Homework 4

Submit via the Canvas page by 11:00pm on November 19, 2015. Homework can be submitted up to 24 hours late with a 10% reduction in points possible. If special circumstances warrant turning in assignments late, please make arrangements in advance.

• ATAP Exercise 10.6.

• ATAP 11.1 and also add an additional part (d): Compute the integral (11.9) for the two choices of contours \(|x|=3\) and \(|x|=3/2\) and compare to the actual error \(f(2)-p(2)\). Confirm that in one case it gives the right result and in the other it does not (and make sure you understand why).

  See http://www.chebfun.org/docs/guide/guide05.html for a discussion of how to compute Cauchy integrals. The function \(f(x)\) defined in the problem is an analytic function that takes the values we want to interpolate at the interpolation points. Any other analytic function taking the same values at these 3 points should result in the same interpolating polynomial. (But a different error at \(x=2\), of course.)

• ATAP 14.2.

• ATAP 15.4. Also plot the Lebesgue function for \(n=10\).

• Exercise 15.A: (additional exercise)

  Confirm using Chebfun that the bound (15.5) holds for the following cases, with \(n=20\):

  • \(f(x) = 1/(1+25x^2)\) with both equally spaced and Chebyshev
  • points
  • \(f(x) = 1/(1+500x^2)\) with both equally spaced and Chebyshev
  • points

  The functions remez and lebesgue may be useful.