How much precision can you squeeze out of a table?
7 hours ago
- #error-analysis
- #numerical-methods
- #interpolation
- Richard Feynman's insight: Deep exploration makes almost anything interesting, including interpolation.
- Interpolation methods: Linear interpolation is basic, but higher-order methods (e.g., cubic, 29th-degree) offer more accuracy.
- Lagrange interpolation theorem provides error bounds, assuming exact tabulated values.
- Error in interpolation mainly comes from gaps between tabulated points, not the precision of values.
- Error bound formula: c h^(n+1) + λδ, where h is spacing, δ is tabulated error, c depends on function derivatives, λ ≥ 1.
- Optimal interpolation order: Choose n such that c h^(n+1) < δ; higher n can be harmful due to exponential growth of λ.
- Examples:
- - Natural logarithms: 4th-order interpolation suffices for precision near tabulated error (10^-15).
- - Sine function: 7th-order interpolation yields 9-digit accuracy.
- - Bessel function J0: 11th-order needed for 4-decimal precision due to wide spacing.
- Modern use: Computers handle table lookups and interpolation, reducing direct human interaction.