Questions for Polynomial Interpolation 1. What is polynomial interpolation? 2. Prove that an interpolating polynomial througn n points is unique. 3. What are the monomial basis? What are their advantages and drawbacks? 4. Describe Lagrange basis. What is their advantage over monomials? 5. What are Newton Polynomials? 6. Given x0 = 1; x1 = −1; x2 = 2; y0 = 6; y1 = 0; y2 = 12; find the interpolating polynomial through these points by: (a) Monomials (b) Lagrange basis (c) Newton basis 7. What are divided differences? 8. Explain the recursive method for computing the divided difference f[x0; : : : ; x3]. 9. Construct the divided differences table for the data 3 x 1 2 0 2 13 5 y 3 4 3 3 and construct the largest order interpolating polynomial. 10. What does the error in polynomial interpolation of a function using equispaced points depend on? 11. Assume a function has all derivatives bounded between [−10; 10]. Will the error in polynomial interpolation icrease or decrease as more points are added? 12. Use Newton's divided differences formula to find the polynomial interpolating f(x) = cos(x) through the equispaced points 0; π=4; π=2. 13. Consider the function f(x) = cos(x) for 0 ≤ x ≤ 2π. We want to find a degree two polynomial which interpolates f(x) at the equispaced points 0; π; 2π. (a) Write down the Lagrange polynomial, p(x), interpolating the function at above points. (b) Use Newton's divided differences formula to find the Newton polynomial, q(x), interpolating the function at above points. (c) What values do the two interpolating polynomials have at x = π=4? Is there a point x where Lagrange polynomial has a different value than the Newton polynomial? That is, is p(x) − q(x) 6= 0 for some x 2 [0; 2π]? (d) With p(x) being the Lagrange polynomial from above, find an upper bound on the error for 0 ≤ x ≤ π using the following interpolation error formula for a polynomial p of degree n which interpolates f in x0; : : : ; xn: n 1 Y f(x) − p(x) = f (n+1)(ξ) (x − x ): (n + 1)! j j=0 1 (e) Find the actual error at x = π=4 and compare with the prediction of the above formula. 14. (a) Newton's divided differences formula to find the polynomial interpolating f(x) = cos(x) through the equispaced points 0; π=4; π=2. (You can use cos(π=4) = 0:7071). (b) With p(x) from above, find an upper bound on the error for 0 ≤ x ≤ π=2 using the interpolation error formula (given below) for a polynomial p of degree n which interpolates f through the equispaced points x0; : : : ; xn: Mhn+1 jf(x) − p(x)j ≤ 4(n + 1) What are M and h in the above equation? (c) Find the actual error at x = π=3 and compare with the prediction of the above formula. Which number is greater? (a) Explain why polynomial interpolation fails with Runge's function. (b) Name two ways to remedy this problem. 15. What is the computational cost for constructing a polynomial using (a) Monomial basis (b) Lagrange basis (c) Newton's basis 16. What is the computational cost for evaluating a polynomial using (a) Monomial basis (b) Lagrange basis (c) Newton's basis 17. For which of functions f1(x) = sin(x); f2(x) = jxj; do we expect polynomial interpolation to work well? Explain. 2.