Archive for August, 2021

Real Derivatives from Imaginary Increments

The solution of many problems requires us to compute derivatives. Complex step differentiation is a method of computing the first derivative of a real function, which circumvents the problem of roundoff error found with typical finite difference approximations.

Rounding error and formula error as functions of step size ${h}$ [Image from Wikimedia Commons].

For finite difference approximations, the choice of step size ${h}$ is crucial: if ${h}$ is too large, the estimate of the derivative is poor, due to truncation error; if ${h}$ is too small, subtraction will cause large rounding errors. The finite difference formulae are ill-conditioned and, if ${h}$ is very small, they produce zero values.

Where it can be applied, complex step differentiation provides a stable and accurate method for computing ${f^\prime(x)}$.

Changing Views on the Age of the Earth

[Image credit: NASA]

In 1650, the Earth was 4654 years old. In 1864 it was 100 million years old. In 1897, the upper limit was revised to 40 million years. Currently, we believe the age to be about 4.5 billion years. What will be the best guess in the year 2050? [TM217 or search for “thatsmaths” at irishtimes.com].

Carnival of Mathematics

The Aperiodical is described on its `About’ page as “a meeting-place for people who already know they like maths and would like to know more”. The Aperiodical coordinates the Carnival of Mathematics (CoM), a monthly blogging roundup hosted on a different blog each month. Generally, the posts describe a collection of interesting recent items on mathematics from around the internet. This month, it is the turn of thatsmaths.com to host CoM.