4 ms·
The central derivative is equal to Im(f(x + j h) / h) where j is the split-complex number which squares to 1, "Im" is the imaginary-part function, and h is a sm
by ogogmad 4y ago
The central derivative is equal to Im(f(x + j h) / h) where j is the split-complex number which squares to 1, "Im" is the imaginary-part function, and h is a small real number. When j is replaced with the dual number epsilon, this instead equals the actual derivative, as is shown in the article. You can also replace j with the complex number i, producing the complex-step method.
Central difference and complex-step are both bad approximations to the actual derivative. Of these, the complex-step method is the most ridiculous because it requires extending a function to act on complex numbers, while not extending it to the closely-related dual numbers.
- vladf 4y ago"Ridiculous" is not a word I'd use... sometimes symbolic derivatives are intractable, in which case complex step derivatives provide high-precision numerical derivatives which are otherwise unattainable. https://vladfeinberg.com/2020/07/05/discrete-residues.html https://vladfeinberg.com/2020/07/05/discrete-residues.html
- ogogmad 4y agoNo. The dual numbers are very similar to the complex numbers and produce exact results. Ridiculous is true and fair. [edit] Also, I wouldn't count using Cauchy's Integral Formula as the complex-step method.