By Nicholas Higham
“The shortest and best way between two truths of the real domain often passes through the imaginary one,” mathematician Jacques Hadamard famously said. Here I will discuss an instance of this maxim that deserves to be better known.
A fundamental tension in numerical analysis is the interplay between truncation errors and rounding errors; this is particularly prevalent in the numerical approximation of derivatives. The forward difference approximation
\[f'(x) \approx...