[CppAD] Arbitrary levels of autodifferentiation

[Brad Bell]

When a function f(x) is recorded, CppAD allows for the computation of arbitrary order derivatives of 
f(x). The multilevel option; e.g., AD< AD<double> > allows one to use arbitrary order derivatives of 
f(x) to express a function g(x) and then take derivatives of g(x).

As you have noticed, in this case the function f(x) would be ADFun< AD<double> > and g(x) would be 
ADFun<double>.

It is possible that you could do what you are asking for using template meta programmings, but I 
have never tried.

I think that the research code stalingrad does automatic differentiation with the derivatives having 
the same type as the original functions; see
	https://github.com/Functional-AutoDiff/STALINGRAD

On 06/03/2018 04:35 PM, Andrew Spielberg wrote:
> Hi all,
> 
> I am trying to come up with a way to get arbitrary levels of AutoDifferentiation using CppAD.  Given 
> a function f : R-->R, I would like to be able to differentiate expressions such as, say:
> 
> g(x) = (f'(x) + f''(x)) * f(x)
> 
> in order to get dg/dx.
> 
> Now, obviously, I can try to rewrite this as a single function just in f whose derivative will be 
> equal to dg/dx, but it that is not always possible.  To this end, I am aware of the multiple levels 
> example:
> 
> https://coin-or.github.io/CppAD/doc/mul_level.cpp.htm 
> <https://coin-or.github.io/CppAD/doc/mul_level.cpp.htm>
> 
> However, as far as I can tell, the number of differentiations here must be known a priori, and the 
> correct number of nestings of ADFun must be chosen up front.  In some cases though (say, when 
> dealing with series of unknown length), that number won't be known up front.
> 
> I am wondering if there is a way to get arbitrary levels of autodifferentiation somehow, so that I 
> can treat gradients not unlike an arithmetic function.  I was thinking, it might be possible with 
> only two levels of ADFun - every time a ADFun<ADFun<T>> is differentiated, it returns an ADFun<T> - 
> and then maybe this can be rewrapped in a new ADFun< > before being reused.  I don't know if this 
> works though.
> 
> Are arbitrary levels possible?  If not, is there another way to achieve what I want?
> 
> 
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