- Cubic Spline:
A simple solution to reducing the complexity of interpolation is to build piecewise-linear approximations, but this can lead to the interpolation function not being “smooth” at the endpoints of the intervals (not differentiable). Using higher-order Hermite polnomials requires knowledge of f’(x) and f’’(x) that may not be available.
a) Explain the concepts of Cubic Spline Interpolation, using both “natural or free” boundary and “clamped” boundary.
Show how the equations are generated to solve the interpolation.
b) Construct the “natural” boundary interpolation for:
- c) Use your answer from part (b) to approximate f(x) and f’(x) and calculate the actual error for
f(x) = ln(ex + 2) for x = 0.25
2. Roots of Unity:
ω is an nth root of unity iff ωn = 1, and
ω is a primitive root of unity iff ωn = 1 AND ωk ≠ 1 for 1 ≤ k < n.
Demonstrate the following:
a) (cos θ + i sin θ)p = cos pθ + i sin pθ
b) cos 2π/n + i sin 2π/n is a primitive nth root of unity
c) If ω is a primitive nth root of unity, then , the complex conjugate of , is also a primitive nth root of unity.
d) if is a root of unity, then a b is also a root of unity.
e) if is a root of unity for all 1 j n (n then
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