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Algebraically, automatic differentiation is the same as adding a nilpotent element e, such that e^2=0 to your algebra. You can continue this pattern out to get
by 33a 11y ago
Algebraically, automatic differentiation is the same as adding a nilpotent element e, such that e^2=0 to your algebra. You can continue this pattern out to get higher order derivatives. For example, if you also add an element f where f^3=0, the coefficient of f is proportional to the second derivative.
- amelius 11y agoSounds interesting. But could you explain this so that people without a postgraduate degree in mathematics can understand this?
- eru 11y agoThe idea might be similar to https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis Edit: Actually, https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis https://en.wikipedia.org/wiki/Smooth_infinitesimal_analysis seems much closer.
- amelius 11y agoInteresting. But... > by denying the law of the excluded middle, e.g., NOT (a ≠ b) does not imply a = b Ouch, that is where my brain starts hurting.