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