Truncation and rounding error bound derivation for finite difference approximation to the first derivative

Question

Finite difference approximation to the first derivative: $$ f'(x)= \frac{f(x+h) - f(x)}{h} $$

Heath's book on Scientific computing Section 1.2.4, Example 1.3 says truncation error for the finite difference approximation is bounded by $\frac{|f''(t)|h}{2}$ for $t$ near $x$, and the rounding error in computing the finite difference is bounded by $\frac{2\epsilon}{h} $

How are these bounds calculated?

Answer 1

• By the linear Taylor expansion with quadratic remainder term, and
• by assuming that the evaluation of $f$ has a seemingly random error bounded in size by $ϵ$.

See Computing differentiation rule with error bound for a plot of the error against the exact derivative value in different approximation schemes for the first derivative.