Dear all,

I am here to ask the help of the community. I am a Physicist, working in the domain of Astrophysics/Cosmology. About a month ago I discovered Julia and I started translating my Python code in this new language: the new code is about 10x faster than the Python optimized code. After this results, I decided I will use Julia for my next research projects.

After this brief introduction, my questions regard the usage of autodifferentiation, since in my project I need to evaluate several numerical derivatives.

## Detailed explanation

In my work I have to evaluate coefficients such as

In this expression there are several terms:

- E(z) , for which I have an analytic expression
- r(z) , which is evaluated as the 1D integral of E(z) (which, for instance, I evaluate using QuadGK.jl)
- W_{i}^{A}(z) , which is composed by analytic functions and/or E(z) and r(z)
- P_{\delta \delta} , which is obtained as the numerical solution of a system of a differential equations ( if one is interested in high precision results) or using some analytical approximations

Altough not explicited stated, all these functions depends on some parameters (the cosmological parameters I am interested in). In order to evaluate the Fisher matrix (the final goal of my project), I need the derivatives of these coefficients C_{i j}^{A B}(\ell) with respect to the aforementioned parameters. Since the integrand is the (numerical) solution of a ODE system and the integration is performed numerically (with Newton-Cotes or Romberg techniques), finite differences techniques are not stable and I employ a technique (described here ) based on linear regression. I am quite satisfied with the numerical stability and precision of this algorithm, but this requires me to evaluate the previous coefficients for 14 values of each parameter. I want to stress that nor the integration interval or the integration variable depends upon the parameters wrt I want to evaluate the derivatives. I started reading something about autodiff and so I asked myself if I could evaluate the derivatives I need using this technique.

- Is it possible to use autodiff with functions which are obtained from numerical integrals?
- Is it possible to use autodiff with functions which are obtained from numerical integrals where the integrand is the numerical solution of a ODE system, or the integrand must be an analytic function? If this is possible, the ODE solver must be written in Julia or could be also a package written in another language (e.g. C)?
- If some of this options are possible, which packages should I consider to implement in my project? Zygote.jl and/or something else?

Furthermore, since my knowledge of autodiff is quite limited, if someone could give me the link ot some good resources, this would be really appreciated.

Thank you for your time:)

Cheers,

Marco Bonici