of Waterloo
• Department: Applied Mathematics
• Course: AMATH242
• Professor: Sverrir

Document Summary

1. (a) find an interpolating polynomial p(x) of degree at most 2 which satis es p(0) = 4, p (0) = 1 and p(1) = 6. Solution: since we have three points, assume we need a second degree polynomial p(x) = ax2 + bx + c. finding its coe cients is straightforward. Write the polynomial p(x) from part (a) in terms of these lagrange polynomials for x as 0, 1 and 2. Solution: the lagrange basis functions associated with any three distinct values x0, x1 and x2 are. L2(x) = (x x1)(x x2) (x0 x1)(x0 x2) (x x0)(x x2) (x1 x0)(x1 x2) (x x0)(x x1) (x2 x0)(x2 x1) 1 (2) and the polynomial pl(x), which is a linear combination of those functions, is pl(x) = f (x0) (x x1)(x x2) (x0 x1)(x0 x2) + f (x1) (x x0)(x x2) (x1 x0)(x1 x2) + f (x2) (x x0)(x x1) (x2 x0)(x2 x1)