Question

In class, we have derived the following 2 bounds for the respective piecewise polynomial interpolation methods (assuming equally spaced data points, at intervals of size h ): - For piecewise linear interpolation ∣f(x)−s(x)∣≤ 1/8∥f ′′∥ [infinity]h²

- For piecewise cubic interpolation ∣f(x)−s(x)∣≤ 9/384 ∥f ′′′′∥ [infinity] ⋅h⁴
Consider the following 2 functions: (a) f(x)=sin(x),x∈[0,2π]. (b) f(x)=sin(10x),x∈[0,2π]. For each of these functions, explain if either the piecewise linear or the piecewise cubic interpolation will lead to a lower error. Is the piecewise cubic approach better under all circumstances?

Answer 1

Piecewise cubic interpolation may lead to a lower error for f(x) = sin(x), while piecewise linear interpolation may be better for f(x) = sin(10x). Overall, piecewise cubic interpolation is usually more accurate.

Step-by-step explanation:

For function f(x) = sin(x), the piecewise cubic interpolation will lead to a lower error. This is because the piecewise cubic interpolation uses a higher degree polynomial to approximate the function, which allows for a more accurate representation. The error bound for piecewise cubic interpolation is smaller than the error bound for piecewise linear interpolation, suggesting that the piecewise cubic approach is better in terms of accuracy.

For function f(x) = sin(10x), the piecewise linear interpolation may actually lead to a lower error. This is because the piecewise cubicy interpolation is more sensitive to variation in the function, and the function sin(10x) has more rapid oscillations compared to sin(x). The piecewise linear interpolation might be able to capture the general trend of the function more effectively.

However, it is important to note that the piecewise cubic interpolation is generally considered to be more accurate and provide better results for most functions.