I. A min-max problem.For n ∈ N+, determine
,
where the minimum is taken over all ai ∈ R and a0 6= 0.
II. Imitate the proof of Chebyshev Theorem.
Let a 1 and denote .
Define
,
where Tn is the Chebyshev polynomial of degree n. Clearly ˆ . Define the max-norm of a function f : R → R as
kfk∞ = max |f(x)|.
x∈[−1,1] Prove
, kpˆn(x)k∞ ≤ kpk∞.
Problems I and II weigh 4 and 6 points, respectively. There is no extra-credit problem for this homework.
The three programming assignments in the next section weigh 10, 5, and 5 points, respectively. The total number of points is thus 30.
2 C++ programming
(a) Implement the Newton formula in a subroutine thatproduces the value of the interpolation polynomial pn(f;x0,x1,...,xn;x) at any real x, where n ∈ N+, x0is are distinct, and f is a function assumed to be available in the form of a subroutine.
(b) Run your routine on the function
for x ∈ [−5,5] using , and n = 2,4,6,8. Plot the polynomials against the exact function to reproduce the plot in the notes that illustrate the Runge phenomenon.
(c) Reuse your subroutine of Newton interpolation to perform Chebyshev interpolation for the function
for x ∈ [−1,1] on the zeros of Chebyshev polynomials Tn with n = 5,10,15,20. Clearly the Runge function f(x) is a scaled version of the function in (b). Plot the interpolating polynomials against the exact function to observe that the Chebyshev interpolation is free of the wide oscillations in the previous homework.
