Answer to Question #141124 in Quantitative Methods for Subhasis

Question

Determine the maximum error in quadratic interpolation at equispaced points ?

Answer 1

Let assume P₁(y) is the linear polynomial interpolating f (y) at y₀ and y₁.

If assumed that f (y) is twice continuously differentiable on an interval [m,n] which contains the points y₀ <y₁.

Then for m ≤ y ≤ n,

f (y)−P₁(y) = (y −y0)((y −y₁)/2) f₀(cy) for some cy between the maximum and minimum of y₀, y₁, and y.

P₁(x) is usually used as an approximation of f (y) for y ∈ [y₀,y₁].

Then for an error bound, |f (y)−P₁(y)|≤ (y −y₀)(y₁ −y).

Easily, with h =y₁ −y₀,

max x0≤x≤x1 (y −y₀)(y₁ −y) = h²/4.

Therefore, |f (y)−P₁(y)|≤ h², y ∈ [y₀,y₁].