MATH221 · Annotated SlidesChapter 3 — Cubic Spline Interpolation

Chapter 3 · Annotated slides

Cubic Spline Interpolation

Cubic spline interpolation: piecewise cubics with continuous first and second derivatives, natural vs. clamped boundary conditions, and solving for the coefficients.

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Cubic spline: Sⱼ(x) = aⱼ + bⱼ(x − xⱼ) + cⱼ(x − xⱼ)² + dⱼ(x − xⱼ)³ on each subinterval; S, S′, S″ continuous.

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Slide 4: Natural Splines

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Natural spline: S″(x₀) = S″(xₙ) = 0.

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Slide 8: Clamped

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Clamped spline: S′(x₀) = f′(x₀), S′(xₙ) = f′(xₙ) (usually more accurate).

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MCQ

A natural cubic spline satisfies at the endpoints:

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