Polynomial Splines: Knots, Continuity, Cubic Forms, and Evaluation

All flashcards

What is a polynomial spline?

A function defined by polynomial pieces on different subintervals of its domain.

How does a smoothing spline differ from an interpolating spline?

A smoothing spline balances closeness to observations with a roughness measure; an interpolating spline is constrained to pass through specified data.

What is Runge's phenomenon?

Oscillatory behavior that can occur in some high-degree polynomial interpolations.

What does $C^1$ continuity at a spline knot require?

The neighboring pieces must match in value and first derivative at the knot.

What is a B-spline?

A spline represented using basis functions; the de Boor algorithm can evaluate it efficiently.

Quiz questions

1. How is the degree of a piecewise polynomial spline determined?Answer: It is the greatest degree among its polynomial pieces.
2. What must match across an interior knot for a spline to be $C^2$ there?Answer: The neighboring pieces' values and first and second derivatives.
3. Which endpoint condition defines a natural cubic spline on $[a,b]$?Answer: $S''(a)=S''(b)=0$
4. Which algorithm is used to efficiently evaluate B-splines?Answer: The de Boor algorithm.
5. What makes a planar parametric curve a spline curve under the stated definition?Answer: Its coordinate functions are splines of the same degree with the same extended knot vector.