Higher-Order Formulas

Numerical Analysis · Axiom Academy

Using more points for better accuracy Consider approximating the derivative with different formulas: The five-point central difference uses two points on each side: The coefficients -1, 8, -8, 1 are chosen to cancel both O(h²) and O(h⁴) error terms, leaving only O(h⁴) error! We want to find coefficients a, b, c, d, e such that: Write Taylor expansions for each term and collect powers of h. The coefficients must satisfy a system of equations that cancels unwanted terms. For the 5-point formula centered at x, symmetry simplifies things: we only need coefficients for f(x±h) and f(x±2h). 4 Common Differentiation Formulas Here are the most commonly used finite difference formulas: 5 Trade-offs and Practical Considerations Much better accuracy for smooth functions Can use larger h (fewer function evaluations for same accuracy) Essential for high-precision scientific computing More function evaluations per derivative Problems at boundaries (need special endpoint formulas) More sensitive to noise in data Requires function to be smooth enough

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