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Differentiation danger
Numerical differentiation amplifies noise and roundoff. Smaller \(h\) is not always better.
Richardson
Richardson extrapolation cancels leading error terms when the asymptotic error model is valid.
Forward/backward differences
One-sided formulas are useful near boundaries but are usually lower order than centered formulas.
Roundoff error
Roundoff can grow like \(u/h\) when differences subtract nearby values.
Central difference
The central difference is symmetric and typically second-order accurate.
Truncation error
Truncation error decreases with \(h\) according to the order of the formula.