I'd like to argue, that when it comes to computers and differentiation, automatic differentiation is the most useful, most important in practice. But often only symbolic and numerical differentiation are mentioned.
What's nice about a discussion of symbolic differentiation is that we can prove a few rules rigorously, and then use those rules to purely mechanically differentiate algebraic expressions we encountered up to trigonometry.
You're right though, in practice, for complex functions expressed as programs, automatic differentiation is superior.
Was it correct? All I know is that I couldn't kill it.
Numerical differentiation comes directly from the definition of derivative and symbolic is what you always did for exercises in your calculus classes.
Automatic differentiation come in two flavours, forward and backward modes. Forward mode is based on dual numbers [1], which is the
quotient of a polynomial ring over the real numbers, by the principal ideal generated by the square on the indeterminate
Another way of thinking of this is to have an element ℇ different from zero, such that ℇ ²= 0 and hand wave the fact that the dual part of the number carry the derivative.
Backward mode builds a graph of computation and doing symbolic differentiation over the graph and compile down the derivative into runable machine code (it could be interpreted, compiled down to IR, neither the form or the execution environment changes the fundamental algorithm).
Maybe they are not really hard, but they are not easy either. Still I think they should be at least mentioned in modern courses.
> Another way of thinking of this is to have an element ℇ different from zero, such that ℇ ²= 0 and hand wave the fact that the dual part of the number carry the derivative.
Aren't these the same way of thinking?
A more common example of this idea is √(-1): R[x]/<x²+1> vs i²+1=0.
That might make more sense in another course, like one on numerical methods, rather than in a programming course that happens to use numerical differentiation as a demonstration of programming techniques and language features.
So in some sense it's maybe not hard, though any attempt to do it without building up some theory (e.g. abstract vector spaces and dual spaces) first will probably come across as magic. On the other hand, magic tricks would be right at home in an intro differential equations class so maybe this would be a perfect addition. Or it can replace Wronskians or something.
Is this a misunderstanding? You use differential equation solvers when all you know is how to calculate the derivative(s) of the function, and you want to get the function itself as the solution. Where would you need numerical differentiation in this?
I think the idea is to use the "tangent linear model" to decide how much importance to give to a particular observation of the initial state.