A student interpolating 25 uniformly spaced data points finds wild oscillations near the endpoints and proposes using a degree-40 polynomial to get a better fit. What is wrong with this reasoning?
AA degree-40 polynomial cannot interpolate more than 40 points
BIncreasing polynomial degree worsens Runge's phenomenon — oscillations near the endpoints grow as degree increases for uniform nodes
CThe student should use Chebyshev nodes instead of a higher-degree polynomial, but the degree itself is not the problem
DHigh-degree polynomials are too slow to compute for practical use
This is exactly Runge's phenomenon: for uniformly spaced nodes, a high-degree global polynomial interpolant oscillates increasingly near the endpoints as degree grows. The solution is NOT to add more degrees — it is to switch to a piecewise approach (like cubic splines) that keeps the polynomial degree low on each interval while enforcing smooth joins between pieces.
Question 2 Multiple Choice
What makes the linear system that arises when constructing a natural cubic spline particularly efficient to solve?
AThe system is diagonal, so each unknown can be solved independently
BThe system is small — it always has exactly 3 unknowns regardless of n
CThe system is tridiagonal, symmetric, and diagonally dominant, allowing O(n) solution via the Thomas algorithm
DThe spline coefficients can be read directly from the data without solving any system
Setting up the interior second-derivative conditions produces a tridiagonal system — each equation involves only three consecutive unknowns. This sparsity, combined with diagonal dominance, makes the Thomas algorithm applicable, solving the system in O(n) time. This is one of the key practical advantages of cubic splines over direct high-degree polynomial interpolation.
Question 3 True / False
A natural cubic spline is called 'natural' because it mimics the shape an elastic rod (drafting spline) takes when forced through data points — it minimizes bending energy.
TTrue
FFalse
Answer: True
This is exactly right. Minimizing bending energy corresponds mathematically to minimizing ∫[S''(x)]² dx, which is precisely what the natural cubic spline achieves when the second derivatives at the endpoints are set to zero. The physical analogy is not decorative — it is the mechanical origin of the construction.
Question 4 True / False
Cubic splines achieve high accuracy primarily by using higher-degree polynomials than simpler interpolants — the cubic degree is what buys smoothness.
TTrue
FFalse
Answer: False
The accuracy and smoothness of cubic splines come from enforcing C² continuity (matching first and second derivatives) at interior nodes, not merely from using cubic polynomials. Each piece is still a low-degree (cubic) polynomial; what distinguishes the spline is the derivative-matching conditions across pieces. You could have piecewise cubics without these conditions and get a much rougher result.
Question 5 Short Answer
Why do cubic splines require boundary conditions, and what are the two most common choices?
Think about your answer, then reveal below.
Model answer: With n+1 nodes, a cubic spline has n cubic pieces, each with 4 coefficients — 4n parameters. The interpolation and C² interior conditions supply 4n−2 constraints, leaving 2 free. Boundary conditions close the system. The natural spline sets S''=0 at both endpoints (zero curvature, minimizing bending). The clamped spline specifies S' at the endpoints using known derivative data.
Without boundary conditions the system is underdetermined — infinitely many cubic splines pass through the same data. The boundary conditions make the solution unique. The natural choice is appropriate when no derivative data is available; the clamped choice is more accurate when endpoint derivative information is known (e.g., from the physical problem generating the data).