16. 11. 2022, Interpolation & Aproximation

1. Interpolate the values of sin(((2π i )/5)) at the points ((2π i )/5) for i=0,1,...,5 by a polynomial of degree 5

2. Sample 6 points on a curve (Subscript[c, 1]) =  {(t^2)+ 3 t, ((3t)/((t^2)+1))}, t ∈ [-3, 0] and interpolate them by a polynomial curve of degree 5.

3. Approximate cos(x) at the points ((2π i)/100), i = 0, ..., 100. (do not use the function LeastSquares :))

4. Find polynomials p, q of degree 2, such that p(0) = 1, p(1) = 2, q(4) = 1,q(5) = -1, p(3) = q(3), p’(3) = q’(3).

5. Consider the function f(x) = sin(x), x∈ [0,2π]. Chose a random point p∈ (0,2π) and compute (C^1) Hermite interpolation of f(x) at the subintervals. Compute the interpolation error by sampling points on f(x) and the resulting polynomials.

6. Consider a Bezier curve (Subscript[c, 2]) given by the following control points: {{0,0},{1,0}, {1,1},{0,1},{-1,1.3},{-1,2},{2,2},{3,0},{1,-1},{0,-1},{-1,-1}}.
For N<10 sample N+1 points on (Subscript[c, 2]). Compute the tangent vectors at these points and interpolate the segments by a Ferguson cubic. Compute the interpolation error by sampling.