Question

Consider a piecewise quadratic polynomial function s(x) that interpolates the N data points (x₁ ,y₁ ),(x₁ ,y₁ ),…,(xₙ ,yₙ ). In each subinterval Iₖ =[xₖ ,xₖ₊₁ ] we define s(x) as a quadratic polynomial sₖ (x)=a₂(k) x₂ +a₁ (k) x+a 0(k)

, such that: - For k=1,2,…,n−2,sₖ (x) should interpolate points (xₖ ,yₖ ), (xₖ₊₁ ,yₖ₊₁ ) and (xₖ₊₁ , yₖ₊₁). - sₙ₋₁ (x) should interpolate (xₙ₋₂ ,yₙ₋₂),(xₙ₋₁ ,yₙ₋₁ ) and (xₙ ,yₙ ). Derive a bound for the interpolation error ∣f(x)−s(x)∣. You may assume that the length hₖ =∣xₖ₊₁ −xₖ ∣ of each subinterval is constant and equal to h.

Answer 1

The question pertains to determining the error bound for a piecewise quadratic polynomial interpolation function and whether six data points suffice for accuracy.

Step-by-step explanation:

The student is asking about a piecewise quadratic polynomial function which interpolates given data points and seeks to understand the error bound of the interpolation. The interpolation error in any subinterval I_k for a function f(x) assuming that f(x) is sufficiently differentiable can typically be bound by using a formula derived from Taylor's theorem, which includes the max value of the nth derivative of f(x) in the interval and the length of the subinterval. However, without the specific function f(x) and its derivatives, we are unable to provide a numerical error bound. When assessing whether a data set like six packages of fruit snacks is sufficient for accurate results, we consider factors such as the variability of the data and the complexity of the function that we are trying to model.